Subgroups — C₃²×#124 #64173

Order 288.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
4	C₃²×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#124	C₃	1	4	yes	maximal, direct product factor
6	C₂³×C₃²	C₂²	1	2	yes	—
7	C₃²×Q₈	—	2	2	no	—
8	C₃²×D₄	—	2	2	no	—
9	C₂×C₃²×C₄	C₂²	1	2	yes	—
10	C₂×C₃²×C₄	C₂²	1	2	yes	—
11	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
12	C₂×C₃×Q₈	C₂×C₃	1	4	yes	complemented
13	C₂×C₃×D₄	C₂×C₃	1	4	yes	—
14	C₃×C₂²⋊C₄	C₂×C₃	1	16	yes	complemented
15	C₃×C₄²	C₂×C₃	1	4	yes	complemented
16	C₂²×C₃²	—	2	2	no	—
17	C₂²×C₃²	—	2	4	no	—
18	C₂²×C₃²	C₂³	1	1	yes	characteristic
19	C₃²×C₄	—	2	2	no	—
20	C₃²×C₄	—	2	2	no	—
21	C₃²×C₄	D₄	1	2	yes	—
22	#124	C₃²	1	1	yes	characteristic, direct product factor, Sylow
23	C₂³×C₃	C₂²×C₃	1	8	yes	—
24	C₃×Q₈	—	2	8	no	—
25	C₃×D₄	—	2	8	no	—
26	C₂×C₃×C₄	C₂²×C₃	1	8	yes	—
27	C₂×C₃×C₄	C₂²×C₃	1	8	yes	—
28	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
29	C₂×C₃²	—	4	2	no	—
30	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
31	C₂×C₃²	#41	1	2	yes	—
32	C₂×Q₈	C₂×C₃²	1	1	yes	characteristic, complemented
33	C₂×D₄	C₂×C₃²	1	1	yes	characteristic
34	C₂²⋊C₄	C₂×C₃²	1	4	yes	complemented
35	C₄²	C₂×C₃²	1	1	yes	characteristic, complemented
36	C₂²×C₃	—	2	8	no	—
37	C₂²×C₃	—	2	16	no	—
38	C₂²×C₃	C₂³×C₃	1	4	yes	—
39	C₃×C₄	—	2	8	no	—
40	C₃×C₄	—	2	8	no	—
41	C₃×C₄	C₃×D₄	1	8	yes	—
42	C₃²	#124	1	1	yes	characteristic, direct product factor, Sylow
43	C₂³	C₂²×C₃²	1	2	yes	—
44	Q₈	—	2	2	no	—
45	D₄	—	2	2	no	—
46	C₂×C₄	C₂²×C₃²	1	2	yes	—
47	C₂×C₄	C₂²×C₃²	1	2	yes	—
48	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
49	C₂×C₃	—	4	8	no	—
50	C₂×C₃	C₂×C₃×D₄	1	4	yes	—
51	C₂×C₃	C₃×#41	1	8	yes	—
52	C₂²	—	2	2	no	—
53	C₂²	—	2	4	no	—
54	C₂²	C₂³×C₃²	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	—	2	2	no	—
57	C₄	C₃²×D₄	1	2	yes	—
58	C₃	C₃×#124	1	4	yes	direct product factor
59	C₂	—	4	2	no	—
60	C₂	C₂×C₃²×D₄	1	1	yes	characteristic
61	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
1. C₂×C₃²×Q₈	C₂	semidirect
3. C₃²×C₂²⋊C₄	C₂	semidirect
4. C₃²×C₄²	C₂	semidirect
5. C₃×#124	C₃	direct (N × H)
12. C₂×C₃×Q₈	C₂×C₃	semidirect
14. C₃×C₂²⋊C₄	C₂×C₃	semidirect
15. C₃×C₄²	C₂×C₃	semidirect
22. #124	C₃²	direct (N × H)
32. C₂×Q₈	C₂×C₃²	semidirect
34. C₂²⋊C₄	C₂×C₃²	semidirect
35. C₄²	C₂×C₃²	semidirect
42. C₃²	#124	direct (N × H)
58. C₃	C₃×#124	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×Q₈	2
C₂×C₃²×D₄	2
C₃²×C₂²⋊C₄	2
C₃²×C₄²	2
C₃×#124	3

Subgroups — C32×#124 #64173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#124 #64173