Subgroups — #101925

Order 512.

322 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#9870	C₂	1	1	yes	maximal, characteristic
2	#9864	C₂	1	1	yes	maximal, characteristic
3	#9602	C₂	1	1	yes	maximal, characteristic
4	#8260	C₂	1	1	yes	maximal, characteristic
5	C₄×#333	C₂	1	1	yes	maximal, characteristic
6	#7031	C₂	1	1	yes	maximal, characteristic
7	#7015	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#1842	C₂²	1	1	yes	characteristic
10	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	C₂×#334	—	2	1	no	—
16	C₂×#333	C₄	1	1	yes	characteristic
17	C₂×#333	C₄	1	1	yes	characteristic
18	C₂×#322	—	2	1	no	—
19	#1296	C₄	1	1	yes	characteristic
20	#1296	C₄	1	1	yes	characteristic
21	#1279	—	2	1	no	—
22	#1279	—	2	1	no	—
23	#1279	—	2	1	no	—
24	#1279	—	2	1	no	—
25	#1277	—	2	1	no	—
26	#1273	—	2	1	no	—
27	C₂×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Q₈	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
38	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	—	2	1	no	—
40	C₂×C₄×C₈	—	2	1	no	—
41	C₂×C₄×C₈	—	2	1	no	—
42	C₂×C₄×C₈	—	2	1	no	—
43	C₂×C₄×C₈	—	2	1	no	—
44	C₂×C₄×C₈	—	2	1	no	—
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	#391	—	2	1	no	—
48	#389	—	2	1	no	—
49	#389	—	2	1	no	—
50	#384	C₂×C₄	1	1	yes	characteristic
51	C₄×C₄⋊C₄	—	2	1	no	—
52	C₄×C₄⋊C₄	—	2	1	no	—
53	C₄³	C₂³	1	1	yes	characteristic
54	#336	—	2	1	no	—
55	#336	—	2	1	no	—
56	#336	—	2	1	no	—
57	#336	—	2	1	no	—
58	#334	—	4	2	no	—
59	#333	—	2	2	no	—
60	#333	—	2	2	no	—
61	#322	—	4	2	no	—
62	C₂²×Q₈	—	2	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂²×C₈	—	2	1	no	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	2	1	no	—
67	C₂²×C₈	—	2	1	no	—
68	C₂²×C₈	—	2	1	no	—
69	C₂²×C₈	—	2	1	no	—
70	C₂²×C₈	—	2	1	no	—
71	C₂²×C₈	—	4	1	no	—
72	C₂²×C₈	—	4	1	no	—
73	C₄⋊Q₈	—	2	1	no	—
74	C₄⋊Q₈	—	2	1	no	—
75	C₄⋊Q₈	—	2	1	no	—
76	C₄⋊Q₈	M₁₆	1	2	yes	—
77	C₄×Q₈	—	8	1	no	—
78	C₄×Q₈	—	8	1	no	—
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	—	2	1	no	—
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	4	1	no	—
85	C₂×C₄²	—	4	1	no	—
86	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄²	#41	1	1	yes	characteristic
90	C₂×C₄²	#41	1	1	yes	characteristic
91	C₂×C₄²	#41	1	1	yes	characteristic
92	C₂×C₄²	#41	1	1	yes	characteristic
93	C₄⋊C₈	—	2	2	no	—
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	2	1	no	—
96	C₄⋊C₈	—	2	1	no	—
97	C₄⋊C₈	—	2	1	no	—
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	—	2	1	no	—
100	C₄⋊C₈	—	2	1	no	—
101	C₄⋊C₈	—	2	1	no	—
102	C₄⋊C₈	—	2	1	no	—
103	C₄⋊C₈	—	2	1	no	—
104	C₄⋊C₈	—	2	1	no	—
105	C₄⋊C₈	—	4	1	no	—
106	C₄⋊C₈	—	4	1	no	—
107	C₄⋊C₈	—	4	1	no	—
108	C₄⋊C₈	—	4	1	no	—
109	C₄⋊C₈	—	4	1	no	—
110	C₄⋊C₈	—	8	1	no	—
111	C₄⋊C₈	M₁₆	1	2	yes	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	—	2	1	no	—
115	C₄×C₈	—	2	1	no	—
116	C₄×C₈	—	2	1	no	—
117	C₄×C₈	—	4	2	no	—
118	C₄×C₈	—	4	1	no	—
119	C₄×C₈	—	4	1	no	—
120	C₄×C₈	—	4	1	no	—
121	C₄×C₈	—	4	1	no	—
122	C₄×C₈	—	4	1	no	—
123	C₄×C₈	—	4	1	no	—
124	C₄×C₈	—	4	1	no	—
125	C₄×C₈	—	4	1	no	—
126	C₄×C₈	SD₁₆	1	2	yes	—
127	#95	—	4	1	no	—
128	#95	—	4	1	no	—
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	4	1	no	—
131	C₂×Q₈	—	8	1	no	—
132	C₂×Q₈	—	8	1	no	—
133	C₂²×C₄	—	4	1	no	—
134	C₂²×C₄	—	4	1	no	—
135	C₂²×C₄	#117	1	1	yes	characteristic
136	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
137	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
138	C₂²×C₄	#121	1	1	yes	characteristic
139	C₂²×C₄	#123	1	1	yes	characteristic
140	C₂²×C₄	#124	1	1	yes	characteristic
141	C₂²×C₄	#126	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	1	no	—
150	C₂×C₈	—	4	1	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	1	no	—
