Subgroups — #99354

Order 512.

370 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23250	C₂	1	1	yes	maximal, characteristic
2	#9637	C₂	1	1	yes	maximal, characteristic
3	#9602	C₂	1	1	yes	maximal, characteristic
4	#9601	C₂	1	1	yes	maximal, characteristic
5	C₄×#333	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⋊Q₈	C₂²	1	1	yes	characteristic
9	#1985	—	2	2	no	—
10	#1939	—	2	1	no	—
11	#1939	—	2	1	no	—
12	#1842	C₂²	1	1	yes	characteristic
13	#1770	—	2	1	no	—
14	#1770	—	2	1	no	—
15	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₄²×C₈	C₂²	1	1	yes	characteristic
20	C₂×#335	—	2	1	no	—
21	C₂×#334	—	2	1	no	—
22	C₂×#333	C₄	1	1	yes	characteristic
23	C₂×#333	C₄	1	1	yes	characteristic
24	C₂×Q₈⋊C₈	—	2	2	no	—
25	#1296	C₄	1	1	yes	characteristic
26	#1296	C₄	1	1	yes	characteristic
27	#1277	—	2	1	no	—
28	#1277	—	2	1	no	—
29	C₂×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Q₈	D₄	1	2	yes	—
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	2	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	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
43	C₂×#97	—	2	2	no	—
44	C₂×C₄×C₈	—	2	1	no	—
45	C₂×C₄×C₈	—	2	1	no	—
46	C₂×C₄×C₈	—	2	1	no	—
47	C₂×C₄×C₈	—	2	1	no	—
48	C₂×C₄×C₈	—	2	1	no	—
49	C₂×C₄×C₈	—	2	1	no	—
50	C₂×C₄×C₈	—	2	1	no	—
51	#391	Q₈	1	2	yes	—
52	#389	—	2	1	no	—
53	#389	—	2	1	no	—
54	#384	C₂×C₄	1	1	yes	characteristic
55	C₄×C₄⋊C₄	—	2	1	no	—
56	C₄×C₄⋊C₄	—	2	1	no	—
57	C₄³	C₂³	1	1	yes	characteristic
58	#336	—	2	2	no	—
59	#336	—	2	2	no	—
60	#336	—	2	1	no	—
61	#336	—	2	1	no	—
62	#336	—	2	1	no	—
63	#336	—	2	1	no	—
64	#335	—	4	2	no	—
65	#334	—	4	2	no	—
66	#333	—	2	2	no	—
67	#333	—	2	2	no	—
68	Q₈⋊C₈	—	4	4	no	—
69	C₂²×Q₈	C₄⋊C₄	1	2	yes	—
70	C₂²×C₈	—	2	1	no	—
71	C₂²×C₈	—	2	1	no	—
72	C₂²×C₈	—	2	1	no	—
73	C₂²×C₈	—	2	1	no	—
74	C₂²×C₈	—	2	1	no	—
75	C₂²×C₈	—	2	1	no	—
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	—	4	1	no	—
79	C₂²×C₈	—	4	1	no	—
80	C₄⋊Q₈	—	2	1	no	—
81	C₄⋊Q₈	—	2	1	no	—
82	C₄⋊Q₈	—	2	1	no	—
83	C₄⋊Q₈	—	2	1	no	—
84	C₄×Q₈	—	4	2	no	—
85	C₄×Q₈	—	4	2	no	—
86	C₂×C₄⋊C₄	—	2	1	no	—
87	C₂×C₄⋊C₄	—	2	1	no	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
92	C₂×C₄²	—	2	2	no	—
93	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₄²	#41	1	1	yes	characteristic
99	C₂×C₄²	#41	1	1	yes	characteristic
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	—	2	1	no	—
108	C₄⋊C₈	—	2	1	no	—
109	C₄⋊C₈	—	2	1	no	—
110	C₄⋊C₈	—	2	1	no	—
111	C₄⋊C₈	—	2	1	no	—
112	C₄⋊C₈	—	4	2	no	—
113	C₄⋊C₈	—	4	1	no	—
114	C₄⋊C₈	—	4	1	no	—
115	C₄⋊C₈	—	4	1	no	—
116	C₄⋊C₈	—	4	1	no	—
117	C₄⋊C₈	—	8	2	no	—
118	C₄⋊C₈	M₁₆	1	2	yes	—
119	C₄⋊C₈	M₁₆	1	2	yes	—
120	C₄⋊C₈	SD₁₆	1	2	yes	—
121	C₄⋊C₈	Q₁₆	1	2	yes	—
122	#97	—	8	2	no	—
123	C₄×C₈	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	2	2	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	—	4	1	no	—
131	C₄×C₈	—	4	1	no	—
132	C₄×C₈	—	4	1	no	—
133	C₄×C₈	—	4	1	no	—
134	C₄×C₈	—	4	1	no	—
135	C₄×C₈	—	4	1	no	—
136	C₄×C₈	—	4	1	no	—
137	#95	—	2	2	no	—
138	#95	—	2	2	no	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
148	C₂²×C₄	#120	1	1	yes	characteristic
149	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
150	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂²×C₄	#123	1	1	yes	characteristic
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	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	—	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	—	4	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	8	1	no	—
