Subgroups — #10694528

Order 768.

406 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#78442	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#70429	C₂	1	2	yes	maximal
4	#69729	C₂	1	1	yes	maximal, characteristic
5	#69471	C₂	1	2	yes	maximal
6	#9749	—	3	1	no	maximal, Sylow
7	#5470	C₂²	1	2	yes	—
8	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	—	2	1	no	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	#5106	—	2	1	no	—
13	#5106	—	2	1	no	—
14	#4925	C₂²	1	2	yes	—
15	#4683	—	2	1	no	—
16	#4683	—	2	1	no	—
17	#4682	C₄	1	1	yes	characteristic, complemented
18	#4682	C₄	1	1	yes	characteristic, complemented
19	#4680	C₄	1	2	yes	complemented
20	C₃×#336	—	2	1	no	—
21	#4484	—	2	4	no	—
22	#4458	—	2	1	no	—
23	#4457	—	2	1	no	—
24	#4452	—	2	4	no	—
25	#3160	—	3	1	no	—
26	C₈×C₄⋊C₄	D₃	1	1	yes	characteristic
27	#1564	—	3	1	no	—
28	#1503	—	3	2	no	—
29	#1481	—	3	2	no	—
30	C₂²×C₃×C₈	—	2	1	no	—
31	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₃×C₄²	—	2	1	no	—
35	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
36	#926	—	2	2	no	—
37	#924	—	2	2	no	—
38	#924	—	2	2	no	—
39	#870	—	2	2	no	—
40	#869	C₂×C₄	1	2	yes	—
41	#869	C₂×C₄	1	2	yes	—
42	C₄×Dic₆	—	2	2	no	—
43	C₄×Dic₆	—	2	2	no	—
44	C₃×C₄⋊C₈	—	2	1	no	—
45	C₃×C₄⋊C₈	D₄	1	2	yes	—
46	C₃×C₄×C₈	—	4	2	no	—
47	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄×C₈	D₄	1	1	yes	characteristic
50	C₃×C₄×C₈	D₄	1	1	yes	characteristic
51	C₃×#95	D₄	1	2	yes	—
52	#830	—	2	2	no	—
53	#818	—	2	1	no	—
54	#818	—	2	1	no	—
55	#818	—	2	1	no	—
56	#818	—	2	1	no	—
57	#817	—	2	2	no	—
58	#817	—	2	2	no	—
59	C₄⋊(C₃⋊C₈)	—	2	2	no	—
60	C₄×C₃⋊C₈	—	2	2	no	—
61	#533	—	3	2	no	—
62	#500	—	3	1	no	—
63	#500	—	3	1	no	—
64	#499	—	6	1	no	—
65	#499	—	6	1	no	—
66	#498	—	3	2	no	—
67	#431	—	3	2	no	—
68	#427	—	6	1	no	—
69	#427	—	6	1	no	—
70	C₂×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
71	C₂×C₄×C₈	—	2	1	no	—
72	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
73	C₄×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
74	#336	—	2	1	no	—
75	#335	—	6	1	no	—
76	#334	—	6	1	no	—
77	#328	—	6	4	no	—
78	Q₈⋊C₈	—	6	4	no	—
79	C₂²×C₃×C₄	—	2	2	no	—
80	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₃×C₄	#41	1	1	yes	characteristic
83	C₂×Dic₆	—	2	2	no	—
84	C₂×Dic₆	—	2	2	no	—
85	C₂×C₃×C₈	—	2	1	no	—
86	C₂×C₃×C₈	—	2	1	no	—
87	C₂×C₃×C₈	—	2	2	no	—
88	C₂×C₃×C₈	—	4	1	no	—
89	C₂×C₃×C₈	—	4	1	no	—
90	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₃×C₈	SD₁₆	1	2	yes	—
95	C₂×C₃×C₈	Q₁₆	1	2	yes	—
96	C₃×C₄⋊C₄	—	2	4	no	—
97	C₃×C₄²	—	4	1	no	—
98	C₃×C₄²	—	4	1	no	—
99	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
100	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
101	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
102	C₃×C₄²	#41	1	1	yes	characteristic
103	#217	—	4	2	no	—
104	C₄⋊Dic₃	—	2	2	no	—
105	C₄⋊Dic₃	—	2	2	no	—
106	C₄⋊Dic₃	—	2	2	no	—
107	C₄⋊Dic₃	—	2	2	no	—
108	Dic₃⋊C₄	—	4	2	no	—
109	Dic₃⋊C₄	—	4	2	no	—
110	C₄×Dic₃	—	4	2	no	—
111	C₂×C₃⋊C₈	—	4	2	no	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	#67	1	1	yes	characteristic
114	C₂²×C₈	#67	1	1	yes	characteristic
115	C₄⋊Q₈	—	3	2	no	—
116	C₄⋊Q₈	—	3	2	no	—
117	#125	—	6	2	no	—
118	C₂²⋊Q₈	—	6	2	no	—
119	C₂²⋊Q₈	—	6	2	no	—
120	C₄×Q₈	—	6	2	no	—
121	C₄×Q₈	—	6	2	no	—
122	#117	—	6	2	no	—
123	C₂×C₄⋊C₄	—	2	1	no	—
124	C₂×C₄²	—	2	1	no	—
125	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
126	#107	—	6	1	no	—
127	#107	—	6	1	no	—
128	#107	—	6	1	no	—
129	#107	—	6	1	no	—
130	#106	—	6	2	no	—
131	#106	—	6	2	no	—
132	C₄⋊C₈	—	2	1	no	—
133	C₄⋊C₈	—	6	2	no	—
134	C₄⋊C₈	D₁₂	1	2	yes	—
135	C₂²⋊C₈	—	6	2	no	—
136	C₄×C₈	—	4	2	no	—
137	C₄×C₈	—	6	2	no	—
138	C₄×C₈	C₄×D₃	1	1	yes	characteristic
139	C₄×C₈	C₄×D₃	1	1	yes	characteristic
140	C₄×C₈	D₁₂	1	1	yes	characteristic
141	C₄×C₈	D₁₂	1	1	yes	characteristic
142	#95	D₁₂	1	2	yes	—
143	C₂³×C₃	#121	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	4	1	no	—
147	C₂×C₃×C₄	—	4	2	no	—
148	C₂×C₃×C₄	—	4	2	no	—
