Subgroups — C₃²×#5689 #13179067

Order 1728.

1460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#897	C₂	1	2	yes	maximal
3	C₃²×#896	C₂	1	2	yes	maximal
4	C₃²×#892	C₂	1	1	yes	maximal, characteristic
5	C₃²×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₃²×#890	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃²×#889	C₂	1	2	yes	maximal, complemented
8	C₃²×#888	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃²×#888	C₂	1	2	yes	maximal, complemented
10	C₃²×#872	C₂	1	1	yes	maximal, characteristic
11	C₃²×#869	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃²×#544	—	3	1	no	maximal
13	C₃×#5689	C₃	1	4	yes	maximal, direct product factor
14	C₂×C₃²×C₄×D₃	C₂²	1	2	yes	—
15	C₂×C₃²×C₄×D₃	C₂²	1	1	yes	characteristic
16	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
17	C₂×C₃²×Dic₆	C₂²	1	2	yes	—
18	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
21	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃³×C₄²	C₂²	1	1	yes	characteristic
23	C₃²×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
24	C₃²×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
25	C₃²×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
28	C₃²×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₃²×C₄⋊Dic₃	C₂²	1	2	yes	—
30	C₃²×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
31	C₃²×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
32	C₃²×Dic₃⋊C₄	C₂²	1	2	yes	—
33	C₃²×Dic₃⋊C₄	C₂²	1	2	yes	—
34	C₃²×Dic₃⋊C₄	C₂²	1	2	yes	—
35	C₃²×C₄×Dic₃	C₂²	1	1	yes	characteristic
36	C₃²×C₄×Dic₃	C₂²	1	2	yes	—
37	C₃²×C₄⋊Q₈	—	3	2	no	—
38	C₃²×C₄⋊Q₈	—	3	1	no	—
39	C₃²×C₄⋊Q₈	—	3	1	no	—
40	C₃²×#125	—	2	4	no	—
41	C₃²×#125	—	3	2	no	—
42	C₃²×#125	—	3	1	no	—
43	C₃²×#125	D₃	1	1	yes	characteristic, complemented
44	C₃²×#125	C₂×C₃	1	4	yes	complemented
45	C₃²×C₂²⋊Q₈	—	3	2	no	—
46	C₃²×C₂²⋊Q₈	—	3	2	no	—
47	C₃²×#117	—	3	1	no	—
48	C₃²×#117	—	3	1	no	—
49	C₂×C₃²×C₄⋊C₄	—	3	1	no	—
50	C₃×#897	C₂×C₃	1	8	yes	—
51	C₃×#896	C₂×C₃	1	8	yes	—
52	C₃×#892	C₂×C₃	1	4	yes	—
53	C₃×D₃×C₄⋊C₄	C₂×C₃	1	4	yes	—
54	C₃×#890	C₂×C₃	1	4	yes	complemented
55	C₃×#889	C₂×C₃	1	8	yes	complemented
56	C₃×#888	C₂×C₃	1	8	yes	complemented
57	C₃×#888	C₂×C₃	1	4	yes	complemented
58	C₃×#872	C₂×C₃	1	4	yes	—
59	C₃×#869	C₂×C₃	1	4	yes	complemented
60	C₂²×C₃²×D₃	C₂³	1	1	yes	characteristic
61	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
62	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
63	C₂×C₃³×C₄	C₂³	1	2	yes	—
64	C₂×C₃³×C₄	C₂³	1	2	yes	—
65	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
66	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
67	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
68	C₂×C₃²×Dic₃	C₂³	1	2	yes	—
69	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
70	C₂×C₃²×Dic₃	C₂³	1	1	yes	characteristic
71	C₃²×C₄×D₃	—	4	2	no	—
72	C₃²×C₄×D₃	Q₈	1	4	yes	complemented
73	C₃²×Dic₆	—	4	2	no	—
