Subgroups — #128677

Order 512.

334 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#12662	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#1311	C₂	1	1	yes	maximal, characteristic
5	C₈×#103	C₂	1	1	yes	maximal, characteristic
6	#7167	C₂	1	1	yes	maximal, characteristic
7	#7151	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
9	Q₈×C₁₆	—	2	2	no	—
10	Q₈×C₁₆	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₁₆	—	2	1	no	—
12	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×#346	—	2	1	no	—
15	C₂×C₄×C₁₆	—	2	1	no	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₄×#103	C₄	1	1	yes	characteristic
20	C₂×Q₈⋊C₈	C₄	1	1	yes	characteristic
21	C₂×C₈²	C₂²	1	1	yes	characteristic
22	#1374	—	2	1	no	—
23	Q₈⋊C₁₆	C₄	1	4	yes	complemented
24	#1311	—	2	2	no	—
25	#1279	C₄	1	1	yes	characteristic
26	#1277	C₄	1	1	yes	characteristic
27	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₁₆	—	2	1	no	—
29	C₂²×C₁₆	—	2	1	no	—
30	C₂²×C₁₆	—	2	1	no	—
31	C₂²×C₁₆	D₄	1	1	yes	characteristic
32	C₂²×C₁₆	D₄	1	1	yes	characteristic
33	C₈×Q₈	—	2	1	no	—
34	C₈×Q₈	—	2	1	no	—
35	C₈×Q₈	C₂×C₄	1	2	yes	—
36	C₈×Q₈	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₈	—	2	1	no	—
38	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×#103	C₈	1	2	yes	—
41	C₂×C₄×C₈	—	2	1	no	—
42	C₂×C₄×C₈	—	2	1	no	—
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
46	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₄⋊C₁₆	—	2	4	no	—
48	C₄⋊C₁₆	—	2	1	no	—
49	C₄⋊C₁₆	—	2	1	no	—
50	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
51	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
52	#346	—	4	2	no	—
53	C₄×C₁₆	—	2	4	no	—
54	C₄×C₁₆	—	2	1	no	—
55	C₄×C₁₆	C₂×C₄	1	2	yes	—
56	C₄×C₁₆	C₂×C₄	1	2	yes	—
57	C₄×C₁₆	D₄	1	2	yes	—
58	#340	C₈	1	2	yes	—
59	#336	—	2	1	no	—
60	Q₈⋊C₈	C₈	1	4	yes	—
61	C₈²	—	2	1	no	—
62	C₈²	—	2	1	no	—
63	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₈	—	2	1	no	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	2	1	no	—
67	C₂²×C₈	C₂×C₈	1	2	yes	—
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₈	#41	1	1	yes	characteristic
70	C₄×Q₈	—	2	1	no	—
71	C₄×Q₈	—	2	1	no	—
72	C₄×Q₈	C₄²	1	2	yes	—
73	C₄×Q₈	C₂×C₈	1	2	yes	—
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
76	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄²	—	2	1	no	—
78	C₂×C₄²	—	2	1	no	—
79	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	—	2	1	no	—
84	C₂×C₁₆	—	2	1	no	—
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	—	4	1	no	—
88	C₂×C₁₆	—	4	1	no	—
89	C₂×C₁₆	—	4	1	no	—
90	C₂×C₁₆	—	4	1	no	—
91	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
92	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
93	C₂×C₁₆	SD₁₆	1	2	yes	—
94	C₂×C₁₆	Q₁₆	1	2	yes	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	2	1	no	—
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	C₄²	1	2	yes	—
100	C₄⋊C₈	C₂×C₈	1	2	yes	—
101	#103	—	2	4	no	—
102	C₄×C₈	—	2	2	no	—
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	2	1	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	4	1	no	—
108	C₄×C₈	—	4	1	no	—
109	C₄×C₈	C₂×C₈	1	2	yes	—
