Subgroups — Q₈×#3708 #11797999

Order 1152.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#3708	C₂	1	3	yes	maximal
3	C₄×#3708	C₂	1	3	yes	maximal
4	C₃²×C₄×Q₈	C₄	1	1	yes	characteristic
5	C₂×#3708	C₂²	1	3	yes	—
6	C₂×#3708	C₂²	1	1	yes	characteristic
7	C₄⋊(C₃⋊(C₃⋊C₈))	C₄	1	3	yes	—
8	C₄×C₃⋊(C₃⋊C₈)	C₂²	1	3	yes	—
9	Q₈×C₃⋊C₈	—	6	2	no	—
10	C₂×C₃²×Q₈	C₈	1	1	yes	characteristic
11	C₃²×C₄⋊C₄	C₈	1	3	yes	—
12	C₃²×C₄²	C₂×C₄	1	3	yes	—
13	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	3	yes	—
14	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	1	yes	characteristic
15	#3708	—	2	3	no	—
16	#3708	Q₈	1	2	yes	direct product factor
17	Q₈×C₁₆	—	9	1	no	maximal, Sylow
18	C₃×C₄×Q₈	—	2	2	no	—
19	C₄⋊(C₃⋊C₈)	—	6	6	no	—
20	C₄×C₃⋊C₈	—	6	6	no	—
21	C₃²×Q₈	C₁₆	1	4	yes	complemented
22	C₂×C₃²×C₄	C₂×C₈	1	3	yes	—
23	C₂×C₃²×C₄	C₂×C₈	1	3	yes	—
24	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
25	C₃⋊(C₃⋊C₈)	—	2	3	no	—
26	C₃⋊(C₃⋊C₈)	C₂×Q₈	1	1	yes	characteristic
27	C₃⋊(C₃⋊C₈)	#41	1	1	yes	characteristic
28	C₈×Q₈	—	9	1	no	—
29	C₄⋊C₁₆	—	9	3	no	—
30	C₄×C₁₆	—	9	3	no	—
31	C₂×C₃×Q₈	—	2	2	no	—
32	C₃×C₄⋊C₄	—	2	6	no	—
33	C₃×C₄²	—	2	6	no	—
34	C₂×C₃⋊C₈	—	6	2	no	—
35	C₂×C₃⋊C₈	—	6	6	no	—
36	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
37	C₃²×C₄	—	2	3	no	—
38	C₃²×C₄	C₂×C₁₆	1	6	yes	—
39	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
40	C₃²×C₄	#131	1	1	yes	characteristic
41	C₄×Q₈	#157	1	1	yes	characteristic
42	C₂×C₁₆	—	9	1	no	—
43	C₂×C₁₆	—	9	3	no	—
44	C₄⋊C₈	—	9	3	no	—
45	C₄×C₈	—	9	3	no	—
46	C₃×Q₈	—	2	8	no	—
47	C₂×C₃×C₄	—	2	2	no	—
48	C₂×C₃×C₄	—	2	6	no	—
49	C₂×C₃×C₄	—	2	6	no	—
50	C₃⋊C₈	—	6	2	no	—
51	C₃⋊C₈	—	6	2	no	—
52	C₃⋊C₈	—	12	6	no	—
53	C₂×C₃²	C₈×Q₈	1	1	yes	characteristic
54	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
55	C₂×C₃²	#504	1	1	yes	characteristic
56	C₂×Q₈	#622	1	1	yes	characteristic
57	C₂×C₈	—	9	1	no	—
58	C₂×C₈	—	9	3	no	—
59	C₄⋊C₄	#622	1	3	yes	—
60	C₄²	C₂×#157	1	3	yes	—
61	C₁₆	—	9	2	no	—
62	C₁₆	—	18	3	no	—
63	C₂²×C₃	—	2	2	no	—
64	C₃×C₄	—	2	12	no	—
65	C₃×C₄	—	2	2	no	—
66	C₃×C₄	—	2	2	no	—
67	C₃×C₄	—	4	6	no	—
68	C₃²	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
69	Q₈	#3708	1	4	yes	direct product factor
70	C₂×C₄	C₂×#622	1	3	yes	—
71	C₂×C₄	C₂×#622	1	3	yes	—
72	C₂×C₄	C₂²×#157	1	1	yes	characteristic
73	C₈	—	9	1	no	—
74	C₈	—	9	1	no	—
75	C₈	—	18	3	no	—
76	C₂×C₃	—	2	2	no	—
77	C₂×C₃	—	2	2	no	—
78	C₂×C₃	—	2	2	no	—
79	C₂²	C₂²×#622	1	1	yes	characteristic
80	C₄	—	2	3	no	—
81	C₄	C₂×#3708	1	6	yes	—
82	C₄	#64289	1	1	yes	characteristic
83	C₄	Q₈×#157	1	1	yes	characteristic
84	C₃	—	2	2	no	—
85	C₂	C₂²×#3708	1	1	yes	characteristic
86	C₂	Q₈×#622	1	1	yes	characteristic
87	C₂	#10593838	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#157
commutator	C₂×C₃²	C₂²×C₁₆
Fitting	C₃²×C₄×Q₈	C₄
socle, prime radical	C₂²×C₃²	C₂²×C₈
cosocle	C₂×C₃⋊(C₃⋊C₈)	C₂³
hypercenter	C₄×Q₈	#157
hypocenter	C₃²	Q₈×C₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. #3708	Q₈	direct (N × H)
21. C₃²×Q₈	C₁₆	semidirect
68. C₃²	Q₈×C₁₆	semidirect
69. Q₈	#3708	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×C₃⋊(C₃⋊C₈)	2
C₄⋊#3708	2
C₄×#3708	2
Q₈×C₁₆	9

Subgroups — Q8×#3708 #11797999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#3708 #11797999