Subgroups — #10693769

Order 768.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#4480	C₂	1	2	yes	maximal
3	#69394	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#67703	C₂	1	2	yes	maximal
6	#9602	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	—	2	1	no	—
10	C₃×C₄³	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
12	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
13	C₂×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
14	C₂×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
17	C₃×#336	—	2	1	no	—
18	#4524	C₄	1	2	yes	—
19	#4480	—	2	4	no	—
20	#1842	—	3	1	no	—
21	C₄×C₄⋊C₈	—	3	1	no	—
22	C₄×C₄⋊C₈	D₃	1	1	yes	characteristic
23	C₂×#334	—	3	2	no	—
24	#1277	—	3	2	no	—
25	C₂²×C₃×C₈	—	2	1	no	—
26	C₂²×C₃×C₈	—	2	1	no	—
27	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
32	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
33	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
34	C₂²×C₃⋊C₈	—	2	2	no	—
35	C₂²×C₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₃⋊C₈	C₂×C₄	1	2	yes	—
37	C₃×C₄⋊C₈	—	2	1	no	—
38	C₃×C₄⋊C₈	—	2	1	no	—
39	C₃×C₄⋊C₈	D₄	1	2	yes	—
40	C₃×C₄⋊C₈	Q₈	1	2	yes	—
41	C₃×C₄×C₈	—	4	2	no	—
42	C₄⋊(C₃⋊C₈)	—	2	2	no	—
43	C₄⋊(C₃⋊C₈)	—	2	2	no	—
44	C₄⋊(C₃⋊C₈)	—	2	2	no	—
45	C₄⋊(C₃⋊C₈)	—	4	2	no	—
46	C₄⋊(C₃⋊C₈)	—	4	2	no	—
47	C₄⋊(C₃⋊C₈)	C₈	1	4	yes	complemented
48	C₄×C₃⋊C₈	—	2	2	no	—
49	C₄×C₃⋊C₈	—	2	2	no	—
50	C₂×C₄⋊C₈	—	3	1	no	—
51	C₂×C₄⋊C₈	—	3	1	no	—
52	C₂×C₄⋊C₈	—	3	2	no	—
53	C₂×C₄⋊C₈	—	6	1	no	—
54	C₂×C₄⋊C₈	—	6	1	no	—
55	C₂×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
56	C₂×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₄×C₈	—	2	1	no	—
58	C₂×C₄×C₈	—	3	2	no	—
59	C₄³	C₂×D₃	1	1	yes	characteristic
60	#336	—	2	1	no	—
61	#336	—	3	2	no	—
62	#334	—	6	4	no	—
63	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₃×C₈	—	2	2	no	—
71	C₂×C₃×C₈	—	2	2	no	—
72	C₂×C₃×C₈	—	4	1	no	—
73	C₂×C₃×C₈	—	4	1	no	—
74	C₂×C₃×C₈	—	4	1	no	—
75	C₂×C₃×C₈	—	4	1	no	—
76	C₃×C₄²	—	2	2	no	—
77	C₃×C₄²	—	2	1	no	—
78	C₃×C₄²	—	2	1	no	—
79	C₃×C₄²	—	2	1	no	—
80	C₃×C₄²	—	2	1	no	—
81	C₃×C₄²	—	4	1	no	—
82	C₃×C₄²	C₂×C₈	1	2	yes	—
83	C₃×C₄²	M₁₆	1	2	yes	—
84	C₃×C₄²	SD₁₆	1	2	yes	—
85	C₃×C₄²	SD₁₆	1	2	yes	—
86	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
87	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
88	C₃×C₄²	#41	1	1	yes	characteristic
89	C₃×C₄²	#41	1	1	yes	characteristic
90	C₂×C₃⋊C₈	—	2	2	no	—
91	C₂×C₃⋊C₈	—	2	2	no	—
92	C₂×C₃⋊C₈	—	2	2	no	—
93	C₂×C₃⋊C₈	—	2	2	no	—
94	C₂×C₃⋊C₈	—	2	2	no	—
95	C₂×C₃⋊C₈	—	4	2	no	—
96	C₂×C₃⋊C₈	—	4	2	no	—
97	C₂×C₃⋊C₈	—	4	2	no	—
98	C₂×C₃⋊C₈	C₂×C₈	1	4	yes	—
99	C₂²×C₈	—	2	1	no	—
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	—	3	2	no	—
102	C₂²×C₈	—	3	2	no	—
103	C₂²×C₈	—	6	2	no	—
104	C₂×C₄²	Dic₆	1	1	yes	characteristic
