Subgroups — #191336

Order 512.

582 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#405	C₂	1	1	yes	maximal, characteristic
2	#8800	C₂	1	2	yes	maximal, complemented
3	#8781	C₂	1	2	yes	maximal, complemented
4	#8731	C₂	1	2	yes	maximal
5	#8602	C₂	1	2	yes	maximal
6	#8267	C₂	1	2	yes	maximal, complemented
7	#8157	C₂	1	2	yes	maximal
8	#8140	C₂	1	2	yes	maximal, complemented
9	C₂×#405	C₄	1	1	yes	characteristic
10	C₂×#405	C₄	1	1	yes	characteristic
11	C₂×#405	C₂²	1	1	yes	characteristic
12	C₂×#405	C₂²	1	1	yes	characteristic
13	C₂×C₄×M₁₆	C₄	1	1	yes	characteristic
14	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
15	C₄×#117	C₂²	1	1	yes	characteristic
16	#1726	C₄	1	1	yes	characteristic
17	#1726	C₄	1	1	yes	characteristic
18	#1726	C₂²	1	1	yes	characteristic
19	#1726	C₂²	1	1	yes	characteristic
20	C₄×#97	C₄	1	1	yes	characteristic
21	C₄×#97	C₂²	1	1	yes	characteristic
22	C₄²×C₈	C₄	1	1	yes	characteristic
23	C₄²×C₈	C₂²	1	1	yes	characteristic
24	#1564	C₂²	1	4	yes	—
25	#1458	C₂²	1	4	yes	—
26	#1452	C₂²	1	4	yes	—
27	#1281	C₂²	1	4	yes	—
28	#1276	C₂²	1	4	yes	—
29	#1275	C₂²	1	4	yes	—
30	#1274	C₂²	1	4	yes	—
31	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×#117	C₂×C₄	1	1	yes	characteristic
36	C₂×#117	C₂³	1	1	yes	characteristic
37	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
38	#431	C₂³	1	8	yes	—
39	#405	—	2	4	no	—
40	#405	—	2	4	no	—
41	#405	—	2	4	no	—
42	#405	—	2	4	no	—
43	C₄×M₁₆	—	2	4	no	—
44	C₂×#97	C₂×C₄	1	1	yes	characteristic
45	C₂×#97	C₂×C₄	1	1	yes	characteristic
46	C₂×#97	C₂×C₄	1	1	yes	characteristic
47	C₂×#97	C₂×C₄	1	1	yes	characteristic
48	C₂×#97	C₂×C₄	1	1	yes	characteristic
49	C₂×#97	C₂×C₄	1	1	yes	characteristic
50	C₂×#97	C₂×C₄	1	1	yes	characteristic
51	C₂×#97	C₂×C₄	1	1	yes	characteristic
52	C₂×#97	C₂³	1	1	yes	characteristic
53	C₂×#97	C₂³	1	1	yes	characteristic
54	C₂×C₄×C₈	—	2	4	no	—
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	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	C₄×C₄⋊C₄	—	2	2	no	—
70	C₄×C₂²⋊C₄	—	2	2	no	—
71	#376	C₂×C₄	1	1	yes	characteristic
72	#376	C₂³	1	1	yes	characteristic
73	C₄³	C₂×C₄	1	1	yes	characteristic
74	C₄³	C₂³	1	1	yes	characteristic
75	#335	—	2	4	no	—
76	#334	—	2	4	no	—
77	#324	—	2	4	no	—
78	#323	—	2	4	no	—
79	#322	—	2	8	no	—
80	C₂³×C₄	—	2	1	no	—
81	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂×M₁₆	—	2	2	no	—
83	C₂×M₁₆	—	2	2	no	—
84	C₂×M₁₆	—	2	2	no	—
85	C₂×M₁₆	—	2	2	no	—
86	C₂×M₁₆	—	2	2	no	—
87	C₂×M₁₆	M₁₆	1	4	yes	—
88	C₂²×C₈	—	2	2	no	—
89	C₂²×C₈	—	2	2	no	—
90	C₂²×C₈	—	2	2	no	—
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	—	2	2	no	—
93	C₂²×C₈	—	2	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	#117	—	2	2	no	—
103	#117	—	2	2	no	—
104	#117	—	2	2	no	—
105	#117	M₁₆	1	4	yes	—
106	C₂×C₄⋊C₄	—	2	1	no	—
107	C₂×C₄⋊C₄	#41	1	2	yes	—
108	C₂×C₂²⋊C₄	—	2	1	no	—
109	C₂×C₂²⋊C₄	#41	1	2	yes	—
110	C₂×C₄²	—	2	1	no	—
111	C₂×C₄²	—	2	2	no	—
112	C₂×C₄²	—	2	2	no	—
113	C₂×C₄²	—	2	2	no	—
114	C₂×C₄²	—	2	2	no	—
115	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄²	C₂⁴	1	1	yes	characteristic
