Subgroups — C₃²×#5612 #13178991

Order 1728.

2380 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₃×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#121	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#938	C₂	1	2	yes	maximal, complemented
5	C₃²×#937	C₂	1	2	yes	maximal, complemented
6	C₃²×#936	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×C₄×#67	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#897	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃²×#893	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃²×#883	C₂	1	2	yes	maximal, complemented
11	C₃²×D₃×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
12	C₃²×#546	—	3	1	no	maximal
13	C₃×#5612	C₃	1	4	yes	maximal, direct product factor
14	C₂³×C₃²×D₃	C₂²	1	2	yes	—
15	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₃³×D₄	C₂²	1	2	yes	complemented
17	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₃²×#67	C₂²	1	1	yes	characteristic
19	C₂×C₃²×#67	C₂²	1	2	yes	—
20	C₂×C₃²×#67	C₂²	1	2	yes	complemented
21	C₃²×D₃×D₄	—	2	4	no	—
22	C₃²×#235	—	2	4	no	—
23	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
24	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
25	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	complemented
26	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	complemented
27	C₂×C₃²×Dic₆	C₂²	1	1	yes	characteristic, complemented
28	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
29	C₃³×C₂²⋊C₄	C₂²	1	2	yes	complemented
30	C₃²×#217	C₂²	1	1	yes	characteristic, complemented
31	C₃²×#217	C₂²	1	2	yes	complemented
32	C₃²×#217	C₂²	1	2	yes	complemented
33	C₃²×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
34	C₃²×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
35	C₃²×(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
36	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	complemented
37	C₃²×Dic₃⋊C₄	C₂²	1	1	yes	characteristic, complemented
38	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic, complemented
39	C₂×C₃²×#41	—	3	1	no	—
40	C₂²×C₃²×D₄	—	3	1	no	—
41	C₂×C₃×D₃×D₄	C₂×C₃	1	4	yes	complemented
42	C₂×C₃×#235	C₂×C₃	1	4	yes	complemented
43	C₃²×#124	—	3	1	no	—
44	C₃²×#123	—	3	2	no	—
45	C₃²×C₂²⋊Q₈	—	3	1	no	—
46	C₃²×#121	—	2	4	no	—
47	C₃²×#121	—	3	2	no	—
48	C₃²×#121	D₃	1	1	yes	characteristic, complemented
49	C₃²×#121	C₂×C₃	1	4	yes	complemented
50	C₃²×#120	—	3	2	no	—
51	C₃²×C₄×D₄	—	3	1	no	—
52	C₃²×C₄×D₄	—	3	1	no	—
53	C₂×C₃²×C₂²⋊C₄	—	3	2	no	—
54	C₃×#938	C₂×C₃	1	8	yes	complemented
55	C₃×#937	C₂×C₃	1	8	yes	complemented
56	C₃×#936	C₂×C₃	1	4	yes	complemented
57	C₃×C₄×#67	C₂×C₃	1	4	yes	complemented
58	C₃×#897	C₂×C₃	1	4	yes	complemented
59	C₃×#893	C₂×C₃	1	4	yes	complemented
60	C₃×#883	C₂×C₃	1	8	yes	complemented
61	C₃×D₃×C₂²⋊C₄	C₂×C₃	1	8	yes	complemented
62	C₂³×C₃³	C₂³	1	2	yes	—
63	C₂³×C₃³	C₂³	1	1	yes	characteristic
64	C₂²×C₃²×D₃	—	2	4	no	—
65	C₂²×C₃²×D₃	—	2	4	no	—
66	C₂²×C₃²×D₃	—	4	2	no	—
67	C₂²×C₃²×D₃	C₂³	1	2	yes	—
