Subgroups — #4547

Order 192.

64 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	#332	—	3	1	no	maximal, Sylow
5	C₂×C₃×Q₈	—	2	1	no	—
6	C₃×C₄⋊C₄	—	2	1	no	—
7	C₃×C₄⋊C₄	C₄	1	2	yes	—
8	C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊C₈	—	2	1	no	—
10	C₂×C₃⋊C₈	—	2	1	no	—
11	C₄⋊Q₈	D₃	1	1	yes	characteristic
12	C₄⋊C₈	—	3	1	no	—
13	C₄⋊C₈	—	3	1	no	—
14	C₃×Q₈	—	4	1	no	—
15	C₃×Q₈	—	4	1	no	—
16	C₂×C₃×C₄	—	2	2	no	—
17	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃×C₄	D₄	1	1	yes	characteristic
19	C₂×C₃×C₄	D₄	1	1	yes	characteristic
20	C₃⋊C₈	—	4	1	no	—
21	C₃⋊C₈	—	4	1	no	—
22	C₂×Q₈	—	2	1	no	—
23	C₂×C₈	—	6	1	no	—
24	C₂×C₈	—	6	1	no	—
25	C₄⋊C₄	—	2	1	no	—
26	C₄⋊C₄	Dic₃	1	2	yes	—
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	—	4	2	no	—
31	C₃×C₄	D₈	1	1	yes	characteristic
32	C₃×C₄	SD₁₆	1	1	yes	characteristic
33	C₃×C₄	SD₁₆	1	1	yes	characteristic
34	C₃×C₄	Q₁₆	1	1	yes	characteristic
35	Q₈	—	4	1	no	—
36	Q₈	—	4	1	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
39	C₂×C₄	#67	1	1	yes	characteristic
40	C₂×C₄	#67	1	1	yes	characteristic
41	C₈	—	12	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	#217	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	4	2	no	—
49	C₄	#213	1	1	yes	characteristic
50	C₄	#214	1	1	yes	characteristic
51	C₄	#215	1	1	yes	characteristic
52	C₄	#216	1	1	yes	characteristic
53	C₃	#332	1	1	yes	characteristic, complemented, Sylow
54	C₂	#832	1	1	yes	characteristic
55	C₂	#835	1	1	yes	characteristic
56	C₂	#836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#217
commutator	C₂×C₃×C₄	C₂×C₄
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	#332
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

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
C₃×C₄⋊Q₈	2
C₄⋊(C₃⋊C₈)	2
#332	3