Subgroups — C₂²×C₃×#397 #10653144

Order 768.

5230 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₂³×C₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#397	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₃×#95	C₂	1	2	yes	maximal, complemented
6	C₂²×C₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#397	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂⁶×C₃	C₂²	1	1	yes	characteristic
9	C₂⁴×C₃×C₄	C₂²	1	2	yes	—
10	C₂⁴×C₃×C₄	C₂²	1	2	yes	—
11	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄⋊C₄	—	2	8	no	—
14	C₂²×C₃×C₄⋊C₄	C₂²	1	12	yes	complemented
15	C₂²×C₃×C₂²⋊C₄	—	2	8	no	—
16	C₂²×C₃×C₂²⋊C₄	—	2	16	no	—
17	C₂²×C₃×C₂²⋊C₄	C₂²	1	24	yes	—
18	C₂²×C₃×C₂²⋊C₄	C₂²	1	12	yes	—
19	C₃×#397	C₂²	1	64	yes	direct product factor
20	C₂×C₃×#95	C₂²	1	24	yes	complemented
21	C₂×C₃×#95	C₂²	1	12	yes	complemented
22	C₂³×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂³×C₂²⋊C₄	C₂×C₃	1	2	yes	—
24	C₂³×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
25	C₂×#397	C₂×C₃	1	24	yes	direct product factor
26	C₂²×#95	C₂×C₃	1	2	yes	complemented
27	C₂²×#95	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂⁵×C₃	—	2	4	no	—
29	C₂⁵×C₃	—	2	4	no	—
30	C₂⁵×C₃	—	2	8	no	—
31	C₂⁵×C₃	—	2	8	no	—
32	C₂⁵×C₃	D₄	1	8	yes	—
33	C₂⁵×C₃	C₂³	1	6	yes	—
34	C₂⁵×C₃	C₂³	1	1	yes	characteristic
35	C₂³×C₃×C₄	—	2	8	no	—
36	C₂³×C₃×C₄	—	2	8	no	—
37	C₂³×C₃×C₄	—	2	8	no	—
38	C₂³×C₃×C₄	—	2	4	no	—
39	C₂³×C₃×C₄	—	2	8	no	—
40	C₂³×C₃×C₄	—	2	8	no	—
41	C₂³×C₃×C₄	—	2	8	no	—
42	C₂³×C₃×C₄	—	2	8	no	—
43	C₂³×C₃×C₄	—	2	8	no	—
44	C₂³×C₃×C₄	D₄	1	8	yes	—
45	C₂³×C₃×C₄	C₂³	1	12	yes	—
46	C₂³×C₃×C₄	C₂³	1	6	yes	—
47	C₂³×C₃×C₄	C₂³	1	12	yes	—
48	C₂³×C₃×C₄	C₂³	1	6	yes	—
49	C₂×C₃×C₄⋊C₄	—	2	48	no	—
50	C₂×C₃×C₄⋊C₄	C₂³	1	16	yes	complemented
51	C₂×C₃×C₂²⋊C₄	—	2	96	no	—
52	C₂×C₃×C₂²⋊C₄	—	2	48	no	—
53	C₂×C₃×C₂²⋊C₄	C₂³	1	32	yes	—
54	C₂×C₃×C₂²⋊C₄	C₂³	1	16	yes	—
55	C₃×#95	C₂³	1	32	yes	complemented
56	C₃×#95	C₂³	1	16	yes	complemented
57	C₂⁶	C₂²×C₃	1	1	yes	characteristic
58	C₂⁴×C₄	C₂²×C₃	1	2	yes	—
59	C₂⁴×C₄	C₂²×C₃	1	2	yes	—
60	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
61	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₄⋊C₄	—	2	8	no	—
63	C₂²×C₄⋊C₄	C₂²×C₃	1	12	yes	complemented
64	C₂²×C₂²⋊C₄	—	2	16	no	—
65	C₂²×C₂²⋊C₄	—	2	8	no	—
66	C₂²×C₂²⋊C₄	C₂²×C₃	1	24	yes	—
67	C₂²×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
68	#397	C₂²×C₃	1	64	yes	direct product factor
69	C₂×#95	C₂²×C₃	1	24	yes	complemented
70	C₂×#95	C₂²×C₃	1	12	yes	complemented
71	C₂⁴×C₃	—	2	32	no	—
72	C₂⁴×C₃	—	2	64	no	—
