Subgroups — #898

Order 96.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
4	(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#126	—	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⋊C₄	—	3	1	no	—
17	C₄⋊C₄	—	3	1	no	—
18	C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
19	C₂²⋊C₄	—	3	1	no	—
20	C₂²⋊C₄	—	3	1	no	—
21	C₂²⋊C₄	—	3	1	no	—
22	C₄²	—	3	1	no	—
23	C₂²×C₃	C₂³	1	1	yes	characteristic
24	C₂×D₃	—	2	1	no	—
25	C₂×D₃	—	2	1	no	—
26	C₂×D₃	—	2	1	no	—
27	C₃×C₄	—	2	1	no	—
28	C₃×C₄	—	2	1	no	—
29	C₃×C₄	—	2	1	no	—
30	Dic₃	—	2	1	no	—
31	Dic₃	—	2	1	no	—
32	Dic₃	—	2	1	no	—
33	C₂³	—	3	1	no	—
34	C₂×C₄	—	3	1	no	—
35	C₂×C₄	—	3	1	no	—
36	C₂×C₄	—	3	1	no	—
37	C₂×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂×C₄	C₂×D₃	1	1	yes	characteristic
39	C₂×C₄	C₂×D₃	1	1	yes	characteristic
40	C₂×C₃	#41	1	1	yes	characteristic
41	C₂×C₃	#41	1	1	yes	characteristic
42	C₂×C₃	#41	1	1	yes	characteristic
43	D₃	—	4	1	no	—
44	C₂²	—	6	1	no	—
45	C₂²	—	6	1	no	—
46	C₂²	—	6	1	no	—
47	C₂²	C₂²×D₃	1	1	yes	characteristic
48	C₄	—	2	1	no	—
49	C₄	—	2	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	6	1	no	—
52	C₄	—	6	1	no	—
53	C₄	—	6	1	no	—
54	C₃	#126	1	1	yes	characteristic, complemented, Sylow
55	C₂	—	12	1	no	—
56	C₂	#235	1	1	yes	characteristic
57	C₂	#237	1	1	yes	characteristic
58	C₂	#239	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	#126
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
1. C₃×C₄⋊C₄	C₂	semidirect
5. C₄⋊Dic₃	C₂	semidirect
6. Dic₃⋊C₄	C₂	semidirect
7. C₄×Dic₃	C₂	semidirect
18. C₄⋊C₄	D₃	semidirect
54. C₃	#126	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄⋊C₄	2
(C₂×D₃)⋊C₄	2
C₄⋊Dic₃	2
Dic₃⋊C₄	2
C₄×Dic₃	2
#126	3