68	C₂²×C₃²×D₃	C₂³	1	1	yes	characteristic
69	C₃³×D₄	—	2	2	no	—
70	C₃³×D₄	—	4	2	no	—
71	C₂×C₃³×C₄	—	2	2	no	—
72	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
73	C₂×C₃³×C₄	C₂³	1	2	yes	—
74	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
75	C₃²×#67	—	2	4	no	—
76	C₃²×#67	—	2	4	no	—
77	C₃²×#67	—	2	2	no	—
78	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
79	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
80	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
81	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
82	C₃²×D₁₂	D₄	1	4	yes	complemented
83	C₃²×C₄×D₃	—	2	4	no	—
84	C₃²×C₄×D₃	—	2	2	no	—
85	C₃²×C₄×D₃	—	4	2	no	—
86	C₃²×Dic₆	—	2	2	no	—
87	C₃×#546	—	3	4	no	—
88	#5612	C₃²	1	1	yes	characteristic, direct product factor
89	C₂⁴×C₃²	—	3	2	no	—
90	C₂³×C₃×D₃	C₂²×C₃	1	8	yes	—
91	C₃²×#41	—	6	4	no	—
92	C₂×C₃²×Q₈	—	3	1	no	—
93	C₂×C₃²×D₄	—	2	4	no	—
94	C₂×C₃²×D₄	—	2	8	no	—
95	C₂×C₃²×D₄	—	3	2	no	—
96	C₂×C₃²×D₄	—	3	2	no	—
97	C₂×C₃²×D₄	—	3	1	no	—
98	C₂×C₃²×D₄	—	3	1	no	—
99	C₂×C₃²×D₄	—	6	4	no	—
100	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
101	C₂×C₃²×D₄	C₂×D₃	1	2	yes	complemented
102	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
103	C₂×C₃²×D₄	C₂²×C₃	1	8	yes	complemented
104	C₂²×C₃²×C₄	—	2	4	no	—
105	C₂²×C₃²×C₄	—	3	1	no	—
106	C₂²×C₃²×C₄	—	3	2	no	—
107	C₂²×C₃²×C₄	—	3	2	no	—
108	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic, complemented
109	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	complemented
110	C₂×C₃×#67	C₂²×C₃	1	8	yes	—
111	C₂×C₃×#67	C₂²×C₃	1	4	yes	—
112	C₂×C₃×#67	C₂²×C₃	1	8	yes	complemented
113	C₃×D₃×D₄	—	2	16	no	—
114	C₃×#235	—	2	16	no	—
115	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
116	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
117	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	complemented
118	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	complemented
119	C₂×C₃×Dic₆	C₂²×C₃	1	4	yes	complemented
120	C₃²×C₄⋊C₄	—	2	4	no	—
121	C₃²×C₄⋊C₄	—	3	1	no	—
122	C₃²×C₄⋊C₄	—	3	2	no	—
123	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
124	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
125	C₃²×C₂²⋊C₄	—	2	8	no	—
126	C₃²×C₂²⋊C₄	—	3	2	no	—
127	C₃²×C₂²⋊C₄	—	3	1	no	—
128	C₃²×C₂²⋊C₄	—	3	2	no	—
129	C₃²×C₂²⋊C₄	—	3	1	no	—
130	C₃²×C₂²⋊C₄	—	3	2	no	—
131	C₃²×C₂²⋊C₄	—	3	2	no	—
132	C₃²×C₂²⋊C₄	C₂×D₃	1	2	yes	complemented
133	C₃²×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
134	C₃²×C₄²	—	3	1	no	—
135	C₃×#217	C₂²×C₃	1	8	yes	complemented
136	C₃×#217	C₂²×C₃	1	8	yes	complemented
137	C₃×#217	C₂²×C₃	1	4	yes	complemented
138	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
139	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
140	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	complemented
141	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	complemented
142	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	complemented
