Subgroups — #135754

Order 512.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12592	C₂	1	1	yes	maximal, characteristic
2	#12008	C₂	1	2	yes	maximal
3	#11999	C₂	1	2	yes	maximal
4	#8780	C₂	1	2	yes	maximal
5	C₂×#346	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	#1726	C₂²	1	1	yes	characteristic
8	#1564	C₂²	1	4	yes	—
9	#1374	C₄	1	4	yes	—
10	#1372	—	2	2	no	—
11	#1371	—	2	2	no	—
12	#1368	—	2	4	no	—
13	#1281	—	2	2	no	—
14	#1274	—	2	2	no	—
15	C₂²×C₁₆	C₂×C₄	1	2	yes	—
16	C₂²×C₁₆	C₂×C₄	1	2	yes	—
17	#431	—	2	4	no	—
18	C₂×#97	D₄	1	1	yes	characteristic
19	C₂×#97	Q₈	1	2	yes	—
20	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
22	#376	D₄	1	1	yes	characteristic
23	#346	D₄	1	2	yes	—
24	#346	Q₈	1	2	yes	—
25	C₄×C₁₆	D₄	1	2	yes	—
26	C₄×C₁₆	Q₈	1	2	yes	—
27	#335	—	2	4	no	—
28	#334	—	2	4	no	—
29	C₂²×C₈	C₄⋊C₄	1	2	yes	—
30	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄²	C₄⋊C₄	1	2	yes	—
33	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
34	C₂×C₁₆	—	2	4	no	—
35	C₂×C₁₆	—	2	4	no	—
36	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
37	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
38	C₄⋊C₈	—	4	4	no	—
39	C₂²⋊C₈	—	4	4	no	—
40	#97	—	2	2	no	—
41	#97	—	4	2	no	—
42	C₄×C₈	—	2	4	no	—
43	C₄×C₈	—	4	4	no	—
44	C₄×C₈	C₂×D₄	1	1	yes	characteristic
45	C₄×C₈	C₂×D₄	1	1	yes	characteristic
46	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
47	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
48	#95	—	2	2	no	—
49	C₂²×C₄	—	2	2	no	—
50	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄	#135	1	2	yes	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	8	4	no	—
57	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₈	#131	1	4	yes	—
62	C₄²	—	2	2	no	—
63	C₄²	—	2	2	no	—
64	C₄²	#127	1	1	yes	characteristic
65	C₄²	C₄⋊Q₈	1	1	yes	characteristic
66	C₄²	#135	1	2	yes	—
67	C₁₆	—	2	4	no	—
68	C₁₆	—	2	4	no	—
69	C₂³	#427	1	1	yes	characteristic
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	4	4	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	#384	1	1	yes	characteristic
74	C₂×C₄	#424	1	2	yes	—
75	C₂×C₄	#427	1	1	yes	characteristic
76	C₂×C₄	#430	1	2	yes	—
77	C₂×C₄	#495	1	2	yes	—
78	C₂×C₄	#499	1	2	yes	—
79	C₈	—	2	1	no	—
80	C₈	—	2	1	no	—
81	C₈	—	4	2	no	—
82	C₈	—	16	4	no	—
83	C₈	#424	1	2	yes	—
84	C₈	#424	1	2	yes	—
85	C₂²	—	2	2	no	—
86	C₂²	#1572	1	2	yes	—
87	C₂²	#1846	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	2	2	no	—
90	C₄	—	8	2	no	—
91	C₄	#1572	1	2	yes	—
92	C₄	#1843	1	1	yes	characteristic
93	C₄	#1849	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	#8811	1	1	yes	characteristic
96	C₂	#12911	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1846
prime radical	C₂³	#427
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#12592	2
#12008	2
#11999	2
#8780	2