Subgroups — #147098

Order 512.

536 conjugacy classes of subgroups.

The subgroup lattice has 317 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2158	C₂	1	1	yes	maximal, characteristic
2	C₂×#2158	C₂	1	1	yes	maximal, characteristic
3	C₂×#2108	C₂	1	1	yes	maximal, characteristic
4	#12567	C₂	1	2	yes	maximal
5	#10871	C₂	1	2	yes	maximal
6	C₂²×M₃₂	C₂²	1	1	yes	characteristic
7	C₂³×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#427	C₂²	1	2	yes	—
9	C₂×#427	C₂²	1	2	yes	—
10	C₂×#405	C₂²	1	1	yes	characteristic
11	#2158	—	2	2	no	—
12	#2158	—	2	1	no	—
13	#2158	—	2	1	no	—
14	#2158	—	2	1	no	—
15	#2158	—	2	1	no	—
16	#2158	C₄	1	4	yes	complemented
17	C₂×#367	—	2	1	no	—
18	C₂×#367	C₄	1	2	yes	complemented
19	C₂×#366	—	2	1	no	—
20	C₂×#366	C₄	1	2	yes	complemented
21	#2108	—	2	2	no	—
22	#2108	C₄	1	4	yes	complemented
23	C₂×#346	—	2	1	no	—
24	C₂×C₄×C₁₆	—	2	1	no	—
25	#1740	—	2	2	no	—
26	#1733	—	2	2	no	—
27	#1386	—	2	4	no	—
28	C₂²×M₁₆	D₄	1	1	yes	characteristic
29	C₂³×C₈	C₂³	1	1	yes	characteristic
30	C₂×#117	—	2	2	no	—
31	C₂×#117	—	2	2	no	—
32	C₂×#117	D₄	1	1	yes	characteristic
33	C₂×M₃₂	—	2	1	no	—
34	C₂×M₃₂	—	2	1	no	—
35	C₂×M₃₂	—	2	1	no	—
36	C₂×M₃₂	C₂×C₄	1	2	yes	—
37	C₂×M₃₂	D₄	1	2	yes	—
38	C₂×M₃₂	Q₈	1	2	yes	—
39	C₂²×C₁₆	—	2	1	no	—
40	C₂²×C₁₆	C₂×C₄	1	2	yes	—
41	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₁₆	C₂×C₄	1	2	yes	—
44	C₂²×C₁₆	D₄	1	1	yes	characteristic
45	C₂²×C₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₁₆	D₄	1	2	yes	—
47	C₂²×C₁₆	D₄	1	1	yes	characteristic
48	C₂²×C₁₆	D₄	1	1	yes	characteristic
49	C₂²×C₁₆	Q₈	1	2	yes	—
50	#427	—	2	2	no	—
51	#427	—	2	2	no	—
52	#427	—	2	2	no	—
53	#427	—	2	2	no	—
54	#427	—	2	2	no	—
55	#427	—	2	2	no	—
56	#427	—	2	2	no	—
57	#427	C₂×C₄	1	4	yes	—
58	C₂×#107	—	2	2	no	—
59	C₂×#107	C₂×C₄	1	4	yes	—
60	C₂×#106	—	2	2	no	—
61	C₂×#106	—	2	2	no	—
62	#405	—	2	1	no	—
63	#405	—	2	1	no	—
64	#405	C₂×C₄	1	2	yes	—
65	#405	Q₈	1	2	yes	—
66	C₂×#97	—	2	1	no	—
67	C₂×C₄×C₈	—	2	1	no	—
68	#367	—	2	4	no	—
69	#367	—	2	2	no	—
70	#367	—	2	2	no	—
71	#366	—	2	4	no	—
72	#366	—	2	2	no	—
73	#366	—	2	2	no	—
74	#346	—	2	4	no	—
75	C₄×C₁₆	—	2	4	no	—
76	#342	—	4	4	no	—
77	#339	—	4	4	no	—
78	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×M₁₆	—	2	1	no	—
80	C₂×M₁₆	—	2	1	no	—
81	C₂×M₁₆	—	2	1	no	—
82	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
83	C₂×M₁₆	D₈	1	2	yes	—
84	C₂×M₁₆	Q₁₆	1	2	yes	—
85	C₂²×C₈	—	2	1	no	—
86	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₈	#41	1	1	yes	characteristic
97	C₂²×C₈	#41	1	1	yes	characteristic
98	C₂²×C₈	#41	1	1	yes	characteristic
99	C₂²×C₈	#41	1	1	yes	characteristic
100	#117	—	2	1	no	—
101	#117	—	2	1	no	—
102	#117	—	2	4	no	—
103	#117	—	4	2	no	—
104	#117	—	4	2	no	—
105	#117	—	4	2	no	—
106	#117	—	4	2	no	—
107	#117	—	4	2	no	—
108	#117	—	4	2	no	—
109	#117	—	4	2	no	—
110	#117	C₄⋊C₄	1	2	yes	—
111	#117	Q₁₆	1	2	yes	—
112	C₂×C₄⋊C₄	—	2	1	no	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	4	2	no	—
115	C₂×C₂²⋊C₄	—	4	2	no	—
116	C₂×C₂²⋊C₄	—	4	2	no	—
117	C₂×C₂²⋊C₄	D₈	1	2	yes	—
118	C₂×C₄²	—	2	1	no	—
119	C₂×C₄²	—	4	2	no	—
120	C₂×C₄²	—	4	2	no	—
121	M₃₂	—	2	2	no	—
122	M₃₂	—	2	2	no	—
123	M₃₂	—	2	2	no	—
