Subgroups — #72320492

Order 1024.

658 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#483760	C₂	1	2	yes	maximal, complemented
2	#483499	C₂	1	1	yes	maximal, characteristic
3	#122963	C₂	1	2	yes	maximal, complemented
4	#122962	C₂	1	2	yes	maximal, complemented
5	#122961	C₂	1	4	yes	maximal, complemented
6	C₂×#7099	C₂	1	1	yes	maximal, characteristic, complemented
7	#122465	C₂	1	2	yes	maximal, complemented
8	#122214	C₂	1	1	yes	maximal, characteristic, complemented
9	#33502	—	2	2	no	—
10	C₄×#508	C₄	1	1	yes	characteristic
11	C₈×#135	C₂²	1	2	yes	—
12	#12062	C₂²	1	4	yes	—
13	#12059	C₂²	1	4	yes	—
14	C₁₆×SD₁₆	—	2	4	no	—
15	#12045	C₄	1	1	yes	characteristic
16	#12045	C₂²	1	1	yes	characteristic
17	#12044	C₄	1	1	yes	characteristic
18	#12044	C₂²	1	1	yes	characteristic
19	#12043	C₄	1	2	yes	—
20	#12043	C₂²	1	2	yes	—
21	#12008	C₄	1	1	yes	characteristic
22	C₂×#1367	C₂²	1	2	yes	—
23	#11621	—	2	2	no	—
24	#11591	C₄	1	2	yes	—
25	#11410	C₂²	1	1	yes	characteristic
26	#11410	C₂²	1	1	yes	characteristic
27	C₂×#1311	C₂²	1	1	yes	characteristic
28	#7099	C₂²	1	2	yes	—
29	#7099	C₂²	1	2	yes	—
30	#7071	C₂²	1	4	yes	—
31	#7070	C₂²	1	4	yes	—
32	#7053	C₂²	1	2	yes	—
33	#7053	C₂²	1	2	yes	—
34	C₂×#508	C₈	1	1	yes	characteristic
35	C₈×#41	—	2	2	no	—
36	C₄×#135	C₂×C₄	1	2	yes	—
37	Q₈×C₁₆	—	2	4	no	—
38	C₄×Q₃₂	C₈	1	1	yes	characteristic
39	C₄×Q₃₂	C₂×C₄	1	1	yes	characteristic
40	C₄×SD₃₂	C₈	1	2	yes	—
41	C₄×SD₃₂	C₂×C₄	1	2	yes	—
42	C₄×D₁₆	C₈	1	1	yes	characteristic
43	C₄×D₁₆	C₂×C₄	1	1	yes	characteristic
44	D₄×C₁₆	—	2	4	no	—
45	D₄×C₁₆	—	4	4	no	—
46	#2162	—	2	1	no	—
47	#2162	—	4	2	no	—
48	#2162	D₄	1	2	yes	—
49	#2158	C₈	1	1	yes	characteristic
50	C₂×C₄⋊C₁₆	—	2	2	no	—
51	C₂×C₄⋊C₁₆	—	4	2	no	—
52	#2138	C₈	1	2	yes	—
53	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
55	C₈×Q₁₆	C₂×C₄	1	2	yes	—
56	C₈×Q₁₆	C₂³	1	2	yes	—
57	C₈×SD₁₆	—	2	2	no	—
58	C₈×SD₁₆	—	2	2	no	—
59	C₈×D₈	C₂×C₄	1	2	yes	—
60	C₈×D₈	C₂³	1	2	yes	—
61	#1564	C₂×C₄	1	2	yes	—
62	#1476	—	2	2	no	—
63	C₂×C₈²	C₂³	1	1	yes	characteristic
64	#1372	C₂×C₄	1	1	yes	characteristic
65	#1372	C₂×C₄	1	1	yes	characteristic
66	#1371	C₂×C₄	1	2	yes	—
67	#1367	—	2	2	no	—
68	#1367	C₂³	1	4	yes	—
69	Q₈⋊C₁₆	—	2	4	no	—
70	#1332	C₂×C₄	1	2	yes	—
71	#1332	C₂×C₄	1	2	yes	—
72	#1331	C₂×C₄	1	2	yes	—
73	#1331	C₂×C₄	1	2	yes	—
74	D₄⋊C₁₆	—	2	4	no	—
75	#1312	C₂³	1	2	yes	—
76	#1312	C₂³	1	2	yes	—
77	#1311	C₂×C₄	1	1	yes	characteristic
