Subgroups — #495454

Order 512.

633 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33418	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₈×#117	C₂	1	1	yes	maximal, characteristic
4	#13163	C₂	1	4	yes	maximal, complemented
5	#12914	C₂	1	4	yes	maximal
6	C₄×#364	C₂	1	4	yes	maximal, complemented
7	C₂²×M₃₂	C₂²	1	2	yes	—
8	C₂×#431	C₄	1	1	yes	characteristic
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₄×#117	C₄	1	1	yes	characteristic
11	#2163	—	2	1	no	—
12	#2163	—	2	1	no	—
13	#2162	—	2	1	no	—
14	#2162	—	2	1	no	—
15	C₂×#364	C₂²	1	8	yes	—
16	#2150	C₂²	1	2	yes	—
17	#2112	C₂²	1	2	yes	—
18	C₄×M₃₂	—	2	1	no	—
19	C₄×M₃₂	—	2	1	no	—
20	C₄×M₃₂	C₂²	1	2	yes	—
21	C₄×M₃₂	C₂²	1	2	yes	—
22	C₄×M₃₂	C₂²	1	2	yes	—
23	C₂×#346	—	2	1	no	—
24	C₂×C₄×C₁₆	—	2	1	no	—
25	C₈×C₄⋊C₄	C₄	1	2	yes	—
26	C₈×C₄⋊C₄	C₂²	1	2	yes	—
27	C₈×C₂²⋊C₄	C₄	1	2	yes	—
28	C₈×C₂²⋊C₄	C₂²	1	2	yes	—
29	#1744	C₄	1	2	yes	—
30	C₄²×C₈	C₂²	1	2	yes	—
31	#1376	C₂²	1	8	yes	—
32	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₈	C₂³	1	1	yes	characteristic
34	C₂×#117	—	2	1	no	—
35	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×M₃₂	—	2	2	no	—
37	C₂×M₃₂	—	2	8	no	—
38	C₂×M₃₂	C₂³	1	4	yes	—
39	C₂×M₃₂	C₂³	1	4	yes	—
40	C₂²×C₁₆	—	2	2	no	—
41	#431	C₈	1	2	yes	—
42	#431	C₈	1	2	yes	—
43	#431	C₈	1	2	yes	—
44	#431	C₈	1	2	yes	—
45	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₈	C₂×C₄	1	2	yes	—
47	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
51	C₂×C₄×C₈	D₄	1	1	yes	characteristic
52	C₂×C₄×C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	D₄	1	1	yes	characteristic
54	C₂×C₄×C₈	D₄	1	1	yes	characteristic
55	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
56	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
57	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂³	1	4	yes	—
60	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₄⋊C₄	—	2	1	no	—
63	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×C₂²⋊C₄	—	2	1	no	—
65	C₄×C₂²⋊C₄	C₂×C₄	1	2	yes	—
66	#376	—	2	1	no	—
67	C₄³	C₂×C₄	1	2	yes	—
68	#364	—	2	16	no	—
69	C₄⋊C₁₆	—	2	2	no	—
70	C₄⋊C₁₆	—	2	2	no	—
71	C₂²⋊C₁₆	—	2	2	no	—
72	C₂²⋊C₁₆	—	2	2	no	—
73	#346	—	2	2	no	—
74	#346	—	2	2	no	—
75	#346	—	2	2	no	—
76	C₄×C₁₆	—	2	2	no	—
77	C₄×C₁₆	—	2	2	no	—
78	C₄×C₁₆	—	2	2	no	—
79	#336	C₂×C₄	1	4	yes	—
80	#336	C₂×C₄	1	4	yes	—
81	C₂³×C₄	M₁₆	1	1	yes	characteristic
82	C₂³×C₄	M₁₆	1	1	yes	characteristic
83	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₈	—	2	4	no	—
85	C₂²×C₈	—	2	4	no	—
86	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₈	C₂²×C₄	1	4	yes	—
93	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₂²×C₈	C₂⁴	1	1	yes	characteristic
99	#117	—	2	8	no	—
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₂²⋊C₄	—	2	2	no	—
102	C₂×C₄²	—	2	2	no	—
103	C₂×C₄²	—	2	2	no	—
104	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
109	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
110	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
111	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
112	C₂×C₄²	M₁₆	1	1	yes	characteristic
113	C₂×C₄²	M₁₆	1	1	yes	characteristic
114	C₂×C₄²	C₂²×C₄	1	4	yes	—
115	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
117	M₃₂	—	2	8	no	—
118	M₃₂	—	2	4	no	—
119	M₃₂	—	2	8	no	—
120	M₃₂	#41	1	8	yes	—
121	C₂×C₁₆	—	2	8	no	—
122	C₂×C₁₆	—	2	4	no	—
123	C₂×C₁₆	—	2	4	no	—
124	C₄⋊C₈	C₂×C₈	1	4	yes	—
125	C₄⋊C₈	C₂×C₈	1	4	yes	—
126	C₂²⋊C₈	C₂×C₈	1	4	yes	—
127	C₂²⋊C₈	C₂×C₈	1	4	yes	—
128	C₄×C₈	—	2	8	no	—
129	C₄×C₈	—	2	8	no	—
130	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
132	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₈	C₂×C₈	1	4	yes	—
139	C₄×C₈	C₂×C₈	1	4	yes	—
140	C₄×C₈	C₂×D₄	1	1	yes	characteristic
141	C₄×C₈	C₂×D₄	1	1	yes	characteristic
