Subgroups — C₃²×#5545 #13178924

Order 1728.

2404 conjugacy classes of subgroups.

The subgroup lattice has 556 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
2	C₃³×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₄×#67	C₂	1	2	yes	maximal, complemented
5	C₂×C₃²×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃²×#893	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#892	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃²×#67⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₃²×D₃×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₃²×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃²×#872	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃²×#518	—	3	1	no	maximal
13	C₃×#5545	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	C₂²×C₃³×C₄	C₂²	1	2	yes	complemented
17	C₂×C₃²×#67	C₂²	1	2	yes	—
18	C₂²×C₃²×Dic₃	C₂²	1	2	yes	complemented
19	C₃²×D₃×D₄	C₄	1	8	yes	complemented
20	C₂×C₃²×D₁₂	C₂²	1	1	yes	characteristic
21	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
22	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	complemented
23	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	complemented
24	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
25	C₃³×C₂²⋊C₄	C₂²	1	2	yes	complemented
26	C₃³×C₄²	C₂²	1	1	yes	characteristic, complemented
27	C₃²×#217	C₂²	1	2	yes	complemented
28	C₃²×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
29	C₃²×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
30	C₃²×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
31	C₃²×(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
32	C₃²×(C₂×D₃)⋊C₄	C₂²	1	2	yes	complemented
33	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
34	C₃²×Dic₃⋊C₄	C₂²	1	1	yes	characteristic, complemented
35	C₃²×Dic₃⋊C₄	C₂²	1	1	yes	characteristic, complemented
36	C₃²×C₄×Dic₃	C₂²	1	2	yes	complemented
37	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic, complemented
38	C₂²×C₃²×D₄	—	3	1	no	—
39	C₂×C₃×D₃×D₄	C₂×C₃	1	4	yes	—
40	C₃²×C₄×D₄	—	2	4	no	—
41	C₃²×C₄×D₄	—	3	1	no	—
42	C₃²×C₄×D₄	—	3	2	no	—
43	C₃²×C₄×D₄	—	3	1	no	—
44	C₃²×C₄×D₄	—	3	2	no	—
45	C₃²×C₄×D₄	—	3	1	no	—
46	C₃²×C₄×D₄	D₃	1	1	yes	characteristic, complemented
47	C₃²×C₄×D₄	C₂×C₃	1	4	yes	complemented
48	C₃²×#117	—	3	1	no	—
49	C₃²×#117	—	3	1	no	—
50	C₂×C₃²×C₂²⋊C₄	—	3	2	no	—
51	C₂×C₃²×C₂²⋊C₄	—	3	2	no	—
52	C₃×D₄×Dic₃	C₂×C₃	1	4	yes	complemented
53	C₃×C₄×#67	C₂×C₃	1	8	yes	complemented
54	C₂×C₃×(C₂×D₃)⋊C₄	C₂×C₃	1	8	yes	complemented
55	C₃×#893	C₂×C₃	1	4	yes	complemented
56	C₃×#892	C₂×C₃	1	4	yes	complemented
57	C₃×#67⋊C₄	C₂×C₃	1	8	yes	complemented
58	C₃×D₃×C₂²⋊C₄	C₂×C₃	1	8	yes	complemented
59	C₃×C₄×D₁₂	C₂×C₃	1	4	yes	complemented
60	C₃×#872	C₂×C₃	1	4	yes	complemented
61	C₂³×C₃³	C₂³	1	2	yes	—
62	C₂²×C₃²×D₃	—	4	2	no	—
63	C₂²×C₃²×D₃	—	4	2	no	—
64	C₂²×C₃²×D₃	C₂×C₄	1	8	yes	complemented
65	C₂²×C₃²×D₃	C₂³	1	1	yes	characteristic
66	C₂²×C₃²×D₃	C₂³	1	2	yes	—
