Subgroups — #191237

Order 512.

551 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20818	C₂	1	1	yes	maximal, characteristic
2	C₂×#1837	C₂	1	1	yes	maximal, characteristic
3	#17323	C₂	1	1	yes	maximal, characteristic
4	#8795	C₂	1	4	yes	maximal, complemented
5	#8792	C₂	1	2	yes	maximal, complemented
6	#8791	C₂	1	2	yes	maximal, complemented
7	#7962	C₂	1	2	yes	maximal
8	#7960	C₂	1	2	yes	maximal
9	#2920	C₂²	1	1	yes	characteristic
10	#2920	C₂²	1	1	yes	characteristic
11	C₂×#423	C₂²	1	2	yes	—
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#2283	C₂²	1	2	yes	—
14	#1846	C₂²	1	2	yes	—
15	#1837	—	2	1	no	—
16	#1837	—	2	1	no	—
17	#1837	C₂²	1	2	yes	—
18	#1837	C₂²	1	2	yes	—
19	#1835	C₂²	1	1	yes	characteristic
20	#1835	C₂²	1	1	yes	characteristic
21	#1808	—	2	1	no	—
22	#1807	—	2	1	no	—
23	C₄×#107	—	2	1	no	—
24	C₄×#106	—	2	1	no	—
25	C₂×#341	—	2	1	no	—
26	C₂×#341	—	2	1	no	—
27	#1452	C₂²	1	2	yes	—
28	#1451	C₄	1	2	yes	complemented
29	#1451	C₄	1	2	yes	complemented
30	#1450	C₂²	1	2	yes	—
31	C₈×M₁₆	C₄	1	2	yes	complemented
32	C₈×M₁₆	C₄	1	2	yes	complemented
33	#1286	C₂²	1	4	yes	—
34	#1286	C₂²	1	4	yes	—
35	#1286	C₂²	1	4	yes	—
36	#1286	C₂²	1	4	yes	—
37	C₂²×M₁₆	—	2	2	no	—
38	C₂³×C₈	D₄	1	1	yes	characteristic
39	C₂³×C₈	D₄	1	1	yes	characteristic
40	C₂²×C₄²	C₂³	1	1	yes	characteristic
41	#432	—	2	1	no	—
42	#432	—	2	1	no	—
43	#432	—	2	1	no	—
44	#432	—	2	1	no	—
45	#431	C₂×C₄	1	2	yes	—
46	#431	C₂×C₄	1	2	yes	—
47	#431	C₂³	1	2	yes	—
48	#431	C₂³	1	2	yes	—
49	C₂×#107	—	2	2	no	—
50	C₂×#106	—	2	2	no	—
51	#423	—	2	1	no	—
52	#423	—	2	1	no	—
53	#423	—	2	2	no	—
54	#423	—	2	2	no	—
55	#423	—	2	2	no	—
56	#423	C₂³	1	4	yes	—
57	C₂×C₄⋊C₈	—	2	2	no	—
58	C₂×C₄⋊C₈	—	2	2	no	—
59	#407	—	2	1	no	—
60	#407	—	2	1	no	—
61	C₄×M₁₆	C₂×C₄	1	2	yes	—
62	C₄×M₁₆	C₂×C₄	1	2	yes	—
63	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×C₈	D₄	1	1	yes	characteristic
68	C₂×C₄×C₈	D₄	1	1	yes	characteristic
69	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
70	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
71	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
75	#383	—	2	2	no	—
76	#382	C₂³	1	2	yes	—
77	#382	C₂³	1	2	yes	—
78	#381	—	2	2	no	—
79	#380	—	2	2	no	—
80	C₄×C₄⋊C₄	—	2	2	no	—
81	C₄×C₂²⋊C₄	—	2	2	no	—
82	#376	—	2	2	no	—
83	#341	—	2	2	no	—
84	#341	—	2	2	no	—
85	#341	—	2	2	no	—
86	#341	—	2	2	no	—
87	#340	—	2	1	no	—
88	#340	—	2	1	no	—
89	#340	—	2	1	no	—
90	#340	—	2	1	no	—
91	#322	C₂×C₄	1	4	yes	—
92	#322	C₂×C₄	1	4	yes	—
93	#322	C₂×C₄	1	4	yes	—
94	C₈²	C₂×C₄	1	4	yes	—
95	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂³×C₄	#41	1	1	yes	characteristic
97	C₂³×C₄	#41	1	1	yes	characteristic
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	2	4	no	—
101	C₂×M₁₆	—	4	2	no	—
102	C₂×M₁₆	—	4	4	no	—
103	C₂×M₁₆	—	4	4	no	—
104	C₂²×C₈	—	4	2	no	—
105	C₂²×C₈	—	4	2	no	—
106	C₂²×C₈	—	4	2	no	—
107	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₂²⋊C₄	—	4	2	no	—
122	C₂×C₄²	—	4	2	no	—
123	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
127	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
128	C₂×C₄²	#41	1	1	yes	characteristic
129	C₂×C₄²	#41	1	1	yes	characteristic
130	C₂×C₄²	#41	1	1	yes	characteristic
131	C₂×C₄²	#41	1	1	yes	characteristic
132	C₂×C₄²	#41	1	1	yes	characteristic
133	C₂×C₄²	#41	1	1	yes	characteristic
134	C₂×C₄²	C₂⁴	1	1	yes	characteristic
135	#107	—	4	4	no	—
136	#106	—	4	4	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	4	no	—
142	C₄⋊C₈	—	2	4	no	—
143	C₄⋊C₈	—	4	2	no	—
144	C₄⋊C₈	—	4	2	no	—
145	C₂²⋊C₈	—	2	2	no	—
146	C₂²⋊C₈	—	2	2	no	—
147	C₂²⋊C₈	—	2	2	no	—
148	C₂²⋊C₈	—	2	2	no	—
149	#97	—	2	2	no	—
150	#97	—	2	2	no	—
151	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
153	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
156	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
159	C₄×C₈	C₂²×C₄	1	4	yes	—
160	C₄×C₈	C₂²×C₄	1	4	yes	—
161	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