78	#1311	C₂×C₄	1	1	yes	characteristic
79	#1311	C₂³	1	1	yes	characteristic
80	#1311	C₂³	1	1	yes	characteristic
81	C₈×C₁₆	—	2	1	no	—
82	C₈×C₁₆	D₄	1	2	yes	—
83	C₂×#135	C₂×C₈	1	2	yes	—
84	C₄×#41	—	2	2	no	—
85	#508	C₁₆	1	8	yes	complemented
86	C₂×Q₃₂	C₂×C₈	1	1	yes	characteristic
87	C₂×SD₃₂	C₂×C₈	1	2	yes	—
88	C₂×D₁₆	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₁₆	—	2	1	no	—
90	C₂²×C₁₆	—	4	4	no	—
91	C₂²×C₁₆	—	4	2	no	—
92	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
93	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
94	C₈×Q₈	—	2	2	no	—
95	C₈×Q₈	—	2	2	no	—
96	C₄×Q₁₆	C₂×C₈	1	2	yes	—
97	C₄×Q₁₆	C₂²×C₄	1	2	yes	—
98	C₄×SD₁₆	—	2	2	no	—
99	C₄×SD₁₆	—	2	2	no	—
100	C₄×D₈	C₂×C₈	1	2	yes	—
101	C₄×D₈	C₂²×C₄	1	2	yes	—
102	C₈×D₄	—	2	2	no	—
103	C₈×D₄	—	2	2	no	—
104	C₈×D₄	—	4	2	no	—
105	C₈×D₄	—	4	2	no	—
106	#431	—	2	2	no	—
107	#431	—	4	2	no	—
108	#427	C₂×C₈	1	2	yes	—
109	#416	—	2	2	no	—
110	C₂×C₄×C₈	—	4	2	no	—
111	C₂×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈	#41	1	1	yes	characteristic
114	#367	C₂×C₈	1	2	yes	—
115	#366	C₂×C₈	1	1	yes	characteristic
116	#366	C₂×C₈	1	1	yes	characteristic
117	C₄⋊C₁₆	—	2	2	no	—
118	C₄⋊C₁₆	—	2	4	no	—
119	C₄⋊C₁₆	—	4	1	no	—
120	C₄⋊C₁₆	—	4	4	no	—
121	C₄⋊C₁₆	—	4	2	no	—
122	#358	C₂×C₈	1	2	yes	—
123	#358	C₂×C₈	1	2	yes	—
124	#357	C₂×C₈	1	2	yes	—
125	#357	C₂×C₈	1	2	yes	—
126	C₂²⋊C₁₆	—	2	2	no	—
127	C₂²⋊C₁₆	—	4	4	no	—
128	C₂²⋊C₁₆	—	4	1	no	—
129	C₄×C₁₆	—	2	1	no	—
130	C₄×C₁₆	—	2	2	no	—
131	C₄×C₁₆	—	4	4	no	—
132	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
133	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
134	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
135	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
136	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
137	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
138	C₄×C₁₆	C₂×D₄	1	4	yes	—
139	#335	C₂²×C₄	1	2	yes	—
140	#335	C₂²×C₄	1	2	yes	—
141	#334	—	2	2	no	—
142	Q₈⋊C₈	—	2	2	no	—
143	Q₈⋊C₈	—	2	2	no	—
144	D₄⋊C₈	—	2	2	no	—
145	D₄⋊C₈	—	2	2	no	—
146	C₈²	C₂²×C₄	1	1	yes	characteristic
147	C₈²	C₂×D₄	1	1	yes	characteristic
148	C₈²	#41	1	1	yes	characteristic
149	C₈²	C₂⁴	1	1	yes	characteristic
150	C₂×#41	—	2	2	no	—
151	#135	—	2	4	no	—
152	#135	C₂×C₁₆	1	8	yes	complemented
153	C₂×Q₁₆	C₂²×C₈	1	2	yes	—
154	C₂×SD₁₆	—	2	2	no	—
155	C₂×D₈	C₂²×C₈	1	2	yes	—
156	C₂²×C₈	—	2	1	no	—
157	C₂²×C₈	—	4	2	no	—
158	C₂²×C₈	—	4	2	no	—
