Subgroups — #4538

Order 192.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#95	C₂	1	1	yes	maximal, characteristic
3	#830	C₂	1	1	yes	maximal, characteristic
4	#328	—	3	1	no	maximal, Sylow
5	C₂×C₃×Q₈	C₄	1	1	yes	characteristic, complemented
6	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×Dic₃	—	2	1	no	—
8	C₃×C₄⋊C₄	—	2	1	no	—
9	C₃×C₄⋊C₄	C₄	1	1	yes	characteristic, complemented
10	C₃×C₂²⋊C₄	—	2	1	no	—
11	C₂×C₃⋊C₈	—	2	1	no	—
12	C₂²⋊Q₈	D₃	1	1	yes	characteristic
13	C₂²⋊C₈	—	3	1	no	—
14	#95	—	3	1	no	—
15	C₂³×C₃	D₄	1	1	yes	characteristic
16	C₃×Q₈	—	4	1	no	—
17	C₂×C₃×C₄	—	2	1	no	—
18	C₂×C₃×C₄	—	2	1	no	—
19	C₂×C₃×C₄	—	4	1	no	—
20	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄	D₄	1	1	yes	characteristic
22	C₂×Dic₃	—	2	2	no	—
23	C₂×Dic₃	—	4	1	no	—
24	C₂×Dic₃	—	4	1	no	—
25	C₃⋊C₈	—	4	1	no	—
26	C₂×Q₈	Dic₃	1	1	yes	characteristic, complemented
27	C₂²×C₄	—	6	1	no	—
28	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
29	C₂×C₈	—	6	1	no	—
30	C₄⋊C₄	—	2	1	no	—
31	C₄⋊C₄	Dic₃	1	1	yes	characteristic, complemented
32	C₂²⋊C₄	—	2	1	no	—
33	C₂²×C₃	—	2	1	no	—
34	C₂²×C₃	—	2	1	no	—
35	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₃	SD₁₆	1	1	yes	characteristic
37	C₂²×C₃	Q₁₆	1	1	yes	characteristic
38	C₃×C₄	—	2	1	no	—
39	C₃×C₄	—	2	1	no	—
40	C₃×C₄	—	4	1	no	—
41	C₃×C₄	—	4	1	no	—
42	Dic₃	—	4	2	no	—
43	C₂³	#67	1	1	yes	characteristic
44	Q₈	—	4	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	4	1	no	—
48	C₂×C₄	—	6	2	no	—
49	C₂×C₄	—	12	1	no	—
50	C₂×C₄	—	12	1	no	—
51	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
52	C₂×C₄	#67	1	1	yes	characteristic
53	C₈	—	12	1	no	—
54	C₂×C₃	—	2	1	no	—
55	C₂×C₃	—	2	1	no	—
56	C₂×C₃	#99	1	1	yes	characteristic
57	C₂×C₃	#103	1	1	yes	characteristic
58	C₂×C₃	#104	1	1	yes	characteristic
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	#215	1	1	yes	characteristic
62	C₂²	#216	1	1	yes	characteristic
63	C₂²	#217	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	2	1	no	—
66	C₄	—	4	1	no	—
67	C₄	—	4	1	no	—
68	C₄	—	12	2	no	—
69	C₃	#328	1	1	yes	characteristic, complemented, Sylow
70	C₂	—	2	1	no	—
71	C₂	—	2	1	no	—
72	C₂	#834	1	1	yes	characteristic
73	C₂	#835	1	1	yes	characteristic
74	C₂	#837	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#217
commutator	C₂×C₃×C₄	C₂×C₄
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₃×C₂²⋊Q₈	C₂
socle	C₂²×C₃	C₂²⋊C₄
prime radical	C₂³×C₃	D₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂²⋊Q₈	D₃
hypocenter	C₃	#328
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
5. C₂×C₃×Q₈	C₄	semidirect
9. C₃×C₄⋊C₄	C₄	semidirect
26. C₂×Q₈	Dic₃	semidirect
31. C₄⋊C₄	Dic₃	semidirect
69. C₃	#328	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₂²⋊Q₈	2
C₃⋊#95	2
#830	2
#328	3