Subgroups — #5193

Order 192.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#828	C₂	1	2	yes	maximal, complemented
4	#826	C₂	1	2	yes	maximal, complemented
5	#812	C₂	1	1	yes	maximal, characteristic, complemented
6	#509	—	3	1	no	maximal, Sylow
7	C₃×#41	—	2	1	no	—
8	C₂×D₁₂	—	2	1	no	—
9	C₃×Q₁₆	C₂²	1	1	yes	characteristic, complemented
10	C₃×SD₁₆	—	2	1	no	—
11	C₃×D₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
13	D₂₄	—	2	1	no	—
14	D₂₄	C₂²	1	2	yes	—
15	C₃⋊C₁₆	C₂²	1	2	yes	complemented
16	#135	D₃	1	1	yes	characteristic, complemented
17	C₂×D₈	—	3	1	no	—
18	SD₃₂	—	3	2	no	—
19	D₁₆	—	3	2	no	—
20	M₃₂	—	3	1	no	—
21	C₂²×D₃	—	4	1	no	—
22	C₃×Q₈	—	2	1	no	—
23	C₃×D₄	—	2	1	no	—
24	C₃×D₄	—	4	1	no	—
25	C₂×C₃×C₄	—	4	1	no	—
26	C₂×C₃×C₄	D₄	1	1	yes	characteristic
27	D₁₂	—	2	2	no	—
28	D₁₂	—	4	1	no	—
29	C₃×C₈	D₄	1	1	yes	characteristic
30	C₃×C₈	C₂³	1	1	yes	characteristic
31	#41	—	2	1	no	—
32	C₂×D₄	—	6	1	no	—
33	Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
34	SD₁₆	—	2	1	no	—
35	D₈	—	3	2	no	—
36	D₈	—	6	1	no	—
37	D₈	C₂×D₃	1	1	yes	characteristic, complemented
38	C₂×C₈	C₂×D₃	1	1	yes	characteristic
39	C₁₆	—	3	2	no	—
40	C₂²×C₃	—	4	1	no	—
41	C₂²×C₃	D₈	1	1	yes	characteristic
42	C₂×D₃	—	4	2	no	—
43	C₂×D₃	—	8	2	no	—
44	C₃×C₄	—	4	1	no	—
45	C₃×C₄	D₈	1	1	yes	characteristic
46	C₃×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂³	—	12	1	no	—
48	Q₈	—	2	1	no	—
49	D₄	—	2	1	no	—
50	D₄	—	4	1	no	—
51	D₄	—	6	2	no	—
52	D₄	—	12	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	#67	1	1	yes	characteristic
55	C₈	#67	1	1	yes	characteristic
56	C₈	C₂²×D₃	1	1	yes	characteristic
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	—	8	1	no	—
59	C₂×C₃	C₂×D₈	1	1	yes	characteristic
60	D₃	—	8	2	no	—
61	C₂²	—	4	1	no	—
62	C₂²	—	12	2	no	—
63	C₂²	—	24	2	no	—
64	C₂²	#213	1	1	yes	characteristic
65	C₄	—	4	1	no	—
66	C₄	#213	1	1	yes	characteristic
67	C₄	C₂×#67	1	1	yes	characteristic
68	C₃	#509	1	1	yes	characteristic, complemented, Sylow
69	C₂	—	2	1	no	—
70	C₂	—	8	1	no	—
71	C₂	—	24	2	no	—
72	C₂	C₂×#213	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#213
commutator, cosocle	C₃×C₈	C₂³
Frattini	C₈	C₂²×D₃
Fitting	C₃×#135	C₂
socle	C₂×C₃	C₂×D₈
hypercenter	#135	D₃
hypocenter	C₃	#509
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₃×#135	C₂	semidirect
2. C₂×D₂₄	C₂	semidirect
3. #828	C₂	semidirect
4. #826	C₂	semidirect
5. #812	C₂	semidirect
9. C₃×Q₁₆	C₂²	semidirect
11. C₃×D₈	C₂²	semidirect
15. C₃⋊C₁₆	C₂²	semidirect
16. #135	D₃	semidirect
33. Q₁₆	C₂×D₃	semidirect
37. D₈	C₂×D₃	semidirect
68. C₃	#509	semidirect

Maximal subgroups

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

subgroup	index
C₃×#135	2
C₂×D₂₄	2
#828	2
#826	2
#812	2
#509	3