Subgroups — #128235

Order 512.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11416	C₂	1	1	yes	maximal, characteristic
2	#11412	C₂	1	1	yes	maximal, characteristic
3	#8141	C₂	1	1	yes	maximal, characteristic
4	#7147	C₂	1	2	yes	maximal
5	#7144	C₂	1	2	yes	maximal
6	#2163	C₂²	1	2	yes	—
7	#2162	C₂²	1	2	yes	—
8	#1746	C₂²	1	1	yes	characteristic
9	#1452	C₂²	1	1	yes	characteristic
10	#1450	C₂²	1	1	yes	characteristic
11	#1313	C₄	1	4	yes	complemented
12	#1312	C₄	1	2	yes	complemented
13	#1311	C₄	1	1	yes	characteristic, complemented
14	C₈×C₁₆	C₄	1	1	yes	characteristic, complemented
15	#1275	C₄	1	2	yes	—
16	#1274	C₄	1	2	yes	—
17	#431	C₂×C₄	1	2	yes	—
18	#431	C₂×C₄	1	2	yes	—
19	C₂×#97	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
21	#376	C₂×C₄	1	1	yes	characteristic
22	C₄⋊C₁₆	—	2	2	no	—
23	C₄⋊C₁₆	—	2	2	no	—
24	C₂²⋊C₁₆	—	2	2	no	—
25	C₂²⋊C₁₆	—	2	2	no	—
26	#346	C₂×C₄	1	4	yes	—
27	C₄×C₁₆	C₂×C₄	1	2	yes	—
28	C₄×C₁₆	C₂×C₄	1	2	yes	—
29	#336	—	2	2	no	—
30	#322	—	2	2	no	—
31	#322	C₂×C₄	1	2	yes	—
32	C₈²	C₂×C₄	1	1	yes	characteristic
33	C₈²	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈	—	2	1	no	—
35	C₂²×C₈	#41	1	2	yes	—
36	C₂×C₄²	—	2	1	no	—
37	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₁₆	—	2	2	no	—
39	C₂×C₁₆	—	2	2	no	—
40	C₂×C₁₆	—	2	4	no	—
41	C₄⋊C₈	—	2	2	no	—
42	C₄⋊C₈	—	2	2	no	—
43	C₂²⋊C₈	—	2	2	no	—
44	C₂²⋊C₈	—	2	2	no	—
45	#97	—	2	2	no	—
46	C₄×C₈	—	2	2	no	—
47	C₄×C₈	C₄²	1	2	yes	—
48	C₄×C₈	C₄²	1	2	yes	—
49	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₄×C₈	#41	1	2	yes	—
52	#95	—	2	2	no	—
53	C₂²×C₄	—	4	1	no	—
54	C₂²×C₄	#117	1	1	yes	characteristic
55	C₂²×C₄	#131	1	2	yes	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	4	2	no	—
59	C₂×C₈	—	4	2	no	—
60	C₂×C₈	—	4	2	no	—
61	C₂×C₈	—	4	1	no	—
62	C₂×C₈	#117	1	2	yes	—
63	C₂×C₈	#117	1	2	yes	—
64	C₄²	—	4	2	no	—
65	C₄²	C₂×C₄²	1	1	yes	characteristic
66	C₄²	#117	1	1	yes	characteristic
67	C₄²	#131	1	2	yes	—
68	C₁₆	—	4	2	no	—
69	C₁₆	—	4	2	no	—
70	C₁₆	—	4	4	no	—
71	C₂³	#433	1	1	yes	characteristic
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	8	1	no	—
75	C₂×C₄	—	8	2	no	—
76	C₂×C₄	#376	1	1	yes	characteristic
77	C₂×C₄	#405	1	2	yes	—
78	C₂×C₄	#433	1	2	yes	—
79	C₂×C₄	#433	1	1	yes	characteristic
80	C₈	—	2	2	no	—
81	C₈	—	2	2	no	—
82	C₈	—	4	2	no	—
83	C₈	—	4	2	no	—
84	C₈	—	8	2	no	—
85	C₈	—	8	1	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	1	no	—
88	C₂²	#1746	1	1	yes	characteristic
89	C₄	—	2	2	no	—
90	C₄	—	2	2	no	—
91	C₄	—	2	1	no	—
92	C₄	—	16	1	no	—
93	C₄	#1747	1	2	yes	—
94	C₂	—	4	1	no	—
95	C₂	#8141	1	1	yes	characteristic
96	C₂	#12597	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#376
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1746
prime radical	C₂³	#433
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #1313	C₄	semidirect
12. #1312	C₄	semidirect
13. #1311	C₄	semidirect
14. C₈×C₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#11416	2
#11412	2
#8141	2
#7147	2
#7144	2