Subgroups — #33324

Order 256.

413 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	C₂×#405	C₂	1	1	yes	maximal, characteristic
3	#2108	C₂	1	8	yes	maximal, complemented
4	C₄×M₃₂	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆	C₄	1	1	yes	characteristic
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂×#117	C₄	1	1	yes	characteristic
8	C₂×M₃₂	C₄	1	16	yes	complemented
9	C₂×M₃₂	C₂²	1	4	yes	—
10	C₂×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₆	C₂²	1	4	yes	—
12	#405	C₄	1	4	yes	—
13	#405	C₂²	1	4	yes	—
14	C₂×#97	C₄	1	2	yes	—
15	C₂×C₄×C₈	C₂²	1	2	yes	—
16	#346	C₂²	1	8	yes	complemented
17	C₄×C₁₆	C₂²	1	8	yes	complemented
18	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆	C₈	1	8	yes	—
20	C₂×M₁₆	C₂×C₄	1	4	yes	—
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₈	C₂×C₄	1	8	yes	—
24	C₂²×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₈	C₂³	1	2	yes	—
27	#117	C₈	1	4	yes	—
28	#117	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄	C₈	1	2	yes	—
30	C₂×C₂²⋊C₄	C₈	1	2	yes	—
31	C₂×C₄²	C₂×C₄	1	2	yes	—
32	M₃₂	C₂×C₄	1	16	yes	—
33	M₃₂	C₂×C₄	1	16	yes	complemented
34	C₂×C₁₆	C₂×C₄	1	16	yes	complemented
35	C₂×C₁₆	C₂³	1	8	yes	—
36	#97	C₂×C₄	1	8	yes	—
37	C₄×C₈	C₂×C₄	1	4	yes	—
38	C₄×C₈	C₂³	1	4	yes	—
39	C₂⁴	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄	C₄²	1	4	yes	—
41	C₂²×C₄	C₂×C₈	1	4	yes	—
42	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄	C₂×C₈	1	2	yes	—
44	C₂²×C₄	C₂×C₈	1	4	yes	—
45	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄	C₂²×C₄	1	2	yes	—
47	M₁₆	C₂×C₈	1	16	yes	—
48	C₂×C₈	C₄²	1	4	yes	—
49	C₂×C₈	C₄²	1	8	yes	—
50	C₂×C₈	C₂×C₈	1	8	yes	—
51	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈	C₂²×C₄	1	2	yes	—
53	C₂×C₈	C₂²×C₄	1	4	yes	—
54	C₂×C₈	C₂²×C₄	1	4	yes	—
55	C₂×C₈	C₂²×C₄	1	4	yes	—
56	C₂×C₈	C₂²×C₄	1	4	yes	—
57	C₂×C₈	C₂⁴	1	1	yes	characteristic
58	C₄⋊C₄	C₂×C₈	1	8	yes	—
59	C₂²⋊C₄	C₂×C₈	1	8	yes	—
60	C₄²	C₂×C₈	1	4	yes	—
61	C₄²	C₂²×C₄	1	4	yes	—
62	C₁₆	C₂²×C₄	1	16	yes	—
63	C₂³	—	4	2	no	—
64	C₂³	C₄×C₈	1	4	yes	—
65	C₂³	C₂²×C₈	1	2	yes	—
66	C₂³	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄	—	4	4	no	—
68	C₂×C₄	C₄×C₈	1	4	yes	—
69	C₂×C₄	C₄×C₈	1	8	yes	—
70	C₂×C₄	C₂×C₄²	1	4	yes	—
71	C₂×C₄	C₂×C₄²	1	2	yes	—
72	C₂×C₄	C₂²×C₈	1	8	yes	—
73	C₂×C₄	C₂²×C₈	1	2	yes	—
74	C₂×C₄	C₂²×C₈	1	4	yes	—
75	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄	C₂²×C₈	1	2	yes	—
77	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
78	C₈	C₂×C₄²	1	2	yes	—
79	C₈	C₂×C₄²	1	4	yes	—
80	C₈	C₂²×C₈	1	8	yes	—
81	C₈	C₂³×C₄	1	2	yes	—
82	C₂²	—	2	2	no	—
83	C₂²	—	2	4	no	—
84	C₂²	—	4	4	no	—
85	C₂²	C₂×C₄×C₈	1	2	yes	—
86	C₂²	C₂×C₄×C₈	1	4	yes	—
87	C₂²	C₂³×C₈	1	1	yes	characteristic
88	C₄	—	2	8	no	—
89	C₄	C₂×C₄×C₈	1	2	yes	—
90	C₄	C₂×C₄×C₈	1	4	yes	—
91	C₄	C₂²×C₄²	1	1	yes	characteristic
92	C₄	C₂³×C₈	1	1	yes	characteristic
93	C₂	—	2	2	no	—
94	C₂	—	2	4	no	—
95	C₂	C₂²×C₄×C₈	1	1	yes	characteristic
96	C₂	#3407	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator	C₂	C₂²×C₄×C₈
socle	C₂²	C₂³×C₈
prime radical	C₂⁴	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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. #2108	C₂	semidirect
4. C₄×M₃₂	C₂	semidirect
8. C₂×M₃₂	C₄	semidirect
16. #346	C₂²	semidirect
17. C₄×C₁₆	C₂²	semidirect
33. M₃₂	C₂×C₄	semidirect
34. C₂×C₁₆	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂	2
C₂×#405	2
#2108	2
C₄×M₃₂	2