110	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₄×C₈	C₂×D₄	1	1	yes	characteristic
113	C₄×C₈	#41	1	1	yes	characteristic
114	#95	—	2	1	no	—
115	C₂×Q₈	—	2	1	no	—
116	C₂×Q₈	—	2	2	no	—
117	C₂×Q₈	C₄×C₈	1	2	yes	—
118	C₂×Q₈	M₃₂	1	4	yes	—
119	C₂²×C₄	—	2	1	no	—
120	C₂²×C₄	—	2	1	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
123	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
124	C₂²×C₄	#131	1	1	yes	characteristic
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	4	1	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	—	4	1	no	—
138	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₈	#117	1	1	yes	characteristic
140	C₂×C₈	C₄×D₄	1	1	yes	characteristic
141	C₂×C₈	C₄×D₄	1	1	yes	characteristic
142	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
143	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
144	C₂×C₈	#135	1	1	yes	characteristic
145	C₂×C₈	#135	1	1	yes	characteristic
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	1	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	C₄×C₈	1	2	yes	—
152	C₄²	—	2	2	no	—
153	C₄²	—	2	2	no	—
154	C₄²	—	4	1	no	—
155	C₄²	—	4	1	no	—
156	C₄²	C₂×C₄²	1	1	yes	characteristic
157	C₄²	C₄×D₄	1	1	yes	characteristic
158	C₄²	C₂²×C₈	1	1	yes	characteristic
159	C₄²	#131	1	1	yes	characteristic
160	C₁₆	—	2	4	no	—
161	C₁₆	—	2	2	no	—
162	C₁₆	—	4	2	no	—
163	C₁₆	—	4	2	no	—
164	C₁₆	#103	1	4	yes	complemented
165	C₂³	C₈×D₄	1	1	yes	characteristic
166	Q₈	—	2	2	no	—
167	Q₈	—	2	4	no	—
168	Q₈	#346	1	4	yes	complemented
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
179	C₂×C₄	#405	1	1	yes	characteristic
180	C₂×C₄	C₈×D₄	1	1	yes	characteristic
181	C₂×C₄	C₈×D₄	1	1	yes	characteristic
182	C₂×C₄	C₈×D₄	1	1	yes	characteristic
183	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
184	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
185	C₂×C₄	#443	1	1	yes	characteristic
186	C₂×C₄	#443	1	1	yes	characteristic
187	C₂×C₄	C₂×M₃₂	1	2	yes	—
188	C₂×C₄	#504	1	2	yes	—
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	4	2	no	—
192	C₈	—	4	2	no	—
193	C₈	—	4	2	no	—
194	C₈	—	4	2	no	—
195	C₈	C₂×#103	1	1	yes	characteristic
196	C₈	#416	1	1	yes	characteristic
197	C₈	C₄×SD₁₆	1	1	yes	characteristic
198	C₈	C₄×Q₁₆	1	1	yes	characteristic
199	C₂²	C₈×SD₁₆	1	1	yes	characteristic
200	C₂²	C₈×Q₁₆	1	1	yes	characteristic
201	C₂²	C₈×C₂²⋊C₄	1	1	yes	characteristic
202	C₂²	#2168	1	2	yes	—
203	C₂²	#2171	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	4	2	no	—
208	C₄	—	8	2	no	—
209	C₄	C₈×SD₁₆	1	1	yes	characteristic
210	C₄	C₈×Q₁₆	1	1	yes	characteristic
211	C₄	C₄×#103	1	1	yes	characteristic
212	C₄	#1764	1	1	yes	characteristic
213	C₄	C₂×#346	1	1	yes	characteristic
214	C₄	#2108	1	1	yes	characteristic
215	C₄	#2168	1	2	yes	—
216	C₂	C₈×#103	1	1	yes	characteristic
217	C₂	#12050	1	2	yes	—
218	C₂	#12053	1	2	yes	—
219	C₂	#12611	1	1	yes	characteristic
220	C₂	#12615	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
10. Q₈×C₁₆	C₄	semidirect
23. Q₈⋊C₁₆	C₄	semidirect
164. C₁₆	#103	semidirect
168. Q₈	#346	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₁₆	2
#12662	2
C₂×Q₈⋊C₁₆	2
C₂×#1311	2
C₈×#103	2
#7167	2
#7151	2