162	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
163	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
164	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
165	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
166	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
167	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
168	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
169	#95	—	2	2	no	—
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	#95	—	2	2	no	—
173	#95	—	4	2	no	—
174	#95	—	4	2	no	—
175	C₂⁴	#124	1	1	yes	characteristic
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	#117	1	1	yes	characteristic
185	C₂²×C₄	#117	1	1	yes	characteristic
186	C₂²×C₄	#117	1	1	yes	characteristic
187	C₂²×C₄	#117	1	1	yes	characteristic
188	C₂²×C₄	#124	1	1	yes	characteristic
189	C₂²×C₄	#124	1	1	yes	characteristic
190	C₂²×C₄	#124	1	1	yes	characteristic
191	C₂²×C₄	#124	1	1	yes	characteristic
192	C₂²×C₄	#124	1	1	yes	characteristic
193	C₂²×C₄	#125	1	1	yes	characteristic
194	C₂²×C₄	#125	1	1	yes	characteristic
195	C₂²×C₄	#125	1	1	yes	characteristic
196	C₂²×C₄	#125	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₂×#41	1	1	yes	characteristic
199	C₂²×C₄	C₂×#41	1	1	yes	characteristic
200	M₁₆	—	4	4	no	—
201	M₁₆	—	4	4	no	—
202	M₁₆	—	4	4	no	—
203	M₁₆	—	4	4	no	—
204	M₁₆	—	8	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	1	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	4	no	—
220	C₂×C₈	—	8	4	no	—
221	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₂²⋊C₄	—	8	4	no	—
234	C₄²	—	8	4	no	—
235	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
236	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
237	C₄²	#117	1	1	yes	characteristic
238	C₄²	#117	1	1	yes	characteristic
239	C₄²	#117	1	1	yes	characteristic
240	C₄²	#117	1	1	yes	characteristic
241	C₄²	#124	1	1	yes	characteristic
242	C₄²	#124	1	1	yes	characteristic
243	C₄²	#125	1	1	yes	characteristic
244	C₄²	#125	1	1	yes	characteristic
245	C₄²	#125	1	1	yes	characteristic
246	C₄²	#125	1	1	yes	characteristic
247	C₄²	C₂³×C₄	1	1	yes	characteristic
248	C₄²	C₂²×Q₈	1	1	yes	characteristic
249	C₄²	C₂×#41	1	1	yes	characteristic
250	C₄²	C₂×#41	1	1	yes	characteristic
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	4	1	no	—
254	C₂³	#382	1	1	yes	characteristic
255	C₂³	#382	1	1	yes	characteristic
256	C₂³	C₂×#124	1	1	yes	characteristic
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	—	8	2	no	—
282	C₂×C₄	#382	1	1	yes	characteristic
283	C₂×C₄	#382	1	1	yes	characteristic
284	C₂×C₄	#382	1	1	yes	characteristic
285	C₂×C₄	#382	1	1	yes	characteristic
286	C₂×C₄	#382	1	1	yes	characteristic
287	C₂×C₄	#382	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
289	C₂×C₄	C₂×#117	1	1	yes	characteristic
290	C₂×C₄	C₂×#117	1	1	yes	characteristic
291	C₂×C₄	C₂×#124	1	1	yes	characteristic
292	C₂×C₄	C₂×#125	1	1	yes	characteristic
293	C₂×C₄	C₂×#125	1	1	yes	characteristic
294	C₂×C₄	#575	1	2	yes	—
295	C₂×C₄	#575	1	2	yes	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	4	4	no	—
305	C₈	—	4	4	no	—
306	C₈	—	8	4	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	4	no	—
315	C₂²	#1833	1	2	yes	—
316	C₂²	C₂×#382	1	1	yes	characteristic
317	C₂²	#3142	1	2	yes	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	16	2	no	—
328	C₄	—	16	2	no	—
329	C₄	#1833	1	2	yes	—
330	C₄	#2911	1	1	yes	characteristic
331	C₄	#2911	1	1	yes	characteristic
332	C₄	#3140	1	2	yes	—
333	C₂	—	2	2	no	—
334	C₂	—	2	2	no	—
335	C₂	—	4	1	no	—
336	C₂	C₂×#1833	1	1	yes	characteristic
337	C₂	C₂×#1833	1	1	yes	characteristic
338	C₂	#20765	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #8795	C₂	semidirect
5. #8792	C₂	semidirect
6. #8791	C₂	semidirect
28. #1451	C₄	semidirect
29. #1451	C₄	semidirect
31. C₈×M₁₆	C₄	semidirect
32. C₈×M₁₆	C₄	semidirect

Maximal subgroups

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

subgroup	index
#20818	2
C₂×#1837	2
#17323	2
#8795	2
#8792	2
#8791	2
#7962	2
#7960	2