73	C₂⁴×C₃	—	2	12	no	—
74	C₂⁴×C₃	—	2	12	no	—
75	C₂⁴×C₃	—	2	24	no	—
76	C₂⁴×C₃	—	2	24	no	—
77	C₂⁴×C₃	—	4	16	no	—
78	C₂⁴×C₃	—	4	16	no	—
79	C₂⁴×C₃	—	4	32	no	—
80	C₂⁴×C₃	C₂×D₄	1	24	yes	—
81	C₂⁴×C₃	C₂×D₄	1	4	yes	—
82	C₂⁴×C₃	C₂×D₄	1	4	yes	—
83	C₂⁴×C₃	#41	1	4	yes	—
84	C₂⁴×C₃	#41	1	8	yes	—
85	C₂⁴×C₃	#41	1	8	yes	—
86	C₂⁴×C₃	C₂⁴	1	4	yes	—
87	C₂⁴×C₃	C₂⁴	1	3	yes	—
88	C₂²×C₃×C₄	—	2	32	no	—
89	C₂²×C₃×C₄	—	2	24	no	—
90	C₂²×C₃×C₄	—	2	24	no	—
91	C₂²×C₃×C₄	—	2	24	no	—
92	C₂²×C₃×C₄	—	2	12	no	—
93	C₂²×C₃×C₄	—	2	24	no	—
94	C₂²×C₃×C₄	—	2	24	no	—
95	C₂²×C₃×C₄	—	2	24	no	—
96	C₂²×C₃×C₄	—	2	24	no	—
97	C₂²×C₃×C₄	—	2	24	no	—
98	C₂²×C₃×C₄	—	4	32	no	—
99	C₂²×C₃×C₄	—	4	32	no	—
100	C₂²×C₃×C₄	—	4	16	no	—
101	C₂²×C₃×C₄	C₂×D₄	1	24	yes	—
102	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
103	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
104	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
105	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
106	C₃×C₄⋊C₄	—	2	32	no	—
107	C₃×C₂²⋊C₄	—	2	64	no	—
108	C₃×C₂²⋊C₄	—	2	32	no	—
109	C₂⁵	—	2	4	no	—
110	C₂⁵	—	2	4	no	—
111	C₂⁵	—	2	8	no	—
112	C₂⁵	—	2	8	no	—
113	C₂⁵	C₃×D₄	1	8	yes	—
114	C₂⁵	C₂³×C₃	1	6	yes	—
115	C₂⁵	C₂³×C₃	1	1	yes	characteristic
116	C₂³×C₄	—	2	8	no	—
117	C₂³×C₄	—	2	8	no	—
118	C₂³×C₄	—	2	8	no	—
119	C₂³×C₄	—	2	4	no	—
120	C₂³×C₄	—	2	8	no	—
121	C₂³×C₄	—	2	8	no	—
122	C₂³×C₄	—	2	8	no	—
123	C₂³×C₄	—	2	8	no	—
124	C₂³×C₄	—	2	8	no	—
125	C₂³×C₄	C₃×D₄	1	8	yes	—
126	C₂³×C₄	C₂³×C₃	1	12	yes	—
127	C₂³×C₄	C₂³×C₃	1	12	yes	—
128	C₂³×C₄	C₂³×C₃	1	6	yes	—
129	C₂³×C₄	C₂³×C₃	1	6	yes	—
130	C₂×C₄⋊C₄	—	2	48	no	—
131	C₂×C₄⋊C₄	C₂³×C₃	1	16	yes	complemented
132	C₂×C₂²⋊C₄	—	2	96	no	—
133	C₂×C₂²⋊C₄	—	2	48	no	—
134	C₂×C₂²⋊C₄	C₂³×C₃	1	32	yes	—
135	C₂×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
136	#95	C₂³×C₃	1	32	yes	complemented
137	#95	C₂³×C₃	1	16	yes	complemented
138	C₂³×C₃	—	2	48	no	—
139	C₂³×C₃	—	2	96	no	—
140	C₂³×C₃	—	2	4	no	—
141	C₂³×C₃	—	2	4	no	—
142	C₂³×C₃	—	2	8	no	—
143	C₂³×C₃	—	2	8	no	—
144	C₂³×C₃	—	4	24	no	—
145	C₂³×C₃	—	4	24	no	—
146	C₂³×C₃	—	4	48	no	—
147	C₂³×C₃	—	4	128	no	—
148	C₂³×C₃	#121	1	16	yes	—
149	C₂³×C₃	#123	1	16	yes	—
150	C₂³×C₃	#123	1	32	yes	—
151	C₂³×C₃	#124	1	16	yes	—
152	C₂³×C₃	#126	1	32	yes	—
153	C₂³×C₃	C₂²×D₄	1	8	yes	—
154	C₂³×C₃	C₂²×D₄	1	6	yes	—
155	C₂³×C₃	C₂²×D₄	1	6	yes	—
156	C₂³×C₃	C₂×#41	1	6	yes	—
157	C₂³×C₃	C₂×#41	1	12	yes	—
158	C₂³×C₃	C₂×#41	1	12	yes	—
159	C₂³×C₃	C₂⁵	1	1	yes	characteristic
160	C₂×C₃×C₄	—	2	48	no	—
161	C₂×C₃×C₄	—	2	8	no	—
162	C₂×C₃×C₄	—	2	8	no	—
