Subgroups — #896

Order 96.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
5	(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²⋊Q₈	—	3	1	no	maximal, Sylow
9	C₂²×D₃	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃	C₂²	1	1	yes	characteristic
16	C₄×D₃	—	2	2	no	—
17	Dic₆	—	2	2	no	—
18	C₂×Q₈	—	3	1	no	—
19	C₂²×C₄	—	3	1	no	—
20	C₄⋊C₄	—	3	1	no	—
21	C₄⋊C₄	—	3	1	no	—
22	C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
23	C₂²⋊C₄	—	3	1	no	—
24	C₂²⋊C₄	—	3	1	no	—
25	C₂²×C₃	C₂³	1	1	yes	characteristic
26	C₂×D₃	—	2	1	no	—
27	C₂×D₃	—	2	1	no	—
28	C₂×D₃	Q₈	1	2	yes	complemented
29	C₃×C₄	—	2	1	no	—
30	C₃×C₄	—	2	1	no	—
31	C₃×C₄	—	2	1	no	—
32	Dic₃	—	2	1	no	—
33	Dic₃	—	2	1	no	—
34	Dic₃	D₄	1	2	yes	—
35	C₂³	—	3	1	no	—
36	Q₈	—	6	2	no	—
37	C₂×C₄	—	3	1	no	—
38	C₂×C₄	—	3	1	no	—
39	C₂×C₄	—	3	1	no	—
40	C₂×C₄	—	6	2	no	—
41	C₂×C₄	C₂×D₃	1	1	yes	characteristic
42	C₂×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₃	#41	1	1	yes	characteristic
47	D₃	—	2	2	no	—
48	C₂²	—	3	2	no	—
49	C₂²	—	6	1	no	—
50	C₂²	—	6	1	no	—
51	C₂²	C₂²×D₃	1	1	yes	characteristic
52	C₄	—	2	1	no	—
53	C₄	—	2	1	no	—
54	C₄	—	2	1	no	—
55	C₄	—	3	2	no	—
56	C₄	—	6	1	no	—
57	C₄	—	6	1	no	—
58	C₃	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
59	C₂	—	6	2	no	—
60	C₂	#235	1	1	yes	characteristic
61	C₂	D₃×D₄	1	1	yes	characteristic
62	C₂	D₃×Q₈	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₆	C₂	semidirect
3. C₃×C₄⋊C₄	C₂	semidirect
6. Dic₃⋊C₄	C₂	semidirect
7. Dic₃⋊C₄	C₂	semidirect
22. C₄⋊C₄	D₃	semidirect
28. C₂×D₃	Q₈	semidirect
58. 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×C₄×D₃	2
C₂×Dic₆	2
C₃×C₄⋊C₄	2
(C₂×D₃)⋊C₄	2
Dic₃⋊C₄	2
C₂²⋊Q₈	3