67	C₃³×D₄	C₂×C₄	1	4	yes	complemented
68	C₂×C₃³×C₄	—	4	2	no	—
69	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
70	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
71	C₂×C₃³×C₄	C₂³	1	2	yes	—
72	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
73	C₃²×#67	C₂×C₄	1	8	yes	complemented
74	C₂×C₃²×Dic₃	—	4	2	no	—
75	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
76	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
77	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
78	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
79	C₃²×D₁₂	C₂×C₄	1	4	yes	complemented
80	C₃²×C₄×D₃	—	4	2	no	—
81	C₃²×C₄×D₃	—	4	2	no	—
82	C₃²×C₄×D₃	C₂×C₄	1	4	yes	complemented
83	C₃×#518	—	3	4	no	—
84	#5545	C₃²	1	1	yes	characteristic, direct product factor
85	C₂⁴×C₃²	—	3	2	no	—
86	C₂³×C₃×D₃	C₂²×C₃	1	8	yes	—
87	C₂×C₃²×D₄	—	2	4	no	—
88	C₂×C₃²×D₄	—	3	2	no	—
89	C₂×C₃²×D₄	—	3	8	no	—
90	C₂×C₃²×D₄	—	3	1	no	—
91	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
92	C₂×C₃²×D₄	C₂²×C₃	1	4	yes	—
93	C₂²×C₃²×C₄	—	2	8	no	—
94	C₂²×C₃²×C₄	—	3	1	no	—
95	C₂²×C₃²×C₄	—	3	2	no	—
96	C₂²×C₃²×C₄	—	3	2	no	—
97	C₂²×C₃²×C₄	—	3	2	no	—
98	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	complemented
99	C₂²×C₃²×C₄	C₂²×C₃	1	8	yes	complemented
100	C₂×C₃×#67	C₂²×C₃	1	8	yes	—
101	C₂²×C₃×Dic₃	C₂²×C₃	1	8	yes	complemented
102	C₃×D₃×D₄	C₃×C₄	1	32	yes	complemented
103	C₂×C₃×D₁₂	C₂²×C₃	1	4	yes	—
104	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
105	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	complemented
106	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	complemented
107	C₃²×C₄⋊C₄	—	2	4	no	—
108	C₃²×C₄⋊C₄	—	3	1	no	—
109	C₃²×C₄⋊C₄	—	3	1	no	—
110	C₃²×C₄⋊C₄	—	3	1	no	—
111	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
112	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	complemented
113	C₃²×C₂²⋊C₄	—	2	8	no	—
114	C₃²×C₂²⋊C₄	—	3	2	no	—
115	C₃²×C₂²⋊C₄	—	3	2	no	—
116	C₃²×C₂²⋊C₄	—	3	1	no	—
117	C₃²×C₂²⋊C₄	—	3	1	no	—
118	C₃²×C₂²⋊C₄	—	3	2	no	—
119	C₃²×C₂²⋊C₄	—	3	2	no	—
120	C₃²×C₂²⋊C₄	C₂×D₃	1	2	yes	complemented
121	C₃²×C₂²⋊C₄	C₂²×C₃	1	8	yes	complemented
122	C₃²×C₄²	—	2	4	no	—
123	C₃²×C₄²	—	3	2	no	—
124	C₃²×C₄²	—	3	1	no	—
125	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic, complemented
126	C₃²×C₄²	C₂²×C₃	1	4	yes	complemented
127	C₃×#217	C₂²×C₃	1	8	yes	complemented
128	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
129	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
130	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
131	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	complemented
132	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	complemented
133	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	complemented
134	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	complemented
135	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	complemented
136	C₃×C₄×Dic₃	C₂²×C₃	1	8	yes	complemented
