Subgroups — C₃×#2176 #72891

Order 384.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×Q₁₆	C₂	1	2	yes	maximal
3	C₃×#367	C₂	1	1	yes	maximal, characteristic
4	C₃×#358	C₂	1	2	yes	maximal
5	C₃×#346	C₂	1	1	yes	maximal, characteristic
6	#2176	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
8	C₃×C₄×Q₈	—	2	2	no	—
9	C₃×Q₃₂	C₄	1	4	yes	complemented
10	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₃×#107	C₂²	1	2	yes	—
13	C₃×#103	—	2	2	no	—
14	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂	C₂×C₃	1	1	yes	characteristic
16	C₄×Q₁₆	C₂×C₃	1	2	yes	—
17	#367	C₂×C₃	1	1	yes	characteristic
18	#358	C₂×C₃	1	2	yes	—
19	#346	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×Q₈	—	2	2	no	—
21	C₃×Q₁₆	—	2	2	no	—
22	C₃×Q₁₆	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₈	D₄	1	1	yes	characteristic
24	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₃×C₄⋊C₄	—	2	2	no	—
26	C₃×C₄⋊C₄	—	4	2	no	—
27	C₃×C₄²	—	4	2	no	—
28	C₃×C₄²	D₄	1	1	yes	characteristic
29	C₃×C₁₆	—	2	1	no	—
30	C₃×C₁₆	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
32	C₄×Q₈	—	2	2	no	—
33	Q₃₂	C₃×C₄	1	4	yes	complemented
34	C₂×C₁₆	C₂²×C₃	1	1	yes	characteristic
35	C₂×C₁₆	C₂²×C₃	1	1	yes	characteristic
36	#107	C₂²×C₃	1	2	yes	—
37	#103	—	2	2	no	—
38	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
39	C₃×Q₈	—	2	4	no	—
40	C₃×Q₈	—	4	2	no	—
41	C₂×C₃×C₄	—	4	2	no	—
42	C₂×C₃×C₄	—	4	2	no	—
43	C₂×C₃×C₄	D₈	1	2	yes	—
44	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
45	C₃×C₈	—	2	1	no	—
46	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₃×C₈	#41	1	1	yes	characteristic
48	C₂×Q₈	—	2	2	no	—
49	Q₁₆	—	2	2	no	—
50	Q₁₆	C₂×C₃×C₄	1	4	yes	—
51	C₂×C₈	C₃×D₄	1	1	yes	characteristic
52	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	4	2	no	—
55	C₄²	—	4	2	no	—
56	C₄²	C₃×D₄	1	1	yes	characteristic
57	C₁₆	—	2	1	no	—
58	C₁₆	C₂×C₃×C₄	1	2	yes	—
59	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	—	4	4	no	—
62	C₃×C₄	—	8	2	no	—
63	C₃×C₄	C₄×D₄	1	1	yes	characteristic
64	C₃×C₄	#135	1	1	yes	characteristic
65	Q₈	—	2	4	no	—
66	Q₈	—	4	2	no	—
67	C₂×C₄	—	4	2	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	C₃×D₈	1	2	yes	—
70	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	C₂²×C₃×C₄	1	1	yes	characteristic
73	C₈	C₃×#41	1	1	yes	characteristic
74	C₂×C₃	C₄×D₈	1	1	yes	characteristic
75	C₂×C₃	#510	1	2	yes	—
76	C₂²	C₂×C₃×D₈	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	4	4	no	—
79	C₄	—	8	2	no	—
80	C₄	C₃×C₄×D₄	1	1	yes	characteristic
81	C₄	C₃×#135	1	1	yes	characteristic
82	C₃	#2176	1	1	yes	characteristic, direct product factor, Sylow
83	C₂	C₃×C₄×D₈	1	1	yes	characteristic
84	C₂	C₃×#510	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	C₂×D₈
commutator	C₈	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₈	C₂³×C₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. #2176	C₃	direct (N × H)
9. C₃×Q₃₂	C₄	semidirect
33. Q₃₂	C₃×C₄	semidirect
82. C₃	#2176	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×Q₃₂	2
C₃×C₄×Q₁₆	2
C₃×#367	2
C₃×#358	2
C₃×#346	2
#2176	3

Subgroups — C3×#2176 #72891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2176 #72891