163	C₂×C₃×C₄	—	2	8	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	8	no	—
166	C₂×C₃×C₄	—	2	8	no	—
167	C₂×C₃×C₄	—	2	8	no	—
168	C₂×C₃×C₄	—	2	8	no	—
169	C₂×C₃×C₄	—	2	8	no	—
170	C₂×C₃×C₄	—	4	48	no	—
171	C₂×C₃×C₄	—	4	48	no	—
172	C₂×C₃×C₄	—	4	24	no	—
173	C₂×C₃×C₄	C₂²×D₄	1	8	yes	—
174	C₂⁴	—	2	32	no	—
175	C₂⁴	—	2	64	no	—
176	C₂⁴	—	2	12	no	—
177	C₂⁴	—	2	12	no	—
178	C₂⁴	—	2	24	no	—
179	C₂⁴	—	2	24	no	—
180	C₂⁴	—	4	16	no	—
181	C₂⁴	—	4	16	no	—
182	C₂⁴	—	4	32	no	—
183	C₂⁴	C₂×C₃×D₄	1	24	yes	—
184	C₂⁴	C₂×C₃×D₄	1	4	yes	—
185	C₂⁴	C₂×C₃×D₄	1	4	yes	—
186	C₂⁴	C₃×#41	1	4	yes	—
187	C₂⁴	C₃×#41	1	8	yes	—
188	C₂⁴	C₃×#41	1	8	yes	—
189	C₂⁴	C₂⁴×C₃	1	4	yes	—
190	C₂⁴	C₂⁴×C₃	1	3	yes	—
191	C₂²×C₄	—	2	32	no	—
192	C₂²×C₄	—	2	24	no	—
193	C₂²×C₄	—	2	24	no	—
194	C₂²×C₄	—	2	24	no	—
195	C₂²×C₄	—	2	12	no	—
196	C₂²×C₄	—	2	24	no	—
197	C₂²×C₄	—	2	24	no	—
198	C₂²×C₄	—	2	24	no	—
199	C₂²×C₄	—	2	24	no	—
200	C₂²×C₄	—	2	24	no	—
201	C₂²×C₄	—	4	32	no	—
202	C₂²×C₄	—	4	32	no	—
203	C₂²×C₄	—	4	16	no	—
204	C₂²×C₄	C₂×C₃×D₄	1	24	yes	—
205	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
206	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
207	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
208	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
209	C₄⋊C₄	—	2	32	no	—
210	C₂²⋊C₄	—	2	64	no	—
211	C₂²⋊C₄	—	2	32	no	—
212	C₂²×C₃	—	2	8	no	—
213	C₂²×C₃	—	2	16	no	—
214	C₂²×C₃	—	4	96	no	—
215	C₂²×C₃	—	4	4	no	—
216	C₂²×C₃	—	4	4	no	—
217	C₂²×C₃	—	4	8	no	—
218	C₂²×C₃	#397	1	64	yes	direct product factor
219	C₂²×C₃	C₂×#121	1	12	yes	—
220	C₂²×C₃	C₂×#123	1	12	yes	—
221	C₂²×C₃	C₂×#123	1	24	yes	—
222	C₂²×C₃	C₂×#124	1	12	yes	—
223	C₂²×C₃	C₂×#126	1	24	yes	—
224	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
225	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
226	C₂²×C₃	C₂²×#41	1	1	yes	characteristic
227	C₂²×C₃	C₂²×#41	1	2	yes	—
228	C₂²×C₃	C₂²×#41	1	2	yes	—
229	C₃×C₄	—	2	8	no	—
230	C₃×C₄	—	4	8	no	—
231	C₃×C₄	—	4	8	no	—
232	C₃×C₄	—	4	4	no	—
233	C₂³	—	2	4	no	—
234	C₂³	—	2	4	no	—
235	C₂³	—	2	8	no	—
236	C₂³	—	2	8	no	—
237	C₂³	—	2	48	no	—
238	C₂³	—	2	96	no	—
239	C₂³	—	4	128	no	—
240	C₂³	—	4	24	no	—
241	C₂³	—	4	24	no	—
242	C₂³	—	4	48	no	—
243	C₂³	C₃×#121	1	16	yes	—
244	C₂³	C₃×#123	1	16	yes	—
245	C₂³	C₃×#123	1	32	yes	—
246	C₂³	C₃×#124	1	16	yes	—
247	C₂³	C₃×#126	1	32	yes	—
248	C₂³	C₂²×C₃×D₄	1	8	yes	—
249	C₂³	C₂²×C₃×D₄	1	6	yes	—
250	C₂³	C₂²×C₃×D₄	1	6	yes	—
251	C₂³	C₂×C₃×#41	1	6	yes	—
252	C₂³	C₂×C₃×#41	1	12	yes	—
253	C₂³	C₂×C₃×#41	1	12	yes	—
254	C₂³	C₂⁵×C₃	1	1	yes	characteristic