143	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	complemented
144	C₂²×C₃³	—	2	1	no	—
145	C₂²×C₃³	—	2	2	no	—
146	C₂²×C₃³	—	2	1	no	—
147	C₂²×C₃³	—	2	2	no	—
148	C₂²×C₃³	—	2	1	no	—
149	C₂²×C₃³	—	2	2	no	—
150	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
151	C₂×C₃²×D₃	—	2	1	no	—
152	C₂×C₃²×D₃	—	2	2	no	—
153	C₂×C₃²×D₃	—	2	2	no	—
154	C₂×C₃²×D₃	—	2	1	no	—
155	C₂×C₃²×D₃	—	4	8	no	—
156	C₂×C₃²×D₃	C₂×D₄	1	4	yes	—
157	C₂×C₃²×D₃	#41	1	2	yes	complemented
158	C₃³×C₄	—	2	1	no	—
159	C₃³×C₄	—	2	2	no	—
160	C₃³×C₄	C₂×D₄	1	2	yes	—
161	C₃²×Dic₃	—	2	2	no	—
162	C₃²×Dic₃	—	2	1	no	—
163	C₃²×Dic₃	—	2	1	no	—
164	C₃²×Dic₃	—	2	1	no	—
165	C₂×C₃×#41	—	3	4	no	—
166	C₂²×C₃×D₄	—	3	4	no	—
167	C₂×D₃×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
168	C₂×#235	C₂×C₃²	1	1	yes	characteristic, complemented
169	C₃×#124	—	3	4	no	—
170	C₃×#123	—	3	8	no	—
171	C₃×C₂²⋊Q₈	—	3	4	no	—
172	C₃×#121	—	2	4	no	—
173	C₃×#121	—	3	8	no	—
174	C₃×#121	C₃×D₃	1	4	yes	complemented
175	C₃×#121	C₂×C₃²	1	1	yes	characteristic, complemented
176	C₃×#120	—	3	8	no	—
177	C₃×C₄×D₄	—	3	4	no	—
178	C₃×C₄×D₄	—	3	4	no	—
179	C₂×C₃×C₂²⋊C₄	—	3	8	no	—
180	#938	C₂×C₃²	1	2	yes	complemented
181	#937	C₂×C₃²	1	2	yes	complemented
182	#936	C₂×C₃²	1	1	yes	characteristic, complemented
183	C₄×#67	C₂×C₃²	1	1	yes	characteristic, complemented
184	#897	C₂×C₃²	1	1	yes	characteristic, complemented
185	#893	C₂×C₃²	1	1	yes	characteristic, complemented
186	#883	C₂×C₃²	1	2	yes	complemented
187	D₃×C₂²⋊C₄	C₂×C₃²	1	2	yes	complemented
188	C₂³×C₃²	—	2	4	no	—
189	C₂³×C₃²	—	2	8	no	—
190	C₂³×C₃²	—	3	1	no	—
191	C₂³×C₃²	—	3	2	no	—
192	C₂³×C₃²	—	6	4	no	—
193	C₂³×C₃²	—	6	4	no	—
194	C₂³×C₃²	—	12	2	no	—
195	C₂³×C₃²	C₂²×D₃	1	2	yes	—
196	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
197	C₂³×C₃²	C₂³×C₃	1	8	yes	—
198	C₂³×C₃²	C₂³×C₃	1	4	yes	—
199	C₂²×C₃×D₃	—	2	16	no	—
200	C₂²×C₃×D₃	—	2	16	no	—
201	C₂²×C₃×D₃	—	4	8	no	—
202	C₂²×C₃×D₃	C₂³×C₃	1	8	yes	—
203	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
204	C₃²×Q₈	—	6	2	no	—
205	C₃²×D₄	—	2	2	no	—
206	C₃²×D₄	—	2	8	no	—
207	C₃²×D₄	—	3	4	no	—
208	C₃²×D₄	—	4	8	no	—
209	C₃²×D₄	—	4	2	no	—
210	C₃²×D₄	—	4	8	no	—
211	C₃²×D₄	—	4	16	no	—
212	C₃²×D₄	—	6	2	no	—
213	C₃²×D₄	—	6	4	no	—
214	C₃²×D₄	—	6	4	no	—
215	C₂×C₃²×C₄	—	2	2	no	—
216	C₂×C₃²×C₄	—	2	4	no	—
217	C₂×C₃²×C₄	—	2	4	no	—
218	C₂×C₃²×C₄	—	2	8	no	—
219	C₂×C₃²×C₄	—	2	8	no	—
220	C₂×C₃²×C₄	—	3	1	no	—
221	C₂×C₃²×C₄	—	3	1	no	—
222	C₂×C₃²×C₄	—	3	2	no	—
223	C₂×C₃²×C₄	—	3	1	no	—
224	C₂×C₃²×C₄	—	4	8	no	—
225	C₂×C₃²×C₄	—	6	4	no	—
226	C₂×C₃²×C₄	—	6	2	no	—
227	C₂×C₃²×C₄	—	12	2	no	—
228	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
229	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
230	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
231	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
232	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
233	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
234	C₃×#67	—	2	16	no	—
