Subgroups — #4482

Order 192.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#869	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#332	—	3	1	no	maximal, Sylow
5	C₂×Dic₆	—	2	1	no	—
6	C₂×C₃×C₈	—	2	1	no	—
7	C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₃	—	2	1	no	—
9	C₄⋊Dic₃	C₄	1	2	yes	—
10	C₂×C₃⋊C₈	—	2	1	no	—
11	C₄⋊Q₈	—	3	1	no	—
12	C₄⋊C₈	—	3	1	no	—
13	C₄⋊C₈	D₃	1	1	yes	characteristic
14	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₄	D₄	1	1	yes	characteristic
16	C₂×C₃×C₄	D₄	1	1	yes	characteristic
17	C₂×Dic₃	—	2	2	no	—
18	Dic₆	—	4	1	no	—
19	Dic₆	—	4	1	no	—
20	C₃×C₈	—	4	1	no	—
21	C₃⋊C₈	—	4	1	no	—
22	C₂×Q₈	—	6	1	no	—
23	C₂×C₈	—	2	1	no	—
24	C₂×C₈	—	6	1	no	—
25	C₄⋊C₄	—	3	2	no	—
26	C₄⋊C₄	—	6	1	no	—
27	C₄²	C₂×D₃	1	1	yes	characteristic
28	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₃×C₄	—	2	1	no	—
30	C₃×C₄	D₈	1	1	yes	characteristic
31	C₃×C₄	SD₁₆	1	1	yes	characteristic
32	C₃×C₄	SD₁₆	1	1	yes	characteristic
33	C₃×C₄	Q₁₆	1	1	yes	characteristic
34	Dic₃	—	4	2	no	—
35	Q₈	—	12	1	no	—
36	Q₈	—	12	1	no	—
37	C₂×C₄	—	6	2	no	—
38	C₂×C₄	C₄×D₃	1	1	yes	characteristic
39	C₂×C₄	D₁₂	1	1	yes	characteristic
40	C₂×C₄	#67	1	1	yes	characteristic
41	C₈	—	4	1	no	—
42	C₈	—	12	1	no	—
43	C₂×C₃	#101	1	1	yes	characteristic
44	C₂×C₃	#102	1	1	yes	characteristic
45	C₂×C₃	#103	1	1	yes	characteristic
46	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	12	2	no	—
49	C₄	#204	1	1	yes	characteristic
50	C₄	Dic₁₂	1	1	yes	characteristic
51	C₄	#213	1	1	yes	characteristic
52	C₄	#215	1	1	yes	characteristic
53	C₃	#332	1	1	yes	characteristic, complemented, Sylow
54	C₂	#809	1	1	yes	characteristic
55	C₂	#816	1	1	yes	characteristic
56	C₂	#824	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂×C₄
Frattini	C₄²	C₂×D₃
Fitting	C₃×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂²
hypercenter	C₄⋊C₈	D₃
hypocenter	C₃	#332
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
53. C₃	#332	semidirect

Maximal subgroups

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

subgroup	index
#869	2
C₃×C₄⋊C₈	2
C₄⋊(C₃⋊C₈)	2
#332	3