Subgroups — #96896

Order 512.

390 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#335	C₂	1	1	yes	maximal, characteristic
2	C₄×#334	C₂	1	1	yes	maximal, characteristic
3	#8260	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₄×#322	C₂	1	1	yes	maximal, characteristic
6	#7015	C₂	1	1	yes	maximal, characteristic
7	#7015	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	C₄²×C₈	C₂²	1	1	yes	characteristic
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	C₂×#335	C₄	1	2	yes	—
16	C₂×#334	C₄	1	2	yes	—
17	C₂×#322	—	2	1	no	—
18	C₂×#322	C₄	1	2	yes	—
19	C₂×#322	C₄	1	2	yes	—
20	C₂×C₈²	—	2	1	no	—
21	#1277	C₄	1	2	yes	—
22	#1277	C₄	1	2	yes	—
23	#1273	—	2	1	no	—
24	#1273	—	2	1	no	—
25	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₈	—	2	1	no	—
30	C₂×C₄×C₈	—	2	1	no	—
31	C₂×C₄×C₈	—	2	1	no	—
32	C₂×C₄×C₈	—	2	1	no	—
33	C₂×C₄×C₈	—	2	1	no	—
34	C₂×C₄×C₈	—	2	1	no	—
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	Q₈	1	2	yes	—
46	C₄³	C₂³	1	1	yes	characteristic
47	#336	—	2	1	no	—
48	#336	—	2	1	no	—
49	#336	—	2	1	no	—
50	#336	—	2	1	no	—
51	#335	—	2	4	no	—
52	#334	—	2	4	no	—
53	#322	—	2	4	no	—
54	#322	—	4	2	no	—
55	#322	C₈	1	8	yes	complemented
56	C₈²	—	4	2	no	—
57	C₂²×C₈	—	2	2	no	—
58	C₂²×C₈	—	2	2	no	—
59	C₂²×C₈	—	2	2	no	—
60	C₂²×C₈	—	2	2	no	—
61	C₂²×C₈	—	2	2	no	—
62	C₂²×C₈	C₄²	1	1	yes	characteristic
63	C₂²×C₈	C₄²	1	1	yes	characteristic
64	C₂²×C₈	C₄²	1	1	yes	characteristic
65	C₂²×C₈	C₄²	1	1	yes	characteristic
66	C₂²×C₈	C₄⋊C₄	1	2	yes	—
67	C₂²×C₈	C₄⋊C₄	1	2	yes	—
68	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₄²	#41	1	1	yes	characteristic
74	C₂×C₄²	#41	1	1	yes	characteristic
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	—	2	2	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	M₁₆	1	4	yes	—
82	C₄⋊C₈	M₁₆	1	4	yes	—
83	C₄×C₈	—	2	4	no	—
84	C₄×C₈	—	2	4	no	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	4	2	no	—
96	C₄×C₈	—	4	1	no	—
97	C₄×C₈	—	4	1	no	—
98	C₄×C₈	—	4	1	no	—
99	C₄×C₈	—	4	1	no	—
100	C₄×C₈	—	4	1	no	—
101	C₄×C₈	—	4	1	no	—
102	C₄×C₈	—	4	1	no	—
103	C₄×C₈	—	4	1	no	—
104	C₄×C₈	C₂×C₈	1	4	yes	—
105	C₄×C₈	C₂×C₈	1	4	yes	—
106	C₄×C₈	C₂×C₈	1	4	yes	—
107	C₄×C₈	M₁₆	1	4	yes	—
108	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
109	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₄	#117	1	1	yes	characteristic
111	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
112	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
113	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
114	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	C₄×C₈	1	2	yes	—
145	C₂×C₈	C₄×C₈	1	2	yes	—
146	C₂×C₈	C₄⋊C₈	1	4	yes	—
147	C₂×C₈	C₄⋊C₈	1	4	yes	—
148	C₄²	—	2	2	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	C₂²×C₈	1	2	yes	—
152	C₄²	C₂×M₁₆	1	2	yes	—
153	C₄²	C₂×M₁₆	1	2	yes	—
154	C₄²	C₂×M₁₆	1	2	yes	—
155	C₄²	#131	1	2	yes	—
156	C₄²	#131	1	2	yes	—
157	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₄²	#137	1	1	yes	characteristic
160	C₄²	#137	1	1	yes	characteristic
161	C₂³	C₄×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
169	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
170	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
171	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
172	C₂×C₄	#405	1	1	yes	characteristic
173	C₂×C₄	#405	1	1	yes	characteristic
174	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
175	C₂×C₄	#428	1	1	yes	characteristic
176	C₂×C₄	#428	1	1	yes	characteristic
177	C₂×C₄	#430	1	2	yes	—
178	C₂×C₄	#431	1	2	yes	—
179	C₂×C₄	#435	1	2	yes	—
180	C₂×C₄	#435	1	2	yes	—
181	C₂×C₄	#440	1	1	yes	characteristic
182	C₂×C₄	#440	1	1	yes	characteristic
183	C₂×C₄	#441	1	1	yes	characteristic
184	C₂×C₄	#442	1	1	yes	characteristic
185	C₂×C₄	#443	1	2	yes	—
186	C₂×C₄	#443	1	2	yes	—
187	C₂×C₄	#446	1	2	yes	—
188	C₂×C₄	#446	1	2	yes	—
189	C₈	—	2	8	no	—
190	C₈	—	2	4	no	—
191	C₈	—	2	4	no	—
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	4	no	—
195	C₈	—	4	4	no	—
196	C₈	—	4	1	no	—
197	C₈	—	4	1	no	—
198	C₈	—	8	2	no	—
199	C₈	—	8	2	no	—
200	C₂²	C₄×C₄⋊C₈	1	1	yes	characteristic
201	C₂²	#1771	1	1	yes	characteristic
202	C₂²	#1771	1	1	yes	characteristic
203	C₂²	#1772	1	1	yes	characteristic
204	C₂²	#1772	1	1	yes	characteristic
205	C₂²	#1776	1	1	yes	characteristic
206	C₂²	#1778	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	C₈×M₁₆	1	1	yes	characteristic
211	C₄	C₈×M₁₆	1	1	yes	characteristic
212	C₄	#1453	1	1	yes	characteristic
213	C₄	#1455	1	1	yes	characteristic
214	C₄	#1565	1	2	yes	—
215	C₄	#1565	1	2	yes	—
216	C₄	#1585	1	2	yes	—
217	C₄	#1586	1	2	yes	—
218	C₄	#1587	1	2	yes	—
219	C₄	#1588	1	2	yes	—
220	C₂	#8262	1	1	yes	characteristic
221	C₂	#8266	1	1	yes	characteristic
222	C₂	#8275	1	1	yes	characteristic
223	C₂	#8288	1	1	yes	characteristic
224	C₂	#8289	1	1	yes	characteristic
225	C₂	#8296	1	1	yes	characteristic
226	C₂	#8296	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	C₄×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
55. #322	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#335	2
C₄×#334	2
#8260	2
C₈×C₄⋊C₈	2
C₄×#322	2
#7015	2