235	C₃×#67	—	2	16	no	—
236	C₃×#67	—	2	8	no	—
237	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
238	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
239	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
240	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
241	C₃×D₁₂	C₃×D₄	1	16	yes	complemented
242	C₃×C₄×D₃	—	2	16	no	—
243	C₃×C₄×D₃	—	2	8	no	—
244	C₃×C₄×D₃	—	4	8	no	—
245	C₃×Dic₆	—	2	8	no	—
246	#546	—	3	1	no	Sylow
247	C₂×C₃³	—	4	1	no	—
248	C₂×C₃³	—	4	2	no	—
249	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₃³	C₂×#41	1	1	yes	characteristic
251	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
252	C₃²×D₃	—	2	2	no	—
253	C₃²×D₃	—	2	4	no	—
254	C₂⁴×C₃	—	3	8	no	—
255	C₂³×D₃	C₂²×C₃²	1	2	yes	—
256	C₃×#41	—	6	16	no	—
257	C₂×C₃×Q₈	—	3	4	no	—
258	C₂×C₃×D₄	—	2	4	no	—
259	C₂×C₃×D₄	—	2	8	no	—
260	C₂×C₃×D₄	—	3	4	no	—
261	C₂×C₃×D₄	—	3	8	no	—
262	C₂×C₃×D₄	—	3	8	no	—
263	C₂×C₃×D₄	—	3	4	no	—
264	C₂×C₃×D₄	—	6	16	no	—
265	C₂×C₃×D₄	C₂×C₃×D₃	1	4	yes	—
266	C₂×C₃×D₄	C₂×C₃×D₃	1	8	yes	complemented
267	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
268	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	complemented
269	C₂²×C₃×C₄	—	2	4	no	—
270	C₂²×C₃×C₄	—	3	8	no	—
271	C₂²×C₃×C₄	—	3	4	no	—
272	C₂²×C₃×C₄	—	3	8	no	—
273	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	complemented
274	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
275	C₂×#67	C₂²×C₃²	1	2	yes	—
276	C₂×#67	C₂²×C₃²	1	1	yes	characteristic
277	C₂×#67	C₂²×C₃²	1	2	yes	complemented
278	D₃×D₄	—	2	4	no	—
279	#235	—	2	4	no	—
280	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
281	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
282	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	complemented
283	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	complemented
284	C₂×Dic₆	C₂²×C₃²	1	1	yes	characteristic, complemented
285	C₃×C₄⋊C₄	—	2	4	no	—
286	C₃×C₄⋊C₄	—	3	4	no	—
287	C₃×C₄⋊C₄	—	3	8	no	—
288	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	complemented
289	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
290	C₃×C₂²⋊C₄	—	2	8	no	—
291	C₃×C₂²⋊C₄	—	3	8	no	—
292	C₃×C₂²⋊C₄	—	3	4	no	—
293	C₃×C₂²⋊C₄	—	3	8	no	—
294	C₃×C₂²⋊C₄	—	3	4	no	—
295	C₃×C₂²⋊C₄	—	3	8	no	—
296	C₃×C₂²⋊C₄	—	3	8	no	—
297	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	8	yes	complemented
298	C₃×C₂²⋊C₄	C₂²×C₃²	1	2	yes	complemented
299	C₃×C₄²	—	3	4	no	—
300	#217	C₂²×C₃²	1	2	yes	complemented
301	#217	C₂²×C₃²	1	2	yes	complemented
302	#217	C₂²×C₃²	1	1	yes	characteristic, complemented
303	(C₂×D₃)⋊C₄	C₂²×C₃²	1	1	yes	characteristic
304	(C₂×D₃)⋊C₄	C₂²×C₃²	1	2	yes	—
305	(C₂×D₃)⋊C₄	C₂²×C₃²	1	2	yes	complemented
306	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	complemented
307	Dic₃⋊C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
308	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic, complemented
309	C₂²×C₃²	—	2	8	no	—
310	C₂²×C₃²	—	2	1	no	—
311	C₂²×C₃²	—	2	2	no	—
