Subgroups — #5722

Order 192.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#944	C₂	1	4	yes	maximal
3	#888	C₂	1	4	yes	maximal
4	#887	C₂	1	4	yes	maximal
5	C₄×Dic₆	C₂	1	2	yes	maximal
6	Q₈²	—	3	1	no	maximal, Sylow
7	C₂×C₃×Q₈	C₂²	1	2	yes	—
8	C₂×Dic₆	C₂²	1	4	yes	—
9	C₂×Dic₆	C₂²	1	2	yes	—
10	C₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃	C₂²	1	2	yes	—
13	Dic₃⋊C₄	C₂²	1	4	yes	—
14	Dic₃⋊C₄	C₂²	1	8	yes	—
15	C₄×Dic₃	C₂²	1	4	yes	—
16	C₄×Dic₃	C₂²	1	4	yes	—
17	C₄⋊Q₈	—	3	4	no	—
18	C₄⋊Q₈	—	3	4	no	—
19	C₄⋊Q₈	D₃	1	1	yes	characteristic
20	C₄×Q₈	—	3	4	no	—
21	C₄×Q₈	—	3	2	no	—
22	C₃×Q₈	—	4	2	no	—
23	C₂×C₃×C₄	C₂³	1	4	yes	—
24	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂×C₃×C₄	C₂³	1	2	yes	—
26	C₂×Dic₃	C₂³	1	4	yes	—
27	C₂×Dic₃	C₂³	1	4	yes	—
28	Dic₆	—	4	4	no	—
29	Dic₆	Q₈	1	8	yes	complemented
30	C₂×Q₈	—	3	2	no	—
31	C₂×Q₈	—	3	4	no	—
32	C₂×Q₈	C₂×D₃	1	2	yes	—
33	C₄⋊C₄	—	3	8	no	—
34	C₄⋊C₄	—	3	4	no	—
35	C₄⋊C₄	—	3	2	no	—
36	C₄⋊C₄	C₂×D₃	1	4	yes	—
37	C₄²	—	3	4	no	—
38	C₄²	—	3	4	no	—
39	C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃	C₂⁴	1	1	yes	characteristic
41	C₃×C₄	—	2	4	no	—
42	C₃×C₄	—	2	1	no	—
43	C₃×C₄	C₂×Q₈	1	4	yes	—
44	Dic₃	—	2	4	no	—
45	Dic₃	C₂×Q₈	1	8	yes	—
46	Q₈	—	3	8	no	—
47	Q₈	—	4	2	no	—
48	Q₈	—	12	4	no	—
49	C₂×C₄	—	3	4	no	—
50	C₂×C₄	—	3	4	no	—
51	C₂×C₄	C₂²×D₃	1	4	yes	—
52	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₃	1	2	yes	—
54	C₂×C₃	C₂²×Q₈	1	2	yes	—
55	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
56	C₂²	C₂³×D₃	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	—	2	4	no	—
59	C₄	—	3	8	no	—
60	C₄	—	6	4	no	—
61	C₄	D₃×Q₈	1	4	yes	—
62	C₃	Q₈²	1	1	yes	characteristic, complemented, Sylow
63	C₂	#1004	1	1	yes	characteristic
64	C₂	C₂×D₃×Q₈	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
29. Dic₆	Q₈	semidirect
62. C₃	Q₈²	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄⋊Q₈	2
#944	2
#888	2
#887	2
C₄×Dic₆	2
Q₈²	3