Subgroups — #9636

Order 256.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1574	C₂	1	2	yes	maximal
2	#1567	C₂	1	1	yes	maximal, characteristic
3	#1564	C₂	1	1	yes	maximal, characteristic
4	#1451	C₂	1	1	yes	maximal, characteristic
5	#1299	C₂	1	2	yes	maximal
6	#431	C₂²	1	2	yes	—
7	#431	C₂²	1	1	yes	characteristic
8	#423	C₂²	1	2	yes	—
9	C₄×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	#335	—	2	2	no	—
12	#335	—	2	1	no	—
13	#335	—	2	1	no	—
14	#335	C₄	1	2	yes	—
15	#334	—	2	2	no	—
16	#334	C₄	1	2	yes	—
17	#322	—	2	1	no	—
18	#322	—	2	1	no	—
19	C₂×M₁₆	—	2	1	no	—
20	C₂×M₁₆	—	2	2	no	—
21	C₂²×C₈	Q₈	1	1	yes	characteristic
22	C₂²×C₈	Q₈	1	1	yes	characteristic
23	C₂×C₄²	C₂³	1	1	yes	characteristic
24	C₄⋊C₈	—	2	2	no	—
25	C₄⋊C₈	—	2	2	no	—
26	C₄⋊C₈	C₂×C₄	1	4	yes	—
27	C₄⋊C₈	D₄	1	1	yes	characteristic
28	C₄⋊C₈	D₄	1	1	yes	characteristic
29	C₂²⋊C₈	—	2	2	no	—
30	C₂²⋊C₈	—	2	1	no	—
31	#97	Q₈	1	2	yes	—
32	C₄×C₈	—	2	1	no	—
33	C₄×C₈	—	2	1	no	—
34	C₄×C₈	—	2	2	no	—
35	C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈	D₄	1	1	yes	characteristic
38	C₄×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄	#41	1	1	yes	characteristic
41	C₂²×C₄	#41	1	1	yes	characteristic
42	M₁₆	—	4	2	no	—
43	M₁₆	—	8	2	no	—
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	—	2	4	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	4	1	no	—
50	C₂×C₈	—	4	2	no	—
51	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₄²	C₂²×C₄	1	1	yes	characteristic
56	C₄²	C₂×D₄	1	1	yes	characteristic
57	C₄²	C₂×D₄	1	1	yes	characteristic
58	C₄²	C₂×Q₈	1	1	yes	characteristic
59	C₂³	#125	1	1	yes	characteristic
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
68	C₂×C₄	#121	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	—	2	1	no	—
74	C₈	—	4	2	no	—
75	C₈	—	4	4	no	—
76	C₈	—	4	1	no	—
77	C₈	—	4	1	no	—
78	C₈	—	8	2	no	—
79	C₈	#108	1	2	yes	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	#389	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	C₂×#108	1	1	yes	characteristic
88	C₄	#430	1	1	yes	characteristic
89	C₄	#444	1	2	yes	—
90	C₄	#472	1	1	yes	characteristic
91	C₄	#473	1	1	yes	characteristic
92	C₂	—	4	1	no	—
93	C₂	#1946	1	1	yes	characteristic
94	C₂	#1950	1	1	yes	characteristic
95	C₂	#1952	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#389
prime radical	C₂³	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#1574	2
#1567	2
#1564	2
#1451	2
#1299	2