Subgroups — #129057

Order 512.

356 conjugacy classes of subgroups.

The subgroup lattice has 188 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	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1312	C₂	1	2	yes	maximal
4	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#7136	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
7	C₂×C₄×C₁₆	C₄	1	2	yes	—
8	C₂×C₄×C₁₆	C₄	1	2	yes	—
9	C₂×C₄×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
13	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₈²	C₂²	1	2	yes	—
16	#1374	C₄	1	2	yes	—
17	#1312	—	2	4	no	—
18	#1273	C₄	1	2	yes	—
19	C₂²×C₁₆	C₂×C₄	1	4	yes	—
20	C₂²×C₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₄⋊C₈	C₈	1	2	yes	—
23	C₂×C₄⋊C₈	C₈	1	2	yes	—
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	2	yes	—
28	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
32	C₄³	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₁₆	—	2	4	no	—
34	C₄×C₁₆	—	2	4	no	—
35	C₄×C₁₆	—	2	4	no	—
36	C₄×C₁₆	—	2	2	no	—
37	C₄×C₁₆	—	2	1	no	—
38	C₄×C₁₆	—	2	1	no	—
39	C₄×C₁₆	—	2	2	no	—
40	C₄×C₁₆	D₄	1	2	yes	—
41	C₄×C₁₆	Q₈	1	2	yes	—
42	#336	C₈	1	2	yes	—
43	#336	C₈	1	2	yes	—
44	C₈²	—	2	2	no	—
45	C₈²	—	2	2	no	—
46	C₂²×C₈	C₄²	1	1	yes	characteristic
47	C₂²×C₈	C₄²	1	1	yes	characteristic
48	C₂²×C₈	C₂×C₈	1	4	yes	—
49	C₂²×C₈	C₂×C₈	1	4	yes	—
50	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄²	C₄²	1	1	yes	characteristic
53	C₂×C₄²	C₄²	1	1	yes	characteristic
54	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
55	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
56	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
57	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
58	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₁₆	—	2	2	no	—
60	C₂×C₁₆	—	2	2	no	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	2	4	no	—
64	C₂×C₁₆	—	2	4	no	—
65	C₂×C₁₆	—	2	4	no	—
66	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
67	C₂×C₁₆	M₁₆	1	4	yes	—
68	C₄⋊C₈	—	2	4	no	—
69	C₄⋊C₈	C₁₆	1	8	yes	complemented
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	2	2	no	—
77	C₄×C₈	—	2	2	no	—
78	C₄×C₈	M₁₆	1	4	yes	—
79	C₄×C₈	M₁₆	1	4	yes	—
80	C₄×C₈	C₂×D₄	1	1	yes	characteristic
81	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₄×C₈	#41	1	1	yes	characteristic
83	C₄×C₈	#41	1	1	yes	characteristic
84	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
85	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
86	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
87	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
88	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
89	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
90	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	#97	1	2	yes	—
101	C₂×C₈	#97	1	2	yes	—
102	C₂×C₈	C₂×C₁₆	1	8	yes	—
103	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₈	#117	1	1	yes	characteristic
105	C₂×C₈	C₄×D₄	1	1	yes	characteristic
106	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
107	C₂×C₈	C₂×M₁₆	1	2	yes	—
108	C₂×C₈	C₂×M₁₆	1	2	yes	—
109	C₂×C₈	C₂×M₁₆	1	2	yes	—
110	C₂×C₈	#131	1	2	yes	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	#97	1	2	yes	—
116	C₄²	#97	1	2	yes	—
117	C₄²	C₂×C₁₆	1	4	yes	—
118	C₄²	M₃₂	1	4	yes	—
119	C₄²	C₄×D₄	1	1	yes	characteristic
120	C₄²	C₄×Q₈	1	1	yes	characteristic
121	C₄²	#131	1	1	yes	characteristic
122	C₄²	#131	1	1	yes	characteristic
123	C₁₆	—	2	4	no	—
124	C₁₆	—	2	4	no	—
125	C₁₆	—	4	4	no	—
126	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	#322	1	2	yes	—
134	C₂×C₄	#322	1	2	yes	—
135	C₂×C₄	C₄×C₁₆	1	2	yes	—
136	C₂×C₄	C₄×C₁₆	1	2	yes	—
137	C₂×C₄	#346	1	2	yes	—
138	C₂×C₄	#346	1	2	yes	—
139	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂×#97	1	1	yes	characteristic
141	C₂×C₄	C₂×#97	1	1	yes	characteristic
142	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
143	C₂×C₄	#405	1	1	yes	characteristic
144	C₂×C₄	#405	1	1	yes	characteristic
145	C₂×C₄	C₈×D₄	1	1	yes	characteristic
146	C₂×C₄	C₈×D₄	1	1	yes	characteristic
147	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
148	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
149	C₂×C₄	C₂²×C₁₆	1	2	yes	—
150	C₂×C₄	C₂×M₃₂	1	2	yes	—
151	C₂×C₄	#504	1	2	yes	—
152	C₂×C₄	#504	1	2	yes	—
153	C₈	—	2	8	no	—
154	C₈	—	2	2	no	—
155	C₈	—	2	2	no	—
156	C₈	—	4	4	no	—
157	C₈	#423	1	2	yes	—
158	C₈	#432	1	2	yes	—
159	C₈	#436	1	2	yes	—
160	C₈	#446	1	2	yes	—
161	C₂²	C₂×#322	1	1	yes	characteristic
162	C₂²	C₈×M₁₆	1	1	yes	characteristic
163	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
164	C₂²	C₂×C₄×C₁₆	1	1	yes	characteristic
165	C₂²	C₂×#346	1	1	yes	characteristic
166	C₂²	#2108	1	1	yes	characteristic
167	C₂²	#2108	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	2	no	—
171	C₄	#1312	1	4	yes	complemented
172	C₄	#1451	1	1	yes	characteristic
173	C₄	#1452	1	1	yes	characteristic
174	C₄	#1767	1	1	yes	characteristic
175	C₄	#1770	1	1	yes	characteristic
176	C₄	D₄×C₁₆	1	2	yes	—
177	C₄	#2169	1	2	yes	—
178	C₄	Q₈×C₁₆	1	2	yes	—
179	C₄	#2183	1	2	yes	—
180	C₂	#8261	1	1	yes	characteristic
181	C₂	C₂×#1312	1	1	yes	characteristic
182	C₂	#11401	1	1	yes	characteristic
183	C₂	C₁₆×M₁₆	1	1	yes	characteristic
184	C₂	#11408	1	1	yes	characteristic
185	C₂	C₁₆×C₄⋊C₄	1	1	yes	characteristic
186	C₂	#12662	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄×C₈	C₂³
commutator	C₂²	C₂×C₄×C₁₆
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
69. C₄⋊C₈	C₁₆	semidirect
171. C₄	#1312	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
C₄²×C₁₆	2
C₂×#1312	2
C₈×C₄⋊C₈	2
#7136	2