159	C₂²×C₈	C₂²×C₈	1	1	yes	characteristic
160	C₂²×C₈	#131	1	1	yes	characteristic
161	C₂²×C₈	AGL(1,ℤ/8)	1	2	yes	—
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	C₄×D₄	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₄×D₄	—	4	2	no	—
167	C₄×D₄	—	4	2	no	—
168	#117	—	2	2	no	—
169	#117	—	4	2	no	—
170	C₂×C₄²	—	4	2	no	—
171	C₂×C₄²	C₄×D₄	1	1	yes	characteristic
172	Q₃₂	C₂×C₁₆	1	4	yes	complemented
173	SD₃₂	C₂×C₁₆	1	8	yes	complemented
174	D₁₆	C₂×C₁₆	1	4	yes	complemented
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	4	no	—
179	C₂×C₁₆	—	4	2	no	—
180	C₂×C₁₆	—	4	2	no	—
181	C₂×C₁₆	—	4	4	no	—
182	C₂×C₁₆	—	16	4	no	—
183	C₂×C₁₆	—	16	2	no	—
184	C₂×C₁₆	C₂×C₁₆	1	4	yes	complemented
185	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
186	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
187	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
188	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
189	#107	C₂²×C₈	1	2	yes	—
190	#107	C₂²×C₈	1	2	yes	—
191	#106	—	2	2	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	4	2	no	—
195	C₄⋊C₈	—	4	2	no	—
196	#103	—	2	2	no	—
197	#103	—	2	2	no	—
198	#102	—	2	2	no	—
199	#102	—	2	2	no	—
200	C₂²⋊C₈	—	4	2	no	—
201	C₂²⋊C₈	—	4	2	no	—
202	C₂²⋊C₈	—	4	2	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	4	2	no	—
205	C₄×C₈	—	4	2	no	—
206	C₄×C₈	—	4	2	no	—
207	C₄×C₈	C₄×D₄	1	1	yes	characteristic
208	C₄×C₈	C₄×D₄	1	1	yes	characteristic
209	C₄×C₈	C₄×D₄	1	1	yes	characteristic
210	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
211	C₄×C₈	#131	1	1	yes	characteristic
212	C₄×C₈	AGL(1,ℤ/8)	1	2	yes	—
213	C₄×C₈	C₂³×C₄	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₄	1	1	yes	characteristic
215	C₄×C₈	C₂×#41	1	1	yes	characteristic
216	#41	—	2	8	no	—
217	#41	—	4	4	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	C₈×D₄	1	1	yes	characteristic
225	C₂²×C₄	#442	1	1	yes	characteristic
226	C₂²×C₄	#444	1	1	yes	characteristic
227	Q₁₆	—	2	2	no	—
228	Q₁₆	C₂²×C₁₆	1	4	yes	—
229	SD₁₆	—	2	8	no	—
230	D₈	—	2	2	no	—
231	D₈	C₂²×C₁₆	1	4	yes	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	8	2	no	—
239	C₂×C₈	—	8	2	no	—
240	C₂×C₈	—	8	1	no	—
241	C₂×C₈	—	16	2	no	—
242	C₂×C₈	C₈×D₄	1	1	yes	characteristic
243	C₂×C₈	C₈×D₄	1	1	yes	characteristic
244	C₂×C₈	#442	1	2	yes	—
245	C₂×C₈	C₂²×C₁₆	1	2	yes	—
246	C₂×C₈	#504	1	2	yes	—
247	C₂×C₈	C₂³×C₈	1	1	yes	characteristic
248	C₂×C₈	C₂×#131	1	1	yes	characteristic
