Subgroups — C₂²×#876 #85676

Order 384.

1320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×(C₂×D₃)⋊C₄	C₂	1	3	yes	maximal
3	C₂²×Dic₃⋊C₄	C₂	1	3	yes	maximal, complemented
4	C₂×#876	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#126	—	3	1	no	maximal, Sylow
6	C₂⁴×D₃	C₂²	1	1	yes	characteristic
7	C₂³×C₃×C₄	C₂²	1	3	yes	—
8	C₂³×Dic₃	C₂²	1	3	yes	—
9	C₂×C₃×C₄²	C₂²	1	12	yes	complemented
10	C₂×(C₂×D₃)⋊C₄	C₂²	1	36	yes	—
11	C₂×Dic₃⋊C₄	C₂²	1	36	yes	complemented
12	#876	C₂²	1	64	yes	direct product factor
13	C₂×#126	—	3	24	no	—
14	C₂²×C₄⋊C₄	—	3	3	no	—
15	C₂²×C₂²⋊C₄	—	3	3	no	—
16	C₂²×C₄²	D₃	1	1	yes	characteristic, complemented
17	C₂⁴×C₃	C₂³	1	1	yes	characteristic
18	C₂³×D₃	—	2	12	no	—
19	C₂³×D₃	C₂³	1	6	yes	—
20	C₂²×C₃×C₄	—	2	12	no	—
21	C₂²×C₃×C₄	C₂³	1	18	yes	—
22	C₂²×Dic₃	—	2	12	no	—
23	C₂²×Dic₃	C₂³	1	18	yes	—
24	C₃×C₄²	C₂³	1	16	yes	complemented
25	(C₂×D₃)⋊C₄	C₂³	1	48	yes	—
26	Dic₃⋊C₄	C₂³	1	48	yes	complemented
27	C₂⁵	—	3	1	no	—
28	C₂³×C₄	—	3	3	no	—
29	C₂³×C₄	C₂×D₃	1	3	yes	—
30	#126	—	3	64	no	—
31	C₂×C₄⋊C₄	—	3	36	no	—
32	C₂×C₂²⋊C₄	—	3	36	no	—
33	C₂×C₄²	C₂×D₃	1	12	yes	complemented
34	C₂³×C₃	#41	1	12	yes	—
35	C₂³×C₃	C₂⁴	1	3	yes	—
36	C₂²×D₃	—	2	36	no	—
37	C₂²×D₃	—	4	16	no	—
38	C₂²×D₃	C₂⁴	1	4	yes	—
39	C₂×C₃×C₄	—	2	36	no	—
40	C₂×C₃×C₄	C₂⁴	1	12	yes	—
41	C₂×Dic₃	—	2	36	no	—
42	C₂×Dic₃	C₂⁴	1	12	yes	—
43	C₂⁴	—	3	6	no	—
44	C₂⁴	—	6	12	no	—
45	C₂⁴	C₂²×D₃	1	1	yes	characteristic
46	C₂²×C₄	—	2	12	no	—
47	C₂²×C₄	—	3	18	no	—
48	C₂²×C₄	—	6	12	no	—
49	C₂²×C₄	C₂²×D₃	1	18	yes	—
50	C₄⋊C₄	—	3	48	no	—
51	C₂²⋊C₄	—	3	48	no	—
52	C₄²	C₂²×D₃	1	16	yes	complemented
53	C₂²×C₃	#126	1	16	yes	complemented
54	C₂²×C₃	C₂×#41	1	18	yes	—
55	C₂²×C₃	C₂⁵	1	1	yes	characteristic
56	C₂×D₃	—	2	12	no	—
57	C₂×D₃	—	4	24	no	—
58	C₃×C₄	—	2	12	no	—
59	Dic₃	—	2	12	no	—
60	C₂³	—	3	4	no	—
61	C₂³	—	6	36	no	—
62	C₂³	—	12	16	no	—
63	C₂³	#235	1	12	yes	—
64	C₂³	C₂³×D₃	1	3	yes	—
65	C₂×C₄	—	2	36	no	—
66	C₂×C₄	—	3	12	no	—
67	C₂×C₄	—	6	36	no	—
68	C₂×C₄	C₂³×D₃	1	12	yes	—
69	C₂×C₃	C₂×#126	1	12	yes	complemented
70	C₂×C₃	C₂²×#41	1	3	yes	—
71	D₃	—	4	4	no	—
72	C₂²	—	6	12	no	—
73	C₂²	—	12	24	no	—
74	C₂²	#876	1	16	yes	direct product factor
75	C₂²	C₂×#235	1	18	yes	—
76	C₂²	C₂⁴×D₃	1	1	yes	characteristic
77	C₄	—	2	12	no	—
78	C₄	—	6	12	no	—
79	C₃	C₂²×#126	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	12	4	no	—
81	C₂	C₂×#876	1	12	yes	direct product factor
82	C₂	C₂²×#235	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle	C₂⁴×C₃	C₂³
prime radical	C₂⁴×D₃	C₂²
hypercenter	C₂²×C₄²	D₃
hypocenter	C₃	C₂²×#126
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₄²	C₂	semidirect
3. C₂²×Dic₃⋊C₄	C₂	semidirect
4. C₂×#876	C₂	direct (N × H)
9. C₂×C₃×C₄²	C₂²	semidirect
11. C₂×Dic₃⋊C₄	C₂²	semidirect
12. #876	C₂²	direct (N × H)
16. C₂²×C₄²	D₃	semidirect
24. C₃×C₄²	C₂³	semidirect
26. Dic₃⋊C₄	C₂³	semidirect
33. C₂×C₄²	C₂×D₃	semidirect
52. C₄²	C₂²×D₃	semidirect
53. C₂²×C₃	#126	semidirect
69. C₂×C₃	C₂×#126	semidirect
74. C₂²	#876	direct (N × H)
79. C₃	C₂²×#126	semidirect
81. C₂	C₂×#876	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×C₄²	2
C₂²×(C₂×D₃)⋊C₄	2
C₂²×Dic₃⋊C₄	2
C₂×#876	2
C₂²×#126	3

Subgroups — C22×#876 #85676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#876 #85676