137	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	complemented
138	C₂²×C₃³	—	2	2	no	—
139	C₂²×C₃³	—	2	2	no	—
140	C₂²×C₃³	C₂²×C₄	1	4	yes	—
141	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
142	C₂×C₃²×D₃	—	2	2	no	—
143	C₂×C₃²×D₃	—	2	2	no	—
144	C₂×C₃²×D₃	—	2	1	no	—
145	C₂×C₃²×D₃	—	2	1	no	—
146	C₂×C₃²×D₃	—	4	8	no	—
147	C₂×C₃²×D₃	C₂²×C₄	1	2	yes	—
148	C₂×C₃²×D₃	C₂²×C₄	1	4	yes	—
149	C₃³×C₄	—	2	1	no	—
150	C₃³×C₄	—	2	1	no	—
151	C₃³×C₄	—	2	2	no	—
152	C₃³×C₄	C₂²×C₄	1	2	yes	—
153	C₃²×Dic₃	—	2	1	no	—
154	C₃²×Dic₃	—	2	2	no	—
155	C₃²×Dic₃	—	2	1	no	—
156	C₃²×Dic₃	C₂²×C₄	1	2	yes	—
157	C₂²×C₃×D₄	—	3	4	no	—
158	C₂×D₃×D₄	C₂×C₃²	1	1	yes	characteristic
159	C₃×C₄×D₄	—	2	4	no	—
160	C₃×C₄×D₄	—	3	4	no	—
161	C₃×C₄×D₄	—	3	8	no	—
162	C₃×C₄×D₄	—	3	4	no	—
163	C₃×C₄×D₄	—	3	8	no	—
164	C₃×C₄×D₄	—	3	4	no	—
165	C₃×C₄×D₄	C₃×D₃	1	4	yes	complemented
166	C₃×C₄×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
167	C₃×#117	—	3	4	no	—
168	C₃×#117	—	3	4	no	—
169	C₂×C₃×C₂²⋊C₄	—	3	8	no	—
170	C₂×C₃×C₂²⋊C₄	—	3	8	no	—
171	D₄×Dic₃	C₂×C₃²	1	1	yes	characteristic, complemented
172	C₄×#67	C₂×C₃²	1	2	yes	complemented
173	C₂×(C₂×D₃)⋊C₄	C₂×C₃²	1	2	yes	complemented
174	#893	C₂×C₃²	1	1	yes	characteristic, complemented
175	#892	C₂×C₃²	1	1	yes	characteristic, complemented
176	#67⋊C₄	C₂×C₃²	1	2	yes	complemented
177	D₃×C₂²⋊C₄	C₂×C₃²	1	2	yes	complemented
178	C₄×D₁₂	C₂×C₃²	1	1	yes	characteristic, complemented
179	#872	C₂×C₃²	1	1	yes	characteristic, complemented
180	C₂³×C₃²	—	2	8	no	—
181	C₂³×C₃²	—	3	1	no	—
182	C₂³×C₃²	—	3	2	no	—
183	C₂³×C₃²	—	3	8	no	—
184	C₂³×C₃²	—	12	2	no	—
185	C₂³×C₃²	—	12	2	no	—
186	C₂³×C₃²	C₂²×D₃	1	2	yes	—
187	C₂³×C₃²	C₂³×C₃	1	8	yes	—
188	C₂²×C₃×D₃	—	4	8	no	—
189	C₂²×C₃×D₃	—	4	8	no	—
190	C₂²×C₃×D₃	C₂×C₃×C₄	1	32	yes	complemented
191	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
192	C₂²×C₃×D₃	C₂³×C₃	1	8	yes	—
193	C₃²×D₄	—	2	16	no	—
194	C₃²×D₄	—	3	8	no	—
195	C₃²×D₄	—	3	4	no	—
196	C₃²×D₄	C₄×D₃	1	4	yes	complemented
197	C₃²×D₄	C₂×C₃×C₄	1	16	yes	complemented
198	C₂×C₃²×C₄	—	2	4	no	—
199	C₂×C₃²×C₄	—	2	4	no	—
200	C₂×C₃²×C₄	—	2	4	no	—
201	C₂×C₃²×C₄	—	2	8	no	—
202	C₂×C₃²×C₄	—	3	1	no	—
203	C₂×C₃²×C₄	—	3	4	no	—
204	C₂×C₃²×C₄	—	3	1	no	—
205	C₂×C₃²×C₄	—	3	2	no	—
206	C₂×C₃²×C₄	—	3	1	no	—
207	C₂×C₃²×C₄	—	4	2	no	—
208	C₂×C₃²×C₄	—	4	8	no	—
209	C₂×C₃²×C₄	—	4	16	no	—
210	C₂×C₃²×C₄	—	12	2	no	—
211	C₂×C₃²×C₄	—	12	2	no	—
212	C₂×C₃²×C₄	—	12	2	no	—
213	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
214	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
215	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
216	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
217	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
218	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
219	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
220	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
221	C₃×#67	C₂×C₃×C₄	1	32	yes	complemented
222	C₂×C₃×Dic₃	—	4	8	no	—
