Subgroups — C₃×#2397 #84261

Order 384.

840 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₄	C₂	1	6	yes	maximal, complemented
2	C₃×#399	C₂	1	4	yes	maximal, complemented
3	C₃×#392	C₂	1	4	yes	maximal, complemented
4	C₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	#2397	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂²×C₃×D₄	C₂²	1	6	yes	—
7	C₂³×C₃×C₄	C₂²	1	4	yes	—
8	C₃×C₄×D₄	—	2	24	no	—
9	C₂×C₃×C₄⋊C₄	C₂²	1	6	yes	—
10	C₂×C₃×C₂²⋊C₄	C₂²	1	12	yes	—
11	C₂×C₃×C₄²	C₂²	1	3	yes	—
12	C₃×#95	C₂²	1	4	yes	—
13	C₂×C₄×D₄	C₂×C₃	1	6	yes	complemented
14	#399	C₂×C₃	1	4	yes	complemented
15	#392	C₂×C₃	1	4	yes	complemented
16	#376	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂⁴×C₃	C₂³	1	4	yes	—
18	C₂×C₃×D₄	—	2	24	no	—
19	C₂×C₃×D₄	—	2	12	no	—
20	C₂²×C₃×C₄	—	2	24	no	—
21	C₂²×C₃×C₄	C₂³	1	4	yes	—
22	C₂²×C₃×C₄	C₂³	1	6	yes	—
23	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₃×C₄⋊C₄	—	2	12	no	—
25	C₃×C₂²⋊C₄	—	2	24	no	—
26	C₃×C₄²	D₄	1	12	yes	complemented
27	C₂²×D₄	C₂²×C₃	1	6	yes	—
28	C₂³×C₄	C₂²×C₃	1	4	yes	—
29	C₄×D₄	—	2	24	no	—
30	C₂×C₄⋊C₄	C₂²×C₃	1	6	yes	—
31	C₂×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
32	C₂×C₄²	C₂²×C₃	1	3	yes	—
33	#95	C₂²×C₃	1	4	yes	—
34	C₂³×C₃	—	2	12	no	—
35	C₂³×C₃	—	2	12	no	—
36	C₂³×C₃	#41	1	8	yes	—
37	C₂³×C₃	C₂⁴	1	1	yes	characteristic
38	C₃×D₄	—	2	48	no	—
39	C₂×C₃×C₄	—	2	12	no	—
40	C₂×C₃×C₄	—	2	12	no	—
41	C₂×C₃×C₄	—	4	16	no	—
42	C₂×C₃×C₄	C₂×D₄	1	12	yes	—
43	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
44	C₂⁴	C₂³×C₃	1	4	yes	—
45	C₂×D₄	—	2	24	no	—
46	C₂×D₄	—	2	12	no	—
47	C₂²×C₄	—	2	24	no	—
48	C₂²×C₄	C₂³×C₃	1	4	yes	—
49	C₂²×C₄	C₂³×C₃	1	6	yes	—
50	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
51	C₄⋊C₄	—	2	12	no	—
52	C₂²⋊C₄	—	2	24	no	—
53	C₄²	C₃×D₄	1	12	yes	complemented
54	C₂²×C₃	—	2	24	no	—
55	C₂²×C₃	—	4	12	no	—
56	C₂²×C₃	—	4	4	no	—
57	C₂²×C₃	C₂²×D₄	1	3	yes	—
58	C₂²×C₃	C₂×#41	1	4	yes	—
59	C₃×C₄	—	2	12	no	—
60	C₃×C₄	—	4	4	no	—
61	C₃×C₄	#127	1	4	yes	—
62	C₂³	—	2	12	no	—
63	C₂³	—	2	12	no	—
64	C₂³	C₃×#41	1	8	yes	—
65	C₂³	C₂⁴×C₃	1	1	yes	characteristic
66	D₄	—	2	48	no	—
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	—	2	12	no	—
69	C₂×C₄	—	4	16	no	—
70	C₂×C₄	C₂×C₃×D₄	1	12	yes	—
71	C₂×C₄	C₂×C₃×D₄	1	6	yes	—
72	C₂×C₃	—	4	8	no	—
73	C₂×C₃	#525	1	6	yes	—
74	C₂×C₃	C₂×#127	1	1	yes	characteristic
75	C₂²	—	2	24	no	—
76	C₂²	—	4	12	no	—
77	C₂²	—	4	4	no	—
78	C₂²	C₂²×C₃×D₄	1	3	yes	—
79	C₂²	C₂×C₃×#41	1	4	yes	—
80	C₄	—	2	12	no	—
81	C₄	—	4	4	no	—
82	C₄	C₃×#127	1	4	yes	—
83	C₃	#2397	1	1	yes	characteristic, direct product factor, Sylow
84	C₂	—	4	8	no	—
85	C₂	C₃×#525	1	6	yes	—
86	C₂	C₂×C₃×#127	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃
socle	C₂³×C₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×D₄	C₂	semidirect
2. C₃×#399	C₂	semidirect
3. C₃×#392	C₂	semidirect
4. C₃×#376	C₂	semidirect
5. #2397	C₃	direct (N × H)
13. C₂×C₄×D₄	C₂×C₃	semidirect
14. #399	C₂×C₃	semidirect
15. #392	C₂×C₃	semidirect
16. #376	C₂×C₃	semidirect
26. C₃×C₄²	D₄	semidirect
53. C₄²	C₃×D₄	semidirect
83. C₃	#2397	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×D₄	2
C₃×#399	2
C₃×#392	2
C₃×#376	2
#2397	3

Subgroups — C3×#2397 #84261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2397 #84261