74	C₃²×Dic₆	—	4	2	no	—
75	C₃×#544	—	3	4	no	—
76	#5689	C₃²	1	1	yes	characteristic, direct product factor
77	C₂×C₃²×Q₈	—	3	2	no	—
78	C₂×C₃²×Q₈	—	3	2	no	—
79	C₂²×C₃²×C₄	—	3	1	no	—
80	C₂²×C₃²×C₄	—	3	2	no	—
81	C₂×C₃×C₄×D₃	C₂²×C₃	1	8	yes	—
82	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
83	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
84	C₂×C₃×Dic₆	C₂²×C₃	1	8	yes	—
85	C₃²×C₄⋊C₄	—	2	4	no	—
86	C₃²×C₄⋊C₄	—	2	4	no	—
87	C₃²×C₄⋊C₄	—	2	8	no	—
88	C₃²×C₄⋊C₄	—	2	8	no	—
89	C₃²×C₄⋊C₄	—	3	2	no	—
90	C₃²×C₄⋊C₄	—	3	1	no	—
91	C₃²×C₄⋊C₄	—	3	1	no	—
92	C₃²×C₄⋊C₄	—	3	1	no	—
93	C₃²×C₄⋊C₄	—	3	1	no	—
94	C₃²×C₄⋊C₄	—	3	2	no	—
95	C₃²×C₄⋊C₄	—	3	2	no	—
96	C₃²×C₄⋊C₄	—	3	2	no	—
97	C₃²×C₄⋊C₄	—	3	2	no	—
98	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
99	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
100	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
101	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
102	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
103	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
104	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
105	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
106	C₃²×C₂²⋊C₄	—	3	1	no	—
107	C₃²×C₂²⋊C₄	—	3	2	no	—
108	C₃²×C₂²⋊C₄	—	3	1	no	—
109	C₃²×C₄²	—	2	4	no	—
110	C₃²×C₄²	—	3	2	no	—
111	C₃²×C₄²	—	3	1	no	—
112	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
113	C₃²×C₄²	C₂²×C₃	1	4	yes	—
114	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
115	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
116	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
117	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
118	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
119	C₃×C₄⋊Dic₃	C₂²×C₃	1	8	yes	—
120	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
121	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
122	C₃×Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
123	C₃×Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
124	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
125	C₃×Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
126	C₃×C₄×Dic₃	C₂²×C₃	1	8	yes	—
127	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
128	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
129	C₂×C₃²×D₃	—	2	1	no	—
130	C₂×C₃²×D₃	—	2	1	no	—
131	C₂×C₃²×D₃	C₂×Q₈	1	2	yes	—
132	C₃³×C₄	—	2	1	no	—
133	C₃³×C₄	—	2	2	no	—
134	C₃³×C₄	—	2	1	no	—
135	C₃³×C₄	—	2	2	no	—
136	C₃³×C₄	C₂×Q₈	1	2	yes	—
137	C₃²×Dic₃	—	2	1	no	—
138	C₃²×Dic₃	—	2	2	no	—
139	C₃²×Dic₃	—	2	2	no	—
140	C₃²×Dic₃	—	2	1	no	—
141	C₃²×Dic₃	C₂×Q₈	1	2	yes	—
142	C₃×C₄⋊Q₈	—	3	8	no	—
143	C₃×C₄⋊Q₈	—	3	4	no	—
144	C₃×C₄⋊Q₈	—	3	4	no	—
145	C₃×#125	—	2	4	no	—
146	C₃×#125	—	3	8	no	—
147	C₃×#125	—	3	4	no	—
148	C₃×#125	C₃×D₃	1	4	yes	complemented
149	C₃×#125	C₂×C₃²	1	1	yes	characteristic, complemented