223	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
224	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
225	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
226	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
227	C₃×D₁₂	C₂×C₃×C₄	1	16	yes	complemented
228	C₃×C₄×D₃	—	4	8	no	—
229	C₃×C₄×D₃	—	4	8	no	—
230	C₃×C₄×D₃	C₂×C₃×C₄	1	16	yes	complemented
231	#518	—	3	1	no	Sylow
232	C₂×C₃³	—	2	4	no	—
233	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
234	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
235	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
236	C₃²×D₃	—	2	2	no	—
237	C₃²×D₃	—	2	4	no	—
238	C₂⁴×C₃	—	3	8	no	—
239	C₂³×D₃	C₂²×C₃²	1	2	yes	—
240	C₂×C₃×D₄	—	2	4	no	—
241	C₂×C₃×D₄	—	3	8	no	—
242	C₂×C₃×D₄	—	3	32	no	—
243	C₂×C₃×D₄	—	3	4	no	—
244	C₂×C₃×D₄	C₂×C₃×D₃	1	4	yes	—
245	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
246	C₂²×C₃×C₄	—	2	8	no	—
247	C₂²×C₃×C₄	—	3	4	no	—
248	C₂²×C₃×C₄	—	3	8	no	—
249	C₂²×C₃×C₄	—	3	8	no	—
250	C₂²×C₃×C₄	—	3	8	no	—
251	C₂²×C₃×C₄	C₂×C₃×D₃	1	8	yes	complemented
252	C₂²×C₃×C₄	C₂²×C₃²	1	2	yes	complemented
253	C₂×#67	C₂²×C₃²	1	2	yes	—
254	C₂²×Dic₃	C₂²×C₃²	1	2	yes	complemented
255	D₃×D₄	C₃²×C₄	1	8	yes	complemented
256	C₂×D₁₂	C₂²×C₃²	1	1	yes	characteristic
257	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
258	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	complemented
259	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	complemented
260	C₃×C₄⋊C₄	—	2	4	no	—
261	C₃×C₄⋊C₄	—	3	4	no	—
262	C₃×C₄⋊C₄	—	3	4	no	—
263	C₃×C₄⋊C₄	—	3	4	no	—
264	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	complemented
265	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
266	C₃×C₂²⋊C₄	—	2	8	no	—
267	C₃×C₂²⋊C₄	—	3	8	no	—
268	C₃×C₂²⋊C₄	—	3	8	no	—
269	C₃×C₂²⋊C₄	—	3	4	no	—
270	C₃×C₂²⋊C₄	—	3	4	no	—
271	C₃×C₂²⋊C₄	—	3	8	no	—
272	C₃×C₂²⋊C₄	—	3	8	no	—
273	C₃×C₂²⋊C₄	C₂×C₃×D₃	1	8	yes	complemented
274	C₃×C₂²⋊C₄	C₂²×C₃²	1	2	yes	complemented
275	C₃×C₄²	—	2	4	no	—
276	C₃×C₄²	—	3	8	no	—
277	C₃×C₄²	—	3	4	no	—
278	C₃×C₄²	C₂×C₃×D₃	1	4	yes	complemented
279	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic, complemented
280	#217	C₂²×C₃²	1	2	yes	complemented
281	(C₂×D₃)⋊C₄	C₂²×C₃²	1	1	yes	characteristic
282	(C₂×D₃)⋊C₄	C₂²×C₃²	1	1	yes	characteristic
283	(C₂×D₃)⋊C₄	C₂²×C₃²	1	2	yes	—
284	(C₂×D₃)⋊C₄	C₂²×C₃²	1	2	yes	complemented
285	(C₂×D₃)⋊C₄	C₂²×C₃²	1	2	yes	complemented
286	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic, complemented
287	Dic₃⋊C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
288	Dic₃⋊C₄	C₂²×C₃²	1	1	yes	characteristic, complemented
289	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic, complemented
290	C₄×Dic₃	C₂²×C₃²	1	2	yes	complemented
291	C₂²×C₃²	—	2	2	no	—
292	C₂²×C₃²	—	2	2	no	—
293	C₂²×C₃²	—	2	4	no	—
294	C₂²×C₃²	—	2	16	no	—
295	C₂²×C₃²	—	2	8	no	—
296	C₂²×C₃²	—	2	8	no	—
297	C₂²×C₃²	—	3	2	no	—
298	C₂²×C₃²	—	3	4	no	—
299	C₂²×C₃²	—	4	8	no	—