312	C₂²×C₃²	—	2	1	no	—
313	C₂²×C₃²	—	2	2	no	—
314	C₂²×C₃²	—	2	1	no	—
315	C₂²×C₃²	—	2	2	no	—
316	C₂²×C₃²	—	2	4	no	—
317	C₂²×C₃²	—	2	4	no	—
318	C₂²×C₃²	—	2	4	no	—
319	C₂²×C₃²	—	2	8	no	—
320	C₂²×C₃²	—	2	4	no	—
321	C₂²×C₃²	—	2	8	no	—
322	C₂²×C₃²	—	2	8	no	—
323	C₂²×C₃²	—	3	4	no	—
324	C₂²×C₃²	—	3	2	no	—
325	C₂²×C₃²	—	4	4	no	—
326	C₂²×C₃²	—	4	4	no	—
327	C₂²×C₃²	—	4	8	no	—
328	C₂²×C₃²	—	4	8	no	—
329	C₂²×C₃²	—	4	8	no	—
330	C₂²×C₃²	—	6	1	no	—
331	C₂²×C₃²	—	6	2	no	—
332	C₂²×C₃²	—	6	1	no	—
333	C₂²×C₃²	—	6	2	no	—
334	C₂²×C₃²	—	12	8	no	—
335	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
336	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
337	C₂×C₃×D₃	—	2	4	no	—
338	C₂×C₃×D₃	—	2	8	no	—
339	C₂×C₃×D₃	—	2	8	no	—
340	C₂×C₃×D₃	—	2	4	no	—
341	C₂×C₃×D₃	—	4	32	no	—
342	C₂×C₃×D₃	C₂×C₃×D₄	1	16	yes	—
343	C₂×C₃×D₃	C₃×#41	1	8	yes	complemented
344	C₃²×C₄	—	2	1	no	—
345	C₃²×C₄	—	2	2	no	—
346	C₃²×C₄	—	2	8	no	—
347	C₃²×C₄	—	2	4	no	—
348	C₃²×C₄	—	2	8	no	—
349	C₃²×C₄	—	4	4	no	—
350	C₃²×C₄	—	4	8	no	—
351	C₃²×C₄	—	6	1	no	—
352	C₃²×C₄	—	6	2	no	—
353	C₃²×C₄	—	6	1	no	—
354	C₃²×C₄	—	6	1	no	—
355	C₃²×C₄	D₃×D₄	1	2	yes	—
356	C₃²×C₄	C₂×C₃×D₄	1	8	yes	—
357	C₃×Dic₃	—	2	8	no	—
358	C₃×Dic₃	—	2	4	no	—
359	C₃×Dic₃	—	2	4	no	—
360	C₃×Dic₃	—	2	4	no	—
361	C₂×#41	—	3	1	no	—
362	C₂²×D₄	—	3	1	no	—
363	#124	—	3	1	no	—
364	#123	—	3	2	no	—
365	C₂²⋊Q₈	—	3	1	no	—
366	#121	—	3	2	no	—
367	#121	C₃²×D₃	1	1	yes	characteristic, complemented
368	#120	—	3	2	no	—
369	C₄×D₄	—	3	1	no	—
370	C₄×D₄	—	3	1	no	—
371	C₂×C₂²⋊C₄	—	3	2	no	—
372	C₃³	#546	1	1	yes	characteristic, complemented, Sylow
373	C₂³×C₃	—	2	4	no	—
374	C₂³×C₃	—	2	8	no	—
375	C₂³×C₃	—	3	4	no	—
376	C₂³×C₃	—	3	8	no	—
377	C₂³×C₃	—	6	16	no	—
378	C₂³×C₃	—	6	16	no	—
379	C₂³×C₃	—	12	8	no	—
380	C₂³×C₃	C₂²×C₃×D₃	1	8	yes	—
381	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
382	C₂³×C₃	C₂³×C₃²	1	2	yes	—
383	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
384	C₂²×D₃	—	2	4	no	—
385	C₂²×D₃	—	2	4	no	—
386	C₂²×D₃	—	4	2	no	—
387	C₂²×D₃	C₂³×C₃²	1	2	yes	—
388	C₂²×D₃	C₂³×C₃²	1	1	yes	characteristic
389	C₃×Q₈	—	6	8	no	—
390	C₃×D₄	—	2	8	no	—
391	C₃×D₄	—	2	2	no	—
392	C₃×D₄	—	3	16	no	—
393	C₃×D₄	—	4	8	no	—
394	C₃×D₄	—	4	8	no	—
395	C₃×D₄	—	4	2	no	—
396	C₃×D₄	—	4	16	no	—
397	C₃×D₄	—	6	8	no	—
398	C₃×D₄	—	6	16	no	—
399	C₃×D₄	—	6	16	no	—
400	C₂×C₃×C₄	—	2	8	no	—
401	C₂×C₃×C₄	—	2	2	no	—
402	C₂×C₃×C₄	—	2	4	no	—
403	C₂×C₃×C₄	—	2	4	no	—
404	C₂×C₃×C₄	—	2	8	no	—
405	C₂×C₃×C₄	—	3	4	no	—
406	C₂×C₃×C₄	—	3	4	no	—
407	C₂×C₃×C₄	—	3	8	no	—
408	C₂×C₃×C₄	—	3	4	no	—
409	C₂×C₃×C₄	—	4	8	no	—
410	C₂×C₃×C₄	—	6	16	no	—
411	C₂×C₃×C₄	—	6	8	no	—
412	C₂×C₃×C₄	—	12	8	no	—
413	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
414	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
415	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
416	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
417	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