105	C₂×C₄²	Dic₆	1	1	yes	characteristic
106	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
107	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
108	C₂×C₄²	#67	1	1	yes	characteristic
109	C₂×C₄²	#67	1	1	yes	characteristic
110	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
111	C₄⋊C₈	—	2	1	no	—
112	C₄⋊C₈	—	2	1	no	—
113	C₄⋊C₈	—	3	4	no	—
114	C₄⋊C₈	—	6	2	no	—
115	C₄⋊C₈	—	6	2	no	—
116	C₄⋊C₈	—	6	2	no	—
117	C₄⋊C₈	—	12	2	no	—
118	C₄⋊C₈	—	12	2	no	—
119	C₄⋊C₈	Dic₆	1	2	yes	—
120	C₄⋊C₈	D₁₂	1	2	yes	—
121	C₄×C₈	—	4	2	no	—
122	C₄×C₈	—	6	2	no	—
123	C₄×C₈	—	6	2	no	—
124	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
135	C₂×C₃×C₄	C₄⋊C₈	1	2	yes	—
136	C₂×C₃×C₄	#106	1	2	yes	—
137	C₂×C₃×C₄	#106	1	2	yes	—
138	C₂×C₃×C₄	#108	1	2	yes	—
139	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
141	C₂×C₃×C₄	#121	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₃×C₄	#125	1	1	yes	characteristic
144	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
147	C₂×C₃×C₄	#131	1	1	yes	characteristic
148	C₂×C₃×C₄	#131	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
151	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₃×C₄	#137	1	1	yes	characteristic
153	C₃×C₈	—	4	2	no	—
154	C₃×C₈	—	4	2	no	—
155	C₃⋊C₈	—	2	4	no	—
156	C₃⋊C₈	—	4	2	no	—
157	C₃⋊C₈	—	8	2	no	—
158	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
159	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
160	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
161	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
162	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
164	C₂²×C₄	C₂×#67	1	1	yes	characteristic
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	3	4	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	6	2	no	—
173	C₂×C₈	—	6	2	no	—
174	C₂×C₈	—	6	2	no	—
175	C₂×C₈	—	6	2	no	—
176	C₂×C₈	—	6	2	no	—
177	C₂×C₈	—	12	2	no	—
178	C₂×C₈	—	12	2	no	—
179	C₂×C₈	—	12	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	4	1	no	—
186	C₄²	D₃×C₈	1	2	yes	—
187	C₄²	ΓL(1,𝔽₂₅)	1	2	yes	—
188	C₄²	#214	1	2	yes	—
189	C₄²	#215	1	2	yes	—
190	C₄²	C₂×Dic₆	1	1	yes	characteristic
191	C₄²	C₂×D₁₂	1	1	yes	characteristic
192	C₄²	#235	1	1	yes	characteristic
193	C₄²	#235	1	1	yes	characteristic
194	C₂²×C₃	#389	1	1	yes	characteristic
195	C₂²×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
196	C₂²×C₃	#424	1	1	yes	characteristic
197	C₂²×C₃	C₂×#106	1	1	yes	characteristic
198	C₂²×C₃	#428	1	1	yes	characteristic
199	C₂²×C₃	C₂×#108	1	1	yes	characteristic
200	C₂²×C₃	#430	1	1	yes	characteristic
201	C₃×C₄	—	2	2	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	4	1	no	—
206	C₃×C₄	#334	1	4	yes	complemented
207	C₃×C₄	C₈×D₄	1	1	yes	characteristic
208	C₃×C₄	#436	1	1	yes	characteristic
209	C₃×C₄	C₈×Q₈	1	1	yes	characteristic
