Subgroups — #4535

Order 192.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊C₈	—	3	1	no	maximal, Sylow
5	C₂×C₃×Q₈	C₄	1	1	yes	characteristic
6	C₃×C₄⋊C₄	—	2	1	no	—
7	C₃×C₄⋊C₄	C₄	1	1	yes	characteristic
8	C₃×C₄²	—	2	1	no	—
9	C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊C₈	—	2	1	no	—
11	C₂×C₃⋊C₈	—	2	1	no	—
12	C₄×Q₈	D₃	1	1	yes	characteristic
13	C₄⋊C₈	—	3	1	no	—
14	C₄×C₈	—	3	1	no	—
15	C₃×Q₈	—	2	1	no	—
16	C₃×Q₈	C₈	1	2	yes	complemented
17	C₂×C₃×C₄	—	2	1	no	—
18	C₂×C₃×C₄	—	2	1	no	—
19	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄	D₄	1	1	yes	characteristic
21	C₂×C₃×C₄	D₄	1	1	yes	characteristic
22	C₃⋊C₈	—	2	2	no	—
23	C₃⋊C₈	—	4	1	no	—
24	C₂×Q₈	Dic₃	1	1	yes	characteristic
25	C₂×C₈	—	6	1	no	—
26	C₂×C₈	—	6	1	no	—
27	C₄⋊C₄	—	2	1	no	—
28	C₄⋊C₄	Dic₃	1	1	yes	characteristic
29	C₄²	—	2	1	no	—
30	C₄²	C₂×D₃	1	1	yes	characteristic
31	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₃×C₄	—	2	2	no	—
33	C₃×C₄	—	2	1	no	—
34	C₃×C₄	—	4	1	no	—
35	C₃×C₄	C₂×C₈	1	1	yes	characteristic
36	C₃×C₄	M₁₆	1	1	yes	characteristic
37	C₃×C₄	SD₁₆	1	1	yes	characteristic
38	C₃×C₄	Q₁₆	1	1	yes	characteristic
39	Q₈	—	2	1	no	—
40	Q₈	C₃⋊C₈	1	2	yes	complemented
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
44	C₂×C₄	#67	1	1	yes	characteristic
45	C₂×C₄	#67	1	1	yes	characteristic
46	C₈	—	6	2	no	—
47	C₈	—	12	1	no	—
48	C₂×C₃	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₃	#103	1	1	yes	characteristic
50	C₂×C₃	#104	1	1	yes	characteristic
51	C₂²	#217	1	1	yes	characteristic
52	C₄	—	2	2	no	—
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
56	C₄	#208	1	1	yes	characteristic
57	C₄	#215	1	1	yes	characteristic
58	C₄	#216	1	1	yes	characteristic
59	C₃	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
60	C₂	#830	1	1	yes	characteristic
61	C₂	#835	1	1	yes	characteristic
62	C₂	#837	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#67
commutator	C₃×C₄	C₂×C₈
Frattini	C₄²	C₂×D₃
Fitting	C₃×C₄×Q₈	C₂
socle, prime radical	C₂²×C₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂²
hypercenter	C₄×Q₈	D₃
hypocenter	C₃	Q₈⋊C₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. C₃×Q₈	C₈	semidirect
40. Q₈	C₃⋊C₈	semidirect
59. C₃	Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈	2
C₄⋊(C₃⋊C₈)	2
C₄×C₃⋊C₈	2
Q₈⋊C₈	3