Subgroups — #946

Order 96.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₃	C₂	1	1	yes	maximal, characteristic
3	(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal
4	C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂²⋊Q₈	—	3	1	no	maximal, Sylow
7	C₂²×D₃	C₂²	1	1	yes	characteristic
8	C₃×Q₈	—	2	2	no	—
9	C₂×C₃×C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₃	C₂²	1	2	yes	—
13	C₄×D₃	—	2	2	no	—
14	C₂×Q₈	D₃	1	1	yes	characteristic, complemented
15	C₂²×C₄	—	3	1	no	—
16	C₄⋊C₄	—	3	2	no	—
17	C₄⋊C₄	—	3	1	no	—
18	C₂²⋊C₄	—	3	2	no	—
19	C₂²×C₃	C₂³	1	1	yes	characteristic
20	C₂×D₃	—	2	1	no	—
21	C₂×D₃	—	2	1	no	—
22	C₂×D₃	Q₈	1	2	yes	complemented
23	C₃×C₄	—	2	2	no	—
24	C₃×C₄	D₄	1	2	yes	—
25	Dic₃	—	2	1	no	—
26	Dic₃	—	2	2	no	—
27	C₂³	—	3	1	no	—
28	Q₈	—	2	2	no	—
29	C₂×C₄	—	3	2	no	—
30	C₂×C₄	—	3	1	no	—
31	C₂×C₄	—	6	2	no	—
32	C₂×C₄	C₂×D₃	1	2	yes	—
33	C₂×C₄	C₂×D₃	1	1	yes	characteristic
34	C₂×C₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₃	#41	1	1	yes	characteristic
37	D₃	—	2	2	no	—
38	C₂²	—	3	2	no	—
39	C₂²	—	6	1	no	—
40	C₂²	—	6	1	no	—
41	C₂²	C₂²×D₃	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	—	6	2	no	—
44	C₄	—	6	1	no	—
45	C₄	#67	1	2	yes	—
46	C₃	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
47	C₂	—	6	2	no	—
48	C₂	D₃×Q₈	1	1	yes	characteristic
49	C₂	#239	1	1	yes	characteristic
50	C₂	C₂×#67	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×Q₈	C₂	semidirect
4. C₄⋊Dic₃	C₂	semidirect
5. Dic₃⋊C₄	C₂	semidirect
14. C₂×Q₈	D₃	semidirect
22. C₂×D₃	Q₈	semidirect
46. C₃	C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×Q₈	2
C₂×C₄×D₃	2
(C₂×D₃)⋊C₄	2
C₄⋊Dic₃	2
Dic₃⋊C₄	2
C₂²⋊Q₈	3