210	C₃×C₄	#446	1	1	yes	characteristic
211	C₃×C₄	#460	1	1	yes	characteristic
212	C₃×C₄	#461	1	1	yes	characteristic
213	C₃×C₄	#475	1	1	yes	characteristic
214	C₃×C₄	#476	1	1	yes	characteristic
215	C₂³	C₂×Dic₃⋊C₄	1	1	yes	characteristic
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	#808	1	2	yes	—
226	C₂×C₄	#808	1	2	yes	—
227	C₂×C₄	Dic₃⋊C₈	1	2	yes	—
228	C₂×C₄	Dic₃⋊C₈	1	2	yes	—
229	C₂×C₄	#822	1	2	yes	—
230	C₂×C₄	C₄×Dic₆	1	1	yes	characteristic
231	C₂×C₄	#869	1	1	yes	characteristic
232	C₂×C₄	#870	1	1	yes	characteristic
233	C₂×C₄	C₄×D₁₂	1	1	yes	characteristic
234	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
235	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
236	C₂×C₄	#901	1	1	yes	characteristic
237	C₂×C₄	#901	1	1	yes	characteristic
238	C₂×C₄	#924	1	1	yes	characteristic
239	C₂×C₄	#930	1	1	yes	characteristic
240	C₂×C₄	C₂×#214	1	1	yes	characteristic
241	C₂×C₄	C₂×#215	1	1	yes	characteristic
242	C₂×C₄	#949	1	1	yes	characteristic
243	C₂×C₄	#951	1	1	yes	characteristic
244	C₈	—	4	2	no	—
245	C₈	—	4	2	no	—
246	C₈	—	6	4	no	—
247	C₈	—	12	2	no	—
248	C₈	—	24	2	no	—
249	C₂×C₃	C₂×#334	1	1	yes	characteristic
250	C₂×C₃	#1565	1	1	yes	characteristic
251	C₂×C₃	#1570	1	1	yes	characteristic
252	C₂×C₃	#1574	1	1	yes	characteristic
253	C₂×C₃	#1939	1	1	yes	characteristic
254	C₂×C₃	#1948	1	1	yes	characteristic
255	C₂×C₃	#1950	1	1	yes	characteristic
256	C₂²	#4928	1	1	yes	characteristic
257	C₂²	C₂×#808	1	1	yes	characteristic
258	C₂²	#4995	1	1	yes	characteristic
259	C₂²	C₂×Dic₃⋊C₈	1	1	yes	characteristic
260	C₂²	#5102	1	1	yes	characteristic
261	C₂²	C₂×#822	1	1	yes	characteristic
262	C₂²	#5124	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	1	no	—
268	C₄	#4480	1	4	yes	complemented
269	C₄	C₈×Dic₆	1	1	yes	characteristic
270	C₄	#4679	1	1	yes	characteristic
271	C₄	C₈×D₁₂	1	1	yes	characteristic
272	C₄	#4688	1	1	yes	characteristic
273	C₄	#5008	1	1	yes	characteristic
274	C₄	#5019	1	1	yes	characteristic
275	C₄	#5020	1	1	yes	characteristic
276	C₄	#5027	1	1	yes	characteristic
277	C₃	#9602	1	1	yes	characteristic, complemented, Sylow
278	C₂	#69467	1	1	yes	characteristic
279	C₂	#69786	1	1	yes	characteristic
280	C₂	#69875	1	1	yes	characteristic
281	C₂	C₂×#4480	1	1	yes	characteristic
282	C₂	#70421	1	1	yes	characteristic
283	C₂	#70746	1	1	yes	characteristic
284	C₂	#70748	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₄⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×C₄⋊C₈	D₃
hypocenter	C₃	#9602
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
47. C₄⋊(C₃⋊C₈)	C₈	semidirect
206. C₃×C₄	#334	semidirect
268. C₄	#4480	semidirect
277. C₃	#9602	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₄⋊C₈	2
C₂×#4480	2
#69394	2
C₄×C₄⋊(C₃⋊C₈)	2
#67703	2
#9602	3