124	M₃₂	C₄⋊C₄	1	4	yes	—
125	C₂×C₁₆	—	2	2	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	C₄²	1	2	yes	—
129	C₂×C₁₆	C₄²	1	2	yes	—
130	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
131	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
132	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
133	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
136	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
139	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
140	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
141	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
142	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
143	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
144	#107	—	2	4	no	—
145	#107	—	2	4	no	—
146	#107	—	2	4	no	—
147	#107	—	4	2	no	—
148	#106	—	2	4	no	—
149	#106	—	4	2	no	—
150	#106	—	4	2	no	—
151	#106	—	4	2	no	—
152	#97	—	2	2	no	—
153	#97	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₂⁴	C₂×D₈	1	1	yes	characteristic
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	4	4	no	—
159	C₂²×C₄	—	8	2	no	—
160	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
162	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
163	C₂²×C₄	#121	1	1	yes	characteristic
164	C₂²×C₄	#121	1	1	yes	characteristic
165	C₂²×C₄	#127	1	1	yes	characteristic
166	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
168	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
169	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
170	C₂²×C₄	#135	1	1	yes	characteristic
171	C₂²×C₄	#135	1	1	yes	characteristic
172	C₂²×C₄	#135	1	1	yes	characteristic
173	C₂²×C₄	#135	1	1	yes	characteristic
174	M₁₆	—	2	2	no	—
175	M₁₆	—	2	2	no	—
176	M₁₆	—	2	2	no	—
177	M₁₆	#107	1	4	yes	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
182	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
184	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂×C₈	#117	1	1	yes	characteristic
190	C₂×C₈	#117	1	1	yes	characteristic
191	C₂×C₈	C₄×D₄	1	1	yes	characteristic
192	C₂×C₈	C₄×D₄	1	1	yes	characteristic
193	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
194	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
195	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
196	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
197	C₂×C₈	#121	1	1	yes	characteristic
198	C₂×C₈	#121	1	1	yes	characteristic
199	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
200	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
201	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
203	C₂×C₈	#124	1	1	yes	characteristic
204	C₂×C₈	#124	1	1	yes	characteristic
205	C₂×C₈	#125	1	1	yes	characteristic
206	C₂×C₈	#125	1	1	yes	characteristic
207	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
208	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	—	4	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	8	2	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	—	8	2	no	—
218	C₂²⋊C₄	—	8	2	no	—
219	C₂²⋊C₄	—	8	2	no	—
220	C₂²⋊C₄	#107	1	4	yes	—
221	C₄²	—	2	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	4	4	no	—
224	C₄²	—	8	2	no	—
225	C₄²	—	8	2	no	—
226	C₄²	—	8	2	no	—
