Subgroups — #512437

Order 512.

514 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2253	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#504	C₂	1	1	yes	maximal, characteristic
3	#33221	C₂	1	2	yes	maximal
4	#13289	C₂	1	2	yes	maximal, complemented
5	#13172	C₂	1	2	yes	maximal, complemented
6	#13171	C₂	1	2	yes	maximal, complemented
7	#12909	C₂	1	1	yes	maximal, characteristic, complemented
8	#12843	C₂	1	2	yes	maximal
9	#12842	C₂	1	2	yes	maximal
10	C₂×#504	C₂²	1	1	yes	characteristic
11	C₂×#504	C₂²	1	1	yes	characteristic
12	C₂×#499	C₂²	1	2	yes	—
13	#3080	—	2	2	no	—
14	C₄×#131	C₂²	1	1	yes	characteristic
15	#2253	—	2	1	no	—
16	#2253	—	2	1	no	—
17	#2253	—	2	1	no	—
18	#2253	—	2	1	no	—
19	C₂×#368	C₂²	1	2	yes	—
20	C₂×#368	C₂²	1	2	yes	—
21	C₂×#367	C₂²	1	2	yes	—
22	C₂×#366	C₂²	1	2	yes	—
23	#2108	—	2	1	no	—
24	#2108	—	2	1	no	—
25	#2108	C₂²	1	1	yes	characteristic
26	#2108	C₂²	1	1	yes	characteristic
27	C₄×M₃₂	—	2	1	no	—
28	C₄×M₃₂	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₁₆	—	2	1	no	—
30	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
31	#1998	—	2	2	no	—
32	#1952	C₂²	1	4	yes	—
33	#1951	—	2	2	no	—
34	#1848	C₂²	1	2	yes	—
35	#1816	C₂²	1	4	yes	—
36	#1815	—	2	2	no	—
37	#1387	C₂²	1	4	yes	—
38	#1387	C₂²	1	4	yes	—
39	C₂×#131	D₄	1	1	yes	characteristic
40	C₂×#131	C₂³	1	1	yes	characteristic
41	C₂×#125	—	2	2	no	—
42	C₄×#41	D₄	1	1	yes	characteristic
43	#504	—	2	2	no	—
44	#504	—	2	2	no	—
45	#504	Q₈	1	4	yes	—
46	#504	Q₈	1	4	yes	—
47	C₂×M₃₂	—	2	1	no	—
48	C₂×M₃₂	—	2	1	no	—
49	C₂×M₃₂	C₂³	1	1	yes	characteristic
50	C₂×M₃₂	C₂³	1	1	yes	characteristic
51	C₂²×C₁₆	—	2	1	no	—
52	C₂²×C₁₆	—	2	1	no	—
53	C₂²×C₁₆	C₂³	1	1	yes	characteristic
54	C₂²×C₁₆	C₂³	1	1	yes	characteristic
55	#499	—	2	2	no	—
56	#499	—	2	2	no	—
57	#499	—	2	2	no	—
58	#499	—	2	2	no	—
59	#479	—	4	4	no	—
60	#478	—	4	4	no	—
61	C₂×#108	—	2	2	no	—
62	C₂×#108	C₂³	1	4	yes	—
63	#428	—	2	4	no	—
64	C₂×#107	C₂³	1	4	yes	—
65	C₂×#106	—	2	2	no	—
66	#423	—	2	2	no	—
67	#417	—	2	4	no	—
68	#405	—	2	1	no	—
69	#405	—	2	1	no	—
70	#405	D₄	1	1	yes	characteristic
71	#405	C₂³	1	1	yes	characteristic
72	C₄×M₁₆	—	2	1	no	—
73	C₄×M₁₆	D₄	1	1	yes	characteristic
74	C₂×C₄×C₈	—	2	1	no	—
75	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
76	#368	—	2	4	no	—
77	#368	—	2	4	no	—
78	#367	—	2	2	no	—
79	#367	—	2	2	no	—
80	#366	—	2	2	no	—
81	#366	—	2	2	no	—
82	#346	—	2	1	no	—
83	#346	—	2	1	no	—
84	#346	—	2	1	no	—
85	C₄×C₁₆	—	2	1	no	—
86	C₄×C₁₆	—	2	1	no	—
87	C₄×C₁₆	—	2	1	no	—
88	C₄×C₁₆	D₄	1	1	yes	characteristic
89	C₄×C₁₆	D₄	1	1	yes	characteristic
90	C₄×C₁₆	D₄	1	1	yes	characteristic
91	C₄×C₁₆	D₄	1	1	yes	characteristic
92	#343	—	2	4	no	—
93	#343	—	2	4	no	—
94	C₂×#41	C₂×D₄	1	1	yes	characteristic
95	#131	—	2	2	no	—
96	#131	—	2	1	no	—
97	#131	—	2	1	no	—
98	#131	Q₁₆	1	4	yes	—
99	#131	C₂×Q₈	1	2	yes	—
100	#131	C₂×Q₈	1	2	yes	—
101	C₂×M₁₆	—	2	1	no	—
102	C₂×M₁₆	—	2	1	no	—
103	C₂×M₁₆	—	2	4	no	—
104	C₂×M₁₆	C₂×D₄	1	1	yes	characteristic
105	C₂×M₁₆	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₈	—	2	1	no	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₈	C₂⁴	1	1	yes	characteristic
110	#125	—	4	2	no	—
111	#125	—	4	2	no	—
112	#125	—	4	4	no	—
113	C₄×Q₈	D₈	1	2	yes	—
114	C₄×D₄	—	2	2	no	—
115	C₄×D₄	D₈	1	2	yes	—
116	#117	—	2	1	no	—
117	#117	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	2	4	no	—
119	C₂×C₄⋊C₄	—	4	4	no	—
120	C₂×C₄²	—	2	1	no	—
121	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
122	M₃₂	—	2	1	no	—
123	M₃₂	—	2	1	no	—
124	M₃₂	—	2	4	no	—
125	M₃₂	—	2	4	no	—
126	M₃₂	C₂×Q₈	1	2	yes	—
127	M₃₂	C₂×Q₈	1	2	yes	—
128	C₂×C₁₆	—	2	1	no	—
129	C₂×C₁₆	—	2	1	no	—
130	C₂×C₁₆	—	2	4	no	—
131	C₂×C₁₆	—	2	4	no	—
132	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
