Subgroups — Q₈×#634 #10596037

Order 576.

636 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃⋊Dic₃	C₂	1	2	yes	maximal
3	#64097	C₂	1	6	yes	maximal
4	#64078	C₂	1	3	yes	maximal
5	C₄×#634	C₂	1	3	yes	maximal
6	Q₈×Dic₆	—	3	4	no	maximal
7	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
8	C₂×#634	C₂²	1	6	yes	—
9	C₂×#634	C₂²	1	1	yes	characteristic
10	C₃²×C₄⋊C₄	C₂²	1	3	yes	—
11	C₃²×C₄²	C₂²	1	3	yes	—
12	C₄⋊(C₃⋊Dic₃)	C₂²	1	3	yes	—
13	C₄⋊(C₃⋊Dic₃)	C₂²	1	6	yes	—
14	#3750	C₂²	1	6	yes	—
15	C₄×C₃⋊Dic₃	C₂²	1	6	yes	—
16	C₃×C₄×Q₈	D₃	1	4	yes	—
17	Q₈×Dic₃	—	3	8	no	—
18	#888	—	3	24	no	—
19	#869	—	3	12	no	—
20	C₄×Dic₆	—	3	12	no	—
21	C₃²×Q₈	Q₈	1	4	yes	complemented
22	C₂×C₃²×C₄	C₂³	1	3	yes	—
23	C₂×C₃²×C₄	C₂³	1	3	yes	—
24	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₂×C₃⋊Dic₃	C₂³	1	6	yes	—
26	C₂×C₃⋊Dic₃	C₂³	1	2	yes	—
27	#634	—	4	6	no	—
28	#634	Q₈	1	4	yes	direct product factor
29	Q₈²	—	9	1	no	Sylow
30	C₂×C₃×Q₈	C₂×D₃	1	4	yes	—
31	C₂×Dic₆	—	3	4	no	—
32	C₂×Dic₆	—	3	24	no	—
33	C₃×C₄⋊C₄	C₂×D₃	1	12	yes	—
34	C₃×C₄²	C₂×D₃	1	12	yes	—
35	C₄⋊Dic₃	—	3	12	no	—
36	C₄⋊Dic₃	—	3	24	no	—
37	Dic₃⋊C₄	—	3	24	no	—
38	C₄×Dic₃	—	3	24	no	—
39	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
40	C₃²×C₄	—	2	3	no	—
41	C₃²×C₄	C₂×Q₈	1	6	yes	—
42	C₃²×C₄	C₂×Q₈	1	2	yes	—
43	C₃⋊Dic₃	—	2	6	no	—
44	C₃⋊Dic₃	C₂×Q₈	1	4	yes	—
45	C₄⋊Q₈	—	9	6	no	—
46	C₄⋊Q₈	—	9	3	no	—
47	C₄×Q₈	—	9	2	no	—
48	C₄×Q₈	—	9	3	no	—
49	C₄×Q₈	C₃⋊D₃	1	1	yes	characteristic
50	C₃×Q₈	Dic₆	1	16	yes	complemented
51	C₂×C₃×C₄	C₂²×D₃	1	12	yes	—
52	C₂×C₃×C₄	C₂²×D₃	1	12	yes	—
53	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
54	C₂×Dic₃	—	3	8	no	—
55	C₂×Dic₃	—	3	24	no	—
56	Dic₆	—	3	16	no	—
57	Dic₆	—	12	24	no	—
58	C₂×C₃²	C₂²×Q₈	1	1	yes	characteristic
59	C₂×C₃²	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×Q₈	—	9	1	no	—
62	C₂×Q₈	—	9	6	no	—
63	C₂×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
64	C₄⋊C₄	—	9	6	no	—
65	C₄⋊C₄	—	9	6	no	—
66	C₄⋊C₄	—	9	3	no	—
67	C₄⋊C₄	C₂×C₃⋊D₃	1	3	yes	—
68	C₄²	—	9	6	no	—
69	C₄²	C₂×C₃⋊D₃	1	3	yes	—
70	C₂²×C₃	C₂³×D₃	1	4	yes	—
71	C₃×C₄	—	2	12	no	—
72	C₃×C₄	C₂×Dic₆	1	24	yes	—
73	C₃×C₄	D₃×Q₈	1	8	yes	—
74	Dic₃	—	3	16	no	—
75	Dic₃	—	6	24	no	—
76	C₃²	Q₈²	1	1	yes	characteristic, complemented, Sylow
77	Q₈	—	9	4	no	—
78	Q₈	—	36	6	no	—
79	Q₈	#634	1	4	yes	direct product factor
80	C₂×C₄	—	9	2	no	—
81	C₂×C₄	—	9	6	no	—
82	C₂×C₄	C₂²×C₃⋊D₃	1	3	yes	—
83	C₂×C₄	C₂²×C₃⋊D₃	1	3	yes	—
84	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
85	C₂×C₃	C₂²×Dic₆	1	4	yes	—
86	C₂×C₃	C₂×D₃×Q₈	1	4	yes	—
87	C₂×C₃	#1009	1	4	yes	—
88	C₂²	C₂³×C₃⋊D₃	1	1	yes	characteristic
89	C₄	—	2	3	no	—
90	C₄	—	9	4	no	—
91	C₄	—	18	6	no	—
92	C₄	C₂×#634	1	6	yes	—
93	C₄	Q₈×C₃⋊D₃	1	2	yes	—
94	C₃	Q₈×Dic₆	1	4	yes	complemented
95	C₂	C₂²×#634	1	1	yes	characteristic
96	C₂	C₂×Q₈×C₃⋊D₃	1	1	yes	characteristic
97	C₂	#64366	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊D₃
commutator, socle, prime radical, cosocle	C₂²×C₃²	C₂⁴
Fitting	C₃²×C₄×Q₈	C₂
hypercenter	C₄×Q₈	C₃⋊D₃
hypocenter	C₃²	Q₈²
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
21. C₃²×Q₈	Q₈	semidirect
28. #634	Q₈	direct (N × H)
50. C₃×Q₈	Dic₆	semidirect
76. C₃²	Q₈²	semidirect
79. Q₈	#634	direct (N × H)
94. C₃	Q₈×Dic₆	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×Q₈	2
Q₈×C₃⋊Dic₃	2
#64097	2
#64078	2
C₄×#634	2
Q₈×Dic₆	3

Subgroups — Q8×#634 #10596037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#634 #10596037