Subgroups — C₂×C₃×C₂²⋊C₈ #5280

Order 192.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₈	C₂	1	2	yes	maximal, complemented
3	C₃×C₂²⋊C₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₂²⋊C₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁴×C₃	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₄	—	2	4	no	—
7	C₂²×C₃×C₄	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄	C₄	1	2	yes	—
9	C₂²×C₃×C₄	C₂²	1	2	yes	—
10	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈	—	2	4	no	—
12	C₂×C₃×C₈	C₂²	1	4	yes	complemented
13	C₂³×C₄	C₂×C₃	1	1	yes	characteristic
14	C₂²×C₈	C₂×C₃	1	2	yes	complemented
15	C₂²⋊C₈	C₂×C₃	1	4	yes	direct product factor
16	C₂³×C₃	—	2	2	no	—
17	C₂³×C₃	—	2	2	no	—
18	C₂³×C₃	C₈	1	4	yes	complemented
19	C₂³×C₃	C₂×C₄	1	2	yes	—
20	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄	—	2	2	no	—
22	C₂×C₃×C₄	—	2	4	no	—
23	C₂×C₃×C₄	—	2	4	no	—
24	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄	D₄	1	4	yes	—
27	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₃×C₈	—	2	4	no	—
29	C₂⁴	C₃×C₄	1	1	yes	characteristic
30	C₂²×C₄	—	2	4	no	—
31	C₂²×C₄	C₃×C₄	1	1	yes	characteristic
32	C₂²×C₄	C₃×C₄	1	2	yes	—
33	C₂²×C₄	C₂²×C₃	1	2	yes	—
34	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic
35	C₂×C₈	—	2	4	no	—
36	C₂×C₈	C₂²×C₃	1	4	yes	complemented
37	C₂²×C₃	—	2	8	no	—
38	C₂²×C₃	—	2	2	no	—
39	C₂²×C₃	—	2	2	no	—
40	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₃	C₂×C₈	1	2	yes	—
42	C₂²×C₃	C₂×C₈	1	4	yes	complemented
43	C₂²×C₃	M₁₆	1	2	yes	—
44	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
45	C₃×C₄	—	2	2	no	—
46	C₃×C₄	C₂²⋊C₄	1	2	yes	—
47	C₃×C₄	C₂×D₄	1	2	yes	—
48	C₂³	—	2	2	no	—
49	C₂³	—	2	2	no	—
50	C₂³	C₃×C₈	1	4	yes	complemented
51	C₂³	C₂×C₃×C₄	1	2	yes	—
52	C₂³	C₂×C₃×C₄	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	C₂×C₃×C₄	1	2	yes	—
57	C₂×C₄	C₂×C₃×C₄	1	1	yes	characteristic
58	C₂×C₄	C₃×D₄	1	4	yes	—
59	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
60	C₈	—	2	4	no	—
61	C₂×C₃	—	2	4	no	—
62	C₂×C₃	C₂²⋊C₈	1	4	yes	direct product factor
63	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₃	C₂×M₁₆	1	1	yes	characteristic
66	C₂²	—	2	8	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₃×C₂²⋊C₄	1	2	yes	—
70	C₂²	C₂×C₃×C₈	1	2	yes	—
71	C₂²	C₂×C₃×C₈	1	4	yes	complemented
72	C₂²	C₃×M₁₆	1	2	yes	—
73	C₂²	C₂²×C₃×C₄	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	C₃×C₂²⋊C₄	1	2	yes	—
76	C₄	C₂×C₃×D₄	1	2	yes	—
77	C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
78	C₂	—	2	4	no	—
79	C₂	C₃×C₂²⋊C₈	1	4	yes	direct product factor
80	C₂	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
81	C₂	C₂²×C₃×C₈	1	1	yes	characteristic
82	C₂	C₂×C₃×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²
commutator	C₂	C₂²×C₃×C₈
Frattini, cosocle	C₂×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₄
prime radical	C₂⁴×C₃	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₈	C₂	semidirect
3. C₃×C₂²⋊C₈	C₂	direct (N × H)
4. C₂×C₂²⋊C₈	C₃	direct (N × H)
12. C₂×C₃×C₈	C₂²	semidirect
14. C₂²×C₈	C₂×C₃	semidirect
15. C₂²⋊C₈	C₂×C₃	direct (N × H)
18. C₂³×C₃	C₈	semidirect
36. C₂×C₈	C₂²×C₃	semidirect
42. C₂²×C₃	C₂×C₈	semidirect
50. C₂³	C₃×C₈	semidirect
62. C₂×C₃	C₂²⋊C₈	direct (N × H)
71. C₂²	C₂×C₃×C₈	semidirect
77. C₃	C₂×C₂²⋊C₈	direct (N × H)
79. C₂	C₃×C₂²⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄	2
C₂²×C₃×C₈	2
C₃×C₂²⋊C₈	2
C₂×C₂²⋊C₈	3

Subgroups — C2×C3×C22⋊C8 #5280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₂²⋊C₈ #5280