Subgroups — C₂²×C₃×#125 #87321

Order 384.

1192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#125	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₃×C₄⋊C₄	C₂	1	4	yes	maximal
3	C₂²×C₃×C₄⋊C₄	C₂	1	2	yes	maximal
4	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
5	C₂²×#125	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂³×C₃×C₄	C₂²	1	2	yes	—
7	C₂³×C₃×C₄	C₂²	1	4	yes	—
8	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₃×#125	C₂²	1	64	yes	direct product factor
10	C₂×C₃×C₄⋊C₄	C₂²	1	48	yes	—
11	C₂×C₃×C₄⋊C₄	C₂²	1	24	yes	—
12	C₂×C₃×C₄²	C₂²	1	12	yes	—
13	C₂×#125	C₂×C₃	1	24	yes	direct product factor
14	C₂²×C₄⋊C₄	C₂×C₃	1	4	yes	—
15	C₂²×C₄⋊C₄	C₂×C₃	1	2	yes	—
16	C₂²×C₄²	C₂×C₃	1	1	yes	characteristic
17	C₂⁴×C₃	C₂³	1	1	yes	characteristic
18	C₂²×C₃×C₄	—	2	8	no	—
19	C₂²×C₃×C₄	—	2	16	no	—
20	C₂²×C₃×C₄	Q₈	1	8	yes	—
21	C₂²×C₃×C₄	C₂³	1	12	yes	—
22	C₂²×C₃×C₄	C₂³	1	24	yes	—
23	C₂²×C₃×C₄	C₂³	1	6	yes	—
24	C₃×C₄⋊C₄	C₂³	1	64	yes	—
25	C₃×C₄⋊C₄	C₂³	1	32	yes	—
26	C₃×C₄²	C₂³	1	16	yes	—
27	C₂³×C₄	C₂²×C₃	1	4	yes	—
28	C₂³×C₄	C₂²×C₃	1	2	yes	—
29	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
30	#125	C₂²×C₃	1	64	yes	direct product factor
31	C₂×C₄⋊C₄	C₂²×C₃	1	48	yes	—
32	C₂×C₄⋊C₄	C₂²×C₃	1	24	yes	—
33	C₂×C₄²	C₂²×C₃	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	—	2	48	no	—
39	C₂×C₃×C₄	C₂×Q₈	1	24	yes	—
40	C₂×C₃×C₄	C₂⁴	1	8	yes	—
41	C₂×C₃×C₄	C₂⁴	1	16	yes	—
42	C₂×C₃×C₄	C₂⁴	1	4	yes	—
43	C₂⁴	C₂³×C₃	1	1	yes	characteristic
44	C₂²×C₄	—	2	16	no	—
45	C₂²×C₄	—	2	8	no	—
46	C₂²×C₄	C₃×Q₈	1	8	yes	—
47	C₂²×C₄	C₂³×C₃	1	12	yes	—
48	C₂²×C₄	C₂³×C₃	1	24	yes	—
49	C₂²×C₄	C₂³×C₃	1	6	yes	—
50	C₄⋊C₄	C₂³×C₃	1	64	yes	—
51	C₄⋊C₄	C₂³×C₃	1	32	yes	—
52	C₄²	C₂³×C₃	1	16	yes	—
53	C₂²×C₃	#125	1	16	yes	direct product factor
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	—	2	16	no	—
59	C₃×C₄	C₂²×Q₈	1	8	yes	—
60	C₂³	C₂×C₃×Q₈	1	4	yes	—
61	C₂³	C₃×#41	1	8	yes	—
62	C₂³	C₂⁴×C₃	1	3	yes	—
63	C₂×C₄	—	2	24	no	—
64	C₂×C₄	—	2	48	no	—
65	C₂×C₄	C₂×C₃×Q₈	1	24	yes	—
66	C₂×C₄	C₂⁴×C₃	1	8	yes	—
67	C₂×C₄	C₂⁴×C₃	1	16	yes	—
68	C₂×C₄	C₂⁴×C₃	1	4	yes	—
69	C₂×C₃	C₂×#125	1	12	yes	direct product factor
70	C₂×C₃	C₂³×Q₈	1	1	yes	characteristic
71	C₂×C₃	C₂²×#41	1	2	yes	—
72	C₂²	C₃×#125	1	16	yes	direct product factor
73	C₂²	C₂²×C₃×Q₈	1	6	yes	—
74	C₂²	C₂×C₃×#41	1	12	yes	—
75	C₂²	C₂⁵×C₃	1	1	yes	characteristic
76	C₄	—	2	8	no	—
77	C₄	—	2	16	no	—
78	C₄	C₂²×C₃×Q₈	1	8	yes	—
79	C₃	C₂²×#125	1	1	yes	characteristic, direct product factor, Sylow
80	C₂	C₂×C₃×#125	1	12	yes	direct product factor
81	C₂	C₂³×C₃×Q₈	1	1	yes	characteristic
82	C₂	C₂²×C₃×#41	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#125	C₂	direct (N × H)
5. C₂²×#125	C₃	direct (N × H)
9. C₃×#125	C₂²	direct (N × H)
13. C₂×#125	C₂×C₃	direct (N × H)
30. #125	C₂²×C₃	direct (N × H)
53. C₂²×C₃	#125	direct (N × H)
69. C₂×C₃	C₂×#125	direct (N × H)
72. C₂²	C₃×#125	direct (N × H)
79. C₃	C₂²×#125	direct (N × H)
80. C₂	C₂×C₃×#125	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×C₃×#125	2
C₂²×C₃×C₄⋊C₄	2
C₂²×C₃×C₄²	2
C₂²×#125	3

Subgroups — C22×C3×#125 #87321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#125 #87321