249	C₂×C₈	C₂×AGL(1,ℤ/8)	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	C₈×D₄	1	1	yes	characteristic
261	C₄²	#442	1	1	yes	characteristic
262	C₄²	#444	1	1	yes	characteristic
263	C₄²	C₂×C₄×D₄	1	1	yes	characteristic
264	C₁₆	—	2	1	no	—
265	C₁₆	—	2	1	no	—
266	C₁₆	—	4	1	no	—
267	C₁₆	—	4	1	no	—
268	C₁₆	—	4	4	no	—
269	C₁₆	—	8	4	no	—
270	C₁₆	—	8	2	no	—
271	C₁₆	C₂²×C₁₆	1	2	yes	—
272	C₁₆	C₂²×C₁₆	1	2	yes	—
273	C₂³	—	4	2	no	—
274	C₂³	#1580	1	1	yes	characteristic
275	Q₈	—	2	4	no	—
276	Q₈	—	4	2	no	—
277	D₄	—	2	4	no	—
278	D₄	—	4	8	no	—
279	D₄	—	4	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	8	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	16	2	no	—
289	C₂×C₄	#1580	1	1	yes	characteristic
290	C₂×C₄	#1580	1	1	yes	characteristic
291	C₂×C₄	#1580	1	1	yes	characteristic
292	C₂×C₄	D₄×C₁₆	1	2	yes	—
293	C₂×C₄	#2171	1	2	yes	—
294	C₂×C₄	C₂×C₈×D₄	1	1	yes	characteristic
295	C₂×C₄	C₂×#442	1	1	yes	characteristic
296	C₂×C₄	C₂×#444	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	2	no	—
300	C₈	—	2	2	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	1	no	—
303	C₈	—	8	2	no	—
304	C₈	—	8	2	no	—
305	C₈	D₄×C₁₆	1	2	yes	—
306	C₈	C₂³×C₁₆	1	1	yes	characteristic
307	C₈	C₂×#504	1	1	yes	characteristic
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	#12062	1	2	yes	—
314	C₂²	C₂×#1580	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	4	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	#12062	1	2	yes	—
322	C₄	C₂×D₄×C₁₆	1	1	yes	characteristic
323	C₄	C₂×#2171	1	1	yes	characteristic
324	C₄	#33469	1	2	yes	—
325	C₂	—	2	2	no	—
326	C₂	—	8	4	no	—
327	C₂	C₂×#12062	1	1	yes	characteristic
328	C₂	#483597	1	1	yes	characteristic
329	C₂	#483601	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#442
commutator	C₈	C₂³×C₁₆
Frattini, cosocle	C₈²	C₂⁴
socle	C₂²	C₂×#1580
prime radical	C₂×#508	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #11410 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #483760	C₂	semidirect
3. #122963	C₂	semidirect
4. #122962	C₂	semidirect
5. #122961	C₂	semidirect
6. C₂×#7099	C₂	semidirect
7. #122465	C₂	semidirect
8. #122214	C₂	semidirect
85. #508	C₁₆	semidirect
152. #135	C₂×C₁₆	semidirect
172. Q₃₂	C₂×C₁₆	semidirect
173. SD₃₂	C₂×C₁₆	semidirect
174. D₁₆	C₂×C₁₆	semidirect
184. C₂×C₁₆	C₂×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#483760	2
#483499	2
#122963	2
#122962	2
#122961	2
C₂×#7099	2
#122465	2
#122214	2