Subgroups — #99352

Order 512.

374 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23153	C₂	1	1	yes	maximal, characteristic
2	#9870	C₂	1	1	yes	maximal, characteristic
3	#9602	C₂	1	1	yes	maximal, characteristic
4	#9602	C₂	1	1	yes	maximal, characteristic
5	C₄×#332	C₂	1	1	yes	maximal, characteristic
6	#7015	C₂	1	1	yes	maximal, characteristic
7	#7015	C₂	1	1	yes	maximal, characteristic
8	C₂×#446	—	2	1	no	—
9	C₄×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#1984	—	2	1	no	—
11	#1939	—	2	1	no	—
12	#1917	—	2	2	no	—
13	#1842	C₂²	1	1	yes	characteristic
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×#334	—	2	1	no	—
21	C₂×#334	—	2	1	no	—
22	C₂×#332	C₄	1	1	yes	characteristic
23	C₂×#332	C₄	1	1	yes	characteristic
24	#1296	C₄	1	1	yes	characteristic
25	#1296	C₄	1	1	yes	characteristic
26	#1279	—	2	2	no	—
27	#1277	—	2	1	no	—
28	#1277	—	2	1	no	—
29	C₂×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Q₈	—	2	1	no	—
31	#446	—	4	2	no	—
32	#446	—	4	2	no	—
33	C₂×C₄⋊C₈	—	2	2	no	—
34	C₂×C₄⋊C₈	—	2	1	no	—
35	C₂×C₄⋊C₈	—	2	1	no	—
36	C₂×C₄⋊C₈	—	2	1	no	—
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	1	yes	characteristic
41	C₂×C₄⋊C₈	D₄	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×#97	—	2	2	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	C₂×C₄×C₈	—	2	1	no	—
52	C₂×C₄×C₈	—	2	1	no	—
53	#391	—	2	1	no	—
54	#389	—	2	1	no	—
55	#389	Q₈	1	2	yes	—
56	#384	C₂×C₄	1	1	yes	characteristic
57	C₄×C₄⋊C₄	—	2	1	no	—
58	C₄×C₄⋊C₄	D₄	1	2	yes	—
59	C₄³	C₂³	1	1	yes	characteristic
60	#336	—	2	2	no	—
61	#336	—	2	2	no	—
62	#336	—	2	1	no	—
63	#336	—	2	1	no	—
64	#336	—	2	1	no	—
65	#336	—	2	1	no	—
66	#334	—	4	2	no	—
67	#334	—	4	2	no	—
68	#332	—	2	2	no	—
69	#332	—	2	2	no	—
70	C₂²×Q₈	—	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	—	2	1	no	—
79	C₂²×C₈	—	4	1	no	—
80	C₂²×C₈	—	4	1	no	—
81	C₄⋊Q₈	—	2	1	no	—
82	C₄⋊Q₈	—	2	1	no	—
83	C₄⋊Q₈	—	2	1	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₄×Q₈	—	4	1	no	—
86	C₄×Q₈	—	4	1	no	—
87	C₄×Q₈	—	4	1	no	—
88	C₄×Q₈	—	4	1	no	—
89	C₂×C₄⋊C₄	—	2	1	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄⋊C₄	—	2	2	no	—
93	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
94	C₂×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
95	C₂×C₄²	—	2	2	no	—
96	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₄²	#41	1	1	yes	characteristic
102	C₂×C₄²	#41	1	1	yes	characteristic
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	—	2	1	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₄⋊C₈	—	2	1	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₄⋊C₈	—	4	4	no	—
117	C₄⋊C₈	—	4	1	no	—
118	C₄⋊C₈	—	4	1	no	—
119	C₄⋊C₈	—	4	1	no	—
120	C₄⋊C₈	—	4	1	no	—
121	C₄⋊C₈	M₁₆	1	2	yes	—
122	C₄⋊C₈	M₁₆	1	2	yes	—
123	C₄⋊C₈	SD₁₆	1	2	yes	—
124	C₄⋊C₈	SD₁₆	1	2	yes	—
125	#97	—	4	4	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	2	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	C₄×C₈	—	4	1	no	—
138	C₄×C₈	—	4	1	no	—
139	C₄×C₈	—	4	1	no	—
140	#95	—	2	2	no	—
141	#95	—	2	2	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	4	1	no	—
144	C₂×Q₈	—	4	1	no	—
145	C₂×Q₈	—	4	1	no	—
146	C₂×Q₈	—	4	1	no	—
147	C₂×Q₈	—	4	1	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
152	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
153	C₂²×C₄	#120	1	1	yes	characteristic
