Subgroups — C₂²×#364 #33399

Order 256.

391 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#364	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈	C₄	1	1	yes	characteristic
5	C₂³×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄²	C₄	1	1	yes	characteristic
7	C₂×M₃₂	—	2	6	no	—
8	C₂×M₃₂	C₂²	1	12	yes	—
9	C₂²×C₁₆	—	2	2	no	—
10	C₂×C₄×C₈	C₄	1	6	yes	—
11	C₂×C₄×C₈	C₂²	1	6	yes	—
12	#364	C₂²	1	16	yes	direct product factor
13	C₂³×C₄	—	2	1	no	—
14	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈	C₈	1	6	yes	—
16	C₂²×C₈	C₈	1	2	yes	—
17	C₂²×C₈	C₂×C₄	1	3	yes	—
18	C₂²×C₈	C₂×C₄	1	6	yes	—
19	C₂²×C₈	D₄	1	3	yes	—
20	C₂²×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	Q₈	1	1	yes	characteristic
22	C₂²×C₈	Q₈	1	3	yes	—
23	C₂²×C₈	C₂³	1	3	yes	—
24	C₂×C₄²	—	2	3	no	—
25	C₂×C₄²	C₂×C₄	1	6	yes	—
26	M₃₂	—	2	12	no	—
27	M₃₂	C₂³	1	8	yes	—
28	C₂×C₁₆	—	2	12	no	—
29	C₄×C₈	C₂×C₄	1	12	yes	—
30	C₄×C₈	C₂³	1	4	yes	—
31	C₂⁴	M₁₆	1	1	yes	characteristic
32	C₂²×C₄	—	2	8	no	—
33	C₂²×C₄	—	2	6	no	—
34	C₂²×C₄	C₄⋊C₄	1	3	yes	—
35	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄	C₂×C₈	1	3	yes	—
37	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄	M₁₆	1	3	yes	—
39	C₂²×C₄	C₂²×C₄	1	3	yes	—
40	C₂×C₈	C₄⋊C₄	1	3	yes	—
41	C₂×C₈	C₄⋊C₄	1	6	yes	—
42	C₂×C₈	C₄⋊C₄	1	3	yes	—
43	C₂×C₈	C₂×C₈	1	12	yes	—
44	C₂×C₈	C₂×C₈	1	12	yes	—
45	C₂×C₈	C₂²×C₄	1	3	yes	—
46	C₂×C₈	C₂²×C₄	1	4	yes	—
47	C₂×C₈	C₂×D₄	1	3	yes	—
48	C₂×C₈	C₂×D₄	1	3	yes	—
49	C₂×C₈	C₂×Q₈	1	3	yes	—
50	C₂×C₈	C₂×Q₈	1	3	yes	—
51	C₂×C₈	C₂⁴	1	1	yes	characteristic
52	C₄²	—	2	6	no	—
53	C₄²	C₂²×C₄	1	4	yes	—
54	C₁₆	—	2	8	no	—
55	C₂³	—	2	4	no	—
56	C₂³	C₄⋊C₈	1	3	yes	—
57	C₂³	C₄⋊C₈	1	1	yes	characteristic
58	C₂³	C₂×M₁₆	1	3	yes	—
59	C₂×C₄	—	2	24	no	—
60	C₂×C₄	—	2	4	no	—
61	C₂×C₄	C₄⋊C₈	1	6	yes	—
62	C₂×C₄	C₄⋊C₈	1	3	yes	—
63	C₂×C₄	C₄⋊C₈	1	3	yes	—
64	C₂×C₄	C₂×C₄⋊C₄	1	3	yes	—
65	C₂×C₄	C₂×C₄⋊C₄	1	3	yes	—
66	C₂×C₄	C₂²×C₈	1	3	yes	—
67	C₂×C₄	C₂²×C₈	1	3	yes	—
68	C₂×C₄	C₂×M₁₆	1	3	yes	—
69	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
70	C₈	C₂×C₄⋊C₄	1	3	yes	—
71	C₈	C₂×C₄⋊C₄	1	3	yes	—
72	C₈	C₂²×C₈	1	8	yes	—
73	C₈	C₂²×D₄	1	1	yes	characteristic
74	C₈	C₂²×Q₈	1	1	yes	characteristic
75	C₂²	—	2	6	no	—
76	C₂²	—	2	6	no	—
77	C₂²	#364	1	4	yes	direct product factor
78	C₂²	C₂×C₄⋊C₈	1	3	yes	—
79	C₂²	C₂×C₄⋊C₈	1	3	yes	—
80	C₂²	C₂²×M₁₆	1	1	yes	characteristic
81	C₄	—	2	8	no	—
82	C₄	C₂×C₄⋊C₈	1	3	yes	—
83	C₄	C₂×C₄⋊C₈	1	3	yes	—
84	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
85	C₄	C₂³×C₈	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	3	no	—
88	C₂	C₂×#364	1	6	yes	direct product factor
89	C₂	C₂²×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₄
commutator	C₄	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂³	C₄⋊C₈
prime radical	C₂⁴	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₂×#364	C₂	direct (N × H)
12. #364	C₂²	direct (N × H)
77. C₂²	#364	direct (N × H)
88. C₂	C₂×#364	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×M₃₂	2
C₂²×C₄×C₈	2
C₂×#364	2

Subgroups — C22×#364 #33399

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#364 #33399