227	C₁₆	—	2	4	no	—
228	C₁₆	#95	1	2	yes	—
229	C₁₆	#95	1	2	yes	—
230	C₁₆	#95	1	2	yes	—
231	C₁₆	#95	1	2	yes	—
232	C₂³	—	4	2	no	—
233	C₂³	C₂×#107	1	1	yes	characteristic
234	C₂³	C₄×D₈	1	1	yes	characteristic
235	C₂³	C₄×D₈	1	1	yes	characteristic
236	C₂³	#466	1	1	yes	characteristic
237	C₂³	#466	1	1	yes	characteristic
238	C₂³	#493	1	1	yes	characteristic
239	C₂³	#500	1	1	yes	characteristic
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	8	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	16	4	no	—
245	C₂×C₄	—	16	4	no	—
246	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
247	C₂×C₄	#380	1	1	yes	characteristic
248	C₂×C₄	#384	1	1	yes	characteristic
249	C₂×C₄	#389	1	1	yes	characteristic
250	C₂×C₄	#389	1	1	yes	characteristic
251	C₂×C₄	#390	1	1	yes	characteristic
252	C₂×C₄	#390	1	1	yes	characteristic
253	C₂×C₄	C₂×#107	1	1	yes	characteristic
254	C₂×C₄	#427	1	1	yes	characteristic
255	C₂×C₄	#427	1	1	yes	characteristic
256	C₂×C₄	C₄×D₈	1	1	yes	characteristic
257	C₂×C₄	C₄×D₈	1	1	yes	characteristic
258	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
259	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
260	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
261	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
262	C₂×C₄	#466	1	1	yes	characteristic
263	C₂×C₄	#466	1	1	yes	characteristic
264	C₂×C₄	#467	1	1	yes	characteristic
265	C₂×C₄	#467	1	1	yes	characteristic
266	C₂×C₄	#467	1	1	yes	characteristic
267	C₂×C₄	#467	1	1	yes	characteristic
268	C₂×C₄	#494	1	1	yes	characteristic
269	C₂×C₄	#495	1	1	yes	characteristic
270	C₂×C₄	#495	1	1	yes	characteristic
271	C₂×C₄	#499	1	1	yes	characteristic
272	C₂×C₄	#499	1	1	yes	characteristic
273	C₂×C₄	#500	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	C₂×#95	1	1	yes	characteristic
276	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
277	C₈	#380	1	1	yes	characteristic
278	C₈	#382	1	1	yes	characteristic
279	C₈	#389	1	1	yes	characteristic
280	C₈	#389	1	1	yes	characteristic
281	C₈	#391	1	1	yes	characteristic
282	C₈	#391	1	1	yes	characteristic
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	C₄×#107	1	1	yes	characteristic
291	C₂²	#1808	1	1	yes	characteristic
292	C₂²	#1845	1	1	yes	characteristic
293	C₂²	#1942	1	1	yes	characteristic
294	C₂²	#1942	1	1	yes	characteristic
295	C₂²	#1970	1	1	yes	characteristic
296	C₂²	#1970	1	1	yes	characteristic
297	C₄	—	4	4	no	—
298	C₄	—	16	4	no	—
299	C₄	—	16	4	no	—
300	C₄	C₄⋊#95	1	1	yes	characteristic
301	C₄	C₄×#107	1	1	yes	characteristic
302	C₄	#1808	1	1	yes	characteristic
303	C₄	#1846	1	1	yes	characteristic
304	C₄	#1942	1	1	yes	characteristic
305	C₄	#1942	1	1	yes	characteristic
306	C₄	#1969	1	1	yes	characteristic
307	C₄	#1969	1	1	yes	characteristic
308	C₂	—	2	1	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	2	1	no	—
314	C₂	#10862	1	1	yes	characteristic
315	C₂	#13243	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1970
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #2158	C₄	semidirect
18. C₂×#367	C₄	semidirect
20. C₂×#366	C₄	semidirect
22. #2108	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2158	2
C₂×#2108	2
#12567	2
#10871	2