Subgroups — C₃×#2273 #84137

Order 384.

912 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₂²⋊C₄	C₂	1	3	yes	maximal
2	C₃×C₄×C₂²⋊C₄	C₂	1	12	yes	maximal, complemented
3	#2273	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂⁵×C₃	C₂²	1	1	yes	characteristic
5	C₂³×C₃×C₄	C₂²	1	6	yes	—
6	C₂×C₃×C₂²⋊C₄	C₄	1	24	yes	complemented
7	C₂×C₃×C₂²⋊C₄	C₂²	1	12	yes	—
8	C₂×C₃×C₄²	C₂²	1	8	yes	complemented
9	C₃×#95	C₂²	1	8	yes	complemented
10	C₂²×C₂²⋊C₄	C₂×C₃	1	3	yes	—
11	C₄×C₂²⋊C₄	C₂×C₃	1	12	yes	complemented
12	C₂⁴×C₃	—	4	4	no	—
13	C₂⁴×C₃	C₂×C₄	1	12	yes	—
14	C₂⁴×C₃	C₂³	1	3	yes	—
15	C₂²×C₃×C₄	—	4	12	no	—
16	C₂²×C₃×C₄	C₂×C₄	1	24	yes	complemented
17	C₂²×C₃×C₄	C₂³	1	12	yes	—
18	C₃×C₂²⋊C₄	C₂×C₄	1	48	yes	complemented
19	C₃×C₄²	—	4	8	no	—
20	C₂⁵	C₂²×C₃	1	1	yes	characteristic
21	C₂³×C₄	C₂²×C₃	1	6	yes	—
22	C₂×C₂²⋊C₄	C₃×C₄	1	24	yes	complemented
23	C₂×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
24	C₂×C₄²	C₂²×C₃	1	8	yes	complemented
25	#95	C₂²×C₃	1	8	yes	complemented
26	C₂³×C₃	—	2	12	no	—
27	C₂³×C₃	—	4	24	no	—
28	C₂³×C₃	C₄²	1	16	yes	complemented
29	C₂³×C₃	C₂²×C₄	1	12	yes	—
30	C₂³×C₃	C₂²×C₄	1	6	yes	—
31	C₂³×C₃	C₂⁴	1	1	yes	characteristic
32	C₂×C₃×C₄	—	2	24	no	—
33	C₂×C₃×C₄	—	4	24	no	—
34	C₂×C₃×C₄	C₂²×C₄	1	24	yes	—
35	C₂⁴	—	4	4	no	—
36	C₂⁴	C₂×C₃×C₄	1	12	yes	—
37	C₂⁴	C₂³×C₃	1	3	yes	—
38	C₂²×C₄	—	4	12	no	—
39	C₂²×C₄	C₂×C₃×C₄	1	24	yes	complemented
40	C₂²×C₄	C₂³×C₃	1	12	yes	—
41	C₂²⋊C₄	C₂×C₃×C₄	1	48	yes	complemented
42	C₄²	—	4	8	no	—
43	C₂²×C₃	—	2	12	no	—
44	C₂²×C₃	—	2	24	no	—
45	C₂²×C₃	—	4	16	no	—
46	C₂²×C₃	C₂×C₄²	1	12	yes	—
47	C₂²×C₃	C₂³×C₄	1	3	yes	—
48	C₂²×C₃	H(5,𝔽₂)	1	4	yes	—
49	C₃×C₄	—	2	24	no	—
50	C₂³	—	2	12	no	—
51	C₂³	—	4	24	no	—
52	C₂³	C₃×C₄²	1	16	yes	complemented
53	C₂³	C₂²×C₃×C₄	1	12	yes	—
54	C₂³	C₂²×C₃×C₄	1	6	yes	—
55	C₂³	C₂⁴×C₃	1	1	yes	characteristic
56	C₂×C₄	—	2	24	no	—
57	C₂×C₄	—	4	24	no	—
58	C₂×C₄	C₂²×C₃×C₄	1	24	yes	—
59	C₂×C₃	—	2	12	no	—
60	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
61	C₂×C₃	#518	1	6	yes	—
62	C₂²	—	2	12	no	—
63	C₂²	—	2	24	no	—
64	C₂²	—	4	16	no	—
65	C₂²	C₂×C₃×C₄²	1	12	yes	—
66	C₂²	C₂³×C₃×C₄	1	3	yes	—
67	C₂²	C₃×H(5,𝔽₂)	1	4	yes	—
68	C₄	—	2	24	no	—
69	C₃	#2273	1	1	yes	characteristic, direct product factor, Sylow
70	C₂	—	2	12	no	—
71	C₂	C₂²×C₃×C₄²	1	1	yes	characteristic
72	C₂	C₃×#518	1	6	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₂²⋊C₄	C₂	semidirect
3. #2273	C₃	direct (N × H)
6. C₂×C₃×C₂²⋊C₄	C₄	semidirect
8. C₂×C₃×C₄²	C₂²	semidirect
9. C₃×#95	C₂²	semidirect
11. C₄×C₂²⋊C₄	C₂×C₃	semidirect
16. C₂²×C₃×C₄	C₂×C₄	semidirect
18. C₃×C₂²⋊C₄	C₂×C₄	semidirect
22. C₂×C₂²⋊C₄	C₃×C₄	semidirect
24. C₂×C₄²	C₂²×C₃	semidirect
25. #95	C₂²×C₃	semidirect
28. C₂³×C₃	C₄²	semidirect
39. C₂²×C₄	C₂×C₃×C₄	semidirect
41. C₂²⋊C₄	C₂×C₃×C₄	semidirect
52. C₂³	C₃×C₄²	semidirect
69. C₃	#2273	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×C₂²⋊C₄	2
C₃×C₄×C₂²⋊C₄	2
#2273	3

Subgroups — C3×#2273 #84137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2273 #84137