154	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂²×C₄	#123	1	1	yes	characteristic
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	4	2	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	—	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	—	4	1	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	8	1	no	—
185	C₂×C₈	—	8	1	no	—
186	C₂×C₈	—	8	1	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄⋊C₄	—	4	1	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	4	2	no	—
206	C₄²	—	4	2	no	—
207	C₄²	C₂×M₁₆	1	1	yes	characteristic
208	C₄²	C₂×M₁₆	1	1	yes	characteristic
209	C₄²	#131	1	1	yes	characteristic
210	C₄²	#131	1	1	yes	characteristic
211	C₄²	C₂×SD₁₆	1	1	yes	characteristic
212	C₄²	C₂×SD₁₆	1	1	yes	characteristic
213	C₄²	#135	1	1	yes	characteristic
214	C₄²	#135	1	1	yes	characteristic
215	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₄²	#137	1	1	yes	characteristic
218	C₄²	#137	1	1	yes	characteristic
219	C₂³	#385	1	1	yes	characteristic
220	Q₈	—	8	1	no	—
221	Q₈	—	8	1	no	—
222	Q₈	—	8	1	no	—
223	Q₈	—	8	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	#423	1	1	yes	characteristic
240	C₂×C₄	#424	1	1	yes	characteristic
241	C₂×C₄	#435	1	1	yes	characteristic
242	C₂×C₄	#435	1	1	yes	characteristic
243	C₂×C₄	#436	1	1	yes	characteristic
244	C₂×C₄	#436	1	1	yes	characteristic
245	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
246	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
247	C₂×C₄	#441	1	1	yes	characteristic
248	C₂×C₄	#442	1	1	yes	characteristic
249	C₂×C₄	#443	1	1	yes	characteristic
250	C₂×C₄	#443	1	1	yes	characteristic
251	C₂×C₄	#444	1	1	yes	characteristic
252	C₂×C₄	#444	1	1	yes	characteristic
253	C₂×C₄	#448	1	1	yes	characteristic
254	C₂×C₄	#449	1	1	yes	characteristic
255	C₂×C₄	#450	1	1	yes	characteristic
256	C₂×C₄	#452	1	1	yes	characteristic
257	C₂×C₄	#454	1	1	yes	characteristic
258	C₂×C₄	#456	1	1	yes	characteristic
259	C₂×C₄	#475	1	1	yes	characteristic
260	C₂×C₄	#476	1	1	yes	characteristic
261	C₂×C₄	#478	1	1	yes	characteristic
262	C₂×C₄	#479	1	1	yes	characteristic
263	C₈	—	4	4	no	—
264	C₈	—	4	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	4	2	no	—
267	C₈	—	4	2	no	—
268	C₈	—	8	1	no	—
269	C₈	—	8	1	no	—
270	C₈	—	8	1	no	—
271	C₈	—	8	1	no	—
272	C₈	—	16	1	no	—
273	C₂²	#1851	1	1	yes	characteristic
274	C₂²	#1857	1	1	yes	characteristic
275	C₂²	#1858	1	1	yes	characteristic
276	C₂²	#1864	1	1	yes	characteristic
277	C₂²	#1865	1	1	yes	characteristic
278	C₂²	#1866	1	1	yes	characteristic
279	C₂²	#1866	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	8	2	no	—
288	C₄	—	8	2	no	—
289	C₄	#1582	1	1	yes	characteristic
290	C₄	#1583	1	1	yes	characteristic
291	C₄	#1586	1	1	yes	characteristic
292	C₄	#1587	1	1	yes	characteristic
293	C₄	#1590	1	1	yes	characteristic
294	C₄	#1590	1	1	yes	characteristic
295	C₄	#1593	1	1	yes	characteristic
296	C₄	#1594	1	1	yes	characteristic
297	C₄	#1626	1	1	yes	characteristic
298	C₄	#1635	1	1	yes	characteristic
299	C₄	#1650	1	1	yes	characteristic
300	C₄	#1651	1	1	yes	characteristic
301	C₂	#8838	1	1	yes	characteristic
302	C₂	#8849	1	1	yes	characteristic
303	C₂	#8850	1	1	yes	characteristic
304	C₂	#8925	1	1	yes	characteristic
305	C₂	#8931	1	1	yes	characteristic
306	C₂	#8932	1	1	yes	characteristic
307	C₂	#8936	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
#23153	2
#9870	2
#9602	2
C₄×#332	2
#7015	2