121	C₄⋊C₈	—	2	8	no	—
122	C₂²⋊C₈	—	2	8	no	—
123	#97	—	2	2	no	—
124	#97	—	2	2	no	—
125	#97	—	2	2	no	—
126	#97	—	2	2	no	—
127	#97	—	2	2	no	—
128	#97	—	2	2	no	—
129	#97	—	2	2	no	—
130	#97	—	2	2	no	—
131	#97	—	2	2	no	—
132	#97	M₁₆	1	4	yes	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	2	2	no	—
136	C₄×C₈	—	2	2	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	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	2	8	no	—
145	C₄×C₈	M₁₆	1	4	yes	—
146	C₄×C₈	#41	1	4	yes	—
147	C₄×C₈	#41	1	4	yes	—
148	#95	—	2	2	no	—
149	#95	—	2	2	no	—
150	C₂⁴	#117	1	1	yes	characteristic
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	4	4	no	—
161	C₂²×C₄	#117	1	1	yes	characteristic
162	C₂²×C₄	#117	1	2	yes	—
163	C₂²×C₄	C₂×M₁₆	1	2	yes	—
164	C₂²×C₄	C₂×M₁₆	1	2	yes	—
165	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
166	C₂²×C₄	C₂×#41	1	2	yes	—
167	M₁₆	—	2	8	no	—
168	M₁₆	—	2	8	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	2	8	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	4	8	no	—
186	C₂×C₈	#117	1	2	yes	—
187	C₂×C₈	#117	1	2	yes	—
188	C₂×C₈	#117	1	2	yes	—
189	C₂×C₈	#117	1	2	yes	—
190	C₂×C₈	C₂×M₁₆	1	2	yes	—
191	C₂×C₈	C₂×M₁₆	1	2	yes	—
192	C₂×C₈	C₂×M₁₆	1	2	yes	—
193	C₂×C₈	C₂×M₁₆	1	2	yes	—
194	C₂×C₈	C₂×M₁₆	1	2	yes	—
195	C₂×C₈	C₂×M₁₆	1	2	yes	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₂²⋊C₄	—	2	4	no	—
199	C₂²⋊C₄	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	4	4	no	—
208	C₄²	—	4	4	no	—
209	C₄²	C₂×M₁₆	1	2	yes	—
210	C₄²	C₂×M₁₆	1	2	yes	—
211	C₄²	C₂×M₁₆	1	2	yes	—
212	C₄²	C₂×M₁₆	1	2	yes	—
213	C₄²	C₂×#41	1	2	yes	—
214	C₄²	C₂×#41	1	2	yes	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	4	2	no	—
218	C₂³	#432	1	2	yes	—
219	C₂³	C₂×#117	1	1	yes	characteristic
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	#432	1	2	yes	—
239	C₂×C₄	C₂×#117	1	1	yes	characteristic
240	C₂×C₄	C₂×#117	1	1	yes	characteristic
241	C₂×C₄	#529	1	4	yes	—
242	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
243	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
244	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
245	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	2	2	no	—
253	C₈	—	8	8	no	—
254	C₈	#432	1	4	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	#2283	1	1	yes	characteristic
261	C₂²	C₂×#432	1	1	yes	characteristic
262	C₂²	#2920	1	1	yes	characteristic
263	C₄	—	2	4	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	4	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	2	no	—
270	C₄	C₂×#432	1	1	yes	characteristic
271	C₄	#2920	1	1	yes	characteristic
272	C₄	#2920	1	1	yes	characteristic
273	C₄	#2920	1	1	yes	characteristic
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	4	2	no	—
277	C₂	#20778	1	1	yes	characteristic
278	C₂	#20787	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #8800	C₂	semidirect
3. #8781	C₂	semidirect
6. #8267	C₂	semidirect
8. #8140	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×#405	2
#8800	2
#8781	2
#8731	2
#8602	2
#8267	2
#8157	2
#8140	2