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	48	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	48	no	—
266	C₂×C₄	—	4	48	no	—
267	C₂×C₄	—	4	24	no	—
268	C₂×C₄	C₂²×C₃×D₄	1	8	yes	—
269	C₂×C₃	—	4	8	no	—
270	C₂×C₃	C₂×#397	1	24	yes	direct product factor
271	C₂×C₃	C₂²×#121	1	1	yes	characteristic
272	C₂×C₃	C₂²×#123	1	1	yes	characteristic
273	C₂×C₃	C₂²×#123	1	2	yes	—
274	C₂×C₃	C₂²×#124	1	1	yes	characteristic
275	C₂×C₃	C₂²×#126	1	2	yes	—
276	C₂²	—	2	8	no	—
277	C₂²	—	2	16	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	96	no	—
282	C₂²	C₃×#397	1	64	yes	direct product factor
283	C₂²	C₂×C₃×#121	1	12	yes	—
284	C₂²	C₂×C₃×#123	1	12	yes	—
285	C₂²	C₂×C₃×#123	1	24	yes	—
286	C₂²	C₂×C₃×#124	1	12	yes	—
287	C₂²	C₂×C₃×#126	1	24	yes	—
288	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
289	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
290	C₂²	C₂²×C₃×#41	1	1	yes	characteristic
291	C₂²	C₂²×C₃×#41	1	2	yes	—
292	C₂²	C₂²×C₃×#41	1	2	yes	—
293	C₄	—	2	8	no	—
294	C₄	—	4	8	no	—
295	C₄	—	4	8	no	—
296	C₄	—	4	4	no	—
297	C₃	C₂²×#397	1	1	yes	characteristic, direct product factor, Sylow
298	C₂	—	4	8	no	—
299	C₂	C₂×C₃×#397	1	24	yes	direct product factor
300	C₂	C₂²×C₃×#121	1	1	yes	characteristic
301	C₂	C₂²×C₃×#123	1	1	yes	characteristic
302	C₂	C₂²×C₃×#123	1	2	yes	—
303	C₂	C₂²×C₃×#124	1	1	yes	characteristic
304	C₂	C₂²×C₃×#126	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
prime radical	C₂⁶×C₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₄⋊C₄	C₂	semidirect
4. C₂×C₃×#397	C₂	direct (N × H)
5. C₂²×C₃×#95	C₂	semidirect
6. C₂²×C₃×#95	C₂	semidirect
7. C₂²×#397	C₃	direct (N × H)
14. C₂²×C₃×C₄⋊C₄	C₂²	semidirect
19. C₃×#397	C₂²	direct (N × H)
20. C₂×C₃×#95	C₂²	semidirect
21. C₂×C₃×#95	C₂²	semidirect
22. C₂³×C₄⋊C₄	C₂×C₃	semidirect
25. C₂×#397	C₂×C₃	direct (N × H)
26. C₂²×#95	C₂×C₃	semidirect
27. C₂²×#95	C₂×C₃	semidirect
50. C₂×C₃×C₄⋊C₄	C₂³	semidirect
55. C₃×#95	C₂³	semidirect
56. C₃×#95	C₂³	semidirect
63. C₂²×C₄⋊C₄	C₂²×C₃	semidirect
68. #397	C₂²×C₃	direct (N × H)
69. C₂×#95	C₂²×C₃	semidirect
70. C₂×#95	C₂²×C₃	semidirect
131. C₂×C₄⋊C₄	C₂³×C₃	semidirect
136. #95	C₂³×C₃	semidirect
137. #95	C₂³×C₃	semidirect
218. C₂²×C₃	#397	direct (N × H)
270. C₂×C₃	C₂×#397	direct (N × H)
282. C₂²	C₃×#397	direct (N × H)
297. C₃	C₂²×#397	direct (N × H)
299. C₂	C₂×C₃×#397	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄⋊C₄	2
C₂³×C₃×C₂²⋊C₄	2
C₂×C₃×#397	2
C₂²×C₃×#95	2
C₂²×#397	3

Subgroups — C22×C3×#397 #10653144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#397 #10653144