Subgroups — #10693801

Order 768.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#335	C₂	1	1	yes	maximal, characteristic
2	#70420	C₂	1	2	yes	maximal
3	C₄×#819	C₂	1	1	yes	maximal, characteristic
4	#69724	C₂	1	1	yes	maximal, characteristic
5	#67706	C₂	1	2	yes	maximal
6	#9608	—	3	1	no	maximal, Sylow
7	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×#819	C₄	1	1	yes	characteristic
10	C₂×#819	C₄	1	1	yes	characteristic
11	#4925	C₂²	1	1	yes	characteristic
12	#4925	C₂²	1	1	yes	characteristic
13	C₄×#208	C₂²	1	2	yes	—
14	C₃×#335	—	2	1	no	—
15	C₃×#335	—	2	1	no	—
16	#4552	C₄	1	2	yes	—
17	#4480	—	2	2	no	—
18	#4479	—	2	2	no	—
19	#4455	—	2	1	no	—
20	C₈×C₃⋊C₈	—	2	1	no	—
21	C₄×#108	—	3	1	no	—
22	#1565	—	3	2	no	—
23	C₂×#335	D₃	1	1	yes	characteristic
24	#1450	—	3	1	no	—
25	#1283	—	3	2	no	—
26	C₂²×C₃×C₈	—	2	2	no	—
27	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
30	C₂×#208	C₂×C₄	1	2	yes	—
31	C₂×#208	C₂×C₄	1	2	yes	—
32	C₃×C₄⋊C₈	—	2	2	no	—
33	C₃×C₄⋊C₈	—	2	2	no	—
34	C₃×C₄×C₈	D₄	1	1	yes	characteristic
35	C₃×C₄×C₈	D₄	1	1	yes	characteristic
36	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
37	C₃×C₄×C₈	Q₈	1	1	yes	characteristic
38	#830	—	2	1	no	—
39	#830	Q₈	1	2	yes	—
40	#819	—	2	2	no	—
41	#819	C₈	1	4	yes	complemented
42	C₄⋊(C₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
44	C₄⋊(C₃⋊C₈)	D₄	1	1	yes	characteristic
45	#803	—	2	2	no	—
46	C₄×C₃⋊C₈	—	2	2	no	—
47	C₄×C₃⋊C₈	—	2	1	no	—
48	C₄×C₃⋊C₈	—	2	1	no	—
49	#431	—	3	1	no	—
50	#431	—	3	1	no	—
51	C₂×#108	—	3	1	no	—
52	C₂×#108	—	3	1	no	—
53	C₂×C₄⋊C₈	C₂×D₃	1	2	yes	—
54	C₄×M₁₆	—	3	2	no	—
55	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	#341	—	3	2	no	—
57	#335	—	2	1	no	—
58	#335	—	2	1	no	—
59	#335	—	6	2	no	—
60	#334	—	6	2	no	—
61	#322	—	6	1	no	—
62	C₈²	—	6	1	no	—
63	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₃×C₄	#41	1	1	yes	characteristic
66	C₂×C₃×C₈	—	2	1	no	—
67	C₂×C₃×C₈	—	2	1	no	—
68	C₂×C₃×C₈	—	4	2	no	—
69	C₂×C₃×C₈	—	8	2	no	—
70	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
75	C₂×C₃×C₈	M₁₆	1	2	yes	—
76	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
78	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
79	C₃×C₄²	#41	1	1	yes	characteristic
80	#208	—	2	2	no	—
81	#208	—	2	2	no	—
82	#208	—	2	2	no	—
83	#208	C₂×C₈	1	4	yes	—
84	C₂×C₃⋊C₈	—	2	2	no	—
85	C₂×C₃⋊C₈	—	2	2	no	—
86	C₂×C₃⋊C₈	—	2	1	no	—
87	C₂×C₃⋊C₈	—	2	1	no	—
88	C₂×C₃⋊C₈	—	2	2	no	—
89	C₂×M₁₆	—	3	2	no	—
90	C₂×M₁₆	—	3	2	no	—
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
93	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
94	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
95	#108	—	3	4	no	—
96	#108	—	6	2	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	3	1	no	—
100	C₄⋊C₈	—	3	1	no	—
101	C₄⋊C₈	—	6	1	no	—
102	C₂²⋊C₈	—	3	2	no	—
103	C₂²⋊C₈	—	6	1	no	—
104	#97	—	6	2	no	—
105	C₄×C₈	—	6	2	no	—
106	C₄×C₈	—	6	1	no	—
107	C₄×C₈	—	6	1	no	—
108	C₄×C₈	Dic₆	1	1	yes	characteristic
109	C₄×C₈	Dic₆	1	1	yes	characteristic
110	C₄×C₈	#67	1	1	yes	characteristic
111	C₄×C₈	#67	1	1	yes	characteristic
112	C₂³×C₃	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	4	1	no	—
116	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
118	C₂×C₃×C₄	#121	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₃×C₄	#125	1	1	yes	characteristic
121	C₂×C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
124	C₂×C₃×C₄	#131	1	1	yes	characteristic
125	C₂×C₃×C₄	#131	1	1	yes	characteristic
126	C₃×C₈	—	2	1	no	—