150	C₃×C₂²⋊Q₈	—	3	8	no	—
151	C₃×C₂²⋊Q₈	—	3	8	no	—
152	C₃×#117	—	3	4	no	—
153	C₃×#117	—	3	4	no	—
154	C₂×C₃×C₄⋊C₄	—	3	4	no	—
155	#897	C₂×C₃²	1	2	yes	—
156	#896	C₂×C₃²	1	2	yes	—
157	#892	C₂×C₃²	1	1	yes	characteristic
158	D₃×C₄⋊C₄	C₂×C₃²	1	1	yes	characteristic
159	#890	C₂×C₃²	1	1	yes	characteristic, complemented
160	#889	C₂×C₃²	1	2	yes	complemented
161	#888	C₂×C₃²	1	1	yes	characteristic, complemented
162	#888	C₂×C₃²	1	2	yes	complemented
163	#872	C₂×C₃²	1	1	yes	characteristic
164	#869	C₂×C₃²	1	1	yes	characteristic, complemented
165	C₂³×C₃²	—	3	1	no	—
166	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
167	C₃²×Q₈	—	12	2	no	—
168	C₃²×Q₈	—	12	2	no	—
169	C₂×C₃²×C₄	—	2	4	no	—
170	C₂×C₃²×C₄	—	2	4	no	—
171	C₂×C₃²×C₄	—	2	4	no	—
172	C₂×C₃²×C₄	—	2	8	no	—
173	C₂×C₃²×C₄	—	2	8	no	—
174	C₂×C₃²×C₄	—	3	1	no	—
175	C₂×C₃²×C₄	—	3	4	no	—
176	C₂×C₃²×C₄	—	3	1	no	—
177	C₂×C₃²×C₄	—	3	1	no	—
178	C₂×C₃²×C₄	—	3	2	no	—
179	C₂×C₃²×C₄	—	3	2	no	—
180	C₂×C₃²×C₄	—	12	2	no	—
181	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
182	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
183	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
184	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
185	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
186	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
187	C₂×C₃²×C₄	C₂³×C₃	1	8	yes	—
188	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
189	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
190	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
191	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
192	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
193	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
194	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
195	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
196	C₃×C₄×D₃	—	4	8	no	—
197	C₃×C₄×D₃	C₃×Q₈	1	16	yes	complemented
198	C₃×Dic₆	—	4	8	no	—
199	C₃×Dic₆	—	4	8	no	—
200	#544	—	3	1	no	Sylow
201	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
202	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
203	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
204	C₃²×D₃	—	2	2	no	—
205	C₂×C₃×Q₈	—	3	8	no	—
206	C₂×C₃×Q₈	—	3	8	no	—
207	C₂²×C₃×C₄	—	3	4	no	—
208	C₂²×C₃×C₄	—	3	8	no	—
209	C₂×C₄×D₃	C₂²×C₃²	1	2	yes	—
210	C₂×C₄×D₃	C₂²×C₃²	1	1	yes	characteristic
211	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
212	C₂×Dic₆	C₂²×C₃²	1	2	yes	—
213	C₃×C₄⋊C₄	—	2	4	no	—
214	C₃×C₄⋊C₄	—	2	4	no	—
215	C₃×C₄⋊C₄	—	2	8	no	—
216	C₃×C₄⋊C₄	—	2	8	no	—
217	C₃×C₄⋊C₄	—	3	8	no	—
218	C₃×C₄⋊C₄	—	3	4	no	—
219	C₃×C₄⋊C₄	—	3	4	no	—
220	C₃×C₄⋊C₄	—	3	8	no	—
221	C₃×C₄⋊C₄	—	3	8	no	—
222	C₃×C₄⋊C₄	—	3	4	no	—
223	C₃×C₄⋊C₄	—	3	4	no	—
224	C₃×C₄⋊C₄	—	3	8	no	—
225	C₃×C₄⋊C₄	—	3	8	no	—
226	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
227	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