180	C₂×C₈	—	8	1	no	—
181	C₂×C₈	—	8	1	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	1	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₄⋊C₄	—	4	1	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	8	2	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	2	no	—
200	C₄²	—	4	2	no	—
201	C₄²	C₂×M₁₆	1	1	yes	characteristic
202	C₄²	C₂×M₁₆	1	1	yes	characteristic
203	C₄²	#131	1	1	yes	characteristic
204	C₄²	#131	1	1	yes	characteristic
205	C₄²	C₂×SD₁₆	1	1	yes	characteristic
206	C₄²	C₂×Q₁₆	1	1	yes	characteristic
207	C₄²	#135	1	1	yes	characteristic
208	C₄²	#135	1	1	yes	characteristic
209	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₄²	#137	1	1	yes	characteristic
211	C₄²	#137	1	1	yes	characteristic
212	C₄²	#137	1	1	yes	characteristic
213	C₂³	#385	1	1	yes	characteristic
214	Q₈	—	4	4	no	—
215	Q₈	—	8	2	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	—	2	1	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	#423	1	1	yes	characteristic
232	C₂×C₄	#424	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	#436	1	1	yes	characteristic
237	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
238	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
239	C₂×C₄	#440	1	1	yes	characteristic
240	C₂×C₄	#441	1	1	yes	characteristic
241	C₂×C₄	#443	1	1	yes	characteristic
242	C₂×C₄	#443	1	1	yes	characteristic
243	C₂×C₄	#444	1	1	yes	characteristic
244	C₂×C₄	#444	1	1	yes	characteristic
245	C₂×C₄	#448	1	1	yes	characteristic
246	C₂×C₄	#449	1	1	yes	characteristic
247	C₂×C₄	#451	1	1	yes	characteristic
248	C₂×C₄	#452	1	1	yes	characteristic
249	C₂×C₄	#453	1	1	yes	characteristic
250	C₂×C₄	#455	1	1	yes	characteristic
251	C₂×C₄	#475	1	1	yes	characteristic
252	C₂×C₄	#477	1	1	yes	characteristic
253	C₂×C₄	#479	1	1	yes	characteristic
254	C₂×C₄	#479	1	1	yes	characteristic
255	C₈	—	4	4	no	—
256	C₈	—	4	2	no	—
257	C₈	—	4	2	no	—
258	C₈	—	4	2	no	—
259	C₈	—	4	2	no	—
260	C₈	—	8	1	no	—
261	C₈	—	8	1	no	—
262	C₈	—	8	1	no	—
263	C₈	—	8	1	no	—
264	C₈	—	16	1	no	—
265	C₂²	#1851	1	1	yes	characteristic
266	C₂²	#1856	1	1	yes	characteristic
267	C₂²	#1858	1	1	yes	characteristic
268	C₂²	#1863	1	1	yes	characteristic
269	C₂²	#1865	1	1	yes	characteristic
270	C₂²	#1866	1	1	yes	characteristic
271	C₂²	#1866	1	1	yes	characteristic
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	—	2	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	8	2	no	—
280	C₄	—	8	2	no	—
281	C₄	#1582	1	1	yes	characteristic
282	C₄	#1584	1	1	yes	characteristic
283	C₄	#1585	1	1	yes	characteristic
284	C₄	#1587	1	1	yes	characteristic
285	C₄	#1590	1	1	yes	characteristic
286	C₄	#1591	1	1	yes	characteristic
287	C₄	#1592	1	1	yes	characteristic
288	C₄	#1593	1	1	yes	characteristic
289	C₄	#1622	1	1	yes	characteristic
290	C₄	#1632	1	1	yes	characteristic
291	C₄	#1645	1	1	yes	characteristic
292	C₄	#1646	1	1	yes	characteristic
293	C₂	#8837	1	1	yes	characteristic
294	C₂	#8848	1	1	yes	characteristic
295	C₂	#8850	1	1	yes	characteristic
296	C₂	#8927	1	1	yes	characteristic
297	C₂	#8931	1	1	yes	characteristic
298	C₂	#8933	1	1	yes	characteristic
299	C₂	#8935	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#385
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
#23250	2
#9637	2
#9602	2
#9601	2
C₄×#333	2
#7015	2