Subgroups — C₂²×#627 #64327

Order 288.

339 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₃	C₂	1	2	yes	maximal
3	C₂×#627	C₂	1	12	yes	maximal, direct product factor
4	C₂²×Dic₆	—	3	2	no	maximal
5	C₂³×C₃²	C₂²	1	1	yes	characteristic
6	C₂×C₃⋊Dic₃	C₂²	1	6	yes	—
7	C₂×C₃×Dic₃	C₂²	1	12	yes	—
8	#627	C₂²	1	16	yes	direct product factor
9	C₂²×C₃×C₄	—	3	2	no	—
10	C₂²×Dic₃	—	3	2	no	—
11	C₂²×Dic₃	—	6	1	no	—
12	C₂²×Dic₃	D₃	1	2	yes	—
13	C₂×Dic₆	—	3	24	no	—
14	C₂²×C₃²	Q₈	1	4	yes	complemented
15	C₂²×C₃²	C₂³	1	3	yes	—
16	C₃⋊Dic₃	C₂³	1	4	yes	—
17	C₃×Dic₃	C₂³	1	8	yes	—
18	C₂²×Q₈	—	9	1	no	Sylow
19	C₂³×C₃	—	2	1	no	—
20	C₂³×C₃	C₂×D₃	1	2	yes	—
21	C₂×C₃×C₄	—	3	12	no	—
22	C₂×Dic₃	—	3	12	no	—
23	C₂×Dic₃	—	6	6	no	—
24	C₂×Dic₃	C₂×D₃	1	12	yes	—
25	Dic₆	—	3	32	no	—
26	C₂×C₃²	C₂×Q₈	1	6	yes	complemented
27	C₂×C₃²	C₂⁴	1	1	yes	characteristic
28	C₂×Q₈	—	9	12	no	—
29	C₂²×C₄	—	3	2	no	—
30	C₂²×C₄	—	9	1	no	—
31	C₂²×C₃	—	2	4	no	—
32	C₂²×C₃	—	2	3	no	—
33	C₂²×C₃	Dic₆	1	8	yes	complemented
34	C₂²×C₃	C₂²×D₃	1	6	yes	—
35	C₃×C₄	—	3	8	no	—
36	Dic₃	—	3	8	no	—
37	Dic₃	—	6	4	no	—
38	Dic₃	C₂²×D₃	1	8	yes	—
39	C₃²	C₂²×Q₈	1	1	yes	characteristic, complemented, Sylow
40	C₂³	D₃²	1	1	yes	characteristic
41	Q₈	—	9	16	no	—
42	C₂×C₄	—	3	12	no	—
43	C₂×C₄	—	9	6	no	—
44	C₂×C₃	—	2	1	no	—
45	C₂×C₃	—	2	6	no	—
46	C₂×C₃	C₂×Dic₆	1	12	yes	complemented
47	C₂×C₃	C₂³×D₃	1	2	yes	—
48	C₂²	#627	1	4	yes	direct product factor
49	C₂²	C₂×D₃²	1	3	yes	—
50	C₄	—	3	8	no	—
51	C₄	—	9	4	no	—
52	C₃	—	2	1	no	—
53	C₃	C₂²×Dic₆	1	2	yes	complemented
54	C₂	C₂×#627	1	6	yes	direct product factor
55	C₂	C₂²×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₃²
commutator, cosocle	C₂×C₃²	C₂⁴
Frattini	C₂	C₂²×D₃²
Fitting, socle, prime radical	C₂³×C₃²	C₂²
hypocenter	C₃²	C₂²×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
3. C₂×#627	C₂	direct (N × H)
8. #627	C₂²	direct (N × H)
14. C₂²×C₃²	Q₈	semidirect
26. C₂×C₃²	C₂×Q₈	semidirect
33. C₂²×C₃	Dic₆	semidirect
39. C₃²	C₂²×Q₈	semidirect
46. C₂×C₃	C₂×Dic₆	semidirect
48. C₂²	#627	direct (N × H)
53. C₃	C₂²×Dic₆	semidirect
54. C₂	C₂×#627	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⋊Dic₃	2
C₂²×C₃×Dic₃	2
C₂×#627	2
C₂²×Dic₆	3

Subgroups — C22×#627 #64327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#627 #64327