228	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
229	C₃×C₄⋊C₄	C₂×C₃×D₃	1	8	yes	—
230	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
231	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
232	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
233	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
234	C₃×C₂²⋊C₄	—	3	4	no	—
235	C₃×C₂²⋊C₄	—	3	8	no	—
236	C₃×C₂²⋊C₄	—	3	4	no	—
237	C₃×C₄²	—	2	4	no	—
238	C₃×C₄²	—	3	8	no	—
239	C₃×C₄²	—	3	4	no	—
240	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
241	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
242	(C₂×D₃)⋊C₄	C₂²×C₃²	1	1	yes	characteristic
243	(C₂×D₃)⋊C₄	C₂²×C₃²	1	1	yes	characteristic
244	(C₂×D₃)⋊C₄	C₂²×C₃²	1	2	yes	—
245	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
246	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
247	C₄⋊Dic₃	C₂²×C₃²	1	1	yes	characteristic
248	C₄⋊Dic₃	C₂²×C₃²	1	2	yes	—
249	Dic₃⋊C₄	C₂²×C₃²	1	1	yes	characteristic
250	Dic₃⋊C₄	C₂²×C₃²	1	1	yes	characteristic
251	Dic₃⋊C₄	C₂²×C₃²	1	2	yes	—
252	Dic₃⋊C₄	C₂²×C₃²	1	2	yes	—
253	Dic₃⋊C₄	C₂²×C₃²	1	2	yes	—
254	C₄×Dic₃	C₂²×C₃²	1	1	yes	characteristic
255	C₄×Dic₃	C₂²×C₃²	1	2	yes	—
256	C₂²×C₃²	—	2	4	no	—
257	C₂²×C₃²	—	3	2	no	—
258	C₂²×C₃²	—	6	1	no	—
259	C₂²×C₃²	—	6	1	no	—
260	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
261	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
262	C₂×C₃×D₃	—	2	4	no	—
263	C₂×C₃×D₃	—	2	4	no	—
264	C₂×C₃×D₃	C₂×C₃×Q₈	1	8	yes	—
265	C₃²×C₄	—	2	2	no	—
266	C₃²×C₄	—	2	1	no	—
267	C₃²×C₄	—	2	2	no	—
268	C₃²×C₄	—	2	1	no	—
269	C₃²×C₄	—	2	8	no	—
270	C₃²×C₄	—	2	4	no	—
271	C₃²×C₄	—	2	8	no	—
272	C₃²×C₄	—	2	8	no	—
273	C₃²×C₄	—	2	4	no	—
274	C₃²×C₄	—	3	2	no	—
275	C₃²×C₄	—	4	8	no	—
276	C₃²×C₄	—	4	4	no	—
277	C₃²×C₄	—	4	4	no	—
278	C₃²×C₄	—	4	8	no	—
279	C₃²×C₄	—	6	2	no	—
280	C₃²×C₄	—	6	1	no	—
281	C₃²×C₄	—	6	2	no	—
282	C₃²×C₄	—	6	1	no	—
283	C₃²×C₄	D₃×Q₈	1	2	yes	—
284	C₃²×C₄	C₂×C₃×Q₈	1	8	yes	—
285	C₃×Dic₃	—	2	4	no	—
286	C₃×Dic₃	—	2	8	no	—
287	C₃×Dic₃	—	2	8	no	—
288	C₃×Dic₃	—	2	4	no	—
289	C₃×Dic₃	C₂×C₃×Q₈	1	8	yes	—
290	C₄⋊Q₈	—	3	2	no	—
291	C₄⋊Q₈	—	3	1	no	—
292	C₄⋊Q₈	—	3	1	no	—
293	#125	—	3	2	no	—
294	#125	—	3	1	no	—
295	#125	C₃²×D₃	1	1	yes	characteristic, complemented
296	C₂²⋊Q₈	—	3	2	no	—
297	C₂²⋊Q₈	—	3	2	no	—
298	#117	—	3	1	no	—
299	#117	—	3	1	no	—
300	C₂×C₄⋊C₄	—	3	1	no	—
301	C₃³	#544	1	1	yes	characteristic, complemented, Sylow
302	C₂³×C₃	—	3	4	no	—
303	C₂²×D₃	C₂³×C₃²	1	1	yes	characteristic
304	C₃×Q₈	—	12	8	no	—
305	C₃×Q₈	—	12	8	no	—
306	C₂×C₃×C₄	—	2	4	no	—
307	C₂×C₃×C₄	—	2	4	no	—
308	C₂×C₃×C₄	—	2	4	no	—
309	C₂×C₃×C₄	—	2	8	no	—
310	C₂×C₃×C₄	—	2	8	no	—
311	C₂×C₃×C₄	—	3	4	no	—
312	C₂×C₃×C₄	—	3	16	no	—
313	C₂×C₃×C₄	—	3	4	no	—
314	C₂×C₃×C₄	—	3	4	no	—
315	C₂×C₃×C₄	—	3	8	no	—
316	C₂×C₃×C₄	—	3	8	no	—
317	C₂×C₃×C₄	—	12	8	no	—
318	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
