Subgroups — #101915

Order 512.

332 conjugacy classes of subgroups.

The subgroup lattice has 300 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	#9673	C₂	1	1	yes	maximal, characteristic
3	#9602	C₂	1	1	yes	maximal, characteristic
4	#8261	C₂	1	1	yes	maximal, characteristic
5	C₄×#332	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×#332	C₄	1	1	yes	characteristic
17	C₂×#332	C₄	1	1	yes	characteristic
18	C₂×Q₈⋊C₈	—	2	1	no	—
19	C₂×Q₈⋊C₈	—	2	1	no	—
20	C₂×#322	—	2	1	no	—
21	#1296	C₄	1	1	yes	characteristic
22	#1296	C₄	1	1	yes	characteristic
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	#332	—	2	2	no	—
60	#332	—	2	2	no	—
61	Q₈⋊C₈	—	4	2	no	—
62	Q₈⋊C₈	—	4	2	no	—
63	#322	—	4	2	no	—
64	C₂²×Q₈	—	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	—	2	1	no	—
72	C₂²×C₈	—	2	1	no	—
73	C₂²×C₈	—	4	1	no	—
74	C₂²×C₈	—	4	1	no	—
75	C₄⋊Q₈	—	2	1	no	—
76	C₄⋊Q₈	—	2	1	no	—
77	C₄⋊Q₈	—	2	1	no	—
78	C₄⋊Q₈	M₁₆	1	2	yes	—
79	C₄×Q₈	—	4	1	no	—
80	C₄×Q₈	—	4	1	no	—
81	C₄×Q₈	—	4	1	no	—
82	C₄×Q₈	—	4	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	2	1	no	—
87	C₂×C₄⋊C₄	—	2	1	no	—
88	C₂×C₄⋊C₄	—	4	1	no	—
89	C₂×C₄²	—	4	1	no	—
90	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄²	#41	1	1	yes	characteristic
94	C₂×C₄²	#41	1	1	yes	characteristic
95	C₂×C₄²	#41	1	1	yes	characteristic
96	C₂×C₄²	#41	1	1	yes	characteristic
97	C₄⋊C₈	—	2	2	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	—	2	1	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	4	1	no	—
108	C₄⋊C₈	—	4	1	no	—
109	C₄⋊C₈	—	4	1	no	—
110	C₄⋊C₈	—	4	1	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₄⋊C₈	—	8	1	no	—
113	C₄⋊C₈	M₁₆	1	2	yes	—
114	C₄⋊C₈	D₈	1	2	yes	—
115	C₄⋊C₈	SD₁₆	1	2	yes	—
116	C₄×C₈	—	2	1	no	—
117	C₄×C₈	—	2	1	no	—
118	C₄×C₈	—	2	1	no	—
119	C₄×C₈	—	4	1	no	—
120	C₄×C₈	—	4	1	no	—
121	C₄×C₈	—	4	2	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	—	4	1	no	—
127	C₄×C₈	—	4	1	no	—
128	C₄×C₈	—	4	1	no	—
129	C₄×C₈	—	4	1	no	—
130	C₄×C₈	SD₁₆	1	2	yes	—
131	#95	—	4	1	no	—
132	#95	—	4	1	no	—
133	C₂×Q₈	—	2	2	no	—
134	C₂×Q₈	—	4	1	no	—
135	C₂×Q₈	—	4	1	no	—
136	C₂×Q₈	—	4	1	no	—
137	C₂×Q₈	—	4	1	no	—
138	C₂×Q₈	—	4	1	no	—
139	C₂²×C₄	—	4	1	no	—
140	C₂²×C₄	—	4	1	no	—
141	C₂²×C₄	#117	1	1	yes	characteristic
142	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
143	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
144	C₂²×C₄	#121	1	1	yes	characteristic
145	C₂²×C₄	#123	1	1	yes	characteristic
146	C₂²×C₄	#124	1	1	yes	characteristic
147	C₂²×C₄	#126	1	1	yes	characteristic
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	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	2	no	—
164	C₂×C₈	—	4	1	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	1	no	—
168	C₂×C₈	—	4	1	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	1	no	—
172	C₂×C₈	—	4	1	no	—
173	C₂×C₈	—	8	1	no	—
174	C₂×C₈	—	8	1	no	—
175	C₂×C₈	—	8	1	no	—
176	C₂×C₈	—	8	1	no	—
177	C₂×C₈	—	8	1	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	8	1	no	—
187	C₄⋊C₄	—	8	1	no	—
188	C₄⋊C₄	—	8	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	4	1	no	—
