Subgroups — #72320554

Order 1024.

574 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#483944	C₂	1	1	yes	maximal, characteristic
2	#483611	C₂	1	2	yes	maximal, complemented
3	#122970	C₂	1	2	yes	maximal, complemented
4	#122967	C₂	1	2	yes	maximal, complemented
5	#122966	C₂	1	1	yes	maximal, characteristic, complemented
6	#122965	C₂	1	1	yes	maximal, characteristic, complemented
7	#122963	C₂	1	1	yes	maximal, characteristic, complemented
8	#122962	C₂	1	1	yes	maximal, characteristic, complemented
9	#122919	C₂	1	1	yes	maximal, characteristic, complemented
10	#122470	C₂	1	2	yes	maximal, complemented
11	#122219	C₂	1	1	yes	maximal, characteristic, complemented
12	#33735	C₄	1	1	yes	characteristic
13	#33483	—	2	2	no	—
14	#17959	C₂²	1	2	yes	—
15	#12086	C₄	1	1	yes	characteristic
16	#12085	C₄	1	1	yes	characteristic
17	#12084	C₄	1	2	yes	—
18	#12064	C₂²	1	2	yes	—
19	#12062	C₂²	1	2	yes	—
20	#12061	C₂²	1	2	yes	—
21	#12059	C₂²	1	2	yes	—
22	#12057	—	2	2	no	—
23	#12056	—	2	2	no	—
24	#12045	C₂²	1	1	yes	characteristic
25	#12044	C₂²	1	1	yes	characteristic
26	#12043	C₂²	1	2	yes	—
27	#12024	C₄	1	1	yes	characteristic
28	#12012	C₂²	1	2	yes	—
29	#11673	C₄	1	2	yes	—
30	#11614	—	2	2	no	—
31	#11411	C₂²	1	1	yes	characteristic
32	#11402	C₂²	1	1	yes	characteristic
33	C₂×#1311	C₂²	1	1	yes	characteristic
34	#7099	C₂²	1	1	yes	characteristic
35	#7099	C₂²	1	1	yes	characteristic
36	#7099	C₂²	1	2	yes	—
37	#7071	C₂²	1	2	yes	—
38	#7071	C₂²	1	2	yes	—
39	#7070	C₂²	1	2	yes	—
40	#7070	C₂²	1	2	yes	—
41	#7053	C₂²	1	1	yes	characteristic
42	#7053	C₂²	1	1	yes	characteristic
43	#7053	C₂²	1	1	yes	characteristic
44	#7053	C₂²	1	1	yes	characteristic
45	#3168	C₂×C₄	1	2	yes	—
46	#2966	—	2	2	no	—
47	#2253	C₈	1	1	yes	characteristic
48	#2249	C₈	1	1	yes	characteristic
49	#2242	C₈	1	2	yes	—
50	#2217	C₈	1	2	yes	—
51	#2216	C₈	1	1	yes	characteristic
52	#2215	C₈	1	1	yes	characteristic
53	#2183	—	2	2	no	—
54	Q₈×C₁₆	—	2	2	no	—
55	C₄×Q₃₂	C₂×C₄	1	1	yes	characteristic
56	C₄×SD₃₂	C₂×C₄	1	2	yes	—
57	C₄×D₁₆	C₂×C₄	1	1	yes	characteristic
58	#2169	—	2	2	no	—
59	#2169	—	4	2	no	—
60	#2168	—	4	2	no	—
61	D₄×C₁₆	—	2	2	no	—
62	#2164	—	4	2	no	—
63	#2162	—	2	1	no	—
64	#2162	—	2	2	no	—
65	#2108	—	4	2	no	—
66	C₄×M₃₂	D₄	1	2	yes	—
67	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
69	#1591	C₂×C₄	1	2	yes	—
70	#1590	—	2	2	no	—
71	#1589	C₂×C₄	1	2	yes	—
72	C₈×Q₁₆	C₂³	1	2	yes	—
73	C₈×SD₁₆	—	2	2	no	—
74	C₈×D₈	C₂³	1	2	yes	—
75	#1568	C₂×C₄	1	2	yes	—
76	#1494	—	2	2	no	—
77	C₂×C₈²	C₂³	1	1	yes	characteristic
