Finite Group Databasebeta

Subgroups — #10709975

Order 768.

272 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2240	C₂	1	1	yes	maximal, characteristic
2	#71384	C₂	1	1	yes	maximal, characteristic, complemented
3	#71362	C₂	1	1	yes	maximal, characteristic, complemented
4	#71288	C₂	1	1	yes	maximal, characteristic
5	#71274	C₂	1	1	yes	maximal, characteristic
6	#69044	C₂	1	1	yes	maximal, characteristic, complemented
7	#67499	C₂	1	1	yes	maximal, characteristic, complemented
8	#12531	—	3	1	no	maximal, Sylow
9	C₃×#500	C₂²	1	1	yes	characteristic
10	C₃×#493	C₂²	1	1	yes	characteristic
11	#5092	—	2	1	no	—
12	#5078	—	2	1	no	—
13	#5076	—	2	1	no	—
14	#5070	—	2	1	no	—
15	#4682	C₂²	1	1	yes	characteristic
16	#4682	C₂²	1	1	yes	characteristic
17	C₃×#357	—	2	2	no	—
18	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	#4562	—	2	2	no	—
20	#4504	—	2	2	no	—
21	#4489	—	2	2	no	—
22	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₈×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#2252	—	3	1	no	—
25	#2252	—	3	1	no	—
26	#2240	—	3	1	no	—
27	#2240	D₃	1	1	yes	characteristic
28	#1713	—	3	1	no	—
29	#1706	—	3	1	no	—
30	#1312	—	3	1	no	—
31	C₂×C₃×D₈	—	2	1	no	—
32	C₂×C₃×D₈	—	2	1	no	—
33	C₃×C₄⋊Q₈	—	2	1	no	—
34	C₃×#127	—	2	1	no	—
35	C₂×#216	—	4	2	no	—
36	C₂×#214	—	4	2	no	—
37	#869	—	2	1	no	—
38	#869	—	2	1	no	—
39	C₂×C₃×C₁₆	D₄	1	2	yes	—
40	C₃×#107	—	2	1	no	—
41	C₃×#107	—	2	1	no	—
42	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
43	#832	—	4	2	no	—
44	#818	—	2	1	no	—
45	#818	—	2	1	no	—
46	#818	—	2	1	no	—
47	#818	—	2	1	no	—
48	C₂×C₃⋊C₁₆	D₄	1	2	yes	—
49	#807	—	4	2	no	—
50	C₄×C₃⋊C₈	D₄	1	1	yes	characteristic
51	C₄×C₃⋊C₈	D₄	1	1	yes	characteristic
52	#500	—	3	1	no	—
53	#500	—	3	1	no	—
54	#500	—	6	1	no	—
55	#500	C₂×D₃	1	1	yes	characteristic
56	#494	—	6	1	no	—
57	#493	C₂×D₃	1	1	yes	characteristic
58	#492	—	6	1	no	—
59	#486	—	6	1	no	—
60	#366	—	6	2	no	—
61	#366	—	6	2	no	—
62	#357	—	2	2	no	—
63	#357	—	6	2	no	—
64	C₄×C₁₆	—	3	1	no	—
65	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
66	C₈²	—	3	1	no	—
67	C₂×C₃×Q₈	—	4	1	no	—
68	C₂×C₃×D₄	—	4	1	no	—
69	C₂×C₃×D₄	—	4	1	no	—
70	C₂×C₃×D₄	—	4	1	no	—
71	C₂×Dic₆	—	4	1	no	—
72	C₂×Dic₆	—	4	1	no	—
73	C₃×D₈	—	4	2	no	—
74	C₃×D₈	—	4	2	no	—
75	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
77	C₃×C₄⋊C₄	—	4	1	no	—
78	C₃×C₄⋊C₄	—	4	1	no	—
79	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
80	#216	—	8	4	no	—
81	#214	—	8	4	no	—
82	C₄⋊Dic₃	—	4	1	no	—
83	C₄⋊Dic₃	—	4	1	no	—
84	C₄⋊Dic₃	—	4	1	no	—
85	C₄⋊Dic₃	—	4	1	no	—
86	C₂×C₃⋊C₈	D₈	1	2	yes	—
87	C₂×C₃⋊C₈	D₈	1	2	yes	—
88	C₃×C₁₆	—	2	2	no	—
89	C₃⋊C₁₆	—	2	2	no	—
90	C₂×Q₁₆	—	12	2	no	—
91	C₂×SD₁₆	—	12	2	no	—
92	C₂×D₈	—	2	1	no	—
93	C₂×D₈	—	2	1	no	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄⋊Q₈	—	6	1	no	—
96	C₄⋊Q₈	—	6	1	no	—
97	#127	—	2	1	no	—
98	C₂×C₁₆	—	3	2	no	—
99	C₂×C₁₆	#67	1	2	yes	—
100	#107	—	2	1	no	—
101	#107	—	2	1	no	—
102	#107	—	6	1	no	—
103	#107	—	6	1	no	—
104	#107	—	6	1	no	—
