Subgroups — #4528

Order 192.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	#830	C₂	1	2	yes	maximal
3	#324	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₄	C₄	1	1	yes	characteristic
5	C₂²×C₃×C₄	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₃×C₄⋊C₄	—	2	2	no	—
8	C₂×C₃⋊C₈	—	2	2	no	—
9	C₂×C₄⋊C₄	D₃	1	1	yes	characteristic
10	C₂²⋊C₈	—	3	2	no	—
11	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₃×C₄	—	2	1	no	—
13	C₂×C₃×C₄	—	2	1	no	—
14	C₂×C₃×C₄	—	2	1	no	—
15	C₂×C₃×C₄	—	4	1	no	—
16	C₂×C₃×C₄	—	4	1	no	—
17	C₂×C₃×C₄	C₈	1	2	yes	complemented
18	C₂×C₃×C₄	D₄	1	2	yes	—
19	C₃⋊C₈	—	4	2	no	—
20	C₂²×C₄	Dic₃	1	1	yes	characteristic
21	C₂²×C₄	Dic₃	1	1	yes	characteristic
22	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
23	C₂×C₈	—	6	2	no	—
24	C₄⋊C₄	—	2	2	no	—
25	C₂²×C₃	—	2	1	no	—
26	C₂²×C₃	—	2	1	no	—
27	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₃	C₂×C₈	1	1	yes	characteristic
29	C₂²×C₃	M₁₆	1	1	yes	characteristic
30	C₃×C₄	—	2	2	no	—
31	C₃×C₄	—	2	2	no	—
32	C₃×C₄	—	4	1	no	—
33	C₂³	C₂×Dic₃	1	1	yes	characteristic
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	4	1	no	—
38	C₂×C₄	—	4	1	no	—
39	C₂×C₄	C₃⋊C₈	1	2	yes	complemented
40	C₂×C₄	#67	1	2	yes	—
41	C₈	—	12	2	no	—
42	C₂×C₃	—	2	1	no	—
43	C₂×C₃	—	2	1	no	—
44	C₂×C₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₃	#99	1	1	yes	characteristic
46	C₂×C₃	#101	1	1	yes	characteristic
47	C₂²	—	2	1	no	—
48	C₂²	—	2	1	no	—
49	C₂²	C₂×C₃⋊C₈	1	1	yes	characteristic
50	C₂²	#208	1	1	yes	characteristic
51	C₂²	#217	1	1	yes	characteristic
52	C₄	—	2	2	no	—
53	C₄	—	2	2	no	—
54	C₄	—	4	1	no	—
55	C₃	#324	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	#830	1	1	yes	characteristic
59	C₂	#834	1	1	yes	characteristic
60	C₂	#836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#217
commutator	C₂²×C₃	C₂×C₈
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₂×C₃×C₄⋊C₄	C₂
socle	C₂²×C₃	C₂²⋊C₄
prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂×C₄⋊C₄	D₃
hypocenter	C₃	#324
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
17. C₂×C₃×C₄	C₈	semidirect
39. C₂×C₄	C₃⋊C₈	semidirect
55. C₃	#324	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄⋊C₄	2
#830	2
#324	3