78	#1372	C₂×C₄	1	1	yes	characteristic
79	#1372	C₂×C₄	1	1	yes	characteristic
80	#1371	C₂×C₄	1	2	yes	—
81	#1370	—	2	2	no	—
82	#1367	C₂³	1	2	yes	—
83	#1367	C₂³	1	2	yes	—
84	Q₈⋊C₁₆	—	2	2	no	—
85	Q₈⋊C₁₆	—	2	2	no	—
86	#1332	C₂×C₄	1	2	yes	—
87	#1332	C₂×C₄	1	2	yes	—
88	#1331	C₂×C₄	1	2	yes	—
89	#1331	C₂×C₄	1	2	yes	—
90	D₄⋊C₁₆	—	2	2	no	—
91	D₄⋊C₁₆	—	2	2	no	—
92	#1312	D₄	1	2	yes	—
93	#1312	C₂³	1	1	yes	characteristic
94	#1312	C₂³	1	1	yes	characteristic
95	#1312	C₂³	1	1	yes	characteristic
96	#1312	C₂³	1	1	yes	characteristic
97	#1311	—	2	1	no	—
98	#1311	C₂×C₄	1	1	yes	characteristic
99	#1311	C₂×C₄	1	1	yes	characteristic
100	#1311	C₂³	1	1	yes	characteristic
101	#1311	C₂³	1	1	yes	characteristic
102	#529	—	2	2	no	—
103	C₂×Q₃₂	C₂×C₈	1	1	yes	characteristic
104	C₂×SD₃₂	C₂×C₈	1	2	yes	—
105	C₂×D₁₆	C₂×C₈	1	1	yes	characteristic
106	C₂×M₃₂	—	2	2	no	—
107	C₂×M₃₂	—	4	2	no	—
108	C₂²×C₁₆	—	2	1	no	—
109	C₂²×C₁₆	—	4	2	no	—
110	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
111	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
112	#499	C₂×C₈	1	2	yes	—
113	#495	C₂×C₈	1	2	yes	—
114	#488	—	2	2	no	—
115	#467	C₂×C₈	1	2	yes	—
116	#466	C₂×C₈	1	2	yes	—
117	#465	—	2	2	no	—
118	#446	—	2	2	no	—
119	C₈×Q₈	—	2	2	no	—
120	C₄×Q₁₆	C₂²×C₄	1	2	yes	—
121	C₄×SD₁₆	—	2	2	no	—
122	C₄×D₈	C₂²×C₄	1	2	yes	—
123	#436	—	2	2	no	—
124	#435	—	4	2	no	—
125	C₈×D₄	—	2	2	no	—
126	C₈×D₄	—	4	2	no	—
127	#433	—	4	2	no	—
128	#432	—	2	2	no	—
129	#405	—	4	2	no	—
130	C₂×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈	#41	1	1	yes	characteristic
133	#367	C₂×C₈	1	2	yes	—
134	#366	C₂×C₈	1	1	yes	characteristic
135	#366	C₂×C₈	1	1	yes	characteristic
136	C₄⋊C₁₆	—	2	2	no	—
137	C₄⋊C₁₆	—	2	2	no	—
138	C₄⋊C₁₆	—	4	1	no	—
139	C₄⋊C₁₆	—	4	2	no	—
140	C₄⋊C₁₆	—	4	2	no	—
141	#358	C₂×C₈	1	2	yes	—
142	#358	C₂×C₈	1	2	yes	—
143	#357	C₂×C₈	1	2	yes	—
144	#357	C₂×C₈	1	2	yes	—
145	C₂²⋊C₁₆	—	4	2	no	—
146	C₂²⋊C₁₆	—	4	2	no	—
147	C₂²⋊C₁₆	—	4	1	no	—
148	C₂²⋊C₁₆	—	4	2	no	—
149	#346	—	2	2	no	—
150	#346	—	4	2	no	—
151	C₄×C₁₆	—	2	1	no	—
152	C₄×C₁₆	—	2	1	no	—
153	C₄×C₁₆	—	2	1	no	—
154	C₄×C₁₆	—	4	2	no	—
155	C₄×C₁₆	—	4	2	no	—
156	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
157	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
158	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
159	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
160	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
161	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
162	C₄×C₁₆	C₂×D₄	1	2	yes	—