133	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
134	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
135	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
136	C₂×C₁₆	C₂×Q₈	1	2	yes	—
137	C₂×C₁₆	C₂×Q₈	1	2	yes	—
138	C₂×C₁₆	#41	1	2	yes	—
139	C₂×C₁₆	#41	1	2	yes	—
140	#108	—	2	4	no	—
141	#108	—	2	4	no	—
142	#108	—	4	4	no	—
143	#107	—	2	4	no	—
144	#107	—	2	4	no	—
145	#107	—	4	4	no	—
146	#106	—	4	2	no	—
147	#106	—	4	2	no	—
148	#106	—	4	4	no	—
149	C₄⋊C₈	—	4	4	no	—
150	#103	—	4	4	no	—
151	#102	—	4	4	no	—
152	#97	—	2	1	no	—
153	#97	—	2	1	no	—
154	#97	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	2	1	no	—
158	C₄×C₈	C₂×D₄	1	1	yes	characteristic
159	C₄×C₈	C₂×D₄	1	1	yes	characteristic
160	C₄×C₈	#41	1	1	yes	characteristic
161	C₄×C₈	#41	1	1	yes	characteristic
162	#41	—	2	1	no	—
163	#41	—	2	1	no	—
164	#41	C₄⋊Q₈	1	2	yes	—
165	#41	C₂×Q₁₆	1	2	yes	—
166	C₂×Q₈	C₂×D₈	1	1	yes	characteristic
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	C₂×D₈	1	1	yes	characteristic
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	4	4	no	—
172	C₂²×C₄	#135	1	2	yes	—
173	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
174	M₁₆	—	2	1	no	—
175	M₁₆	—	2	1	no	—
176	M₁₆	—	2	2	no	—
177	M₁₆	—	2	2	no	—
178	M₁₆	—	2	4	no	—
179	M₁₆	—	4	4	no	—
180	M₁₆	—	4	4	no	—
181	M₁₆	C₄⋊Q₈	1	2	yes	—
182	M₁₆	C₂×Q₁₆	1	2	yes	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	4	4	no	—
189	C₂×C₈	#121	1	1	yes	characteristic
190	C₂×C₈	#121	1	1	yes	characteristic
191	C₂×C₈	C₂×Q₁₆	1	2	yes	—
192	C₂×C₈	#135	1	2	yes	—
193	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
195	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
196	C₂×C₈	C₂×#41	1	1	yes	characteristic
197	C₂×C₈	C₂×#41	1	1	yes	characteristic
198	C₂×C₈	C₂×#41	1	1	yes	characteristic
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	8	4	no	—
203	C₄⋊C₄	—	8	4	no	—
204	C₄⋊C₄	C₂×D₈	1	2	yes	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	2	no	—
209	C₄²	#121	1	1	yes	characteristic
210	C₄²	#121	1	1	yes	characteristic
211	C₄²	C₂×D₈	1	2	yes	—
212	C₁₆	—	2	2	no	—
213	C₁₆	—	2	2	no	—
214	C₁₆	C₂²⋊Q₈	1	2	yes	—
215	C₁₆	C₂²⋊Q₈	1	2	yes	—
216	C₁₆	C₂²⋊Q₈	1	2	yes	—
217	C₁₆	C₂²⋊Q₈	1	2	yes	—
218	C₂³	—	2	1	no	—
219	C₂³	C₂×#135	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	#500	1	2	yes	—
222	D₄	—	2	2	no	—
223	D₄	—	2	2	no	—
224	D₄	—	2	1	no	—
225	D₄	#500	1	2	yes	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	8	4	no	—
233	C₂×C₄	—	16	4	no	—
234	C₂×C₄	#466	1	2	yes	—
235	C₂×C₄	#466	1	2	yes	—
236	C₂×C₄	C₂×#121	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
238	C₂×C₄	#532	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
240	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
241	C₂×C₄	C₂×#135	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	8	4	no	—
245	C₈	#467	1	2	yes	—
246	C₈	#467	1	2	yes	—
247	C₈	C₂×C₂²⋊Q₈	1	1	yes	characteristic
248	C₈	C₂×C₂²⋊Q₈	1	1	yes	characteristic
249	C₈	#533	1	1	yes	characteristic
250	C₈	#533	1	1	yes	characteristic
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	C₂×#466	1	1	yes	characteristic
255	C₂²	#3160	1	1	yes	characteristic
256	C₂²	#3170	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	16	4	no	—
262	C₄	#2857	1	1	yes	characteristic
263	C₄	C₂×#467	1	1	yes	characteristic
264	C₄	C₂×#500	1	1	yes	characteristic
265	C₄	#3169	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	4	2	no	—
269	C₂	#33209	1	1	yes	characteristic
270	C₂	#33617	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#2253	C₂	semidirect
4. #13289	C₂	semidirect
5. #13172	C₂	semidirect
6. #13171	C₂	semidirect
7. #12909	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2253	2
C₄×#504	2
#33221	2
#13289	2
#13172	2
#13171	2
#12909	2
#12843	2
#12842	2