319	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
320	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
321	C₂×C₃×C₄	C₂²×C₃×D₃	1	8	yes	—
322	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
323	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
324	C₂×C₃×C₄	C₂³×C₃²	1	2	yes	—
325	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
326	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
327	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
328	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
329	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
330	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
331	C₂×Dic₃	C₂³×C₃²	1	2	yes	—
332	C₂×Dic₃	C₂³×C₃²	1	1	yes	characteristic
333	C₄×D₃	—	4	2	no	—
334	C₄×D₃	C₃²×Q₈	1	4	yes	complemented
335	Dic₆	—	4	2	no	—
336	Dic₆	—	4	2	no	—
337	C₂×C₃²	—	2	4	no	—
338	C₂×C₃²	—	2	4	no	—
339	C₂×C₃²	—	2	4	no	—
340	C₂×C₃²	—	6	2	no	—
341	C₂×C₃²	C₂×D₃×Q₈	1	1	yes	characteristic
342	C₂×C₃²	#1009	1	1	yes	characteristic
343	C₂×C₃²	#1010	1	1	yes	characteristic
344	C₂×C₃²	C₂²×C₃×Q₈	1	4	yes	—
345	C₂×C₃²	C₃×H(5,𝔽₂)	1	4	yes	—
346	C₂×C₃²	C₃×2₋¹⁺⁴	1	4	yes	—
347	C₃×D₃	—	2	8	no	—
348	C₂×Q₈	—	3	2	no	—
349	C₂×Q₈	—	3	2	no	—
350	C₂²×C₄	—	3	1	no	—
351	C₂²×C₄	—	3	2	no	—
352	C₄⋊C₄	—	3	2	no	—
353	C₄⋊C₄	—	3	1	no	—
354	C₄⋊C₄	—	3	1	no	—
355	C₄⋊C₄	—	3	1	no	—
356	C₄⋊C₄	—	3	1	no	—
357	C₄⋊C₄	—	3	2	no	—
358	C₄⋊C₄	—	3	2	no	—
359	C₄⋊C₄	—	3	2	no	—
360	C₄⋊C₄	—	3	2	no	—
361	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
362	C₄⋊C₄	C₂×C₃²×D₃	1	1	yes	characteristic
363	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
364	C₄⋊C₄	C₂×C₃²×D₃	1	2	yes	—
365	C₂²⋊C₄	—	3	1	no	—
366	C₂²⋊C₄	—	3	2	no	—
367	C₂²⋊C₄	—	3	1	no	—
368	C₄²	—	3	2	no	—
369	C₄²	—	3	1	no	—
370	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic
371	C₂²×C₃	—	2	4	no	—
372	C₂²×C₃	—	3	8	no	—
373	C₂²×C₃	—	6	4	no	—
374	C₂²×C₃	—	6	4	no	—
375	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
376	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
377	C₂×D₃	—	2	1	no	—
378	C₂×D₃	—	2	1	no	—
379	C₂×D₃	C₂×C₃²×Q₈	1	2	yes	—
380	C₃×C₄	—	2	8	no	—
381	C₃×C₄	—	2	4	no	—
382	C₃×C₄	—	2	8	no	—
383	C₃×C₄	—	2	4	no	—
384	C₃×C₄	—	2	1	no	—
385	C₃×C₄	—	2	2	no	—
386	C₃×C₄	—	2	2	no	—
387	C₃×C₄	—	2	1	no	—
388	C₃×C₄	—	2	8	no	—
389	C₃×C₄	—	3	8	no	—
390	C₃×C₄	—	4	8	no	—
391	C₃×C₄	—	4	4	no	—
392	C₃×C₄	—	4	4	no	—
393	C₃×C₄	—	4	8	no	—
394	C₃×C₄	—	6	8	no	—
395	C₃×C₄	—	6	4	no	—
396	C₃×C₄	—	6	8	no	—
397	C₃×C₄	—	6	4	no	—
398	C₃×C₄	C₃×D₃×Q₈	1	8	yes	—
399	C₃×C₄	C₂×C₃²×Q₈	1	2	yes	—
400	Dic₃	—	2	1	no	—
401	Dic₃	—	2	2	no	—
402	Dic₃	—	2	2	no	—
403	Dic₃	—	2	1	no	—
404	Dic₃	C₂×C₃²×Q₈	1	2	yes	—
405	C₃²	—	2	4	no	—
406	C₃²	#5689	1	1	yes	characteristic, direct product factor
407	C₃²	C₃×#544	1	4	yes	complemented
408	C₂³	—	3	1	no	—
409	Q₈	—	12	2	no	—
410	Q₈	—	12	2	no	—
