Subgroups — #134601

Order 512.

402 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#494	C₂	1	1	yes	maximal, characteristic
2	#12879	C₂	1	1	yes	maximal, characteristic
3	C₂×#1362	C₂	1	2	yes	maximal
4	#8806	C₂	1	1	yes	maximal, characteristic
5	#7274	C₂	1	2	yes	maximal
6	C₂×#494	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₁₆	—	2	1	no	—
8	C₂×C₄×Q₁₆	C₄	1	2	yes	—
9	C₄×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	#1969	—	2	1	no	—
12	#1969	C₄	1	2	yes	—
13	#1956	—	2	1	no	—
14	#1845	C₂²	1	1	yes	characteristic
15	#1842	—	2	1	no	—
16	C₄²×C₈	C₂²	1	1	yes	characteristic
17	C₂×#335	—	2	1	no	—
18	C₂×#335	C₂²	1	2	yes	—
19	#1374	—	2	1	no	—
20	#1374	—	2	1	no	—
21	#1362	—	2	4	no	—
22	C₂²×Q₁₆	—	2	1	no	—
23	C₂²×Q₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊Q₈	—	2	1	no	—
25	C₂×C₄×Q₈	—	2	2	no	—
26	C₂²×C₁₆	—	2	2	no	—
27	#494	—	2	1	no	—
28	#494	—	2	1	no	—
29	#494	D₄	1	2	yes	—
30	#494	Q₈	1	2	yes	—
31	C₄×Q₁₆	—	4	4	no	—
32	C₄×Q₁₆	—	4	4	no	—
33	C₂×#107	—	2	1	no	—
34	C₂×#107	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₈	—	2	2	no	—
36	C₂×C₄⋊C₈	—	4	1	no	—
37	C₂×C₄⋊C₈	—	4	1	no	—
38	C₂×#103	—	2	2	no	—
39	C₂×#103	—	2	2	no	—
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
46	#391	—	2	2	no	—
47	#389	—	4	1	no	—
48	#389	—	4	1	no	—
49	#384	—	2	1	no	—
50	C₄×C₄⋊C₄	—	4	1	no	—
51	C₄×C₄⋊C₄	—	4	1	no	—
52	C₄³	D₄	1	1	yes	characteristic
53	C₄⋊C₁₆	—	2	2	no	—
54	C₄⋊C₁₆	—	2	2	no	—
55	#335	—	2	2	no	—
56	#335	—	2	2	no	—
57	#335	—	4	2	no	—
58	C₂²×Q₈	—	2	2	no	—
59	C₂×Q₁₆	—	2	2	no	—
60	C₂×Q₁₆	—	2	2	no	—
61	C₂×Q₁₆	—	2	2	no	—
62	C₂×Q₁₆	—	4	2	no	—
63	C₂×Q₁₆	—	4	2	no	—
64	C₂×Q₁₆	—	4	2	no	—
65	C₂×Q₁₆	—	4	2	no	—
66	C₂×Q₁₆	C₄⋊C₄	1	4	yes	—
67	C₂²×C₈	—	4	2	no	—
68	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
72	C₄⋊Q₈	—	2	2	no	—
73	C₄⋊Q₈	—	2	2	no	—
74	C₄⋊Q₈	—	4	1	no	—
75	C₄⋊Q₈	—	4	1	no	—
76	C₄×Q₈	—	4	4	no	—
77	C₄×Q₈	—	4	4	no	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄⋊C₄	—	4	1	no	—
80	C₂×C₄⋊C₄	—	4	1	no	—
81	C₂×C₄⋊C₄	—	4	2	no	—
82	C₂×C₄⋊C₄	—	4	1	no	—
83	C₂×C₄⋊C₄	—	4	1	no	—
84	C₂×C₄²	—	4	2	no	—
85	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄²	D₈	1	1	yes	characteristic
88	C₂×C₄²	D₈	1	1	yes	characteristic
89	C₂×C₄²	SD₁₆	1	1	yes	characteristic
90	C₂×C₄²	SD₁₆	1	1	yes	characteristic
91	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
92	C₂×C₁₆	—	4	2	no	—
93	C₂×C₁₆	—	4	2	no	—
94	C₂×C₁₆	—	4	2	no	—
95	#107	—	4	2	no	—
96	#107	—	4	2	no	—
97	C₄⋊C₈	—	4	2	no	—
98	C₄⋊C₈	—	4	2	no	—
99	C₄⋊C₈	—	8	2	no	—
100	C₄⋊C₈	—	8	2	no	—
101	#103	—	4	4	no	—
102	#103	—	4	4	no	—
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	—	2	2	no	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	—	4	1	no	—
109	C₄×C₈	C₂×D₄	1	1	yes	characteristic
110	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
111	C₄×C₈	#41	1	1	yes	characteristic
112	C₄×C₈	#41	1	1	yes	characteristic
113	#95	—	4	2	no	—
114	#95	—	4	2	no	—
115	C₂×Q₈	—	2	4	no	—
116	C₂×Q₈	—	4	2	no	—
117	C₂×Q₈	—	4	2	no	—
118	C₂×Q₈	—	4	2	no	—
119	C₂×Q₈	—	4	2	no	—
