Subgroups — #72286838

Order 1024.

380 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#483499	C₂	1	1	yes	maximal, characteristic
2	#149502	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#7099	C₂	1	1	yes	maximal, characteristic, complemented
4	#93758	C₂	1	2	yes	maximal, complemented
5	#93757	C₂	1	2	yes	maximal, complemented
6	C₄×#508	C₄	1	1	yes	characteristic
7	C₈×#135	—	2	1	no	—
8	C₄×M₆₄	—	2	1	no	—
9	#12045	C₄	1	1	yes	characteristic
10	#12045	C₂²	1	1	yes	characteristic
11	#12044	C₄	1	1	yes	characteristic
12	#12044	C₂²	1	1	yes	characteristic
13	#12043	—	2	1	no	—
14	#12043	—	2	1	no	—
15	#12008	C₄	1	1	yes	characteristic
16	C₂×#1367	—	2	1	no	—
17	#11591	—	2	1	no	—
18	C₂×#1311	C₂²	1	1	yes	characteristic
19	#7332	—	2	1	no	—
20	#7332	C₂²	1	2	yes	—
21	#7099	—	2	1	no	—
22	#7099	C₂²	1	2	yes	—
23	C₂×#508	C₈	1	1	yes	characteristic
24	C₈×#41	—	4	1	no	—
25	C₄×#135	—	2	1	no	—
26	C₂×M₆₄	—	2	2	no	—
27	C₄×Q₃₂	C₈	1	1	yes	characteristic
28	C₄×Q₃₂	C₂×C₄	1	1	yes	characteristic
29	C₄×SD₃₂	—	2	1	no	—
30	C₄×SD₃₂	—	2	1	no	—
31	C₄×D₁₆	C₈	1	1	yes	characteristic
32	C₄×D₁₆	C₂×C₄	1	1	yes	characteristic
33	#2158	C₈	1	1	yes	characteristic
34	C₂×C₄⋊C₁₆	—	4	1	no	—
35	#2138	—	2	1	no	—
36	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₆	D₄	1	1	yes	characteristic
38	C₈×Q₁₆	—	2	1	no	—
39	C₈×Q₁₆	—	2	1	no	—
40	C₈×SD₁₆	—	4	1	no	—
41	C₈×SD₁₆	—	4	1	no	—
42	C₈×D₈	—	2	1	no	—
43	C₈×D₈	—	2	1	no	—
44	#1564	—	2	1	no	—
45	#1476	—	4	1	no	—
46	C₂×C₈²	D₄	1	1	yes	characteristic
47	#1397	—	2	2	no	—
48	C₄×C₃₂	—	2	2	no	—
49	#1372	C₂×C₄	1	1	yes	characteristic
50	#1372	C₂×C₄	1	1	yes	characteristic
51	#1371	—	2	1	no	—
52	#1367	—	2	2	no	—
53	#1367	—	4	1	no	—
54	#1332	—	2	1	no	—
55	#1332	—	2	1	no	—
56	#1331	—	2	1	no	—
57	#1331	—	2	1	no	—
58	#1311	C₂×C₄	1	1	yes	characteristic
59	#1311	D₄	1	1	yes	characteristic
60	#1311	D₄	1	1	yes	characteristic
61	#1311	C₂³	1	1	yes	characteristic
62	C₂×#135	—	2	1	no	—
63	C₄×#41	—	4	1	no	—
64	#508	—	2	2	no	—
65	#508	C₁₆	1	4	yes	complemented
66	C₂×Q₃₂	C₂×C₈	1	1	yes	characteristic
67	C₂×SD₃₂	—	2	1	no	—
68	C₂×D₁₆	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₁₆	—	2	1	no	—
70	C₂²×C₁₆	—	8	1	no	—
71	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₁₆	M₁₆	1	1	yes	characteristic
73	C₈×Q₈	—	4	1	no	—
74	C₈×Q₈	—	4	1	no	—
75	C₄×Q₁₆	—	2	1	no	—
76	C₄×Q₁₆	—	2	1	no	—
77	C₄×SD₁₆	—	4	1	no	—
78	C₄×SD₁₆	—	4	1	no	—
79	C₄×D₈	—	2	1	no	—
80	C₄×D₈	—	2	1	no	—
81	C₈×D₄	—	4	1	no	—
82	C₈×D₄	—	4	1	no	—
83	C₈×D₄	—	8	1	no	—
84	C₈×D₄	—	8	1	no	—
85	#431	—	4	1	no	—
86	#431	—	8	1	no	—
87	#427	—	2	1	no	—
88	#416	—	4	1	no	—
89	C₂×C₄×C₈	—	8	1	no	—
90	C₂×C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₈	D₈	1	1	yes	characteristic
92	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
93	M₆₄	—	4	4	no	—
94	C₂×C₃₂	—	2	4	no	—
95	#367	—	2	1	no	—
96	#366	C₂×C₈	1	1	yes	characteristic
97	#366	C₂×C₈	1	1	yes	characteristic