300	C₂²×C₃²	—	4	8	no	—
301	C₂²×C₃²	—	6	2	no	—
302	C₂²×C₃²	—	6	1	no	—
303	C₂²×C₃²	—	6	2	no	—
304	C₂²×C₃²	—	6	1	no	—
305	C₂²×C₃²	—	12	8	no	—
306	C₂²×C₃²	C₂×C₄×D₃	1	4	yes	—
307	C₂²×C₃²	C₂²×C₃×C₄	1	16	yes	—
308	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
309	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
310	C₂×C₃×D₃	—	2	8	no	—
311	C₂×C₃×D₃	—	2	8	no	—
312	C₂×C₃×D₃	—	2	4	no	—
313	C₂×C₃×D₃	—	2	4	no	—
314	C₂×C₃×D₃	—	4	32	no	—
315	C₂×C₃×D₃	C₂²×C₃×C₄	1	8	yes	—
316	C₂×C₃×D₃	C₂²×C₃×C₄	1	16	yes	—
317	C₃²×C₄	—	2	8	no	—
318	C₃²×C₄	—	2	1	no	—
319	C₃²×C₄	—	2	2	no	—
320	C₃²×C₄	—	2	1	no	—
321	C₃²×C₄	—	2	4	no	—
322	C₃²×C₄	—	2	8	no	—
323	C₃²×C₄	—	2	4	no	—
324	C₃²×C₄	—	2	8	no	—
325	C₃²×C₄	—	3	2	no	—
326	C₃²×C₄	—	4	4	no	—
327	C₃²×C₄	—	4	8	no	—
328	C₃²×C₄	—	6	1	no	—
329	C₃²×C₄	—	6	2	no	—
330	C₃²×C₄	—	6	1	no	—
331	C₃²×C₄	C₂×C₄×D₃	1	2	yes	—
332	C₃²×C₄	C₂²×C₃×C₄	1	8	yes	—
333	C₃×Dic₃	—	2	4	no	—
334	C₃×Dic₃	—	2	8	no	—
335	C₃×Dic₃	—	2	4	no	—
336	C₃×Dic₃	C₂²×C₃×C₄	1	8	yes	—
337	C₂²×D₄	—	3	1	no	—
338	C₄×D₄	—	3	1	no	—
339	C₄×D₄	—	3	2	no	—
340	C₄×D₄	—	3	1	no	—
341	C₄×D₄	—	3	2	no	—
342	C₄×D₄	—	3	1	no	—
343	C₄×D₄	C₃²×D₃	1	1	yes	characteristic, complemented
344	#117	—	3	1	no	—
345	#117	—	3	1	no	—
346	C₂×C₂²⋊C₄	—	3	2	no	—
347	C₂×C₂²⋊C₄	—	3	2	no	—
348	C₃³	#518	1	1	yes	characteristic, complemented, Sylow
349	C₂³×C₃	—	2	8	no	—
350	C₂³×C₃	—	3	4	no	—
351	C₂³×C₃	—	3	8	no	—
352	C₂³×C₃	—	3	32	no	—
353	C₂³×C₃	—	12	8	no	—
354	C₂³×C₃	—	12	8	no	—
355	C₂³×C₃	C₂²×C₃×D₃	1	8	yes	—
356	C₂³×C₃	C₂³×C₃²	1	2	yes	—
357	C₂²×D₃	—	4	2	no	—
358	C₂²×D₃	—	4	2	no	—
359	C₂²×D₃	C₂×C₃²×C₄	1	8	yes	complemented
360	C₂²×D₃	C₂³×C₃²	1	1	yes	characteristic
361	C₂²×D₃	C₂³×C₃²	1	2	yes	—
362	C₃×D₄	—	2	16	no	—
363	C₃×D₄	—	3	32	no	—
364	C₃×D₄	—	3	16	no	—
365	C₃×D₄	C₃×C₄×D₃	1	16	yes	complemented
366	C₃×D₄	C₂×C₃²×C₄	1	4	yes	complemented
367	C₂×C₃×C₄	—	2	4	no	—
368	C₂×C₃×C₄	—	2	4	no	—
369	C₂×C₃×C₄	—	2	4	no	—
370	C₂×C₃×C₄	—	2	8	no	—
371	C₂×C₃×C₄	—	3	4	no	—
372	C₂×C₃×C₄	—	3	16	no	—
373	C₂×C₃×C₄	—	3	4	no	—
374	C₂×C₃×C₄	—	3	4	no	—
375	C₂×C₃×C₄	—	3	8	no	—
376	C₂×C₃×C₄	—	4	8	no	—
377	C₂×C₃×C₄	—	4	2	no	—
378	C₂×C₃×C₄	—	4	16	no	—
379	C₂×C₃×C₄	—	12	8	no	—
380	C₂×C₃×C₄	—	12	8	no	—
381	C₂×C₃×C₄	—	12	8	no	—
382	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
383	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
384	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
385	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
386	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
387	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
388	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
389	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
390	#67	C₂×C₃²×C₄	1	8	yes	complemented
391	C₂×Dic₃	—	4	2	no	—
392	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
393	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
394	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
