Subgroups — C₂²×#368 #33412

Order 256.

331 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108	C₂	1	2	yes	maximal
3	C₂×#368	C₂	1	12	yes	maximal, direct product factor
4	C₂²×M₁₆	—	2	2	no	—
5	C₂³×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆	C₄	1	2	yes	—
7	C₂²×C₁₆	C₄	1	6	yes	—
8	C₂²×C₁₆	C₂²	1	6	yes	—
9	C₂×#108	—	2	6	no	—
10	C₂×#108	C₂²	1	12	yes	—
11	#368	C₂²	1	16	yes	direct product factor
12	C₂³×C₄	D₄	1	1	yes	characteristic
13	C₂×M₁₆	—	2	12	no	—
14	C₂×M₁₆	—	4	6	no	—
15	C₂²×C₈	—	4	2	no	—
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	3	yes	—
18	C₂²×C₈	D₄	1	3	yes	—
19	C₂²×C₈	Q₈	1	1	yes	characteristic
20	C₂²×C₈	Q₈	1	3	yes	—
21	C₂²×C₈	C₂³	1	3	yes	—
22	C₂×C₁₆	C₂×C₄	1	12	yes	—
23	C₂×C₁₆	C₂×C₄	1	12	yes	—
24	C₂×C₁₆	C₂³	1	4	yes	—
25	#108	—	2	12	no	—
26	#108	C₂³	1	8	yes	—
27	C₂⁴	Q₁₆	1	1	yes	characteristic
28	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
29	C₂²×C₄	C₄⋊C₄	1	3	yes	—
30	C₂²×C₄	D₈	1	3	yes	—
31	C₂²×C₄	D₈	1	1	yes	characteristic
32	C₂²×C₄	Q₁₆	1	3	yes	—
33	C₂²×C₄	C₂×D₄	1	3	yes	—
34	M₁₆	—	2	8	no	—
35	M₁₆	—	4	12	no	—
36	C₂×C₈	—	4	12	no	—
37	C₂×C₈	C₄⋊C₄	1	6	yes	—
38	C₂×C₈	C₄⋊C₄	1	3	yes	—
39	C₂×C₈	C₄⋊C₄	1	3	yes	—
40	C₂×C₈	C₂²×C₄	1	3	yes	—
41	C₂×C₈	C₂²×C₄	1	3	yes	—
42	C₂×C₈	C₂×D₄	1	3	yes	—
43	C₂×C₈	C₂×Q₈	1	3	yes	—
44	C₂×C₈	C₂×Q₈	1	3	yes	—
45	C₂×C₈	C₂⁴	1	1	yes	characteristic
46	C₁₆	C₂²×C₄	1	8	yes	—
47	C₂³	—	2	4	no	—
48	C₂³	#107	1	3	yes	—
49	C₂³	#107	1	1	yes	characteristic
50	C₂³	C₂×Q₁₆	1	3	yes	—
51	C₂×C₄	#107	1	6	yes	—
52	C₂×C₄	#107	1	3	yes	—
53	C₂×C₄	#107	1	3	yes	—
54	C₂×C₄	C₂×C₄⋊C₄	1	3	yes	—
55	C₂×C₄	C₂×C₄⋊C₄	1	3	yes	—
56	C₂×C₄	C₂×D₈	1	3	yes	—
57	C₂×C₄	C₂×D₈	1	3	yes	—
58	C₂×C₄	C₂×Q₁₆	1	3	yes	—
59	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
60	C₈	—	4	8	no	—
61	C₈	C₂×C₄⋊C₄	1	3	yes	—
62	C₈	C₂×C₄⋊C₄	1	3	yes	—
63	C₈	C₂³×C₄	1	1	yes	characteristic
64	C₈	C₂²×Q₈	1	1	yes	characteristic
65	C₂²	—	2	6	no	—
66	C₂²	—	2	6	no	—
67	C₂²	#368	1	4	yes	direct product factor
68	C₂²	C₂×#107	1	3	yes	—
69	C₂²	C₂×#107	1	3	yes	—
70	C₂²	C₂²×Q₁₆	1	1	yes	characteristic
71	C₄	C₂×#107	1	3	yes	—
72	C₄	C₂×#107	1	3	yes	—
73	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₄	C₂²×D₈	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	2	3	no	—
77	C₂	C₂×#368	1	6	yes	direct product factor
78	C₂	C₂²×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₈
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂³	#107
prime radical	C₂⁴	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ 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×#368	C₂	direct (N × H)
11. #368	C₂²	direct (N × H)
67. C₂²	#368	direct (N × H)
77. C₂	C₂×#368	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×C₁₆	2
C₂²×#108	2
C₂×#368	2

Subgroups — C22×#368 #33412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#368 #33412