418	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
419	#67	—	2	4	no	—
420	#67	—	2	4	no	—
421	#67	—	2	2	no	—
422	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
423	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
424	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
425	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
426	D₁₂	C₃²×D₄	1	4	yes	complemented
427	C₄×D₃	—	2	4	no	—
428	C₄×D₃	—	2	2	no	—
429	C₄×D₃	—	4	2	no	—
430	Dic₆	—	2	2	no	—
431	C₂×C₃²	—	2	4	no	—
432	C₂×C₃²	—	2	4	no	—
433	C₂×C₃²	—	2	4	no	—
434	C₂×C₃²	—	4	8	no	—
435	C₂×C₃²	—	4	1	no	—
436	C₂×C₃²	—	4	2	no	—
437	C₂×C₃²	—	4	4	no	—
438	C₂×C₃²	—	4	8	no	—
439	C₂×C₃²	—	6	4	no	—
440	C₂×C₃²	—	6	2	no	—
441	C₂×C₃²	—	8	8	no	—
442	C₂×C₃²	C₂×D₃×D₄	1	1	yes	characteristic
443	C₂×C₃²	#1004	1	1	yes	characteristic
444	C₂×C₃²	D₃×#41	1	1	yes	characteristic
445	C₂×C₃²	C₂²×C₃×D₄	1	4	yes	—
446	C₂×C₃²	C₂×C₃×#41	1	4	yes	—
447	C₂×C₃²	C₃×H(5,𝔽₂)	1	4	yes	—
448	C₃×D₃	—	2	8	no	—
449	C₃×D₃	—	2	16	no	—
450	C₂⁴	—	3	2	no	—
451	#41	—	6	4	no	—
452	C₂×Q₈	—	3	1	no	—
453	C₂×D₄	—	3	1	no	—
454	C₂×D₄	—	3	2	no	—
455	C₂×D₄	—	3	2	no	—
456	C₂×D₄	—	3	1	no	—
457	C₂×D₄	—	6	4	no	—
458	C₂×D₄	C₂×C₃²×D₃	1	1	yes	characteristic
459	C₂×D₄	C₂×C₃²×D₃	1	2	yes	complemented
460	C₂²×C₄	—	3	2	no	—
461	C₂²×C₄	—	3	1	no	—
462	C₂²×C₄	—	3	2	no	—
463	C₂²×C₄	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
464	C₄⋊C₄	—	3	1	no	—
465	C₄⋊C₄	—	3	2	no	—
466	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
467	C₂²⋊C₄	—	3	2	no	—
468	C₂²⋊C₄	—	3	1	no	—
469	C₂²⋊C₄	—	3	2	no	—
470	C₂²⋊C₄	—	3	1	no	—
471	C₂²⋊C₄	—	3	2	no	—
472	C₂²⋊C₄	—	3	2	no	—
473	C₂²⋊C₄	C₂×C₃²×D₃	1	2	yes	complemented
474	C₄²	—	3	1	no	—
475	C₂²×C₃	—	2	8	no	—
476	C₂²×C₃	—	2	4	no	—
477	C₂²×C₃	—	2	8	no	—
478	C₂²×C₃	—	2	4	no	—
479	C₂²×C₃	—	2	8	no	—
480	C₂²×C₃	—	2	4	no	—
481	C₂²×C₃	—	2	8	no	—
482	C₂²×C₃	—	2	1	no	—
483	C₂²×C₃	—	2	2	no	—
484	C₂²×C₃	—	2	1	no	—
485	C₂²×C₃	—	2	2	no	—
486	C₂²×C₃	—	2	1	no	—
487	C₂²×C₃	—	2	2	no	—
488	C₂²×C₃	—	2	4	no	—
489	C₂²×C₃	—	3	16	no	—
490	C₂²×C₃	—	3	8	no	—
491	C₂²×C₃	—	4	4	no	—
492	C₂²×C₃	—	4	4	no	—
493	C₂²×C₃	—	4	8	no	—
494	C₂²×C₃	—	4	8	no	—
495	C₂²×C₃	—	4	8	no	—
496	C₂²×C₃	—	6	4	no	—
497	C₂²×C₃	—	6	8	no	—
498	C₂²×C₃	—	6	4	no	—
499	C₂²×C₃	—	6	8	no	—
500	C₂²×C₃	—	12	32	no	—
501	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
502	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
503	C₂×D₃	—	2	1	no	—
504	C₂×D₃	—	2	2	no	—
505	C₂×D₃	—	2	2	no	—
506	C₂×D₃	—	2	1	no	—
507	C₂×D₃	—	4	8	no	—
508	C₂×D₃	C₂×C₃²×D₄	1	4	yes	—
509	C₂×D₃	C₃²×#41	1	2	yes	complemented
510	C₃×C₄	—	2	4	no	—
511	C₃×C₄	—	2	8	no	—
512	C₃×C₄	—	2	1	no	—
513	C₃×C₄	—	2	2	no	—
514	C₃×C₄	—	2	8	no	—
515	C₃×C₄	—	4	4	no	—
516	C₃×C₄	—	4	8	no	—
517	C₃×C₄	—	6	4	no	—
518	C₃×C₄	—	6	8	no	—
519	C₃×C₄	—	6	4	no	—
520	C₃×C₄	—	6	4	no	—
521	C₃×C₄	C₃×D₃×D₄	1	8	yes	—
522	C₃×C₄	C₂×C₃²×D₄	1	2	yes	—
523	Dic₃	—	2	2	no	—
524	Dic₃	—	2	1	no	—
525	Dic₃	—	2	1	no	—
526	Dic₃	—	2	1	no	—
527	C₃²	—	2	4	no	—