98	C₄⋊C₁₆	—	4	2	no	—
99	C₄⋊C₁₆	—	8	1	no	—
100	#358	—	2	1	no	—
101	#358	—	2	1	no	—
102	#357	—	2	1	no	—
103	#357	—	2	1	no	—
104	C₄×C₁₆	—	2	1	no	—
105	C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
108	C₄×C₁₆	M₁₆	1	1	yes	characteristic
109	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
110	C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
111	#335	—	2	1	no	—
112	#335	—	2	1	no	—
113	#334	—	4	1	no	—
114	Q₈⋊C₈	—	4	1	no	—
115	Q₈⋊C₈	—	4	1	no	—
116	D₄⋊C₈	—	4	1	no	—
117	D₄⋊C₈	—	4	1	no	—
118	C₈²	C₂²⋊C₄	1	1	yes	characteristic
119	C₈²	D₈	1	1	yes	characteristic
120	C₈²	SD₁₆	1	1	yes	characteristic
121	C₈²	C₂×D₄	1	1	yes	characteristic
122	C₂×#41	—	4	1	no	—
123	#135	—	2	4	no	—
124	#135	—	4	2	no	—
125	C₂×Q₁₆	—	2	1	no	—
126	C₂×SD₁₆	—	4	1	no	—
127	C₂×D₈	—	2	1	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	—	8	1	no	—
131	C₂²×C₈	—	8	1	no	—
132	C₂²×C₈	C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₈	#104	1	1	yes	characteristic
134	C₄×Q₈	—	4	1	no	—
135	C₄×Q₈	—	4	1	no	—
136	C₄×D₄	—	4	1	no	—
137	C₄×D₄	—	4	1	no	—
138	C₄×D₄	—	8	1	no	—
139	C₄×D₄	—	8	1	no	—
140	#117	—	4	1	no	—
141	#117	—	8	1	no	—
142	C₂×C₄²	—	8	1	no	—
143	C₂×C₄²	#102	1	1	yes	characteristic
144	Q₃₂	—	2	1	no	—
145	Q₃₂	C₂×C₁₆	1	2	yes	complemented
146	SD₃₂	—	2	4	no	—
147	D₁₆	—	2	1	no	—
148	D₁₆	C₂×C₁₆	1	2	yes	complemented
149	C₂×C₁₆	—	2	4	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	2	1	no	—
153	C₂×C₁₆	—	8	2	no	—
154	C₂×C₁₆	—	16	2	no	—
155	C₂×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
156	C₂×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
157	C₂×C₁₆	C₂×C₁₆	1	2	yes	—
158	C₂×C₁₆	M₃₂	1	2	yes	—
159	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
160	C₂×C₁₆	C₂×M₁₆	1	1	yes	characteristic
161	#107	—	2	1	no	—
162	#107	—	2	1	no	—
163	#106	—	4	1	no	—
164	C₄⋊C₈	—	4	1	no	—
165	C₄⋊C₈	—	4	1	no	—
166	C₄⋊C₈	—	8	1	no	—
167	C₄⋊C₈	—	8	1	no	—
168	#103	—	4	1	no	—
169	#103	—	4	1	no	—
170	#102	—	4	1	no	—
171	#102	—	4	1	no	—
172	C₂²⋊C₈	—	8	1	no	—
173	C₂²⋊C₈	—	8	1	no	—
174	C₂²⋊C₈	—	8	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	—	8	1	no	—
178	C₄×C₈	—	8	1	no	—
179	C₄×C₈	—	8	1	no	—
180	C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
181	C₄×C₈	#102	1	1	yes	characteristic
182	C₄×C₈	#102	1	1	yes	characteristic
183	C₄×C₈	#102	1	1	yes	characteristic
184	C₄×C₈	#104	1	1	yes	characteristic
185	C₄×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
186	C₄×C₈	C₂×D₈	1	1	yes	characteristic
187	C₄×C₈	C₂×SD₁₆	1	1	yes	characteristic
188	C₃₂	—	4	4	no	—
189	#41	—	4	4	no	—
190	#41	—	8	2	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	8	1	no	—
194	C₂²×C₄	—	8	1	no	—
195	C₂²×C₄	—	8	1	no	—
196	C₂²×C₄	—	8	1	no	—
197	C₂²×C₄	D₄⋊C₈	1	1	yes	characteristic
198	C₂²×C₄	#360	1	1	yes	characteristic
199	C₂²×C₄	#361	1	1	yes	characteristic
200	Q₁₆	—	2	2	no	—
201	Q₁₆	—	4	1	no	—
202	SD₁₆	—	4	4	no	—