156	C₂×C₈	—	4	1	no	—
157	C₂×C₈	—	4	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	4	1	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	8	1	no	—
168	C₂×C₈	—	8	1	no	—
169	C₂×C₈	—	8	1	no	—
170	C₂×C₈	—	8	1	no	—
171	C₂×C₈	—	8	1	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	1	no	—
177	C₄⋊C₄	—	4	1	no	—
178	C₄⋊C₄	—	4	1	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	8	1	no	—
181	C₄⋊C₄	—	8	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	4	1	no	—
189	C₄²	—	8	1	no	—
190	C₄²	—	8	1	no	—
191	C₄²	C₂×M₁₆	1	1	yes	characteristic
192	C₄²	C₂×M₁₆	1	1	yes	characteristic
193	C₄²	#131	1	1	yes	characteristic
194	C₄²	#131	1	1	yes	characteristic
195	C₄²	C₂×SD₁₆	1	1	yes	characteristic
196	C₄²	#135	1	1	yes	characteristic
197	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₄²	#137	1	1	yes	characteristic
199	C₄²	#137	1	1	yes	characteristic
200	C₄²	#137	1	1	yes	characteristic
201	C₄²	#137	1	1	yes	characteristic
202	C₄²	#137	1	1	yes	characteristic
203	C₂³	#388	1	1	yes	characteristic
204	Q₈	—	8	2	no	—
205	Q₈	—	8	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	—	8	1	no	—
218	C₂×C₄	—	8	1	no	—
219	C₂×C₄	—	8	1	no	—
220	C₂×C₄	—	8	1	no	—
221	C₂×C₄	#432	1	1	yes	characteristic
222	C₂×C₄	#433	1	1	yes	characteristic
223	C₂×C₄	#435	1	1	yes	characteristic
224	C₂×C₄	#435	1	1	yes	characteristic
225	C₂×C₄	#435	1	1	yes	characteristic
226	C₂×C₄	#436	1	1	yes	characteristic
227	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
228	C₂×C₄	#440	1	1	yes	characteristic
229	C₂×C₄	#441	1	1	yes	characteristic
230	C₂×C₄	#441	1	1	yes	characteristic
231	C₂×C₄	#443	1	1	yes	characteristic
232	C₂×C₄	#443	1	1	yes	characteristic
233	C₂×C₄	#443	1	1	yes	characteristic
234	C₂×C₄	#444	1	1	yes	characteristic
235	C₂×C₄	#461	1	1	yes	characteristic
236	C₂×C₄	#464	1	1	yes	characteristic
237	C₂×C₄	#468	1	1	yes	characteristic
238	C₂×C₄	#470	1	1	yes	characteristic
239	C₂×C₄	#471	1	1	yes	characteristic
240	C₂×C₄	#472	1	1	yes	characteristic
241	C₂×C₄	#483	1	1	yes	characteristic
242	C₂×C₄	#485	1	1	yes	characteristic
243	C₂×C₄	#489	1	1	yes	characteristic
244	C₂×C₄	#491	1	1	yes	characteristic
245	C₈	—	2	4	no	—
246	C₈	—	4	2	no	—
247	C₈	—	4	2	no	—
248	C₈	—	8	2	no	—
249	C₈	—	8	1	no	—
250	C₈	—	8	1	no	—
251	C₈	—	8	1	no	—
252	C₈	—	8	1	no	—
253	C₈	—	8	1	no	—
254	C₈	—	16	1	no	—
255	C₂²	#1924	1	1	yes	characteristic
256	C₂²	#1927	1	1	yes	characteristic
257	C₂²	#1929	1	1	yes	characteristic
258	C₂²	#1931	1	1	yes	characteristic
259	C₂²	#1932	1	1	yes	characteristic
260	C₂²	#1935	1	1	yes	characteristic
261	C₂²	#1937	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	4	1	no	—
269	C₄	—	8	2	no	—
270	C₄	—	16	1	no	—
271	C₄	#1583	1	1	yes	characteristic
272	C₄	#1585	1	1	yes	characteristic
273	C₄	#1587	1	1	yes	characteristic
274	C₄	#1587	1	1	yes	characteristic
275	C₄	#1587	1	1	yes	characteristic
276	C₄	#1588	1	1	yes	characteristic
277	C₄	#1590	1	1	yes	characteristic
278	C₄	#1593	1	1	yes	characteristic
279	C₄	#1691	1	1	yes	characteristic
280	C₄	#1695	1	1	yes	characteristic
281	C₄	#1698	1	1	yes	characteristic
282	C₄	#1704	1	1	yes	characteristic
283	C₂	#9547	1	1	yes	characteristic
284	C₂	#9552	1	1	yes	characteristic
285	C₂	#9559	1	1	yes	characteristic
286	C₂	#9575	1	1	yes	characteristic
287	C₂	#9579	1	1	yes	characteristic
288	C₂	#9586	1	1	yes	characteristic
289	C₂	#9591	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#388
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Maximal subgroups

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

subgroup	index
#9870	2
#9864	2
#9602	2
#8260	2
C₄×#333	2
#7031	2
#7015	2