528	C₃²	#5612	1	1	yes	characteristic, direct product factor
529	C₃²	C₃×#546	1	4	yes	complemented
530	C₂³	—	3	1	no	—
531	C₂³	—	3	2	no	—
532	C₂³	—	6	4	no	—
533	C₂³	—	6	4	no	—
534	C₂³	—	12	2	no	—
535	C₂³	C₂²×C₃²×D₃	1	2	yes	—
536	C₂³	C₂²×C₃²×D₃	1	1	yes	characteristic
537	Q₈	—	6	2	no	—
538	D₄	—	2	2	no	—
539	D₄	—	3	4	no	—
540	D₄	—	4	2	no	—
541	D₄	—	6	4	no	—
542	D₄	—	6	4	no	—
543	D₄	—	6	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	3	1	no	—
546	C₂×C₄	—	3	1	no	—
547	C₂×C₄	—	3	2	no	—
548	C₂×C₄	—	3	1	no	—
549	C₂×C₄	—	6	4	no	—
550	C₂×C₄	—	6	2	no	—
551	C₂×C₄	—	12	2	no	—
552	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
553	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
554	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
555	C₂×C₃	—	2	4	no	—
556	C₂×C₃	—	2	4	no	—
557	C₂×C₃	—	2	4	no	—
558	C₂×C₃	—	4	8	no	—
559	C₂×C₃	—	4	8	no	—
560	C₂×C₃	—	4	4	no	—
561	C₂×C₃	—	4	2	no	—
562	C₂×C₃	—	4	1	no	—
563	C₂×C₃	—	6	8	no	—
564	C₂×C₃	—	6	16	no	—
565	C₂×C₃	—	8	8	no	—
566	C₂×C₃	C₂×C₃×D₃×D₄	1	4	yes	—
567	C₂×C₃	C₃×#1004	1	4	yes	—
568	C₂×C₃	C₃×D₃×#41	1	4	yes	—
569	C₂×C₃	C₂²×C₃²×D₄	1	1	yes	characteristic
570	C₂×C₃	C₂×C₃²×#41	1	1	yes	characteristic
571	C₂×C₃	C₃²×H(5,𝔽₂)	1	1	yes	characteristic
572	D₃	—	2	2	no	—
573	D₃	—	2	4	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	1	no	—
579	C₂²	—	2	2	no	—
580	C₂²	—	3	4	no	—
581	C₂²	—	3	2	no	—
582	C₂²	—	6	1	no	—
583	C₂²	—	6	2	no	—
584	C₂²	—	6	1	no	—
585	C₂²	—	6	2	no	—
586	C₂²	—	12	8	no	—
587	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
588	C₄	—	2	1	no	—
589	C₄	—	2	2	no	—
590	C₄	—	6	1	no	—
591	C₄	—	6	2	no	—
592	C₄	—	6	1	no	—
593	C₄	—	6	1	no	—
594	C₄	C₃²×D₃×D₄	1	2	yes	—
595	C₃	—	2	4	no	—
596	C₃	C₃×#5612	1	4	yes	direct product factor
597	C₃	C₃²×#546	1	1	yes	characteristic, complemented
598	C₂	—	4	2	no	—
599	C₂	—	4	1	no	—
600	C₂	—	6	2	no	—
601	C₂	—	6	4	no	—
602	C₂	C₂×C₃²×D₃×D₄	1	1	yes	characteristic
603	C₂	C₃²×#1004	1	1	yes	characteristic
604	C₂	C₃²×D₃×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×#121	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃²×#121	D₃
hypocenter	C₃	C₃²×#546
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×#121

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₃²×D₃×D₄	C₂	semidirect
2. C₂×C₃²×#235	C₂	semidirect
3. C₃³×#121	C₂	semidirect
4. C₃²×#938	C₂	semidirect
5. C₃²×#937	C₂	semidirect
6. C₃²×#936	C₂	semidirect
7. C₃²×C₄×#67	C₂	semidirect
8. C₃²×#897	C₂	semidirect
9. C₃²×#893	C₂	semidirect
10. C₃²×#883	C₂	semidirect
11. C₃²×D₃×C₂²⋊C₄	C₂	semidirect
13. C₃×#5612	C₃	direct (N × H)
16. C₂×C₃³×D₄	C₂²	semidirect
17. C₂²×C₃³×C₄	C₂²	semidirect
20. C₂×C₃²×#67	C₂²	semidirect
25. C₂×C₃²×C₄×D₃	C₂²	semidirect
26. C₂×C₃²×C₄×D₃	C₂²	semidirect
27. C₂×C₃²×Dic₆	C₂²	semidirect
28. C₃³×C₄⋊C₄	C₂²	semidirect
29. C₃³×C₂²⋊C₄	C₂²	semidirect
30. C₃²×#217	C₂²	semidirect
31. C₃²×#217	C₂²	semidirect
32. C₃²×#217	C₂²	semidirect
35. C₃²×(C₂×D₃)⋊C₄	C₂²	semidirect
36. C₃²×C₄⋊Dic₃	C₂²	semidirect