395	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
396	D₁₂	C₂×C₃²×C₄	1	4	yes	complemented
397	C₄×D₃	—	4	2	no	—
398	C₄×D₃	—	4	2	no	—
399	C₄×D₃	C₂×C₃²×C₄	1	4	yes	complemented
400	C₂×C₃²	—	2	4	no	—
401	C₂×C₃²	—	2	4	no	—
402	C₂×C₃²	—	2	4	no	—
403	C₂×C₃²	—	2	4	no	—
404	C₂×C₃²	—	2	16	no	—
405	C₂×C₃²	—	4	16	no	—
406	C₂×C₃²	—	6	4	no	—
407	C₂×C₃²	—	6	2	no	—
408	C₂×C₃²	C₂²×C₄×D₃	1	1	yes	characteristic
409	C₂×C₃²	#1004	1	1	yes	characteristic
410	C₂×C₃²	#1009	1	1	yes	characteristic
411	C₂×C₃²	C₂³×C₃×C₄	1	4	yes	—
412	C₂×C₃²	C₃×H(5,𝔽₂)	1	4	yes	—
413	C₂×C₃²	C₃×H(5,𝔽₂)	1	4	yes	—
414	C₃×D₃	—	2	8	no	—
415	C₃×D₃	—	2	16	no	—
416	C₂⁴	—	3	2	no	—
417	C₂×D₄	—	3	2	no	—
418	C₂×D₄	—	3	8	no	—
419	C₂×D₄	—	3	1	no	—
420	C₂×D₄	C₂×C₃²×D₃	1	1	yes	characteristic
421	C₂²×C₄	—	3	1	no	—
422	C₂²×C₄	—	3	2	no	—
423	C₂²×C₄	—	3	2	no	—
424	C₂²×C₄	—	3	2	no	—
425	C₂²×C₄	C₂×C₃²×D₃	1	2	yes	complemented
426	C₄⋊C₄	—	3	1	no	—
427	C₄⋊C₄	—	3	1	no	—
428	C₄⋊C₄	—	3	1	no	—
429	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
430	C₂²⋊C₄	—	3	2	no	—
431	C₂²⋊C₄	—	3	2	no	—
432	C₂²⋊C₄	—	3	1	no	—
433	C₂²⋊C₄	—	3	1	no	—
434	C₂²⋊C₄	—	3	2	no	—
435	C₂²⋊C₄	—	3	2	no	—
436	C₂²⋊C₄	C₂×C₃²×D₃	1	2	yes	complemented
437	C₄²	—	3	2	no	—
438	C₄²	—	3	1	no	—
439	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
440	C₂²×C₃	—	2	8	no	—
441	C₂²×C₃	—	2	8	no	—
442	C₂²×C₃	—	2	2	no	—
443	C₂²×C₃	—	2	2	no	—
444	C₂²×C₃	—	2	4	no	—
445	C₂²×C₃	—	2	16	no	—
446	C₂²×C₃	—	3	8	no	—
447	C₂²×C₃	—	3	16	no	—
448	C₂²×C₃	—	4	8	no	—
449	C₂²×C₃	—	4	8	no	—
450	C₂²×C₃	—	6	8	no	—
451	C₂²×C₃	—	6	4	no	—
452	C₂²×C₃	—	6	8	no	—
453	C₂²×C₃	—	6	4	no	—
454	C₂²×C₃	—	12	32	no	—
455	C₂²×C₃	C₂×C₃×C₄×D₃	1	16	yes	—
456	C₂²×C₃	C₂²×C₃²×C₄	1	4	yes	—
457	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
458	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
459	C₂×D₃	—	2	2	no	—
460	C₂×D₃	—	2	2	no	—
461	C₂×D₃	—	2	1	no	—
462	C₂×D₃	—	2	1	no	—
463	C₂×D₃	—	4	8	no	—
464	C₂×D₃	C₂²×C₃²×C₄	1	2	yes	—
465	C₂×D₃	C₂²×C₃²×C₄	1	4	yes	—
466	C₃×C₄	—	2	8	no	—
467	C₃×C₄	—	2	4	no	—
468	C₃×C₄	—	2	8	no	—
469	C₃×C₄	—	2	4	no	—
470	C₃×C₄	—	2	1	no	—
471	C₃×C₄	—	2	2	no	—
472	C₃×C₄	—	2	1	no	—
473	C₃×C₄	—	2	8	no	—
474	C₃×C₄	—	3	8	no	—
475	C₃×C₄	—	4	8	no	—
476	C₃×C₄	—	4	4	no	—
477	C₃×C₄	—	6	4	no	—
478	C₃×C₄	—	6	8	no	—
479	C₃×C₄	—	6	4	no	—
480	C₃×C₄	C₂×C₃×C₄×D₃	1	8	yes	—
481	C₃×C₄	C₂²×C₃²×C₄	1	2	yes	—
482	Dic₃	—	2	1	no	—
483	Dic₃	—	2	2	no	—
484	Dic₃	—	2	1	no	—
485	Dic₃	C₂²×C₃²×C₄	1	2	yes	—
486	C₃²	—	2	4	no	—
487	C₃²	#5545	1	1	yes	characteristic, direct product factor
488	C₃²	C₃×#518	1	4	yes	complemented
489	C₂³	—	3	1	no	—
490	C₂³	—	3	2	no	—
491	C₂³	—	3	8	no	—
492	C₂³	—	12	2	no	—
493	C₂³	—	12	2	no	—
494	C₂³	C₂²×C₃²×D₃	1	2	yes	—
495	D₄	—	3	8	no	—
496	D₄	—	3	4	no	—
497	D₄	C₃²×C₄×D₃	1	4	yes	complemented
498	C₂×C₄	—	3	1	no	—
499	C₂×C₄	—	3	4	no	—
500	C₂×C₄	—	3	1	no	—
501	C₂×C₄	—	3	2	no	—
502	C₂×C₄	—	3	1	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	12	2	no	—