142	C₄×C₈	C₂×D₄	1	1	yes	characteristic
143	C₄×C₈	C₂×D₄	1	1	yes	characteristic
144	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
145	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
146	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
147	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
148	#95	—	2	2	no	—
149	#95	—	2	2	no	—
150	C₂⁴	C₂×M₁₆	1	1	yes	characteristic
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
157	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
158	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
159	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
160	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
162	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
167	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
168	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
169	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
170	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
171	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
172	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
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	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×C₈	C₂²×C₈	1	8	yes	—
193	C₂×C₈	#131	1	4	yes	—
194	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
196	C₂×C₈	C₂×#41	1	4	yes	—
197	C₄⋊C₄	—	2	8	no	—
198	C₂²⋊C₄	—	2	8	no	—
199	C₄²	—	2	8	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	4	4	no	—
203	C₄²	C₄⋊C₈	1	1	yes	characteristic
204	C₄²	C₄⋊C₈	1	1	yes	characteristic
205	C₄²	C₄⋊C₈	1	1	yes	characteristic
206	C₄²	C₄⋊C₈	1	1	yes	characteristic
207	C₄²	C₄⋊C₈	1	1	yes	characteristic
208	C₄²	C₄⋊C₈	1	1	yes	characteristic
209	C₄²	C₄⋊C₈	1	1	yes	characteristic
210	C₄²	C₄⋊C₈	1	1	yes	characteristic
211	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
213	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
214	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
215	C₄²	C₂²×C₈	1	1	yes	characteristic
216	C₄²	C₂²×C₈	1	1	yes	characteristic
217	C₄²	C₂²×C₈	1	1	yes	characteristic
218	C₄²	C₂²×C₈	1	1	yes	characteristic
219	C₁₆	—	2	8	no	—
220	C₁₆	—	4	4	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	4	1	no	—
224	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
225	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
226	C₂³	C₂²×M₁₆	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	#364	1	4	yes	complemented
249	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
252	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₄×#41	1	2	yes	—
261	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
262	C₂×C₄	C₂×#131	1	2	yes	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	8	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	#424	1	2	yes	—
269	C₈	#424	1	2	yes	—
270	C₈	#525	1	2	yes	—
271	C₈	#533	1	2	yes	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	4	no	—
280	C₂²	C₂×#364	1	2	yes	—
281	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
282	C₂²	#2965	1	2	yes	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	8	no	—
291	C₄	C₂×#364	1	2	yes	—
292	C₄	C₂×#364	1	2	yes	—
293	C₄	#2153	1	2	yes	—
294	C₄	#2296	1	1	yes	characteristic
295	C₄	C₂×#424	1	1	yes	characteristic
296	C₄	C₈×#41	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	—	2	2	no	—
299	C₂	—	4	1	no	—
300	C₂	#21349	1	1	yes	characteristic
301	C₂	C₂²×#364	1	1	yes	characteristic
302	C₂	C₂×#2153	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #13163	C₂	semidirect
6. C₄×#364	C₂	semidirect
248. C₂×C₄	#364	semidirect

Maximal subgroups

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

subgroup	index
#33418	2
C₂×C₄×M₃₂	2
C₈×#117	2
#13163	2
#12914	2
C₄×#364	2