Subgroups — C₃×#2657 #84521

Order 384.

844 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#127	C₂	1	3	yes	maximal, complemented
2	C₂×C₃×#124	C₂	1	3	yes	maximal, complemented
3	C₃×#392	C₂	1	8	yes	maximal, complemented
4	C₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	#2657	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
7	C₂²×C₃×D₄	C₂²	1	3	yes	—
8	C₂²×C₃×D₄	C₂²	1	12	yes	—
9	C₃×#127	—	2	12	no	—
10	C₃×#124	—	2	12	no	—
11	C₂×C₃×C₂²⋊C₄	C₂²	1	12	yes	—
12	C₂×C₃×C₄²	C₂²	1	3	yes	—
13	C₃×#95	C₂²	1	4	yes	—
14	C₂×#127	C₂×C₃	1	3	yes	complemented
15	C₂×#124	C₂×C₃	1	3	yes	complemented
16	#392	C₂×C₃	1	8	yes	complemented
17	#376	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂⁴×C₃	C₂³	1	6	yes	—
19	C₂×C₃×Q₈	—	2	6	no	—
20	C₂×C₃×D₄	—	2	6	no	—
21	C₂×C₃×D₄	—	2	24	no	—
22	C₂×C₃×D₄	—	2	12	no	—
23	C₂×C₃×D₄	—	4	24	no	—
24	C₂²×C₃×C₄	C₂³	1	2	yes	—
25	C₂²×C₃×C₄	C₂³	1	6	yes	—
26	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₃×C₂²⋊C₄	—	2	24	no	—
28	C₃×C₄²	D₄	1	12	yes	complemented
29	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
30	C₂²×D₄	C₂²×C₃	1	3	yes	—
31	C₂²×D₄	C₂²×C₃	1	12	yes	—
32	#127	—	2	12	no	—
33	#124	—	2	12	no	—
34	C₂×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
35	C₂×C₄²	C₂²×C₃	1	3	yes	—
36	#95	C₂²×C₃	1	4	yes	—
37	C₂³×C₃	—	2	6	no	—
38	C₂³×C₃	—	2	12	no	—
39	C₂³×C₃	—	2	24	no	—
40	C₂³×C₃	C₂⁴	1	1	yes	characteristic
41	C₃×Q₈	—	4	4	no	—
42	C₃×D₄	—	4	48	no	—
43	C₃×D₄	—	4	12	no	—
44	C₂×C₃×C₄	—	2	12	no	—
45	C₂×C₃×C₄	—	2	6	no	—
46	C₂×C₃×C₄	C₂×D₄	1	12	yes	—
47	C₂×C₃×C₄	C₂×D₄	1	6	yes	—
48	C₂⁴	C₂³×C₃	1	6	yes	—
49	C₂×Q₈	—	2	6	no	—
50	C₂×D₄	—	2	6	no	—
51	C₂×D₄	—	2	24	no	—
52	C₂×D₄	—	2	12	no	—
53	C₂×D₄	—	4	24	no	—
54	C₂²×C₄	C₂³×C₃	1	2	yes	—
55	C₂²×C₄	C₂³×C₃	1	6	yes	—
56	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
57	C₂²⋊C₄	—	2	24	no	—
58	C₄²	C₃×D₄	1	12	yes	complemented
59	C₂²×C₃	—	4	6	no	—
60	C₂²×C₃	—	4	24	no	—
61	C₂²×C₃	—	4	12	no	—
62	C₂²×C₃	C₂²×D₄	1	3	yes	—
63	C₂²×C₃	H(5,𝔽₂)	1	4	yes	—
64	C₃×C₄	—	2	12	no	—
65	C₃×C₄	—	4	2	no	—
66	C₃×C₄	#127	1	4	yes	—
67	C₂³	—	2	6	no	—
68	C₂³	—	2	12	no	—
69	C₂³	—	2	24	no	—
70	C₂³	C₂⁴×C₃	1	1	yes	characteristic
71	Q₈	—	4	4	no	—
72	D₄	—	4	48	no	—
73	D₄	—	4	12	no	—
74	C₂×C₄	—	2	12	no	—
75	C₂×C₄	—	2	6	no	—
76	C₂×C₄	C₂×C₃×D₄	1	12	yes	—
77	C₂×C₄	C₂×C₃×D₄	1	6	yes	—
78	C₂×C₃	—	8	6	no	—
79	C₂×C₃	C₂×#127	1	1	yes	characteristic
80	C₂×C₃	#535	1	6	yes	—
81	C₂²	—	4	6	no	—
82	C₂²	—	4	24	no	—
83	C₂²	—	4	12	no	—
84	C₂²	C₂²×C₃×D₄	1	3	yes	—
85	C₂²	C₃×H(5,𝔽₂)	1	4	yes	—
86	C₄	—	2	12	no	—
87	C₄	—	4	2	no	—
88	C₄	C₃×#127	1	4	yes	—
89	C₃	#2657	1	1	yes	characteristic, direct product factor, Sylow
90	C₂	—	8	6	no	—
91	C₂	C₂×C₃×#127	1	1	yes	characteristic
92	C₂	C₃×#535	1	6	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#127	C₂	semidirect
2. C₂×C₃×#124	C₂	semidirect
3. C₃×#392	C₂	semidirect
4. C₃×#376	C₂	semidirect
5. #2657	C₃	direct (N × H)
14. C₂×#127	C₂×C₃	semidirect
15. C₂×#124	C₂×C₃	semidirect
16. #392	C₂×C₃	semidirect
17. #376	C₂×C₃	semidirect
28. C₃×C₄²	D₄	semidirect
58. C₄²	C₃×D₄	semidirect
89. C₃	#2657	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×#127	2
C₂×C₃×#124	2
C₃×#392	2
C₃×#376	2
#2657	3

Subgroups — C3×#2657 #84521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2657 #84521