Subgroups — C₂×C₃×#519 #87310

Order 384.

1488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#519	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃×C₄×Q₈	C₂	1	8	yes	maximal, complemented
4	C₂×C₃×#117	C₂	1	6	yes	maximal
5	C₂×#519	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂²×C₃×Q₈	C₄	1	16	yes	complemented
7	C₂²×C₃×Q₈	C₂²	1	4	yes	—
8	C₂²×C₃×Q₈	C₂²	1	8	yes	—
9	C₂³×C₃×C₄	C₂²	1	3	yes	—
10	C₃×C₄×Q₈	C₂²	1	64	yes	complemented
11	C₃×#117	C₂²	1	48	yes	—
12	C₂×C₃×C₄⋊C₄	C₂²	1	12	yes	—
13	C₂×C₃×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₄²	C₂²	1	12	yes	—
15	C₂³×Q₈	C₂×C₃	1	1	yes	characteristic
16	#519	C₂×C₃	1	16	yes	direct product factor
17	C₂×C₄×Q₈	C₂×C₃	1	8	yes	complemented
18	C₂×#117	C₂×C₃	1	6	yes	—
19	C₂⁴×C₃	C₂³	1	1	yes	characteristic
20	C₂×C₃×Q₈	C₂×C₄	1	32	yes	—
21	C₂×C₃×Q₈	C₂×C₄	1	64	yes	complemented
22	C₂×C₃×Q₈	C₂³	1	16	yes	—
23	C₂²×C₃×C₄	C₂×C₄	1	24	yes	—
24	C₂²×C₃×C₄	C₂³	1	8	yes	—
25	C₂²×C₃×C₄	C₂³	1	12	yes	—
26	C₂²×C₃×C₄	C₂³	1	6	yes	—
27	C₃×C₄⋊C₄	C₂³	1	48	yes	—
28	C₃×C₂²⋊C₄	C₂³	1	16	yes	—
29	C₃×C₄²	C₂³	1	48	yes	—
30	C₂²×Q₈	C₃×C₄	1	16	yes	complemented
31	C₂²×Q₈	C₂²×C₃	1	4	yes	—
32	C₂²×Q₈	C₂²×C₃	1	8	yes	—
33	C₂³×C₄	C₂²×C₃	1	3	yes	—
34	C₄×Q₈	C₂²×C₃	1	64	yes	complemented
35	#117	C₂²×C₃	1	48	yes	—
36	C₂×C₄⋊C₄	C₂²×C₃	1	12	yes	—
37	C₂×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
38	C₂×C₄²	C₂²×C₃	1	12	yes	—
39	C₂³×C₃	—	2	4	no	—
40	C₂³×C₃	C₂²×C₄	1	4	yes	—
41	C₂³×C₃	C₂⁴	1	2	yes	—
42	C₂³×C₃	C₂⁴	1	1	yes	characteristic
43	C₃×Q₈	C₂²×C₄	1	64	yes	—
44	C₂×C₃×C₄	—	2	16	no	—
45	C₂×C₃×C₄	C₂²×C₄	1	24	yes	—
46	C₂×C₃×C₄	C₂²×C₄	1	48	yes	—
47	C₂×C₃×C₄	C₂⁴	1	16	yes	—
48	C₂×C₃×C₄	C₂⁴	1	12	yes	—
49	C₂⁴	C₂³×C₃	1	1	yes	characteristic
50	C₂×Q₈	C₂×C₃×C₄	1	32	yes	—
51	C₂×Q₈	C₂×C₃×C₄	1	64	yes	complemented
52	C₂×Q₈	C₂³×C₃	1	16	yes	—
53	C₂²×C₄	C₂×C₃×C₄	1	24	yes	—
54	C₂²×C₄	C₂³×C₃	1	8	yes	—
55	C₂²×C₄	C₂³×C₃	1	6	yes	—
56	C₂²×C₄	C₂³×C₃	1	12	yes	—
57	C₄⋊C₄	C₂³×C₃	1	48	yes	—
58	C₂²⋊C₄	C₂³×C₃	1	16	yes	—
59	C₄²	C₂³×C₃	1	48	yes	—
60	C₂²×C₃	—	2	8	no	—
61	C₂²×C₃	—	2	4	no	—
62	C₂²×C₃	C₂³×C₄	1	4	yes	—
63	C₂²×C₃	C₂³×C₄	1	2	yes	—
64	C₂²×C₃	2₋¹⁺⁴	1	4	yes	—
65	C₂²×C₃	C₂⁵	1	1	yes	characteristic
66	C₃×C₄	—	2	16	no	—
67	C₃×C₄	C₂³×C₄	1	24	yes	—
68	C₂³	—	2	4	no	—
69	C₂³	C₂²×C₃×C₄	1	4	yes	—
70	C₂³	C₂⁴×C₃	1	2	yes	—
71	C₂³	C₂⁴×C₃	1	1	yes	characteristic
72	Q₈	C₂²×C₃×C₄	1	64	yes	—
73	C₂×C₄	—	2	16	no	—
74	C₂×C₄	C₂²×C₃×C₄	1	24	yes	—
75	C₂×C₄	C₂²×C₃×C₄	1	48	yes	—
76	C₂×C₄	C₂⁴×C₃	1	16	yes	—
77	C₂×C₄	C₂⁴×C₃	1	12	yes	—
78	C₂×C₃	—	2	4	no	—
79	C₂×C₃	#519	1	4	yes	direct product factor
80	C₂×C₃	C₂⁴×C₄	1	1	yes	characteristic
81	C₂×C₃	C₂×2₋¹⁺⁴	1	2	yes	—
82	C₂²	—	2	4	no	—
83	C₂²	—	2	8	no	—
84	C₂²	C₂³×C₃×C₄	1	4	yes	—
85	C₂²	C₂³×C₃×C₄	1	2	yes	—
86	C₂²	C₃×2₋¹⁺⁴	1	4	yes	—
87	C₂²	C₂⁵×C₃	1	1	yes	characteristic
88	C₄	—	2	16	no	—
89	C₄	C₂³×C₃×C₄	1	24	yes	—
90	C₃	C₂×#519	1	1	yes	characteristic, direct product factor, Sylow
91	C₂	—	2	4	no	—
92	C₂	C₃×#519	1	4	yes	direct product factor
93	C₂	C₂⁴×C₃×C₄	1	1	yes	characteristic
94	C₂	C₂×C₃×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator	C₂	C₂⁴×C₃×C₄
Frattini, cosocle	C₂²	C₂⁵×C₃
prime radical	C₂⁴×C₃	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#519	C₂	direct (N × H)
3. C₂×C₃×C₄×Q₈	C₂	semidirect
5. C₂×#519	C₃	direct (N × H)
6. C₂²×C₃×Q₈	C₄	semidirect
10. C₃×C₄×Q₈	C₂²	semidirect
16. #519	C₂×C₃	direct (N × H)
17. C₂×C₄×Q₈	C₂×C₃	semidirect
21. C₂×C₃×Q₈	C₂×C₄	semidirect
30. C₂²×Q₈	C₃×C₄	semidirect
34. C₄×Q₈	C₂²×C₃	semidirect
51. C₂×Q₈	C₂×C₃×C₄	semidirect
79. C₂×C₃	#519	direct (N × H)
90. C₃	C₂×#519	direct (N × H)
92. C₂	C₃×#519	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃×Q₈	2
C₃×#519	2
C₂×C₃×C₄×Q₈	2
C₂×C₃×#117	2
C₂×#519	3

Subgroups — C2×C3×#519 #87310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#519 #87310