Subgroups — C₂²×#798 #81999

Order 384.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#798	C₂	1	12	yes	maximal, direct product factor
5	C₂²×M₃₂	—	3	1	no	maximal, Sylow
6	C₂²×C₄×D₃	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊C₈	C₄	1	1	yes	characteristic
9	C₂×D₃×C₈	C₄	1	6	yes	—
10	C₂×D₃×C₈	C₂²	1	6	yes	—
11	C₂×C₃×C₁₆	C₂²	1	6	yes	complemented
12	C₂×C₃⋊C₁₆	C₂²	1	6	yes	complemented
13	#798	C₂²	1	16	yes	direct product factor
14	C₂³×C₈	—	3	1	no	—
15	C₂×M₃₂	—	3	12	no	—
16	C₂²×C₁₆	—	3	1	no	—
17	C₂²×C₁₆	D₃	1	1	yes	characteristic, complemented
18	C₂³×D₃	C₈	1	1	yes	characteristic
19	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₃	C₈	1	1	yes	characteristic
21	C₂×C₄×D₃	C₈	1	6	yes	—
22	C₂×C₄×D₃	C₂×C₄	1	6	yes	—
23	C₂×C₃×C₈	C₂×C₄	1	3	yes	—
24	C₂×C₃×C₈	C₂³	1	3	yes	—
25	C₂×C₃⋊C₈	C₂×C₄	1	6	yes	—
26	D₃×C₈	C₂×C₄	1	12	yes	—
27	D₃×C₈	C₂³	1	4	yes	—
28	C₃×C₁₆	C₂³	1	4	yes	complemented
29	C₃⋊C₁₆	C₂³	1	4	yes	complemented
30	C₂³×C₄	—	3	1	no	—
31	C₂²×C₈	—	3	1	no	—
32	C₂²×C₈	—	3	6	no	—
33	C₂²×C₈	—	3	6	no	—
34	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
35	M₃₂	—	3	16	no	—
36	C₂×C₁₆	—	3	6	no	—
37	C₂×C₁₆	C₂×D₃	1	6	yes	complemented
38	C₂³×C₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×D₃	—	2	4	no	—
40	C₂²×D₃	C₂×C₈	1	6	yes	—
41	C₂×C₃×C₄	C₂×C₈	1	3	yes	—
42	C₂×C₃×C₄	C₂²×C₄	1	3	yes	—
43	C₂×Dic₃	C₂×C₈	1	6	yes	—
44	C₄×D₃	C₂×C₈	1	12	yes	—
45	C₄×D₃	C₂²×C₄	1	4	yes	—
46	C₃×C₈	C₂²×C₄	1	3	yes	—
47	C₃×C₈	C₂⁴	1	1	yes	characteristic
48	C₃⋊C₈	C₂²×C₄	1	4	yes	—
49	C₂⁴	—	3	1	no	—
50	C₂²×C₄	—	3	1	no	—
51	C₂²×C₄	—	3	6	no	—
52	C₂²×C₄	—	3	6	no	—
53	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
54	C₂×C₈	—	3	6	no	—
55	C₂×C₈	—	3	12	no	—
56	C₂×C₈	—	3	4	no	—
57	C₂×C₈	C₄×D₃	1	3	yes	—
58	C₂×C₈	C₂²×D₃	1	3	yes	—
59	C₁₆	—	3	4	no	—
60	C₁₆	C₂²×D₃	1	4	yes	complemented
61	C₂²×C₃	M₃₂	1	4	yes	complemented
62	C₂²×C₃	C₂²×C₈	1	3	yes	—
63	C₂×D₃	—	2	12	no	—
64	C₂×D₃	C₂²×C₈	1	4	yes	—
65	C₃×C₄	C₂²×C₈	1	3	yes	—
66	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
67	Dic₃	C₂²×C₈	1	4	yes	—
68	C₂³	—	3	6	no	—
69	C₂³	—	6	4	no	—
70	C₂³	D₃×C₈	1	1	yes	characteristic
71	C₂×C₄	—	3	6	no	—
72	C₂×C₄	—	3	12	no	—
73	C₂×C₄	—	3	4	no	—
74	C₂×C₄	D₃×C₈	1	3	yes	—
75	C₂×C₄	C₂×C₄×D₃	1	3	yes	—
76	C₈	—	3	4	no	—
77	C₈	C₂×C₄×D₃	1	3	yes	—
78	C₈	C₂³×D₃	1	1	yes	characteristic
79	C₂×C₃	C₂×M₃₂	1	6	yes	complemented
80	C₂×C₃	C₂³×C₈	1	1	yes	characteristic
81	D₃	—	2	4	no	—
82	C₂²	—	3	4	no	—
83	C₂²	—	6	12	no	—
84	C₂²	#798	1	4	yes	direct product factor
85	C₂²	C₂×D₃×C₈	1	3	yes	—
86	C₄	—	3	4	no	—
87	C₄	C₂×D₃×C₈	1	3	yes	—
88	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
89	C₃	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
90	C₂	—	6	4	no	—
91	C₂	C₂×#798	1	6	yes	direct product factor
92	C₂	C₂²×D₃×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₃
commutator	C₂×C₃	C₂³×C₈
Frattini	C₈	C₂³×D₃
Fitting	C₂²×C₃×C₁₆	C₂
socle	C₂³×C₃	C₂×C₈
prime radical	C₂³×D₃	C₈
cosocle	C₃×C₈	C₂⁴
hypercenter	C₂²×C₁₆	D₃
hypocenter	C₃	C₂²×M₃₂
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₁₆	C₂	semidirect
3. C₂²×C₃⋊C₁₆	C₂	semidirect
4. C₂×#798	C₂	direct (N × H)
11. C₂×C₃×C₁₆	C₂²	semidirect
12. C₂×C₃⋊C₁₆	C₂²	semidirect
13. #798	C₂²	direct (N × H)
17. C₂²×C₁₆	D₃	semidirect
28. C₃×C₁₆	C₂³	semidirect
29. C₃⋊C₁₆	C₂³	semidirect
37. C₂×C₁₆	C₂×D₃	semidirect
60. C₁₆	C₂²×D₃	semidirect
61. C₂²×C₃	M₃₂	semidirect
79. C₂×C₃	C₂×M₃₂	semidirect
84. C₂²	#798	direct (N × H)
89. C₃	C₂²×M₃₂	semidirect
91. C₂	C₂×#798	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₃×C₈	2
C₂²×C₃×C₁₆	2
C₂²×C₃⋊C₁₆	2
C₂×#798	2
C₂²×M₃₂	3

Subgroups — C22×#798 #81999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#798 #81999