505	C₂×C₄	—	12	2	no	—
506	C₂×C₄	—	12	2	no	—
507	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
508	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
509	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
510	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
511	C₂×C₃	—	2	16	no	—
512	C₂×C₃	—	2	4	no	—
513	C₂×C₃	—	2	4	no	—
514	C₂×C₃	—	2	4	no	—
515	C₂×C₃	—	2	4	no	—
516	C₂×C₃	—	4	16	no	—
517	C₂×C₃	—	6	8	no	—
518	C₂×C₃	—	6	16	no	—
519	C₂×C₃	C₂²×C₃×C₄×D₃	1	4	yes	—
520	C₂×C₃	C₃×#1004	1	4	yes	—
521	C₂×C₃	C₃×#1009	1	4	yes	—
522	C₂×C₃	C₂³×C₃²×C₄	1	1	yes	characteristic
523	C₂×C₃	C₃²×H(5,𝔽₂)	1	1	yes	characteristic
524	C₂×C₃	C₃²×H(5,𝔽₂)	1	1	yes	characteristic
525	D₃	—	2	2	no	—
526	D₃	—	2	4	no	—
527	C₂²	—	2	2	no	—
528	C₂²	—	2	2	no	—
529	C₂²	—	3	2	no	—
530	C₂²	—	3	4	no	—
531	C₂²	—	6	2	no	—
532	C₂²	—	6	1	no	—
533	C₂²	—	6	2	no	—
534	C₂²	—	6	1	no	—
535	C₂²	—	12	8	no	—
536	C₂²	C₂×C₃²×C₄×D₃	1	4	yes	—
537	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
538	C₄	—	2	1	no	—
539	C₄	—	2	2	no	—
540	C₄	—	2	1	no	—
541	C₄	—	3	2	no	—
542	C₄	—	6	1	no	—
543	C₄	—	6	2	no	—
544	C₄	—	6	1	no	—
545	C₄	C₂×C₃²×C₄×D₃	1	2	yes	—
546	C₃	—	2	4	no	—
547	C₃	C₃×#5545	1	4	yes	direct product factor
548	C₃	C₃²×#518	1	1	yes	characteristic, complemented
549	C₂	—	2	4	no	—
550	C₂	—	6	2	no	—
551	C₂	—	6	4	no	—
552	C₂	C₂²×C₃²×C₄×D₃	1	1	yes	characteristic
553	C₂	C₃²×#1004	1	1	yes	characteristic
554	C₂	C₃²×#1009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator	C₂×C₃	C₂³×C₃²×C₄
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×C₄×D₄	C₂
socle	C₂²×C₃³	C₂⁴
prime radical	C₂×C₃²×D₃×D₄	C₂
cosocle	C₂²×C₃	C₂⁴×C₃²
hypercenter	C₃²×C₄×D₄	D₃
hypocenter	C₃	C₃²×#518
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₃³×C₄×D₄	C₂	semidirect
3. C₃²×D₄×Dic₃	C₂	semidirect
4. C₃²×C₄×#67	C₂	semidirect
5. C₂×C₃²×(C₂×D₃)⋊C₄	C₂	semidirect
6. C₃²×#893	C₂	semidirect
7. C₃²×#892	C₂	semidirect
8. C₃²×#67⋊C₄	C₂	semidirect
9. C₃²×D₃×C₂²⋊C₄	C₂	semidirect
10. C₃²×C₄×D₁₂	C₂	semidirect
11. C₃²×#872	C₂	semidirect
13. C₃×#5545	C₃	direct (N × H)
16. C₂²×C₃³×C₄	C₂²	semidirect
18. C₂²×C₃²×Dic₃	C₂²	semidirect
19. C₃²×D₃×D₄	C₄	semidirect
22. C₂×C₃²×C₄×D₃	C₂²	semidirect
23. C₂×C₃²×C₄×D₃	C₂²	semidirect
24. C₃³×C₄⋊C₄	C₂²	semidirect
25. C₃³×C₂²⋊C₄	C₂²	semidirect
26. C₃³×C₄²	C₂²	semidirect
27. C₃²×#217	C₂²	semidirect
31. C₃²×(C₂×D₃)⋊C₄	C₂²	semidirect
32. C₃²×(C₂×D₃)⋊C₄	C₂²	semidirect
33. C₃²×C₄⋊Dic₃	C₂²	semidirect
34. C₃²×Dic₃⋊C₄	C₂²	semidirect
35. C₃²×Dic₃⋊C₄	C₂²	semidirect
36. C₃²×C₄×Dic₃	C₂²	semidirect
37. C₃²×C₄×Dic₃	C₂²	semidirect
46. C₃²×C₄×D₄	D₃	semidirect
47. C₃²×C₄×D₄	C₂×C₃	semidirect
52. C₃×D₄×Dic₃	C₂×C₃	semidirect
53. C₃×C₄×#67	C₂×C₃	semidirect
54. C₂×C₃×(C₂×D₃)⋊C₄	C₂×C₃	semidirect
55. C₃×#893	C₂×C₃	semidirect
56. C₃×#892	C₂×C₃	semidirect
57. C₃×#67⋊C₄	C₂×C₃	semidirect
58. C₃×D₃×C₂²⋊C₄	C₂×C₃	semidirect
59. C₃×C₄×D₁₂	C₂×C₃	semidirect