196	C₄²	—	8	1	no	—
197	C₄²	—	8	1	no	—
198	C₄²	C₂×M₁₆	1	1	yes	characteristic
199	C₄²	C₂×M₁₆	1	1	yes	characteristic
200	C₄²	#131	1	1	yes	characteristic
201	C₄²	#131	1	1	yes	characteristic
202	C₄²	C₂×D₈	1	1	yes	characteristic
203	C₄²	C₂×SD₁₆	1	1	yes	characteristic
204	C₄²	C₂×SD₁₆	1	1	yes	characteristic
205	C₄²	#135	1	1	yes	characteristic
206	C₄²	#135	1	1	yes	characteristic
207	C₄²	#135	1	1	yes	characteristic
208	C₄²	#137	1	1	yes	characteristic
209	C₄²	#137	1	1	yes	characteristic
210	C₂³	#388	1	1	yes	characteristic
211	Q₈	—	4	2	no	—
212	Q₈	—	4	2	no	—
213	Q₈	—	8	1	no	—
214	Q₈	—	8	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	1	no	—
226	C₂×C₄	—	8	1	no	—
227	C₂×C₄	—	8	1	no	—
228	C₂×C₄	—	8	1	no	—
229	C₂×C₄	—	8	1	no	—
230	C₂×C₄	#432	1	1	yes	characteristic
231	C₂×C₄	#433	1	1	yes	characteristic
232	C₂×C₄	#435	1	1	yes	characteristic
233	C₂×C₄	#435	1	1	yes	characteristic
234	C₂×C₄	#435	1	1	yes	characteristic
235	C₂×C₄	#436	1	1	yes	characteristic
236	C₂×C₄	C₄×D₈	1	1	yes	characteristic
237	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
238	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
239	C₂×C₄	#441	1	1	yes	characteristic
240	C₂×C₄	#443	1	1	yes	characteristic
241	C₂×C₄	#444	1	1	yes	characteristic
242	C₂×C₄	#444	1	1	yes	characteristic
243	C₂×C₄	#444	1	1	yes	characteristic
244	C₂×C₄	#461	1	1	yes	characteristic
245	C₂×C₄	#464	1	1	yes	characteristic
246	C₂×C₄	#465	1	1	yes	characteristic
247	C₂×C₄	#466	1	1	yes	characteristic
248	C₂×C₄	#471	1	1	yes	characteristic
249	C₂×C₄	#473	1	1	yes	characteristic
250	C₂×C₄	#483	1	1	yes	characteristic
251	C₂×C₄	#485	1	1	yes	characteristic
252	C₂×C₄	#486	1	1	yes	characteristic
253	C₂×C₄	#488	1	1	yes	characteristic
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	4	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	4	2	no	—
259	C₈	—	8	2	no	—
260	C₈	—	8	1	no	—
261	C₈	—	8	1	no	—
262	C₈	—	8	1	no	—
263	C₈	—	16	1	no	—
264	C₂²	#1924	1	1	yes	characteristic
265	C₂²	#1927	1	1	yes	characteristic
266	C₂²	#1929	1	1	yes	characteristic
267	C₂²	#1930	1	1	yes	characteristic
268	C₂²	#1932	1	1	yes	characteristic
269	C₂²	#1934	1	1	yes	characteristic
270	C₂²	#1938	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	8	2	no	—
279	C₄	—	16	1	no	—
280	C₄	#1581	1	1	yes	characteristic
281	C₄	#1581	1	1	yes	characteristic
282	C₄	#1582	1	1	yes	characteristic
283	C₄	#1583	1	1	yes	characteristic
284	C₄	#1583	1	1	yes	characteristic
285	C₄	#1587	1	1	yes	characteristic
286	C₄	#1590	1	1	yes	characteristic
287	C₄	#1593	1	1	yes	characteristic
288	C₄	#1671	1	1	yes	characteristic
289	C₄	#1673	1	1	yes	characteristic
290	C₄	#1680	1	1	yes	characteristic
291	C₄	#1682	1	1	yes	characteristic
292	C₂	#9547	1	1	yes	characteristic
293	C₂	#9551	1	1	yes	characteristic
294	C₂	#9558	1	1	yes	characteristic
295	C₂	#9565	1	1	yes	characteristic
296	C₂	#9583	1	1	yes	characteristic
297	C₂	#9587	1	1	yes	characteristic
298	C₂	#9590	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
#9673	2
#9602	2
#8261	2
C₄×#332	2
#7031	2
#7015	2