Subgroups — C₂²×#870 #85669

Order 384.

1192 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
2	C₂²×C₄⋊Dic₃	C₂	1	2	yes	maximal
3	C₂²×Dic₃⋊C₄	C₂	1	4	yes	maximal
4	C₂×#870	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#125	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₂²	1	2	yes	—
7	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂³×Dic₃	C₂²	1	4	yes	—
9	C₂×C₃×C₄²	C₂²	1	12	yes	—
10	C₂×C₄⋊Dic₃	C₂²	1	24	yes	—
11	C₂×Dic₃⋊C₄	C₂²	1	48	yes	—
12	#870	C₂²	1	64	yes	direct product factor
13	C₂×#125	—	3	24	no	—
14	C₂²×C₄⋊C₄	—	3	4	no	—
15	C₂²×C₄⋊C₄	—	3	2	no	—
16	C₂²×C₄²	D₃	1	1	yes	characteristic
17	C₂⁴×C₃	C₂³	1	1	yes	characteristic
18	C₂²×C₃×C₄	—	2	8	no	—
19	C₂²×C₃×C₄	Q₈	1	8	yes	—
20	C₂²×C₃×C₄	C₂³	1	12	yes	—
21	C₂²×C₃×C₄	C₂³	1	6	yes	—
22	C₂²×Dic₃	—	2	16	no	—
23	C₂²×Dic₃	C₂³	1	24	yes	—
24	C₃×C₄²	C₂³	1	16	yes	—
25	C₄⋊Dic₃	C₂³	1	32	yes	—
26	Dic₃⋊C₄	C₂³	1	64	yes	—
27	C₂³×C₄	—	3	4	no	—
28	C₂³×C₄	C₂×D₃	1	2	yes	—
29	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
30	#125	—	3	64	no	—
31	C₂×C₄⋊C₄	—	3	48	no	—
32	C₂×C₄⋊C₄	—	3	24	no	—
33	C₂×C₄²	C₂×D₃	1	12	yes	—
34	C₂³×C₃	C₂×Q₈	1	4	yes	—
35	C₂³×C₃	#41	1	8	yes	—
36	C₂³×C₃	C₂⁴	1	3	yes	—
37	C₂×C₃×C₄	—	2	24	no	—
38	C₂×C₃×C₄	C₂×Q₈	1	24	yes	—
39	C₂×C₃×C₄	C₂⁴	1	8	yes	—
40	C₂×C₃×C₄	C₂⁴	1	4	yes	—
41	C₂×Dic₃	—	2	48	no	—
42	C₂×Dic₃	C₂⁴	1	16	yes	—
43	C₂⁴	C₂²×D₃	1	1	yes	characteristic
44	C₂²×C₄	—	2	8	no	—
45	C₂²×C₄	—	3	24	no	—
46	C₂²×C₄	—	6	16	no	—
47	C₂²×C₄	Dic₆	1	8	yes	—
48	C₂²×C₄	C₂²×D₃	1	12	yes	—
49	C₂²×C₄	C₂²×D₃	1	6	yes	—
50	C₄⋊C₄	—	3	64	no	—
51	C₄⋊C₄	—	3	32	no	—
52	C₄²	C₂²×D₃	1	16	yes	—
53	C₂²×C₃	#125	1	16	yes	complemented
54	C₂²×C₃	C₂²×Q₈	1	6	yes	—
55	C₂²×C₃	C₂×#41	1	12	yes	—
56	C₂²×C₃	C₂⁵	1	1	yes	characteristic
57	C₃×C₄	—	2	8	no	—
58	C₃×C₄	C₂²×Q₈	1	8	yes	—
59	Dic₃	—	2	16	no	—
60	C₂³	C₂×Dic₆	1	4	yes	—
61	C₂³	#235	1	8	yes	—
62	C₂³	C₂³×D₃	1	3	yes	—
63	C₂×C₄	—	2	24	no	—
64	C₂×C₄	—	3	16	no	—
65	C₂×C₄	—	6	48	no	—
66	C₂×C₄	C₂×Dic₆	1	24	yes	—
67	C₂×C₄	C₂³×D₃	1	8	yes	—
68	C₂×C₄	C₂³×D₃	1	4	yes	—
69	C₂×C₃	C₂×#125	1	12	yes	complemented
70	C₂×C₃	C₂³×Q₈	1	1	yes	characteristic
71	C₂×C₃	C₂²×#41	1	2	yes	—
72	C₂²	#870	1	16	yes	direct product factor
73	C₂²	C₂²×Dic₆	1	6	yes	—
74	C₂²	C₂×#235	1	12	yes	—
75	C₂²	C₂⁴×D₃	1	1	yes	characteristic
76	C₄	—	2	8	no	—
77	C₄	—	6	16	no	—
78	C₄	C₂²×Dic₆	1	8	yes	—
79	C₃	C₂²×#125	1	1	yes	characteristic, complemented, Sylow
80	C₂	C₂×#870	1	12	yes	direct product factor
81	C₂	C₂³×Dic₆	1	1	yes	characteristic
82	C₂	C₂²×#235	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle, prime radical	C₂⁴×C₃	C₂³
hypercenter	C₂²×C₄²	D₃
hypocenter	C₃	C₂²×#125
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
4. C₂×#870	C₂	direct (N × H)
12. #870	C₂²	direct (N × H)
53. C₂²×C₃	#125	semidirect
69. C₂×C₃	C₂×#125	semidirect
72. C₂²	#870	direct (N × H)
79. C₃	C₂²×#125	semidirect
80. C₂	C₂×#870	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×C₄⋊Dic₃	2
C₂²×Dic₃⋊C₄	2
C₂×#870	2
C₂²×#125	3

Subgroups — C22×#870 #85669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#870 #85669