Subgroups — C₃²×#123 #64172

Order 288.

234 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₃²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×#123	C₃	1	4	yes	maximal, direct product factor
7	C₂³×C₃²	C₂²	1	1	yes	characteristic
8	C₂³×C₃²	C₂²	1	1	yes	characteristic
9	C₃²×D₄	—	2	2	no	—
10	C₂×C₃²×C₄	—	2	2	no	—
11	C₂×C₃²×C₄	C₂²	1	2	yes	—
12	C₂×C₃²×C₄	C₂²	1	2	yes	—
13	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×D₄	C₂×C₃	1	4	yes	—
15	C₂²×C₃×C₄	C₂×C₃	1	4	yes	complemented
16	C₃×C₄⋊C₄	C₂×C₃	1	8	yes	complemented
17	C₃×C₂²⋊C₄	C₂×C₃	1	4	yes	complemented
18	C₃×C₂²⋊C₄	C₂×C₃	1	8	yes	complemented
19	C₂²×C₃²	—	2	2	no	—
20	C₂²×C₃²	—	2	1	no	—
21	C₂²×C₃²	—	2	2	no	—
22	C₂²×C₃²	D₄	1	2	yes	—
23	C₂²×C₃²	C₂³	1	1	yes	characteristic
24	C₃²×C₄	—	2	2	no	—
25	C₃²×C₄	—	2	1	no	—
26	C₃²×C₄	—	2	2	no	—
27	#123	C₃²	1	1	yes	characteristic, direct product factor, Sylow
28	C₂³×C₃	C₂²×C₃	1	4	yes	—
29	C₂³×C₃	C₂²×C₃	1	4	yes	—
30	C₃×D₄	—	2	8	no	—
31	C₂×C₃×C₄	—	2	8	no	—
32	C₂×C₃×C₄	C₂²×C₃	1	8	yes	—
33	C₂×C₃×C₄	C₂²×C₃	1	8	yes	—
34	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
35	C₂×C₃²	—	2	2	no	—
36	C₂×C₃²	—	4	1	no	—
37	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
38	C₂×C₃²	#41	1	2	yes	—
39	C₂×D₄	C₂×C₃²	1	1	yes	characteristic
40	C₂²×C₄	C₂×C₃²	1	1	yes	characteristic, complemented
41	C₄⋊C₄	C₂×C₃²	1	2	yes	complemented
42	C₂²⋊C₄	C₂×C₃²	1	1	yes	characteristic, complemented
43	C₂²⋊C₄	C₂×C₃²	1	2	yes	complemented
44	C₂²×C₃	—	2	8	no	—
45	C₂²×C₃	—	2	4	no	—
46	C₂²×C₃	—	2	8	no	—
47	C₂²×C₃	C₃×D₄	1	8	yes	—
48	C₂²×C₃	C₂³×C₃	1	4	yes	—
49	C₃×C₄	—	2	8	no	—
50	C₃×C₄	—	2	4	no	—
51	C₃×C₄	—	2	8	no	—
52	C₃²	#123	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³	C₂²×C₃²	1	1	yes	characteristic
54	C₂³	C₂²×C₃²	1	1	yes	characteristic
55	D₄	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	C₂²×C₃²	1	2	yes	—
58	C₂×C₄	C₂²×C₃²	1	2	yes	—
59	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
60	C₂×C₃	—	2	8	no	—
61	C₂×C₃	—	4	4	no	—
62	C₂×C₃	C₂×C₃×D₄	1	4	yes	—
63	C₂×C₃	C₃×#41	1	8	yes	—
64	C₂²	—	2	2	no	—
65	C₂²	—	2	1	no	—
66	C₂²	—	2	2	no	—
67	C₂²	C₃²×D₄	1	2	yes	—
68	C₂²	C₂³×C₃²	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	2	2	no	—
72	C₃	C₃×#123	1	4	yes	direct product factor
73	C₂	—	2	2	no	—
74	C₂	—	4	1	no	—
75	C₂	C₂×C₃²×D₄	1	1	yes	characteristic
76	C₂	C₃²×#41	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃²	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃²
prime radical	C₂×C₃²×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×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×C₄⋊C₄	C₂	semidirect
4. C₃²×C₂²⋊C₄	C₂	semidirect
5. C₃²×C₂²⋊C₄	C₂	semidirect
6. C₃×#123	C₃	direct (N × H)
15. C₂²×C₃×C₄	C₂×C₃	semidirect
16. C₃×C₄⋊C₄	C₂×C₃	semidirect
17. C₃×C₂²⋊C₄	C₂×C₃	semidirect
18. C₃×C₂²⋊C₄	C₂×C₃	semidirect
27. #123	C₃²	direct (N × H)
40. C₂²×C₄	C₂×C₃²	semidirect
41. C₄⋊C₄	C₂×C₃²	semidirect
42. C₂²⋊C₄	C₂×C₃²	semidirect
43. C₂²⋊C₄	C₂×C₃²	semidirect
52. C₃²	#123	direct (N × H)
72. C₃	C₃×#123	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₄	2
C₂²×C₃²×C₄	2
C₃²×C₄⋊C₄	2
C₃²×C₂²⋊C₄	2
C₃×#123	3

Subgroups — C32×#123 #64172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#123 #64172