149	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	#121	1	1	yes	characteristic
153	C₂×C₃×C₄	#124	1	1	yes	characteristic
154	C₂×C₃×C₄	#127	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
157	C₂×C₃×C₄	#137	1	1	yes	characteristic
158	C₂×C₃×C₄	#137	1	1	yes	characteristic
159	C₂×Dic₃	—	4	2	no	—
160	C₂×Dic₃	—	4	2	no	—
161	Dic₆	—	4	4	no	—
162	Dic₆	—	8	2	no	—
163	C₃×C₈	—	2	1	no	—
164	C₃×C₈	—	2	1	no	—
165	C₃×C₈	—	4	2	no	—
166	C₃×C₈	#103	1	2	yes	—
167	C₃×C₈	#103	1	2	yes	—
168	C₃⋊C₈	—	8	2	no	—
169	C₂×Q₈	—	6	2	no	—
170	C₂×Q₈	—	6	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
173	C₂²×C₄	#235	1	1	yes	characteristic
174	C₂²×C₄	C₂×#67	1	1	yes	characteristic
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	12	2	no	—
181	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
182	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
183	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
184	C₂×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
185	C₂×C₈	#214	1	2	yes	—
186	C₂×C₈	#216	1	2	yes	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	6	2	no	—
189	C₄⋊C₄	—	6	2	no	—
190	C₄⋊C₄	—	6	2	no	—
191	C₄⋊C₄	—	6	2	no	—
192	C₄⋊C₄	—	12	2	no	—
193	C₄⋊C₄	—	12	2	no	—
194	C₂²⋊C₄	—	12	2	no	—
195	C₄²	—	4	1	no	—
196	C₄²	—	4	1	no	—
197	C₄²	—	12	2	no	—
198	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
199	C₄²	C₂×D₁₂	1	1	yes	characteristic
200	C₄²	C₂×D₁₂	1	1	yes	characteristic
201	C₄²	#235	1	1	yes	characteristic
202	C₂²×C₃	—	2	1	no	—
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	#390	1	1	yes	characteristic
205	C₂²×C₃	#461	1	1	yes	characteristic
206	C₂²×C₃	#462	1	1	yes	characteristic
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	4	4	no	—
211	C₃×C₄	C₂×#103	1	1	yes	characteristic
212	C₃×C₄	#419	1	1	yes	characteristic
213	C₃×C₄	#443	1	2	yes	—
214	C₃×C₄	#461	1	1	yes	characteristic
215	C₃×C₄	#462	1	1	yes	characteristic
216	Dic₃	—	8	2	no	—
217	Dic₃	—	8	2	no	—
218	C₂³	#930	1	1	yes	characteristic
219	Q₈	—	12	4	no	—
220	Q₈	—	24	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	12	2	no	—
227	C₂×C₄	—	12	2	no	—
228	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
229	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
230	C₂×C₄	#874	1	1	yes	characteristic
231	C₂×C₄	#875	1	1	yes	characteristic
232	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
233	C₂×C₄	#930	1	1	yes	characteristic
234	C₂×C₄	C₂×#214	1	1	yes	characteristic
235	C₂×C₄	C₂×#216	1	1	yes	characteristic
236	C₂×C₄	#951	1	1	yes	characteristic
237	C₂×C₄	#951	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	2	no	—
241	C₈	—	24	2	no	—
242	C₈	#810	1	2	yes	—
243	C₈	#810	1	2	yes	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	#1974	1	1	yes	characteristic
247	C₂×C₃	#1976	1	1	yes	characteristic
248	C₂×C₃	#1981	1	1	yes	characteristic
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	#4939	1	1	yes	characteristic
252	C₂²	#5019	1	1	yes	characteristic
253	C₂²	#5031	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	—	24	2	no	—
259	C₄	—	24	2	no	—
260	C₄	#4700	1	2	yes	—
261	C₄	C₂×#810	1	1	yes	characteristic
262	C₄	#5004	1	1	yes	characteristic
263	C₄	#5019	1	1	yes	characteristic
264	C₄	#5031	1	1	yes	characteristic
265	C₃	#9749	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	#69486	1	1	yes	characteristic
269	C₂	#69795	1	1	yes	characteristic
270	C₂	#69897	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#930
commutator	C₃×C₄²	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₈×C₄⋊C₄	C₂
socle	C₂²×C₃	#390
prime radical	C₂³×C₃	#121
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₈×C₄⋊C₄	D₃
hypocenter	C₃	#9749
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
17. #4682	C₄	semidirect
18. #4682	C₄	semidirect
19. #4680	C₄	semidirect
265. C₃	#9749	semidirect

Maximal subgroups

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

subgroup	index
#78442	2
C₃×C₈×C₄⋊C₄	2
#70429	2
#69729	2
#69471	2
#9749	3