411	C₂×C₄	—	3	1	no	—
412	C₂×C₄	—	3	4	no	—
413	C₂×C₄	—	3	1	no	—
414	C₂×C₄	—	3	1	no	—
415	C₂×C₄	—	3	2	no	—
416	C₂×C₄	—	3	2	no	—
417	C₂×C₄	—	12	2	no	—
418	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
419	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
420	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
421	C₂×C₄	C₂²×C₃²×D₃	1	2	yes	—
422	C₂×C₄	C₂²×C₃²×D₃	1	1	yes	characteristic
423	C₂×C₃	—	2	4	no	—
424	C₂×C₃	—	2	4	no	—
425	C₂×C₃	—	2	4	no	—
426	C₂×C₃	—	6	8	no	—
427	C₂×C₃	C₂×C₃×D₃×Q₈	1	4	yes	—
428	C₂×C₃	C₃×#1009	1	4	yes	—
429	C₂×C₃	C₃×#1010	1	4	yes	—
430	C₂×C₃	C₂²×C₃²×Q₈	1	1	yes	characteristic
431	C₂×C₃	C₃²×H(5,𝔽₂)	1	1	yes	characteristic
432	C₂×C₃	C₃²×2₋¹⁺⁴	1	1	yes	characteristic
433	D₃	—	2	2	no	—
434	C₂²	—	3	2	no	—
435	C₂²	—	6	1	no	—
436	C₂²	—	6	1	no	—
437	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
438	C₄	—	2	2	no	—
439	C₄	—	2	1	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	1	no	—
442	C₄	—	3	2	no	—
443	C₄	—	6	2	no	—
444	C₄	—	6	1	no	—
445	C₄	—	6	2	no	—
446	C₄	—	6	1	no	—
447	C₄	C₃²×D₃×Q₈	1	2	yes	—
448	C₃	—	2	4	no	—
449	C₃	C₃×#5689	1	4	yes	direct product factor
450	C₃	C₃²×#544	1	1	yes	characteristic, complemented
451	C₂	—	6	2	no	—
452	C₂	C₂×C₃²×D₃×Q₈	1	1	yes	characteristic
453	C₂	C₃²×#1009	1	1	yes	characteristic
454	C₂	C₃²×#1010	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×#125	C₂
socle	C₂²×C₃³	C₂⁴
prime radical	C₂²×C₃²×D₃	C₂³
hypercenter	C₃²×#125	D₃
hypocenter	C₃	C₃²×#544
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×#125

Complements

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

N	H ≅ G/N	type
1. C₃³×#125	C₂	semidirect
6. C₃²×#890	C₂	semidirect
7. C₃²×#889	C₂	semidirect
8. C₃²×#888	C₂	semidirect
9. C₃²×#888	C₂	semidirect
11. C₃²×#869	C₂	semidirect
13. C₃×#5689	C₃	direct (N × H)
43. C₃²×#125	D₃	semidirect
44. C₃²×#125	C₂×C₃	semidirect
54. C₃×#890	C₂×C₃	semidirect
55. C₃×#889	C₂×C₃	semidirect
56. C₃×#888	C₂×C₃	semidirect
57. C₃×#888	C₂×C₃	semidirect
59. C₃×#869	C₂×C₃	semidirect
72. C₃²×C₄×D₃	Q₈	semidirect
76. #5689	C₃²	direct (N × H)
148. C₃×#125	C₃×D₃	semidirect
149. C₃×#125	C₂×C₃²	semidirect
159. #890	C₂×C₃²	semidirect
160. #889	C₂×C₃²	semidirect
161. #888	C₂×C₃²	semidirect
162. #888	C₂×C₃²	semidirect
164. #869	C₂×C₃²	semidirect
197. C₃×C₄×D₃	C₃×Q₈	semidirect
295. #125	C₃²×D₃	semidirect
301. C₃³	#544	semidirect
334. C₄×D₃	C₃²×Q₈	semidirect
406. C₃²	#5689	direct (N × H)
407. C₃²	C₃×#544	semidirect
449. C₃	C₃×#5689	direct (N × H)
450. C₃	C₃²×#544	semidirect

Maximal subgroups

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

subgroup	index
C₃³×#125	2
C₃²×#897	2
C₃²×#896	2
C₃²×#892	2
C₃²×D₃×C₄⋊C₄	2
C₃²×#890	2
C₃²×#889	2
C₃²×#888	2
C₃²×#872	2
C₃²×#869	2
C₃²×#544	3
C₃×#5689	3

Subgroups — C32×#5689 #13179067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#5689 #13179067