Subgroups — C₂×C₃×#504 #85270

Order 384.

368 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₃×#504	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×M₃₂	C₂	1	3	yes	maximal, complemented
4	C₂²×C₃×C₁₆	C₂	1	3	yes	maximal, complemented
5	C₂×#504	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×#41	C₄	1	1	yes	characteristic
7	C₃×#131	C₄	1	4	yes	—
8	C₃×#131	C₂²	1	4	yes	—
9	C₂×C₃×M₁₆	C₄	1	3	yes	—
10	C₂²×C₃×C₈	C₂²	1	3	yes	—
11	C₃×M₃₂	C₂²	1	12	yes	complemented
12	C₂×C₃×C₁₆	C₂²	1	3	yes	—
13	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₁₆	C₂²	1	12	yes	complemented
15	C₂×#131	C₂×C₃	1	1	yes	characteristic
16	#504	C₂×C₃	1	8	yes	direct product factor
17	C₂×M₃₂	C₂×C₃	1	3	yes	complemented
18	C₂²×C₁₆	C₂×C₃	1	3	yes	complemented
19	C₃×#41	C₈	1	4	yes	—
20	C₃×#41	C₂×C₄	1	4	yes	—
21	C₂×C₃×Q₈	C₈	1	1	yes	characteristic
22	C₂×C₃×D₄	C₈	1	3	yes	—
23	C₂²×C₃×C₄	C₂×C₄	1	3	yes	—
24	C₃×M₁₆	C₂×C₄	1	12	yes	—
25	C₂×C₃×C₈	C₂×C₄	1	3	yes	—
26	C₂×C₃×C₈	C₂×C₄	1	6	yes	—
27	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₈	C₂³	1	6	yes	—
29	C₃×C₁₆	C₂³	1	6	yes	—
30	C₃×C₁₆	C₂³	1	2	yes	—
31	C₂×#41	C₃×C₄	1	1	yes	characteristic
32	#131	C₃×C₄	1	4	yes	—
33	#131	C₂²×C₃	1	4	yes	—
34	C₂×M₁₆	C₃×C₄	1	3	yes	—
35	C₂²×C₈	C₂²×C₃	1	3	yes	—
36	M₃₂	C₂²×C₃	1	12	yes	complemented
37	C₂×C₁₆	C₂²×C₃	1	3	yes	—
38	C₂×C₁₆	C₂²×C₃	1	1	yes	characteristic
39	C₂×C₁₆	C₂²×C₃	1	12	yes	complemented
40	C₂³×C₃	C₂×C₈	1	3	yes	—
41	C₃×Q₈	C₂×C₈	1	4	yes	—
42	C₃×D₄	C₂×C₈	1	12	yes	—
43	C₂×C₃×C₄	C₂×C₈	1	3	yes	—
44	C₂×C₃×C₄	C₂×C₈	1	6	yes	—
45	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
47	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₈	C₂²×C₄	1	6	yes	—
49	C₃×C₈	C₂⁴	1	1	yes	characteristic
50	#41	C₃×C₈	1	4	yes	—
51	#41	C₂×C₃×C₄	1	4	yes	—
52	C₂×Q₈	C₃×C₈	1	1	yes	characteristic
53	C₂×D₄	C₃×C₈	1	3	yes	—
54	C₂²×C₄	C₂×C₃×C₄	1	3	yes	—
55	M₁₆	C₂×C₃×C₄	1	12	yes	—
56	C₂×C₈	C₂×C₃×C₄	1	3	yes	—
57	C₂×C₈	C₂×C₃×C₄	1	6	yes	—
58	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
59	C₂×C₈	C₂³×C₃	1	6	yes	—
60	C₁₆	C₂³×C₃	1	2	yes	—
61	C₁₆	C₂³×C₃	1	6	yes	—
62	C₂²×C₃	—	2	6	no	—
63	C₂²×C₃	C₂²×C₈	1	6	yes	—
64	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
65	C₃×C₄	C₂²×C₈	1	1	yes	characteristic
66	C₃×C₄	C₂²×C₈	1	6	yes	—
67	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
68	C₂³	C₂×C₃×C₈	1	3	yes	—
69	Q₈	C₂×C₃×C₈	1	4	yes	—
70	D₄	C₂×C₃×C₈	1	12	yes	—
71	C₂×C₄	C₂×C₃×C₈	1	3	yes	—
72	C₂×C₄	C₂×C₃×C₈	1	6	yes	—
73	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
74	C₂×C₄	C₂²×C₃×C₄	1	6	yes	—
75	C₈	C₂²×C₃×C₄	1	1	yes	characteristic
76	C₈	C₂²×C₃×C₄	1	6	yes	—
77	C₈	C₂⁴×C₃	1	1	yes	characteristic
78	C₂×C₃	—	2	6	no	—
79	C₂×C₃	#504	1	2	yes	direct product factor
80	C₂×C₃	C₂³×C₈	1	1	yes	characteristic
81	C₂²	—	2	6	no	—
82	C₂²	C₂²×C₃×C₈	1	6	yes	—
83	C₂²	C₂²×C₃×C₈	1	1	yes	characteristic
84	C₄	C₂²×C₃×C₈	1	1	yes	characteristic
85	C₄	C₂²×C₃×C₈	1	6	yes	—
86	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
87	C₃	C₂×#504	1	1	yes	characteristic, direct product factor, Sylow
88	C₂	—	2	6	no	—
89	C₂	C₃×#504	1	2	yes	direct product factor
90	C₂	C₂³×C₃×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₁₆	C₂²
commutator	C₂	C₂³×C₃×C₈
Frattini, cosocle	C₈	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₈
prime radical	C₂×C₃×#41	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#504	C₂	direct (N × H)
3. C₂×C₃×M₃₂	C₂	semidirect
4. C₂²×C₃×C₁₆	C₂	semidirect
5. C₂×#504	C₃	direct (N × H)
11. C₃×M₃₂	C₂²	semidirect
14. C₂×C₃×C₁₆	C₂²	semidirect
16. #504	C₂×C₃	direct (N × H)
17. C₂×M₃₂	C₂×C₃	semidirect
18. C₂²×C₁₆	C₂×C₃	semidirect
36. M₃₂	C₂²×C₃	semidirect
39. C₂×C₁₆	C₂²×C₃	semidirect
79. C₂×C₃	#504	direct (N × H)
87. C₃	C₂×#504	direct (N × H)
89. C₂	C₃×#504	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×#131	2
C₃×#504	2
C₂×C₃×M₃₂	2
C₂²×C₃×C₁₆	2
C₂×#504	3

Subgroups — C2×C3×#504 #85270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#504 #85270