Subgroups — C₃×#2276 #84140

Order 384.

628 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Q₈	C₂	1	3	yes	maximal
2	C₃×C₄×C₄⋊C₄	C₂	1	9	yes	maximal
3	C₃×#376	C₂	1	3	yes	maximal
4	#2276	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
6	C₃×C₄×Q₈	C₄	1	24	yes	complemented
7	C₂×C₃×C₄⋊C₄	C₂²	1	9	yes	—
8	C₂×C₃×C₄²	C₂²	1	6	yes	—
9	C₂×C₃×C₄²	C₂²	1	9	yes	—
10	C₃×#95	C₂²	1	1	yes	characteristic
11	C₃×#95	C₂²	1	9	yes	—
12	C₂×C₄×Q₈	C₂×C₃	1	3	yes	—
13	C₄×C₄⋊C₄	C₂×C₃	1	9	yes	—
14	#376	C₂×C₃	1	3	yes	—
15	C₂×C₃×Q₈	C₂×C₄	1	12	yes	—
16	C₂²×C₃×C₄	C₂³	1	3	yes	—
17	C₂²×C₃×C₄	C₂³	1	9	yes	—
18	C₂²×C₃×C₄	C₂³	1	3	yes	—
19	C₃×C₄⋊C₄	C₂×C₄	1	36	yes	—
20	C₃×C₄²	—	4	6	no	—
21	C₃×C₄²	C₂×C₄	1	36	yes	—
22	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
23	C₄×Q₈	C₃×C₄	1	24	yes	complemented
24	C₂×C₄⋊C₄	C₂²×C₃	1	9	yes	—
25	C₂×C₄²	C₂²×C₃	1	6	yes	—
26	C₂×C₄²	C₂²×C₃	1	9	yes	—
27	#95	C₂²×C₃	1	1	yes	characteristic
28	#95	C₂²×C₃	1	9	yes	—
29	C₂³×C₃	C₂⁴	1	1	yes	characteristic
30	C₃×Q₈	C₄²	1	16	yes	complemented
31	C₂×C₃×C₄	—	2	9	no	—
32	C₂×C₃×C₄	—	2	6	no	—
33	C₂×C₃×C₄	—	2	9	no	—
34	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
35	C₂×C₃×C₄	C₂²×C₄	1	18	yes	—
36	C₂×C₃×C₄	C₂²×C₄	1	18	yes	—
37	C₂×Q₈	C₂×C₃×C₄	1	12	yes	—
38	C₂²×C₄	C₂³×C₃	1	3	yes	—
39	C₂²×C₄	C₂³×C₃	1	9	yes	—
40	C₂²×C₄	C₂³×C₃	1	3	yes	—
41	C₄⋊C₄	C₂×C₃×C₄	1	36	yes	—
42	C₄²	—	4	6	no	—
43	C₄²	C₂×C₃×C₄	1	36	yes	—
44	C₂²×C₃	C₂³×C₄	1	3	yes	—
45	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
46	C₂²×C₃	2₋¹⁺⁴	1	3	yes	—
47	C₃×C₄	—	2	6	no	—
48	C₃×C₄	—	2	18	no	—
49	C₃×C₄	C₂×C₄²	1	12	yes	—
50	C₂³	C₂⁴×C₃	1	1	yes	characteristic
51	Q₈	C₃×C₄²	1	16	yes	complemented
52	C₂×C₄	—	2	9	no	—
53	C₂×C₄	—	2	6	no	—
54	C₂×C₄	—	2	9	no	—
55	C₂×C₄	C₂²×C₃×C₄	1	6	yes	—
56	C₂×C₄	C₂²×C₃×C₄	1	18	yes	—
57	C₂×C₄	C₂²×C₃×C₄	1	18	yes	—
58	C₂×C₃	C₂²×C₄²	1	1	yes	characteristic
59	C₂×C₃	#519	1	3	yes	—
60	C₂×C₃	#520	1	3	yes	—
61	C₂²	C₂³×C₃×C₄	1	3	yes	—
62	C₂²	C₃×H(5,𝔽₂)	1	1	yes	characteristic
63	C₂²	C₃×2₋¹⁺⁴	1	3	yes	—
64	C₄	—	2	6	no	—
65	C₄	—	2	18	no	—
66	C₄	C₂×C₃×C₄²	1	12	yes	—
67	C₃	#2276	1	1	yes	characteristic, direct product factor, Sylow
68	C₂	C₂²×C₃×C₄²	1	1	yes	characteristic
69	C₂	C₃×#519	1	3	yes	—
70	C₂	C₃×#520	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂⁴
commutator	C₂	C₂²×C₃×C₄²
Frattini, cosocle	C₂³	C₂⁴×C₃
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
4. #2276	C₃	direct (N × H)
6. C₃×C₄×Q₈	C₄	semidirect
23. C₄×Q₈	C₃×C₄	semidirect
30. C₃×Q₈	C₄²	semidirect
51. Q₈	C₃×C₄²	semidirect
67. C₃	#2276	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×C₄×Q₈	2
C₃×C₄×C₄⋊C₄	2
C₃×#376	2
#2276	3

Subgroups — C3×#2276 #84140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2276 #84140