105	#107	—	6	1	no	—
106	#107	—	12	2	no	—
107	#102	—	12	2	no	—
108	C₄×C₈	—	3	1	no	—
109	C₄×C₈	—	3	1	no	—
110	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
111	C₂³×C₃	—	8	1	no	—
112	C₃×Q₈	—	8	2	no	—
113	C₃×D₄	—	8	1	no	—
114	C₃×D₄	—	8	1	no	—
115	C₃×D₄	—	8	2	no	—
116	C₃×D₄	—	8	2	no	—
117	C₂×C₃×C₄	—	8	1	no	—
118	C₂×C₃×C₄	#127	1	1	yes	characteristic
119	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
121	C₂×Dic₃	—	8	1	no	—
122	C₂×Dic₃	—	8	1	no	—
123	Dic₆	—	8	2	no	—
124	Dic₆	—	8	2	no	—
125	C₃×C₈	#137	1	2	yes	—
126	C₃×C₈	#137	1	2	yes	—
127	C₃⋊C₈	—	2	2	no	—
128	C₃⋊C₈	D₁₆	1	4	yes	—
129	C₂×Q₈	—	4	1	no	—
130	C₂×Q₈	—	12	1	no	—
131	C₂×Q₈	—	12	1	no	—
132	C₂×D₄	—	4	1	no	—
133	C₂×D₄	—	4	1	no	—
134	C₂×D₄	—	4	1	no	—
135	Q₁₆	—	24	4	no	—
136	SD₁₆	—	24	4	no	—
137	D₈	—	4	2	no	—
138	D₈	—	4	2	no	—
139	C₂×C₈	—	3	2	no	—
140	C₂×C₈	—	3	2	no	—
141	C₂×C₈	D₃×D₄	1	1	yes	characteristic
142	C₂×C₈	C₂×#67	1	1	yes	characteristic
143	C₄⋊C₄	—	4	1	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	12	1	no	—
146	C₄⋊C₄	—	12	1	no	—
147	C₄⋊C₄	—	12	1	no	—
148	C₄⋊C₄	—	12	1	no	—
149	C₄²	D₃×D₄	1	1	yes	characteristic
150	C₁₆	—	2	2	no	—
151	C₁₆	—	6	2	no	—
152	C₂²×C₃	—	16	1	no	—
153	C₂²×C₃	—	16	1	no	—
154	C₂²×C₃	—	16	1	no	—
155	C₂²×C₃	#493	1	1	yes	characteristic
156	C₃×C₄	—	16	1	no	—
157	C₃×C₄	#497	1	1	yes	characteristic
158	C₃×C₄	#497	1	1	yes	characteristic
159	C₃×C₄	C₂×D₁₆	1	2	yes	—
160	C₃×C₄	#510	1	2	yes	—
161	Dic₃	—	16	1	no	—
162	Dic₃	—	16	1	no	—
163	C₂³	—	8	1	no	—
164	Q₈	—	8	2	no	—
165	Q₈	—	24	2	no	—
166	Q₈	—	24	2	no	—
167	D₄	—	8	2	no	—
168	D₄	—	8	1	no	—
169	D₄	—	8	2	no	—
170	D₄	—	8	1	no	—
171	C₂×C₄	—	8	1	no	—
172	C₂×C₄	—	24	1	no	—
173	C₂×C₄	—	24	1	no	—
174	C₂×C₄	D₃×D₈	1	1	yes	characteristic
175	C₂×C₄	D₃×D₈	1	1	yes	characteristic
176	C₂×C₄	#940	1	1	yes	characteristic
177	C₈	—	3	4	no	—
178	C₈	—	6	2	no	—
179	C₈	#915	1	2	yes	—
180	C₈	#951	1	2	yes	—
181	C₂×C₃	—	32	1	no	—
182	C₂×C₃	#1722	1	1	yes	characteristic
183	C₂×C₃	#2246	1	1	yes	characteristic
184	C₂×C₃	#2251	1	1	yes	characteristic
185	C₂²	—	16	1	no	—
186	C₂²	—	16	1	no	—
187	C₂²	—	16	1	no	—
188	C₂²	#5151	1	1	yes	characteristic
189	C₄	—	16	1	no	—
190	C₄	—	48	1	no	—
191	C₄	—	48	1	no	—
192	C₄	D₃×D₁₆	1	2	yes	—
193	C₄	#4915	1	2	yes	—
194	C₄	#5060	1	1	yes	characteristic
195	C₄	#5167	1	1	yes	characteristic
196	C₃	#12531	1	1	yes	characteristic, complemented, Sylow
197	C₂	—	32	1	no	—
198	C₂	#71195	1	1	yes	characteristic
199	C₂	#71464	1	1	yes	characteristic
200	C₂	#71482	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#5151
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#2240	C₂
socle	C₂²×C₃	#493
prime radical	C₃×#493	C₂²
hypercenter	#2240	D₃
hypocenter	C₃	#12531
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
2. #71384	C₂	semidirect
3. #71362	C₂	semidirect
6. #69044	C₂	semidirect
7. #67499	C₂	semidirect
196. C₃	#12531	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2240	2
#71384	2
#71362	2
#71288	2
#71274	2
#69044	2
#67499	2
#12531	3