Subgroups — #129042

Order 512.

364 conjugacy classes of subgroups.

The subgroup lattice has 174 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	1	yes	maximal, characteristic
2	C₄×#346	C₂	1	1	yes	maximal, characteristic
3	C₂×#1311	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#7136	C₂	1	1	yes	maximal, characteristic
7	#7136	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₂×#346	C₄	1	2	yes	—
10	C₂×#346	C₄	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₈	C₄	1	2	yes	—
16	C₄²×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1374	C₄	1	2	yes	—
20	#1311	—	2	2	no	—
21	C₈×C₁₆	—	2	2	no	—
22	#1273	C₄	1	2	yes	—
23	C₂²×C₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊C₈	C₈	1	4	yes	—
28	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
36	C₄³	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₁₆	C₈	1	8	yes	complemented
38	#346	—	2	4	no	—
39	#346	C₈	1	8	yes	complemented
40	C₄×C₁₆	—	2	1	no	—
41	C₄×C₁₆	—	2	1	no	—
42	C₄×C₁₆	—	2	1	no	—
43	C₄×C₁₆	—	2	1	no	—
44	C₄×C₁₆	—	2	1	no	—
45	C₄×C₁₆	—	2	1	no	—
46	C₄×C₁₆	D₄	1	2	yes	—
47	C₄×C₁₆	Q₈	1	2	yes	—
48	#336	C₈	1	4	yes	—
49	C₈²	—	2	1	no	—
50	C₈²	—	2	1	no	—
51	C₈²	—	2	1	no	—
52	C₈²	—	2	1	no	—
53	C₂²×C₈	C₄²	1	1	yes	characteristic
54	C₂²×C₈	C₄²	1	1	yes	characteristic
55	C₂²×C₈	C₂×C₈	1	4	yes	—
56	C₂²×C₈	C₂×C₈	1	4	yes	—
57	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄²	C₄²	1	1	yes	characteristic
60	C₂×C₄²	C₄²	1	1	yes	characteristic
61	C₂×C₄²	C₂×C₈	1	2	yes	—
62	C₂×C₄²	C₂×C₈	1	2	yes	—
63	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	2	2	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	2	2	no	—
70	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
71	C₂×C₁₆	C₂×C₈	1	4	yes	—
72	C₂×C₁₆	C₂×C₈	1	4	yes	—
73	C₂×C₁₆	C₂×C₈	1	4	yes	—
74	C₂×C₁₆	C₂×C₈	1	4	yes	—
75	C₂×C₁₆	M₁₆	1	4	yes	—
76	C₄⋊C₈	—	2	8	no	—
77	C₄×C₈	—	2	2	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	—	2	2	no	—
83	C₄×C₈	—	2	2	no	—
84	C₄×C₈	C₂×C₈	1	4	yes	—
85	C₄×C₈	C₂×C₈	1	4	yes	—
86	C₄×C₈	C₂×D₄	1	1	yes	characteristic
87	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
88	C₄×C₈	#41	1	1	yes	characteristic
89	C₄×C₈	#41	1	1	yes	characteristic
90	C₂²×C₄	C₄×C₈	1	2	yes	—
91	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
92	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
93	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
94	C₂²×C₄	C₂²×C₈	1	2	yes	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	C₄×C₈	1	2	yes	—
105	C₂×C₈	C₄×C₈	1	2	yes	—
106	C₂×C₈	M₃₂	1	8	yes	—
107	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₈	#117	1	1	yes	characteristic
109	C₂×C₈	C₄×D₄	1	1	yes	characteristic
110	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
111	C₂×C₈	C₂²×C₈	1	2	yes	—
112	C₂×C₈	C₂²×C₈	1	2	yes	—
113	C₂×C₈	C₂×M₁₆	1	2	yes	—
114	C₂×C₈	#131	1	2	yes	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	2	4	no	—
119	C₄²	C₄×C₈	1	2	yes	—
120	C₄²	C₄×C₈	1	2	yes	—
121	C₄²	C₄×D₄	1	1	yes	characteristic
122	C₄²	C₄×Q₈	1	1	yes	characteristic
123	C₄²	#131	1	2	yes	—
124	C₁₆	—	2	4	no	—
125	C₁₆	—	2	4	no	—
126	C₁₆	—	2	4	no	—
127	C₁₆	C₄⋊C₈	1	8	yes	complemented
128	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	C₈²	1	2	yes	—
135	C₂×C₄	C₈²	1	2	yes	—
136	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
139	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
140	C₂×C₄	#405	1	1	yes	characteristic
141	C₂×C₄	#405	1	1	yes	characteristic
142	C₂×C₄	C₈×D₄	1	2	yes	—
143	C₂×C₄	C₈×Q₈	1	2	yes	—
144	C₂×C₄	C₂×M₃₂	1	4	yes	—
145	C₂×C₄	#504	1	4	yes	—
146	C₈	—	2	8	no	—
147	C₈	—	2	2	no	—
148	C₈	—	2	2	no	—
149	C₈	—	4	4	no	—
150	C₈	C₂×C₄⋊C₈	1	2	yes	—
151	C₈	#431	1	2	yes	—
152	C₈	C₈×D₄	1	2	yes	—
153	C₈	C₈×Q₈	1	2	yes	—
154	C₂²	C₂×C₈²	1	1	yes	characteristic
155	C₂²	C₈×M₁₆	1	1	yes	characteristic
156	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
157	C₂²	C₄×M₃₂	1	2	yes	—
158	C₂²	#2108	1	2	yes	—
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	2	no	—
162	C₄	C₈×M₁₆	1	1	yes	characteristic
163	C₄	#1450	1	1	yes	characteristic
164	C₄	C₄×C₄⋊C₈	1	1	yes	characteristic
165	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
166	C₄	#2168	1	4	yes	—
167	C₄	#2183	1	4	yes	—
168	C₂	C₈×C₄⋊C₈	1	1	yes	characteristic
169	C₂	C₈×M₃₂	1	1	yes	characteristic
170	C₂	#11399	1	1	yes	characteristic
171	C₂	#11409	1	2	yes	—
172	C₂	#12663	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄×C₈	C₂³
commutator	C₂²	C₂×C₈²
socle, prime radical	C₂³	C₂×C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
37. C₄⋊C₁₆	C₈	semidirect
39. #346	C₈	semidirect
127. C₁₆	C₄⋊C₈	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
C₄×#346	2
C₂×#1311	2
C₂×C₈×C₁₆	2
C₈×C₄⋊C₈	2
#7136	2