Subgroups — C₂×C₃²×C₂²⋊C₄ #64164

Order 288.

396 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₄	C₂	1	2	yes	maximal, complemented
3	C₃²×C₂²⋊C₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₂²⋊C₄	C₃	1	4	yes	maximal, direct product factor
5	C₂³×C₃²	—	2	4	no	—
6	C₂³×C₃²	C₄	1	4	yes	complemented
7	C₂³×C₃²	C₂²	1	2	yes	—
8	C₂³×C₃²	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄	—	2	4	no	—
10	C₂×C₃²×C₄	C₂²	1	4	yes	complemented
11	C₂⁴×C₃	C₂×C₃	1	4	yes	—
12	C₂²×C₃×C₄	C₂×C₃	1	8	yes	complemented
13	C₃×C₂²⋊C₄	C₂×C₃	1	16	yes	direct product factor
14	C₂²×C₃²	—	2	8	no	—
15	C₂²×C₃²	—	2	4	no	—
16	C₂²×C₃²	C₂×C₄	1	2	yes	—
17	C₂²×C₃²	C₂×C₄	1	4	yes	complemented
18	C₂²×C₃²	D₄	1	4	yes	—
19	C₂²×C₃²	C₂³	1	1	yes	characteristic
20	C₃²×C₄	—	2	4	no	—
21	C₂×C₂²⋊C₄	C₃²	1	1	yes	characteristic, direct product factor, Sylow
22	C₂³×C₃	—	2	16	no	—
23	C₂³×C₃	C₃×C₄	1	16	yes	complemented
24	C₂³×C₃	C₂²×C₃	1	8	yes	—
25	C₂³×C₃	C₂²×C₃	1	4	yes	—
26	C₂×C₃×C₄	—	2	16	no	—
27	C₂×C₃×C₄	C₂²×C₃	1	16	yes	complemented
28	C₂×C₃²	—	2	4	no	—
29	C₂×C₃²	C₂²⋊C₄	1	4	yes	direct product factor
30	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₃²	C₂×D₄	1	2	yes	—
32	C₂⁴	C₂×C₃²	1	1	yes	characteristic
33	C₂²×C₄	C₂×C₃²	1	2	yes	complemented
34	C₂²⋊C₄	C₂×C₃²	1	4	yes	direct product factor
35	C₂²×C₃	—	2	32	no	—
36	C₂²×C₃	—	2	16	no	—
37	C₂²×C₃	C₂×C₃×C₄	1	8	yes	—
38	C₂²×C₃	C₂×C₃×C₄	1	16	yes	complemented
39	C₂²×C₃	C₃×D₄	1	16	yes	—
40	C₂²×C₃	C₂³×C₃	1	4	yes	—
41	C₃×C₄	—	2	16	no	—
42	C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
43	C₂³	—	2	4	no	—
44	C₂³	C₃²×C₄	1	4	yes	complemented
45	C₂³	C₂²×C₃²	1	2	yes	—
46	C₂³	C₂²×C₃²	1	1	yes	characteristic
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	C₂²×C₃²	1	4	yes	complemented
49	C₂×C₃	—	2	16	no	—
50	C₂×C₃	C₃×C₂²⋊C₄	1	16	yes	direct product factor
51	C₂×C₃	C₂²×C₃×C₄	1	4	yes	—
52	C₂×C₃	C₂×C₃×D₄	1	8	yes	—
53	C₂²	—	2	8	no	—
54	C₂²	—	2	4	no	—
55	C₂²	C₂×C₃²×C₄	1	2	yes	—
56	C₂²	C₂×C₃²×C₄	1	4	yes	complemented
57	C₂²	C₃²×D₄	1	4	yes	—
58	C₂²	C₂³×C₃²	1	1	yes	characteristic
59	C₄	—	2	4	no	—
60	C₃	C₂×C₃×C₂²⋊C₄	1	4	yes	direct product factor
61	C₂	—	2	4	no	—
62	C₂	C₃²×C₂²⋊C₄	1	4	yes	direct product factor
63	C₂	C₂²×C₃²×C₄	1	1	yes	characteristic
64	C₂	C₂×C₃²×D₄	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃²	C₂²
commutator	C₂	C₂²×C₃²×C₄
Frattini, cosocle	C₂²	C₂³×C₃²
prime radical	C₂⁴×C₃²	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₄	C₂	semidirect
3. C₃²×C₂²⋊C₄	C₂	direct (N × H)
4. C₂×C₃×C₂²⋊C₄	C₃	direct (N × H)
6. C₂³×C₃²	C₄	semidirect
10. C₂×C₃²×C₄	C₂²	semidirect
12. C₂²×C₃×C₄	C₂×C₃	semidirect
13. C₃×C₂²⋊C₄	C₂×C₃	direct (N × H)
17. C₂²×C₃²	C₂×C₄	semidirect
21. C₂×C₂²⋊C₄	C₃²	direct (N × H)
23. C₂³×C₃	C₃×C₄	semidirect
27. C₂×C₃×C₄	C₂²×C₃	semidirect
29. C₂×C₃²	C₂²⋊C₄	direct (N × H)
33. C₂²×C₄	C₂×C₃²	semidirect
34. C₂²⋊C₄	C₂×C₃²	direct (N × H)
38. C₂²×C₃	C₂×C₃×C₄	semidirect
42. C₃²	C₂×C₂²⋊C₄	direct (N × H)
44. C₂³	C₃²×C₄	semidirect
48. C₂×C₄	C₂²×C₃²	semidirect
50. C₂×C₃	C₃×C₂²⋊C₄	direct (N × H)
56. C₂²	C₂×C₃²×C₄	semidirect
60. C₃	C₂×C₃×C₂²⋊C₄	direct (N × H)
62. C₂	C₃²×C₂²⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃²	2
C₂²×C₃²×C₄	2
C₃²×C₂²⋊C₄	2
C₂×C₃×C₂²⋊C₄	3

Subgroups — C2×C32×C22⋊C4 #64164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₂²⋊C₄ #64164