60. C₃×#872	C₂×C₃	semidirect
64. C₂²×C₃²×D₃	C₂×C₄	semidirect
67. C₃³×D₄	C₂×C₄	semidirect
73. C₃²×#67	C₂×C₄	semidirect
79. C₃²×D₁₂	C₂×C₄	semidirect
82. C₃²×C₄×D₃	C₂×C₄	semidirect
84. #5545	C₃²	direct (N × H)
98. C₂²×C₃²×C₄	C₂×D₃	semidirect
99. C₂²×C₃²×C₄	C₂²×C₃	semidirect
101. C₂²×C₃×Dic₃	C₂²×C₃	semidirect
102. C₃×D₃×D₄	C₃×C₄	semidirect
105. C₂×C₃×C₄×D₃	C₂²×C₃	semidirect
106. C₂×C₃×C₄×D₃	C₂²×C₃	semidirect
111. C₃²×C₄⋊C₄	C₂×D₃	semidirect
112. C₃²×C₄⋊C₄	C₂²×C₃	semidirect
120. C₃²×C₂²⋊C₄	C₂×D₃	semidirect
121. C₃²×C₂²⋊C₄	C₂²×C₃	semidirect
125. C₃²×C₄²	C₂×D₃	semidirect
126. C₃²×C₄²	C₂²×C₃	semidirect
127. C₃×#217	C₂²×C₃	semidirect
131. C₃×(C₂×D₃)⋊C₄	C₂²×C₃	semidirect
132. C₃×(C₂×D₃)⋊C₄	C₂²×C₃	semidirect
133. C₃×C₄⋊Dic₃	C₂²×C₃	semidirect
134. C₃×Dic₃⋊C₄	C₂²×C₃	semidirect
135. C₃×Dic₃⋊C₄	C₂²×C₃	semidirect
136. C₃×C₄×Dic₃	C₂²×C₃	semidirect
137. C₃×C₄×Dic₃	C₂²×C₃	semidirect
165. C₃×C₄×D₄	C₃×D₃	semidirect
166. C₃×C₄×D₄	C₂×C₃²	semidirect
171. D₄×Dic₃	C₂×C₃²	semidirect
172. C₄×#67	C₂×C₃²	semidirect
173. C₂×(C₂×D₃)⋊C₄	C₂×C₃²	semidirect
174. #893	C₂×C₃²	semidirect
175. #892	C₂×C₃²	semidirect
176. #67⋊C₄	C₂×C₃²	semidirect
177. D₃×C₂²⋊C₄	C₂×C₃²	semidirect
178. C₄×D₁₂	C₂×C₃²	semidirect
179. #872	C₂×C₃²	semidirect
190. C₂²×C₃×D₃	C₂×C₃×C₄	semidirect
196. C₃²×D₄	C₄×D₃	semidirect
197. C₃²×D₄	C₂×C₃×C₄	semidirect
221. C₃×#67	C₂×C₃×C₄	semidirect
227. C₃×D₁₂	C₂×C₃×C₄	semidirect
230. C₃×C₄×D₃	C₂×C₃×C₄	semidirect
251. C₂²×C₃×C₄	C₂×C₃×D₃	semidirect
252. C₂²×C₃×C₄	C₂²×C₃²	semidirect
254. C₂²×Dic₃	C₂²×C₃²	semidirect
255. D₃×D₄	C₃²×C₄	semidirect
258. C₂×C₄×D₃	C₂²×C₃²	semidirect
259. C₂×C₄×D₃	C₂²×C₃²	semidirect
264. C₃×C₄⋊C₄	C₂×C₃×D₃	semidirect
265. C₃×C₄⋊C₄	C₂²×C₃²	semidirect
273. C₃×C₂²⋊C₄	C₂×C₃×D₃	semidirect
274. C₃×C₂²⋊C₄	C₂²×C₃²	semidirect
278. C₃×C₄²	C₂×C₃×D₃	semidirect
279. C₃×C₄²	C₂²×C₃²	semidirect
280. #217	C₂²×C₃²	semidirect
284. (C₂×D₃)⋊C₄	C₂²×C₃²	semidirect
285. (C₂×D₃)⋊C₄	C₂²×C₃²	semidirect
286. C₄⋊Dic₃	C₂²×C₃²	semidirect
287. Dic₃⋊C₄	C₂²×C₃²	semidirect
288. Dic₃⋊C₄	C₂²×C₃²	semidirect
289. C₄×Dic₃	C₂²×C₃²	semidirect
290. C₄×Dic₃	C₂²×C₃²	semidirect
343. C₄×D₄	C₃²×D₃	semidirect
348. C₃³	#518	semidirect
359. C₂²×D₃	C₂×C₃²×C₄	semidirect
365. C₃×D₄	C₃×C₄×D₃	semidirect
366. C₃×D₄	C₂×C₃²×C₄	semidirect
390. #67	C₂×C₃²×C₄	semidirect
396. D₁₂	C₂×C₃²×C₄	semidirect
399. C₄×D₃	C₂×C₃²×C₄	semidirect
425. C₂²×C₄	C₂×C₃²×D₃	semidirect
429. C₄⋊C₄	C₂×C₃²×D₃	semidirect
436. C₂²⋊C₄	C₂×C₃²×D₃	semidirect
439. C₄²	C₂×C₃²×D₃	semidirect
487. C₃²	#5545	direct (N × H)
488. C₃²	C₃×#518	semidirect
497. D₄	C₃²×C₄×D₃	semidirect
547. C₃	C₃×#5545	direct (N × H)
548. C₃	C₃²×#518	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
C₃³×C₄×D₄	2
C₃²×D₄×Dic₃	2
C₃²×C₄×#67	2
C₂×C₃²×(C₂×D₃)⋊C₄	2
C₃²×#893	2
C₃²×#892	2
C₃²×#67⋊C₄	2
C₃²×D₃×C₂²⋊C₄	2
C₃²×C₄×D₁₂	2
C₃²×#872	2
C₃²×#518	3
C₃×#5545	3

Subgroups — C32×#5545 #13178924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5545 #13178924