Subgroups — #109390

Order 512.

601 conjugacy classes of subgroups.

The subgroup lattice has 329 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	C₄×#427	C₂	1	1	yes	maximal, characteristic
3	#17323	C₂	1	1	yes	maximal, characteristic
4	#8052	C₂	1	2	yes	maximal
5	#7940	C₂	1	2	yes	maximal
6	#2920	C₂²	1	1	yes	characteristic
7	#2920	C₂²	1	1	yes	characteristic
8	C₂×#427	—	2	1	no	—
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	#2283	C₂²	1	2	yes	—
11	C₄×#117	C₂²	1	2	yes	—
12	#1846	C₄	1	2	yes	complemented
13	#1835	C₄	1	2	yes	complemented
14	#1808	—	2	1	no	—
15	#1807	—	2	1	no	—
16	C₄×#107	—	2	1	no	—
17	C₄×#107	—	2	1	no	—
18	C₄×#107	C₄	1	2	yes	complemented
19	C₄×#106	—	2	1	no	—
20	C₄×#106	—	2	1	no	—
21	C₄×#106	C₄	1	2	yes	complemented
22	#1735	—	2	1	no	—
23	#1452	—	2	1	no	—
24	#1451	—	2	1	no	—
25	#1451	C₄	1	2	yes	complemented
26	#1450	—	2	1	no	—
27	C₈×M₁₆	—	2	1	no	—
28	C₈×M₁₆	C₄	1	2	yes	complemented
29	#1297	—	2	4	no	—
30	#1284	—	2	4	no	—
31	C₂³×C₈	D₄	1	1	yes	characteristic
32	C₂³×C₈	D₄	1	1	yes	characteristic
33	C₂×#117	—	2	2	no	—
34	C₂²×C₄²	C₂³	1	1	yes	characteristic
35	#432	—	2	1	no	—
36	#432	—	2	1	no	—
37	#432	—	2	1	no	—
38	#432	Q₈	1	2	yes	—
39	#431	—	2	1	no	—
40	#431	—	2	1	no	—
41	#431	—	2	1	no	—
42	#431	C₂×C₄	1	2	yes	—
43	#427	—	2	8	no	—
44	C₂×#107	—	2	2	no	—
45	C₂×#107	—	2	2	no	—
46	C₂×#106	—	2	2	no	—
47	C₂×#106	—	2	2	no	—
48	#423	—	2	1	no	—
49	#423	Q₈	1	2	yes	—
50	#407	D₄	1	2	yes	—
51	#407	D₄	1	2	yes	—
52	C₄×M₁₆	—	2	1	no	—
53	C₄×M₁₆	C₂×C₄	1	2	yes	—
54	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈	D₄	1	1	yes	characteristic
61	C₂×C₄×C₈	D₄	1	1	yes	characteristic
62	C₂×C₄×C₈	D₄	1	1	yes	characteristic
63	C₂×C₄×C₈	D₄	1	1	yes	characteristic
64	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
65	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
66	#383	—	2	2	no	—
67	#382	C₂×C₄	1	4	yes	—
68	#381	—	2	2	no	—
69	#380	—	2	2	no	—
70	C₄×C₄⋊C₄	—	2	2	no	—
71	C₄×C₄⋊C₄	—	2	2	no	—
72	C₄×C₄⋊C₄	C₂×C₄	1	4	yes	—
73	C₄×C₂²⋊C₄	—	2	2	no	—
74	C₄×C₂²⋊C₄	—	2	2	no	—
75	C₄×C₂²⋊C₄	—	2	2	no	—
76	#376	—	2	2	no	—
77	#376	—	2	2	no	—
78	C₄³	—	2	2	no	—
79	#340	—	2	2	no	—
80	#340	—	2	2	no	—
81	#340	—	2	2	no	—
82	#340	—	2	2	no	—
83	#322	—	2	2	no	—
84	#322	—	2	2	no	—
85	#322	—	2	2	no	—
86	C₈²	—	2	2	no	—
87	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
92	C₂²×C₈	—	2	4	no	—
93	C₂²×C₈	—	4	2	no	—
94	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
106	#117	—	2	8	no	—
107	#117	—	4	4	no	—
108	C₂×C₄⋊C₄	—	2	4	no	—
109	C₂×C₄⋊C₄	—	2	4	no	—
110	C₂×C₂²⋊C₄	—	4	2	no	—
111	C₂×C₂²⋊C₄	—	4	2	no	—
112	C₂×C₄²	—	2	4	no	—
113	C₂×C₄²	—	4	2	no	—
114	C₂×C₄²	—	4	2	no	—
115	C₂×C₄²	—	4	2	no	—
116	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
122	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
123	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
124	C₂×C₄²	#41	1	1	yes	characteristic
125	C₂×C₄²	#41	1	1	yes	characteristic
126	C₂×C₄²	#41	1	1	yes	characteristic
127	C₂×C₄²	#41	1	1	yes	characteristic
128	#107	—	2	8	no	—
129	#107	—	4	4	no	—
130	#106	—	2	8	no	—
131	#106	—	4	4	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₂²⋊C₈	—	2	2	no	—
137	C₂²⋊C₈	—	2	2	no	—
138	C₂²⋊C₈	—	2	2	no	—
139	C₂²⋊C₈	C₄⋊C₄	1	4	yes	—
140	#97	—	2	2	no	—
141	#97	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	C₄²	1	1	yes	characteristic
145	C₄×C₈	C₄²	1	1	yes	characteristic
146	C₄×C₈	C₄²	1	1	yes	characteristic
147	C₄×C₈	C₄²	1	1	yes	characteristic
148	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₈	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
160	#95	—	2	4	no	—
161	#95	—	2	4	no	—
162	#95	—	2	4	no	—
163	#95	—	4	2	no	—
164	#95	—	4	2	no	—