120	C₂×Q₈	—	4	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	#102	1	1	yes	characteristic
124	C₂²×C₄	#102	1	1	yes	characteristic
125	C₂²×C₄	#102	1	1	yes	characteristic
126	C₂²×C₄	#102	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
129	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
130	Q₁₆	—	4	4	no	—
131	Q₁₆	—	4	4	no	—
132	Q₁₆	—	4	4	no	—
133	Q₁₆	—	4	4	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	8	2	no	—
142	C₂×C₈	—	8	2	no	—
143	C₂×C₈	—	8	2	no	—
144	C₂×C₈	#100	1	2	yes	—
145	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₈	#117	1	1	yes	characteristic
147	C₂×C₈	#121	1	1	yes	characteristic
148	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
149	C₂×C₈	#135	1	2	yes	—
150	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	8	2	no	—
158	C₄⋊C₄	—	8	4	no	—
159	C₄⋊C₄	—	8	1	no	—
160	C₄⋊C₄	—	8	1	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	4	1	no	—
169	C₄²	—	8	4	no	—
170	C₄²	SD₃₂	1	2	yes	—
171	C₄²	Q₃₂	1	2	yes	—
172	C₄²	#121	1	1	yes	characteristic
173	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
174	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₄²	#137	1	1	yes	characteristic
176	C₁₆	—	8	2	no	—
177	C₂³	C₂×#102	1	1	yes	characteristic
178	Q₈	—	4	4	no	—
179	Q₈	—	4	4	no	—
180	Q₈	—	8	2	no	—
181	Q₈	—	8	2	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	#358	1	2	yes	—
200	C₂×C₄	#358	1	2	yes	—
201	C₂×C₄	#362	1	2	yes	—
202	C₂×C₄	#380	1	1	yes	characteristic
203	C₂×C₄	C₂×#100	1	1	yes	characteristic
204	C₂×C₄	#413	1	1	yes	characteristic
205	C₂×C₄	#416	1	1	yes	characteristic
206	C₂×C₄	#417	1	1	yes	characteristic
207	C₂×C₄	#418	1	1	yes	characteristic
208	C₂×C₄	#459	1	1	yes	characteristic
209	C₂×C₄	#461	1	1	yes	characteristic
210	C₂×C₄	#474	1	1	yes	characteristic
211	C₂×C₄	#476	1	1	yes	characteristic
212	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
213	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
214	C₂×C₄	#509	1	1	yes	characteristic
215	C₂×C₄	#510	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	4	1	no	—
220	C₈	—	16	2	no	—
221	C₈	#463	1	2	yes	—
222	C₈	#478	1	2	yes	—
223	C₂²	#1540	1	1	yes	characteristic
224	C₂²	#1545	1	1	yes	characteristic
225	C₂²	#1802	1	1	yes	characteristic
226	C₂²	C₂×#358	1	1	yes	characteristic
227	C₂²	#2139	1	1	yes	characteristic
228	C₂²	C₂×#362	1	1	yes	characteristic
229	C₂²	#2148	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	8	4	no	—
237	C₄	—	16	2	no	—
238	C₄	#1551	1	1	yes	characteristic
239	C₄	#1557	1	1	yes	characteristic
240	C₄	#1800	1	1	yes	characteristic
241	C₄	#1804	1	1	yes	characteristic
242	C₄	#2207	1	1	yes	characteristic
243	C₄	#2209	1	1	yes	characteristic
244	C₄	#2225	1	1	yes	characteristic
245	C₄	#2227	1	1	yes	characteristic
246	C₂	#8588	1	1	yes	characteristic
247	C₂	#11949	1	1	yes	characteristic
248	C₂	#11954	1	1	yes	characteristic
249	C₂	#11962	1	1	yes	characteristic
250	C₂	#11966	1	1	yes	characteristic
251	C₂	#12812	1	1	yes	characteristic
252	C₂	#12819	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₄×#494	2
#12879	2
C₂×#1362	2
#8806	2
#7274	2