163	C₄×C₁₆	C₂×D₄	1	2	yes	—
164	#335	C₂²×C₄	1	2	yes	—
165	#335	C₂²×C₄	1	2	yes	—
166	#334	—	2	2	no	—
167	Q₈⋊C₈	—	2	2	no	—
168	Q₈⋊C₈	—	2	2	no	—
169	D₄⋊C₈	—	2	2	no	—
170	D₄⋊C₈	—	2	2	no	—
171	C₈²	C₂²×C₄	1	1	yes	characteristic
172	C₈²	C₂×D₄	1	1	yes	characteristic
173	C₈²	#41	1	1	yes	characteristic
174	C₈²	C₂⁴	1	1	yes	characteristic
175	C₂×Q₁₆	C₂²×C₈	1	2	yes	—
176	C₂×SD₁₆	—	2	2	no	—
177	C₂×D₈	C₂²×C₈	1	2	yes	—
178	C₂×M₁₆	—	4	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	4	2	no	—
181	C₂²×C₈	C₂²×C₈	1	1	yes	characteristic
182	C₂²×C₈	#131	1	1	yes	characteristic
183	C₂²×C₈	AGL(1,ℤ/8)	1	2	yes	—
184	#126	—	4	2	no	—
185	#125	—	2	2	no	—
186	#124	—	2	2	no	—
187	#123	—	4	2	no	—
188	C₂²⋊Q₈	—	2	2	no	—
189	#121	—	2	2	no	—
190	C₄×Q₈	—	2	2	no	—
191	C₄×D₄	—	2	2	no	—
192	C₄×D₄	—	4	2	no	—
193	#117	—	4	2	no	—
194	C₂×C₄²	C₄×D₄	1	1	yes	characteristic
195	Q₃₂	—	2	2	no	—
196	SD₃₂	—	4	2	no	—
197	D₁₆	—	2	2	no	—
198	M₃₂	—	8	2	no	—
199	M₃₂	—	8	4	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	—	2	1	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	2	2	no	—
205	C₂×C₁₆	—	4	1	no	—
206	C₂×C₁₆	—	4	2	no	—
207	C₂×C₁₆	—	4	1	no	—
208	C₂×C₁₆	—	4	1	no	—
209	C₂×C₁₆	—	4	2	no	—
210	C₂×C₁₆	—	4	2	no	—
211	C₂×C₁₆	—	8	4	no	—
212	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
213	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
214	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
215	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
216	#107	C₂²×C₈	1	2	yes	—
217	#107	C₂²×C₈	1	2	yes	—
218	#106	—	2	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	2	2	no	—
221	C₄⋊C₈	—	4	2	no	—
222	C₄⋊C₈	—	4	2	no	—
223	#103	—	2	2	no	—
224	#103	—	2	2	no	—
225	#102	—	2	2	no	—
226	#102	—	2	2	no	—
227	C₂²⋊C₈	—	4	2	no	—
228	C₂²⋊C₈	—	4	2	no	—
229	C₂²⋊C₈	—	4	2	no	—
230	#97	—	4	2	no	—
231	#97	—	4	2	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	4	2	no	—
234	C₄×C₈	C₄×D₄	1	1	yes	characteristic
235	C₄×C₈	C₄×D₄	1	1	yes	characteristic
236	C₄×C₈	C₄×D₄	1	1	yes	characteristic
237	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
238	C₄×C₈	#131	1	1	yes	characteristic
239	C₄×C₈	AGL(1,ℤ/8)	1	2	yes	—
240	C₄×C₈	C₂³×C₄	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₄	1	1	yes	characteristic
242	C₄×C₈	C₂×#41	1	1	yes	characteristic
243	C₂×Q₈	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₈×D₄	1	1	yes	characteristic
248	C₂²×C₄	#442	1	1	yes	characteristic
249	C₂²×C₄	#444	1	1	yes	characteristic
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	2	2	no	—
