Subgroups — C₂²×#215 #5807

Order 192.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#215	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×SD₁₆	—	3	1	no	maximal, Sylow
6	C₂³×D₃	—	2	1	no	—
7	C₂×C₃×Q₈	—	2	3	no	—
8	C₂×C₃×Q₈	C₂²	1	6	yes	complemented
9	C₂²×C₃×C₄	—	2	1	no	—
10	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂×D₁₂	—	2	3	no	—
12	C₂×D₁₂	C₂²	1	6	yes	—
13	#215	C₂²	1	16	yes	direct product factor
14	C₂×C₃⋊C₈	C₂²	1	6	yes	complemented
15	C₂²×Q₈	D₃	1	1	yes	characteristic, complemented
16	C₂²×D₄	—	3	1	no	—
17	C₂×SD₁₆	—	3	12	no	—
18	C₂²×C₈	—	3	1	no	—
19	C₂³×C₃	D₄	1	1	yes	characteristic
20	C₂²×D₃	—	2	6	no	—
21	C₂²×D₃	—	4	4	no	—
22	C₃×Q₈	—	2	6	no	—
23	C₃×Q₈	C₂³	1	4	yes	complemented
24	C₂×C₃×C₄	—	2	6	no	—
25	C₂×C₃×C₄	D₄	1	3	yes	—
26	C₂×C₃×C₄	C₂³	1	3	yes	—
27	D₁₂	—	2	6	no	—
28	D₁₂	C₂³	1	4	yes	—
29	C₃⋊C₈	C₂³	1	4	yes	complemented
30	C₂⁴	—	6	1	no	—
31	C₂×Q₈	—	2	3	no	—
32	C₂×Q₈	C₂×D₃	1	6	yes	complemented
33	C₂×D₄	—	3	6	no	—
34	C₂×D₄	—	6	3	no	—
35	C₂²×C₄	—	2	1	no	—
36	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
37	SD₁₆	—	3	16	no	—
38	C₂×C₈	—	3	6	no	—
39	C₂²×C₃	SD₁₆	1	4	yes	complemented
40	C₂²×C₃	C₂×D₄	1	3	yes	—
41	C₂×D₃	—	2	4	no	—
42	C₂×D₃	—	4	12	no	—
43	C₃×C₄	—	2	4	no	—
44	C₃×C₄	C₂×D₄	1	3	yes	—
45	C₃×C₄	C₂⁴	1	1	yes	characteristic
46	C₂³	—	6	6	no	—
47	C₂³	—	12	4	no	—
48	C₂³	#67	1	1	yes	characteristic
49	Q₈	—	2	6	no	—
50	Q₈	C₂²×D₃	1	4	yes	complemented
51	D₄	—	3	4	no	—
52	D₄	—	6	6	no	—
53	C₂×C₄	—	2	6	no	—
54	C₂×C₄	#67	1	3	yes	—
55	C₂×C₄	C₂²×D₃	1	3	yes	—
56	C₈	—	3	4	no	—
57	C₂×C₃	C₂×SD₁₆	1	6	yes	complemented
58	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
59	D₃	—	4	4	no	—
60	C₂²	—	6	4	no	—
61	C₂²	—	12	12	no	—
62	C₂²	#215	1	4	yes	direct product factor
63	C₂²	C₂×#67	1	3	yes	—
64	C₄	—	2	4	no	—
65	C₄	C₂×#67	1	3	yes	—
66	C₄	C₂³×D₃	1	1	yes	characteristic
67	C₃	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	12	4	no	—
69	C₂	C₂×#215	1	6	yes	direct product factor
70	C₂	C₂²×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#67
commutator, cosocle	C₃×C₄	C₂⁴
Frattini	C₄	C₂³×D₃
Fitting	C₂²×C₃×Q₈	C₂
socle	C₂³×C₃	D₄
prime radical	C₂²×D₁₂	C₂
hypercenter	C₂²×Q₈	D₃
hypocenter	C₃	C₂²×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Q₈	C₂	semidirect
3. C₂×#215	C₂	direct (N × H)
4. C₂²×C₃⋊C₈	C₂	semidirect
8. C₂×C₃×Q₈	C₂²	semidirect
13. #215	C₂²	direct (N × H)
14. C₂×C₃⋊C₈	C₂²	semidirect
15. C₂²×Q₈	D₃	semidirect
23. C₃×Q₈	C₂³	semidirect
29. C₃⋊C₈	C₂³	semidirect
32. C₂×Q₈	C₂×D₃	semidirect
39. C₂²×C₃	SD₁₆	semidirect
50. Q₈	C₂²×D₃	semidirect
57. C₂×C₃	C₂×SD₁₆	semidirect
62. C₂²	#215	direct (N × H)
67. C₃	C₂²×SD₁₆	semidirect
69. C₂	C₂×#215	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈	2
C₂²×D₁₂	2
C₂×#215	2
C₂²×C₃⋊C₈	2
C₂²×SD₁₆	3

Subgroups — C22×#215 #5807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#215 #5807