165	#95	—	4	2	no	—
166	C₂⁴	#127	1	1	yes	characteristic
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	4	no	—
172	C₂²×C₄	—	4	4	no	—
173	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
179	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
180	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
182	C₂²×C₄	#121	1	1	yes	characteristic
183	C₂²×C₄	#121	1	1	yes	characteristic
184	C₂²×C₄	#121	1	1	yes	characteristic
185	C₂²×C₄	#121	1	1	yes	characteristic
186	C₂²×C₄	#124	1	1	yes	characteristic
187	C₂²×C₄	#124	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
191	M₁₆	—	4	4	no	—
192	M₁₆	—	4	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	#95	1	1	yes	characteristic
207	C₂×C₈	#95	1	1	yes	characteristic
208	C₂×C₈	#95	1	1	yes	characteristic
209	C₂×C₈	#95	1	1	yes	characteristic
210	C₂×C₈	#95	1	1	yes	characteristic
211	C₂×C₈	#95	1	1	yes	characteristic
212	C₂×C₈	#95	1	1	yes	characteristic
213	C₂×C₈	#95	1	1	yes	characteristic
214	C₂×C₈	#104	1	4	yes	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₂²⋊C₄	—	4	8	no	—
220	C₂²⋊C₄	—	8	4	no	—
221	C₄²	—	4	8	no	—
222	C₄²	—	8	4	no	—
223	C₄²	—	8	4	no	—
224	C₄²	—	8	4	no	—
225	C₄²	C₂×C₄²	1	1	yes	characteristic
226	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
227	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
228	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
229	C₄²	#117	1	1	yes	characteristic
230	C₄²	#117	1	1	yes	characteristic
231	C₄²	C₄×Q₈	1	1	yes	characteristic
232	C₄²	C₄×Q₈	1	1	yes	characteristic
233	C₄²	C₄×Q₈	1	1	yes	characteristic
234	C₄²	C₄×Q₈	1	1	yes	characteristic
235	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
236	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
237	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
238	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
239	C₄²	#125	1	1	yes	characteristic
240	C₄²	#125	1	1	yes	characteristic
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	4	1	no	—
244	C₂³	#384	1	1	yes	characteristic
245	C₂³	#390	1	1	yes	characteristic
246	C₂³	#390	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	8	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	—	8	4	no	—
272	C₂×C₄	C₂×#95	1	1	yes	characteristic
273	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
274	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
275	C₂×C₄	#380	1	1	yes	characteristic
276	C₂×C₄	#380	1	1	yes	characteristic
277	C₂×C₄	#382	1	1	yes	characteristic
278	C₂×C₄	#389	1	1	yes	characteristic
279	C₂×C₄	#389	1	1	yes	characteristic
280	C₂×C₄	#389	1	1	yes	characteristic
281	C₂×C₄	#389	1	1	yes	characteristic
282	C₂×C₄	#391	1	1	yes	characteristic
283	C₂×C₄	#391	1	1	yes	characteristic
284	C₂×C₄	C₂×#104	1	2	yes	—
285	C₂×C₄	#421	1	2	yes	—
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	4	4	no	—
293	C₈	—	4	4	no	—
294	C₈	#339	1	2	yes	—
295	C₈	#339	1	2	yes	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	4	4	no	—
304	C₂²	C₄⋊#95	1	1	yes	characteristic
305	C₂²	#1976	1	2	yes	—
306	C₂²	#1980	1	2	yes	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	8	8	no	—
317	C₄	—	16	4	no	—
318	C₄	C₂×#339	1	1	yes	characteristic
319	C₄	#1733	1	1	yes	characteristic
320	C₄	#1812	1	2	yes	—
321	C₄	#1986	1	2	yes	—
322	C₂	—	2	2	no	—
323	C₂	—	2	2	no	—
324	C₂	—	4	1	no	—
325	C₂	#10851	1	1	yes	characteristic
326	C₂	#10866	1	1	yes	characteristic
327	C₂	#10871	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
12. #1846	C₄	semidirect
13. #1835	C₄	semidirect
18. C₄×#107	C₄	semidirect
21. C₄×#106	C₄	semidirect
25. #1451	C₄	semidirect
28. 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
C₄×#427	2
#17323	2
#8052	2
#7940	2