203	D₈	—	2	2	no	—
204	D₈	—	4	1	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	8	1	no	—
212	C₂×C₈	—	8	1	no	—
213	C₂×C₈	—	8	1	no	—
214	C₂×C₈	—	16	2	no	—
215	C₂×C₈	—	16	2	no	—
216	C₂×C₈	D₄⋊C₈	1	1	yes	characteristic
217	C₂×C₈	D₄⋊C₈	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊C₁₆	1	2	yes	—
219	C₂×C₈	#350	1	2	yes	—
220	C₂×C₈	C₂×C₂²⋊C₈	1	1	yes	characteristic
221	C₂×C₈	C₂×#104	1	1	yes	characteristic
222	C₄⋊C₄	—	4	1	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	8	1	no	—
225	C₄⋊C₄	—	8	1	no	—
226	C₂²⋊C₄	—	8	1	no	—
227	C₂²⋊C₄	—	8	1	no	—
228	C₂²⋊C₄	—	8	1	no	—
229	C₄²	—	8	1	no	—
230	C₄²	—	8	1	no	—
231	C₄²	—	8	1	no	—
232	C₄²	D₄⋊C₈	1	1	yes	characteristic
233	C₄²	#360	1	1	yes	characteristic
234	C₄²	#361	1	1	yes	characteristic
235	C₄²	C₂×#102	1	1	yes	characteristic
236	C₁₆	—	2	2	no	—
237	C₁₆	—	2	2	no	—
238	C₁₆	—	2	1	no	—
239	C₁₆	—	2	1	no	—
240	C₁₆	—	16	2	no	—
241	C₁₆	C₂²⋊C₁₆	1	2	yes	—
242	C₁₆	C₂²×C₁₆	1	1	yes	characteristic
243	C₁₆	C₂×M₃₂	1	1	yes	characteristic
244	C₂³	—	8	1	no	—
245	C₂³	#1333	1	1	yes	characteristic
246	Q₈	—	4	2	no	—
247	Q₈	—	8	1	no	—
248	D₄	—	4	2	no	—
249	D₄	—	8	4	no	—
250	D₄	—	8	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	8	1	no	—
256	C₂×C₄	—	8	1	no	—
257	C₂×C₄	—	16	1	no	—
258	C₂×C₄	—	16	1	no	—
259	C₂×C₄	—	16	2	no	—
260	C₂×C₄	D₄⋊C₁₆	1	2	yes	—
261	C₂×C₄	#1333	1	1	yes	characteristic
262	C₂×C₄	#1333	1	1	yes	characteristic
263	C₂×C₄	#1333	1	1	yes	characteristic
264	C₂×C₄	#1336	1	2	yes	—
265	C₂×C₄	C₂×D₄⋊C₈	1	1	yes	characteristic
266	C₂×C₄	C₂×#360	1	1	yes	characteristic
267	C₂×C₄	C₂×#361	1	1	yes	characteristic
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	4	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	16	1	no	—
275	C₈	—	16	1	no	—
276	C₈	D₄⋊C₁₆	1	2	yes	—
277	C₈	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
278	C₈	C₂×#350	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	8	2	no	—
282	C₂²	—	16	1	no	—
283	C₂²	—	16	1	no	—
284	C₂²	#7070	1	2	yes	—
285	C₂²	C₂×#1333	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	8	2	no	—
290	C₄	—	16	1	no	—
291	C₄	—	16	1	no	—
292	C₄	#7070	1	2	yes	—
293	C₄	#7389	1	2	yes	—
294	C₄	C₂×D₄⋊C₁₆	1	1	yes	characteristic
295	C₄	C₂×#1336	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	16	2	no	—
298	C₂	C₂×#7070	1	1	yes	characteristic
299	C₂	C₂×#7389	1	1	yes	characteristic
300	C₂	#150074	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#1333
commutator	C₁₆	C₂²×C₁₆
Frattini, cosocle	#1311	C₂³
socle	C₂²	C₂×#1333
prime radical	C₂×#508	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1311 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #149502	C₂	semidirect
3. C₂×#7099	C₂	semidirect
4. #93758	C₂	semidirect
5. #93757	C₂	semidirect
65. #508	C₁₆	semidirect
145. Q₃₂	C₂×C₁₆	semidirect
148. D₁₆	C₂×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#483499	2
#149502	2
C₂×#7099	2
#93758	2
#93757	2