Subgroups — #129117

Order 512.

318 conjugacy classes of subgroups.

The subgroup lattice has 161 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×#1370	C₂	1	1	yes	maximal, characteristic
3	C₂×#1367	C₂	1	1	yes	maximal, characteristic
4	C₈×#97	C₂	1	1	yes	maximal, characteristic
5	#7151	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₁₆	—	2	2	no	—
9	#1744	C₄	1	1	yes	characteristic
10	C₄×#97	C₄	1	1	yes	characteristic
11	C₄²×C₈	C₂²	1	1	yes	characteristic
12	C₂×#322	C₄	1	1	yes	characteristic
13	C₂×#322	C₄	1	1	yes	characteristic
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	C₂×C₈²	C₂²	1	1	yes	characteristic
16	#1374	—	2	2	no	—
17	#1370	C₄	1	4	yes	complemented
18	#1367	C₄	1	4	yes	complemented
19	C₂²×C₁₆	—	2	2	no	—
20	C₂²×C₁₆	—	2	2	no	—
21	C₂×#97	C₈	1	2	yes	—
22	C₂×#97	C₈	1	2	yes	—
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	2	yes	—
29	C₂×C₄×C₈	Q₈	1	2	yes	—
30	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
31	C₄³	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₁₆	—	2	2	no	—
33	C₄⋊C₁₆	—	2	2	no	—
34	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
35	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
36	C₄×C₁₆	—	4	4	no	—
37	#336	—	2	2	no	—
38	#322	—	2	2	no	—
39	#322	C₈	1	4	yes	—
40	C₈²	—	2	1	no	—
41	C₈²	—	2	1	no	—
42	C₈²	C₂×C₄	1	2	yes	—
43	C₈²	C₂×C₄	1	2	yes	—
44	C₂²×C₈	—	2	1	no	—
45	C₂²×C₈	—	2	1	no	—
46	C₂²×C₈	C₄⋊C₄	1	2	yes	—
47	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₈	C₂×C₈	1	2	yes	—
49	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄²	C₄⋊C₄	1	2	yes	—
53	C₂×C₄²	C₂×C₈	1	2	yes	—
54	C₂×C₄²	M₁₆	1	2	yes	—
55	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₁₆	—	2	4	no	—
57	C₂×C₁₆	—	2	4	no	—
58	C₂×C₁₆	—	4	2	no	—
59	C₂×C₁₆	—	4	2	no	—
60	C₂×C₁₆	—	4	2	no	—
61	C₂×C₁₆	—	4	2	no	—
62	#97	—	2	4	no	—
63	#97	C₁₆	1	8	yes	complemented
64	C₄×C₈	—	2	2	no	—
65	C₄×C₈	—	2	2	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	4	2	no	—
71	C₄×C₈	C₄²	1	2	yes	—
72	C₄×C₈	C₄²	1	2	yes	—
73	C₄×C₈	C₂×C₈	1	2	yes	—
74	C₄×C₈	C₂×C₈	1	2	yes	—
75	C₄×C₈	C₂×C₈	1	2	yes	—
76	C₄×C₈	M₁₆	1	2	yes	—
77	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₄×C₈	#41	1	1	yes	characteristic
80	C₄×C₈	#41	1	1	yes	characteristic
81	C₂²×C₄	C₄⋊C₈	1	4	yes	—
82	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
84	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	C₄×C₈	1	2	yes	—
96	C₂×C₈	C₄×C₈	1	2	yes	—
97	C₂×C₈	C₄⋊C₈	1	4	yes	—
98	C₂×C₈	C₂×C₁₆	1	4	yes	—
99	C₂×C₈	C₂×C₁₆	1	4	yes	—
100	C₂×C₈	C₄×D₄	1	1	yes	characteristic
101	C₂×C₈	C₄×D₄	1	1	yes	characteristic
102	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₈	#137	1	2	yes	—
106	C₄²	—	2	2	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	4	2	no	—
109	C₄²	C₂×C₁₆	1	4	yes	—
110	C₄²	M₃₂	1	4	yes	—
111	C₄²	C₂×C₄²	1	1	yes	characteristic
112	C₄²	#117	1	1	yes	characteristic
113	C₄²	C₂²×C₈	1	1	yes	characteristic
114	C₄²	C₂×M₁₆	1	1	yes	characteristic
115	C₁₆	—	4	4	no	—
116	C₁₆	—	4	4	no	—
117	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₄⋊C₁₆	1	8	yes	—
122	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
124	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
125	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
126	C₂×C₄	#428	1	2	yes	—
127	C₂×C₄	#430	1	2	yes	—
128	C₂×C₄	#431	1	1	yes	characteristic
129	C₂×C₄	#431	1	1	yes	characteristic
130	C₂×C₄	C₂²×C₁₆	1	2	yes	—
131	C₂×C₄	C₂×M₃₂	1	2	yes	—
132	C₈	—	2	4	no	—
133	C₈	—	2	2	no	—
134	C₈	—	2	4	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	4	2	no	—
138	C₈	—	4	2	no	—
139	C₈	C₄×C₁₆	1	4	yes	complemented
140	C₈	#440	1	2	yes	—
141	C₈	#441	1	1	yes	characteristic
142	C₈	#442	1	1	yes	characteristic
143	C₂²	#1565	1	2	yes	—
144	C₂²	C₄×C₄⋊C₈	1	1	yes	characteristic
145	C₂²	C₂×C₄⋊C₁₆	1	2	yes	—
146	C₂²	#2153	1	2	yes	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	8	no	—
150	C₄	#1565	1	2	yes	—
151	C₄	#1776	1	1	yes	characteristic
152	C₄	#1778	1	1	yes	characteristic
153	C₄	C₂×C₄×C₁₆	1	1	yes	characteristic
154	C₄	C₄×M₃₂	1	1	yes	characteristic
155	C₄	#2162	1	2	yes	—
156	C₂	#8275	1	1	yes	characteristic
157	C₂	#12015	1	4	yes	—
158	C₂	C₄×C₄⋊C₁₆	1	1	yes	characteristic
159	C₂	#12656	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ 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
17. #1370	C₄	semidirect
18. #1367	C₄	semidirect
63. #97	C₁₆	semidirect
139. C₈	C₄×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₁₆	2
C₂×#1370	2
C₂×#1367	2
C₈×#97	2
#7151	2