Subgroups — #889

Order 96.

56 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
2	C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic
5	Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic
6	Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
8	#125	—	3	1	no	maximal, Sylow
9	C₂×C₃×C₄	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×Dic₃	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	—	3	1	no	—
19	C₄⋊C₄	—	3	1	no	—
20	C₄⋊C₄	—	3	1	no	—
21	C₄⋊C₄	D₃	1	1	yes	characteristic
22	C₄²	—	3	1	no	—
23	C₂²×C₃	C₂³	1	1	yes	characteristic
24	C₃×C₄	—	2	1	no	—
25	C₃×C₄	—	2	1	no	—
26	C₃×C₄	—	2	1	no	—
27	Dic₃	—	2	1	no	—
28	Dic₃	—	2	1	no	—
29	Dic₃	—	2	1	no	—
30	Dic₃	Q₈	1	2	yes	—
31	C₂×C₄	—	3	1	no	—
32	C₂×C₄	—	3	1	no	—
33	C₂×C₄	—	3	1	no	—
34	C₂×C₄	—	3	1	no	—
35	C₂×C₄	C₂×D₃	1	1	yes	characteristic
36	C₂×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂×C₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₃	#41	1	1	yes	characteristic
40	C₂×C₃	#41	1	1	yes	characteristic
41	C₂²	C₂²×D₃	1	1	yes	characteristic
42	C₄	—	2	1	no	—
43	C₄	—	2	1	no	—
44	C₄	—	2	1	no	—
45	C₄	—	3	2	no	—
46	C₄	—	6	1	no	—
47	C₄	—	6	1	no	—
48	C₄	—	6	1	no	—
49	C₃	#125	1	1	yes	characteristic, complemented, Sylow
50	C₂	#235	1	1	yes	characteristic
51	C₂	#237	1	1	yes	characteristic
52	C₂	D₃×Q₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃
commutator, socle, prime radical, cosocle	C₂²×C₃	C₂³
Fitting	C₃×C₄⋊C₄	C₂
hypercenter	C₄⋊C₄	D₃
hypocenter	C₃	#125
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
49. C₃	#125	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
C₄⋊Dic₃	2
Dic₃⋊C₄	2
C₄×Dic₃	2
#125	3