127	C₃×C₈	—	2	1	no	—
128	C₃×C₈	—	8	2	no	—
129	C₃×C₈	C₄⋊C₈	1	2	yes	—
130	C₃×C₈	C₄⋊C₈	1	2	yes	—
131	C₃⋊C₈	—	2	4	no	—
132	C₃⋊C₈	—	4	2	no	—
133	C₃⋊C₈	—	4	2	no	—
134	C₃⋊C₈	—	4	1	no	—
135	C₃⋊C₈	—	4	1	no	—
136	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
137	C₂²×C₄	#237	1	1	yes	characteristic
138	C₂²×C₄	D₃×Q₈	1	1	yes	characteristic
139	M₁₆	—	3	4	no	—
140	M₁₆	—	6	2	no	—
141	M₁₆	—	6	2	no	—
142	M₁₆	—	6	2	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	6	2	no	—
147	C₂×C₈	—	6	2	no	—
148	C₂×C₈	—	6	1	no	—
149	C₂×C₈	—	6	1	no	—
150	C₂×C₈	—	6	2	no	—
151	C₂×C₈	—	8	2	no	—
152	C₂×C₈	D₃×C₈	1	2	yes	—
153	C₂×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
154	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
155	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
156	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
157	C₂×C₈	Dic₃⋊C₄	1	1	yes	characteristic
158	C₄²	C₂×Dic₆	1	1	yes	characteristic
159	C₄²	D₃×D₄	1	1	yes	characteristic
160	C₄²	#239	1	1	yes	characteristic
161	C₄²	C₂×#67	1	1	yes	characteristic
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	2	1	no	—
164	C₂²×C₃	#389	1	1	yes	characteristic
165	C₂²×C₃	C₈×Q₈	1	1	yes	characteristic
166	C₂²×C₃	#446	1	1	yes	characteristic
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
171	C₃×C₄	#424	1	1	yes	characteristic
172	C₃×C₄	C₈×D₄	1	1	yes	characteristic
173	C₃×C₄	#436	1	1	yes	characteristic
174	C₃×C₄	#472	1	1	yes	characteristic
175	C₃×C₄	#473	1	1	yes	characteristic
176	C₂³	#887	1	1	yes	characteristic
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	#888	1	1	yes	characteristic
181	C₂×C₄	#890	1	1	yes	characteristic
182	C₂×C₄	#893	1	1	yes	characteristic
183	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
184	C₂×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
185	C₂×C₄	#907	1	1	yes	characteristic
186	C₂×C₄	#907	1	1	yes	characteristic
187	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
188	C₂×C₄	#938	1	1	yes	characteristic
189	C₂×C₄	#946	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	6	4	no	—
193	C₈	—	8	2	no	—
194	C₈	—	12	2	no	—
195	C₈	—	12	1	no	—
196	C₈	—	12	1	no	—
197	C₈	—	12	2	no	—
198	C₈	Dic₃⋊C₈	1	2	yes	—
199	C₈	Dic₃⋊C₈	1	2	yes	—
200	C₂×C₃	—	2	1	no	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	#1939	1	1	yes	characteristic
203	C₂×C₃	#1952	1	1	yes	characteristic
204	C₂×C₃	#1953	1	1	yes	characteristic
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	#4830	1	1	yes	characteristic
208	C₂²	#4831	1	1	yes	characteristic
209	C₂²	#4972	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	#4834	1	1	yes	characteristic
214	C₄	#4838	1	1	yes	characteristic
215	C₄	C₂×Dic₃⋊C₈	1	1	yes	characteristic
216	C₄	#5119	1	1	yes	characteristic
217	C₄	#5160	1	1	yes	characteristic
218	C₄	#5192	1	1	yes	characteristic
219	C₃	#9608	1	1	yes	characteristic, complemented, Sylow
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#70468	1	1	yes	characteristic
223	C₂	#70625	1	1	yes	characteristic
224	C₂	#70706	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#938
commutator	C₂×C₃×C₄	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×#335	C₂
socle	C₂²×C₃	#389
prime radical	C₂³×C₃	C₄×Q₈
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×#335	D₃
hypocenter	C₃	#9608
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
41. #819	C₈	semidirect
219. C₃	#9608	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#335	2
#70420	2
C₄×#819	2
#69724	2
#67706	2
#9608	3