Subgroups — C₂×C₃×#367 #72872

Order 384.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#107	C₂	1	2	yes	maximal
3	C₃×#367	C₂	1	4	yes	maximal, direct product factor
4	C₂×#367	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₄⋊C₄	—	2	2	no	—
7	C₂×C₃×C₁₆	C₄	1	4	yes	complemented
8	C₂×C₃×C₁₆	C₂²	1	2	yes	—
9	C₃×#107	—	2	2	no	—
10	C₃×#107	C₂²	1	4	yes	—
11	C₂²×C₁₆	C₂×C₃	1	1	yes	characteristic
12	C₂×#107	C₂×C₃	1	2	yes	—
13	#367	C₂×C₃	1	4	yes	direct product factor
14	C₂²×C₃×C₄	—	4	2	no	—
15	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
16	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₃×C₈	D₄	1	1	yes	characteristic
18	C₂×C₃×C₈	Q₈	1	2	yes	—
19	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
20	C₃×C₄⋊C₄	—	2	4	no	—
21	C₃×C₄⋊C₄	—	4	2	no	—
22	C₃×C₁₆	C₂×C₄	1	4	yes	complemented
23	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
24	C₂×C₄⋊C₄	—	2	2	no	—
25	C₂×C₁₆	C₃×C₄	1	4	yes	complemented
26	C₂×C₁₆	C₂²×C₃	1	2	yes	—
27	#107	—	2	2	no	—
28	#107	C₂²×C₃	1	4	yes	—
29	C₂³×C₃	D₈	1	1	yes	characteristic
30	C₂×C₃×C₄	—	4	4	no	—
31	C₂×C₃×C₄	—	8	4	no	—
32	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
33	C₂×C₃×C₄	D₈	1	1	yes	characteristic
34	C₂×C₃×C₄	Q₁₆	1	2	yes	—
35	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
36	C₃×C₈	C₄⋊C₄	1	2	yes	—
37	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄	—	4	2	no	—
40	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
41	C₂×C₈	C₂×C₃×C₄	1	2	yes	—
42	C₂×C₈	C₃×D₄	1	1	yes	characteristic
43	C₂×C₈	C₃×Q₈	1	2	yes	—
44	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
45	C₄⋊C₄	—	2	4	no	—
46	C₄⋊C₄	—	4	2	no	—
47	C₁₆	C₂×C₃×C₄	1	4	yes	complemented
48	C₂²×C₃	#107	1	2	yes	—
49	C₂²×C₃	SD₃₂	1	4	yes	—
50	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
51	C₃×C₄	—	8	4	no	—
52	C₃×C₄	#107	1	2	yes	—
53	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
55	C₂³	C₃×D₈	1	1	yes	characteristic
56	C₂×C₄	—	4	4	no	—
57	C₂×C₄	—	8	4	no	—
58	C₂×C₄	C₃×C₄⋊C₄	1	2	yes	—
59	C₂×C₄	C₃×D₈	1	1	yes	characteristic
60	C₂×C₄	C₃×Q₁₆	1	2	yes	—
61	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
62	C₈	C₃×C₄⋊C₄	1	2	yes	—
63	C₈	C₂²×C₃×C₄	1	1	yes	characteristic
64	C₈	C₂×C₃×Q₈	1	1	yes	characteristic
65	C₂×C₃	#367	1	4	yes	direct product factor
66	C₂×C₃	C₂×#107	1	1	yes	characteristic
67	C₂×C₃	C₂×SD₃₂	1	2	yes	—
68	C₂²	C₃×#107	1	2	yes	—
69	C₂²	C₃×SD₃₂	1	4	yes	—
70	C₂²	C₂×C₃×D₈	1	1	yes	characteristic
71	C₄	—	8	4	no	—
72	C₄	C₃×#107	1	2	yes	—
73	C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
74	C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
75	C₃	C₂×#367	1	1	yes	characteristic, direct product factor, Sylow
76	C₂	C₃×#367	1	4	yes	direct product factor
77	C₂	C₂×C₃×#107	1	1	yes	characteristic
78	C₂	C₂×C₃×SD₃₂	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃×#367	C₂	direct (N × H)
4. C₂×#367	C₃	direct (N × H)
7. C₂×C₃×C₁₆	C₄	semidirect
13. #367	C₂×C₃	direct (N × H)
22. C₃×C₁₆	C₂×C₄	semidirect
25. C₂×C₁₆	C₃×C₄	semidirect
47. C₁₆	C₂×C₃×C₄	semidirect
65. C₂×C₃	#367	direct (N × H)
75. C₃	C₂×#367	direct (N × H)
76. C₂	C₃×#367	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×C₁₆	2
C₂×C₃×#107	2
C₃×#367	2
C₂×#367	3

Subgroups — C2×C3×#367 #72872

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#367 #72872