252	SD₁₆	—	8	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	2	2	no	—
255	M₁₆	—	8	4	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	8	2	no	—
265	C₂×C₈	—	8	2	no	—
266	C₂×C₈	—	8	1	no	—
267	C₂×C₈	C₈×D₄	1	1	yes	characteristic
268	C₂×C₈	C₈×D₄	1	1	yes	characteristic
269	C₂×C₈	#442	1	2	yes	—
270	C₂×C₈	C₂³×C₈	1	1	yes	characteristic
271	C₂×C₈	C₂×#131	1	1	yes	characteristic
272	C₂×C₈	C₂×AGL(1,ℤ/8)	1	2	yes	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₄²	—	4	2	no	—
285	C₄²	C₈×D₄	1	1	yes	characteristic
286	C₄²	#442	1	1	yes	characteristic
287	C₄²	#444	1	1	yes	characteristic
288	C₄²	C₂×C₄×D₄	1	1	yes	characteristic
289	C₁₆	—	2	2	no	—
290	C₁₆	—	2	1	no	—
291	C₁₆	—	2	1	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	4	1	no	—
294	C₁₆	—	4	2	no	—
295	C₁₆	—	4	2	no	—
296	C₁₆	—	8	2	no	—
297	C₁₆	—	8	2	no	—
298	C₁₆	—	8	1	no	—
299	C₁₆	—	8	1	no	—
300	C₁₆	#504	1	2	yes	—
301	C₂³	—	4	2	no	—
302	C₂³	#1580	1	1	yes	characteristic
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	16	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	#1580	1	1	yes	characteristic
318	C₂×C₄	#1580	1	1	yes	characteristic
319	C₂×C₄	#1580	1	1	yes	characteristic
320	C₂×C₄	C₂×C₈×D₄	1	1	yes	characteristic
321	C₂×C₄	C₂×#442	1	1	yes	characteristic
322	C₂×C₄	C₂×#444	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	—	4	1	no	—
329	C₈	—	4	1	no	—
330	C₈	—	8	2	no	—
331	C₈	—	8	2	no	—
332	C₈	C₂×#504	1	1	yes	characteristic
333	C₈	#3407	1	1	yes	characteristic
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	C₂×#1580	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₄	#33438	1	1	yes	characteristic
349	C₄	#33443	1	1	yes	characteristic
350	C₄	#33468	1	2	yes	—
351	C₂	—	4	1	no	—
352	C₂	—	16	2	no	—
353	C₂	#483585	1	1	yes	characteristic
354	C₂	#483589	1	1	yes	characteristic
355	C₂	#483601	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₈²	C₂⁴
socle	C₂²	C₂×#1580
prime radical	#2215	C₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #11402 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #483611	C₂	semidirect
3. #122970	C₂	semidirect
4. #122967	C₂	semidirect
5. #122966	C₂	semidirect
6. #122965	C₂	semidirect
7. #122963	C₂	semidirect
8. #122962	C₂	semidirect
9. #122919	C₂	semidirect
10. #122470	C₂	semidirect
11. #122219	C₂	semidirect

Maximal subgroups

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

subgroup	index
#483944	2
#483611	2
#122970	2
#122967	2
#122966	2
#122965	2
#122963	2
#122962	2
#122919	2
#122470	2
#122219	2