37. C₃²×Dic₃⋊C₄	C₂²	semidirect
38. C₃²×C₄×Dic₃	C₂²	semidirect
41. C₂×C₃×D₃×D₄	C₂×C₃	semidirect
42. C₂×C₃×#235	C₂×C₃	semidirect
48. C₃²×#121	D₃	semidirect
49. C₃²×#121	C₂×C₃	semidirect
54. C₃×#938	C₂×C₃	semidirect
55. C₃×#937	C₂×C₃	semidirect
56. C₃×#936	C₂×C₃	semidirect
57. C₃×C₄×#67	C₂×C₃	semidirect
58. C₃×#897	C₂×C₃	semidirect
59. C₃×#893	C₂×C₃	semidirect
60. C₃×#883	C₂×C₃	semidirect
61. C₃×D₃×C₂²⋊C₄	C₂×C₃	semidirect
82. C₃²×D₁₂	D₄	semidirect
88. #5612	C₃²	direct (N × H)
101. C₂×C₃²×D₄	C₂×D₃	semidirect
103. C₂×C₃²×D₄	C₂²×C₃	semidirect
108. C₂²×C₃²×C₄	C₂×D₃	semidirect
109. C₂²×C₃²×C₄	C₂²×C₃	semidirect
112. C₂×C₃×#67	C₂²×C₃	semidirect
117. C₂×C₃×C₄×D₃	C₂²×C₃	semidirect
118. C₂×C₃×C₄×D₃	C₂²×C₃	semidirect
119. C₂×C₃×Dic₆	C₂²×C₃	semidirect
123. C₃²×C₄⋊C₄	C₂×D₃	semidirect
124. C₃²×C₄⋊C₄	C₂²×C₃	semidirect
132. C₃²×C₂²⋊C₄	C₂×D₃	semidirect
133. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
135. C₃×#217	C₂²×C₃	semidirect
136. C₃×#217	C₂²×C₃	semidirect
137. C₃×#217	C₂²×C₃	semidirect
140. C₃×(C₂×D₃)⋊C₄	C₂²×C₃	semidirect
141. C₃×C₄⋊Dic₃	C₂²×C₃	semidirect
142. C₃×Dic₃⋊C₄	C₂²×C₃	semidirect
143. C₃×C₄×Dic₃	C₂²×C₃	semidirect
157. C₂×C₃²×D₃	#41	semidirect
167. C₂×D₃×D₄	C₂×C₃²	semidirect
168. C₂×#235	C₂×C₃²	semidirect
174. C₃×#121	C₃×D₃	semidirect
175. C₃×#121	C₂×C₃²	semidirect
180. #938	C₂×C₃²	semidirect
181. #937	C₂×C₃²	semidirect
182. #936	C₂×C₃²	semidirect
183. C₄×#67	C₂×C₃²	semidirect
184. #897	C₂×C₃²	semidirect
185. #893	C₂×C₃²	semidirect
186. #883	C₂×C₃²	semidirect
187. D₃×C₂²⋊C₄	C₂×C₃²	semidirect
241. C₃×D₁₂	C₃×D₄	semidirect
266. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
268. C₂×C₃×D₄	C₂²×C₃²	semidirect
273. C₂²×C₃×C₄	C₂×C₃×D₃	semidirect
274. C₂²×C₃×C₄	C₂²×C₃²	semidirect
277. C₂×#67	C₂²×C₃²	semidirect
282. C₂×C₄×D₃	C₂²×C₃²	semidirect
283. C₂×C₄×D₃	C₂²×C₃²	semidirect
284. C₂×Dic₆	C₂²×C₃²	semidirect
288. C₃×C₄⋊C₄	C₂×C₃×D₃	semidirect
289. C₃×C₄⋊C₄	C₂²×C₃²	semidirect
297. C₃×C₂²⋊C₄	C₂×C₃×D₃	semidirect
298. C₃×C₂²⋊C₄	C₂²×C₃²	semidirect
300. #217	C₂²×C₃²	semidirect
301. #217	C₂²×C₃²	semidirect
302. #217	C₂²×C₃²	semidirect
305. (C₂×D₃)⋊C₄	C₂²×C₃²	semidirect
306. C₄⋊Dic₃	C₂²×C₃²	semidirect
307. Dic₃⋊C₄	C₂²×C₃²	semidirect
308. C₄×Dic₃	C₂²×C₃²	semidirect
343. C₂×C₃×D₃	C₃×#41	semidirect
367. #121	C₃²×D₃	semidirect
372. C₃³	#546	semidirect
426. D₁₂	C₃²×D₄	semidirect
459. C₂×D₄	C₂×C₃²×D₃	semidirect
463. C₂²×C₄	C₂×C₃²×D₃	semidirect
466. C₄⋊C₄	C₂×C₃²×D₃	semidirect
473. C₂²⋊C₄	C₂×C₃²×D₃	semidirect
509. C₂×D₃	C₃²×#41	semidirect
528. C₃²	#5612	direct (N × H)
529. C₃²	C₃×#546	semidirect
596. C₃	C₃×#5612	direct (N × H)
597. C₃	C₃²×#546	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃²×D₃×D₄	2
C₂×C₃²×#235	2
C₃³×#121	2
C₃²×#938	2
C₃²×#937	2
C₃²×#936	2
C₃²×C₄×#67	2
C₃²×#897	2
C₃²×#893	2
C₃²×#883	2
C₃²×D₃×C₂²⋊C₄	2
C₃²×#546	3
C₃×#5612	3

Subgroups — C32×#5612 #13178991

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5612 #13178991