Subgroups — C₂×C₃×#498 #85021

Order 384.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₃×#498	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×#106	C₂	1	4	yes	maximal
4	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#498	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂²×C₃×Q₈	—	2	2	no	—
7	C₂²×C₃×C₈	C₂²	1	2	yes	—
8	C₃×C₄⋊Q₈	—	2	4	no	—
9	C₃×C₄⋊Q₈	C₂²	1	8	yes	—
10	C₂×C₃×C₄⋊C₄	—	2	2	no	—
11	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₃×#106	C₂²	1	16	yes	—
14	C₃×C₄×C₈	C₂²	1	4	yes	—
15	C₂×C₄⋊Q₈	C₂×C₃	1	2	yes	—
16	#498	C₂×C₃	1	8	yes	direct product factor
17	C₂×#106	C₂×C₃	1	4	yes	—
18	C₂×C₄×C₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×Q₈	—	2	8	no	—
20	C₂×C₃×Q₈	—	4	8	no	—
21	C₂²×C₃×C₄	—	2	4	no	—
22	C₂²×C₃×C₄	D₄	1	2	yes	—
23	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₂×C₃×C₈	Q₈	1	8	yes	complemented
25	C₂×C₃×C₈	C₂³	1	4	yes	—
26	C₃×C₄⋊C₄	—	2	8	no	—
27	C₃×C₄⋊C₄	—	2	4	no	—
28	C₃×C₄⋊C₄	C₂³	1	8	yes	—
29	C₃×C₄²	D₄	1	2	yes	—
30	C₃×C₄²	C₂³	1	2	yes	—
31	C₂²×Q₈	—	2	2	no	—
32	C₂²×C₈	C₂²×C₃	1	2	yes	—
33	C₄⋊Q₈	—	2	4	no	—
34	C₄⋊Q₈	C₂²×C₃	1	8	yes	—
35	C₂×C₄⋊C₄	—	2	2	no	—
36	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
37	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
38	#106	C₂²×C₃	1	16	yes	—
39	C₄×C₈	C₂²×C₃	1	4	yes	—
40	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
41	C₃×Q₈	—	4	16	no	—
42	C₂×C₃×C₄	—	2	8	no	—
43	C₂×C₃×C₄	—	4	8	no	—
44	C₂×C₃×C₄	SD₁₆	1	8	yes	—
45	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
47	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
48	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
49	C₃×C₈	C₂×Q₈	1	8	yes	complemented
50	C₂×Q₈	—	2	8	no	—
51	C₂×Q₈	—	4	8	no	—
52	C₂²×C₄	—	2	4	no	—
53	C₂²×C₄	C₃×D₄	1	2	yes	—
54	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
55	C₂×C₈	C₃×Q₈	1	8	yes	complemented
56	C₂×C₈	C₂³×C₃	1	4	yes	—
57	C₄⋊C₄	—	2	8	no	—
58	C₄⋊C₄	—	2	4	no	—
59	C₄⋊C₄	C₂³×C₃	1	8	yes	—
60	C₄²	C₃×D₄	1	2	yes	—
61	C₄²	C₂³×C₃	1	2	yes	—
62	C₂²×C₃	C₄⋊Q₈	1	2	yes	—
63	C₂²×C₃	C₂×SD₁₆	1	4	yes	—
64	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
65	C₃×C₄	—	4	8	no	—
66	C₃×C₄	C₄⋊Q₈	1	2	yes	—
67	C₃×C₄	C₂×SD₁₆	1	8	yes	—
68	C₃×C₄	C₂²×Q₈	1	2	yes	—
69	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
70	Q₈	—	4	16	no	—
71	C₂×C₄	—	2	8	no	—
72	C₂×C₄	—	4	8	no	—
73	C₂×C₄	C₃×SD₁₆	1	8	yes	—
74	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
75	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
76	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
77	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
78	C₈	C₂×C₃×Q₈	1	8	yes	complemented
79	C₂×C₃	#498	1	4	yes	direct product factor
80	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
81	C₂×C₃	C₂²×SD₁₆	1	2	yes	—
82	C₂²	C₃×C₄⋊Q₈	1	2	yes	—
83	C₂²	C₂×C₃×SD₁₆	1	4	yes	—
84	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
85	C₄	—	4	8	no	—
86	C₄	C₃×C₄⋊Q₈	1	2	yes	—
87	C₄	C₂×C₃×SD₁₆	1	8	yes	—
88	C₄	C₂²×C₃×Q₈	1	2	yes	—
89	C₃	C₂×#498	1	1	yes	characteristic, direct product factor, Sylow
90	C₂	C₃×#498	1	4	yes	direct product factor
91	C₂	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
92	C₂	C₂²×C₃×SD₁₆	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#498	C₂	direct (N × H)
5. C₂×#498	C₃	direct (N × H)
16. #498	C₂×C₃	direct (N × H)
24. C₂×C₃×C₈	Q₈	semidirect
49. C₃×C₈	C₂×Q₈	semidirect
55. C₂×C₈	C₃×Q₈	semidirect
78. C₈	C₂×C₃×Q₈	semidirect
79. C₂×C₃	#498	direct (N × H)
89. C₃	C₂×#498	direct (N × H)
90. C₂	C₃×#498	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⋊Q₈	2
C₃×#498	2
C₂×C₃×#106	2
C₂×C₃×C₄×C₈	2
C₂×#498	3

Subgroups — C2×C3×#498 #85021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#498 #85021