Subgroups — #484130

Order 512.

414 conjugacy classes of subgroups.

The subgroup lattice has 263 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33735	C₂	1	1	yes	maximal, characteristic
2	#18087	C₂	1	2	yes	maximal, complemented
3	#12086	C₂	1	1	yes	maximal, characteristic, complemented
4	#12085	C₂	1	1	yes	maximal, characteristic, complemented
5	#12071	C₂	1	2	yes	maximal, complemented
6	#12070	C₂	1	2	yes	maximal, complemented
7	#12045	C₂	1	1	yes	maximal, characteristic, complemented
8	#12044	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#1372	C₂	1	1	yes	maximal, characteristic, complemented
10	#11648	C₂	1	2	yes	maximal, complemented
11	#11408	C₂	1	1	yes	maximal, characteristic, complemented
12	#3168	C₂²	1	2	yes	—
13	#2994	—	2	2	no	—
14	#2253	C₄	1	1	yes	characteristic
15	#2249	C₄	1	1	yes	characteristic
16	#2242	C₄	1	2	yes	—
17	#2217	C₄	1	2	yes	—
18	#2216	C₄	1	1	yes	characteristic
19	#2215	C₄	1	1	yes	characteristic
20	C₄×Q₃₂	C₂²	1	1	yes	characteristic
21	C₄×SD₃₂	C₂²	1	2	yes	—
22	C₄×D₁₆	C₂²	1	1	yes	characteristic
23	#2162	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
25	#1591	C₂²	1	2	yes	—
26	#1589	C₂²	1	2	yes	—
27	#1588	—	2	2	no	—
28	#1585	—	2	2	no	—
29	C₈×Q₁₆	C₂²	1	2	yes	—
30	C₈×D₈	C₂²	1	2	yes	—
31	C₂×#335	C₂²	1	2	yes	—
32	#1483	—	2	2	no	—
33	C₈×M₁₆	C₂²	1	1	yes	characteristic
34	#1372	C₂²	1	1	yes	characteristic
35	#1372	C₂²	1	2	yes	—
36	#1372	C₂²	1	1	yes	characteristic
37	#1332	C₂²	1	2	yes	—
38	#1332	C₂²	1	2	yes	—
39	#1331	C₂²	1	2	yes	—
40	#1331	C₂²	1	2	yes	—
41	#1312	C₂²	1	1	yes	characteristic
42	#1312	C₂²	1	1	yes	characteristic
43	#1311	C₂²	1	1	yes	characteristic
44	#1311	C₂²	1	1	yes	characteristic
45	#529	—	2	2	no	—
46	C₂×Q₃₂	C₂×C₄	1	1	yes	characteristic
47	C₂×SD₃₂	C₂×C₄	1	2	yes	—
48	C₂×D₁₆	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
51	#499	C₂×C₄	1	2	yes	—
52	#495	C₂×C₄	1	2	yes	—
53	#488	—	2	2	no	—
54	#467	C₂×C₄	1	2	yes	—
55	#466	C₂×C₄	1	2	yes	—
56	#465	—	2	2	no	—
57	#446	—	2	2	no	—
58	C₈×Q₈	—	2	2	no	—
59	C₄×Q₁₆	C₂³	1	2	yes	—
60	C₄×SD₁₆	—	2	2	no	—
61	C₄×D₈	C₂³	1	2	yes	—
62	#436	—	2	2	no	—
63	#435	—	4	2	no	—
64	C₈×D₄	—	2	2	no	—
65	C₈×D₄	—	4	2	no	—
66	#433	—	4	2	no	—
67	#431	D₄	1	1	yes	characteristic
68	#424	—	4	2	no	—
69	C₂×C₄⋊C₈	—	2	2	no	—
70	C₄×M₁₆	D₄	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
72	#367	C₂×C₄	1	2	yes	—
73	#366	C₂×C₄	1	1	yes	characteristic
74	#366	C₂×C₄	1	1	yes	characteristic
75	C₄⋊C₁₆	—	2	1	no	—
76	#358	C₂×C₄	1	2	yes	—
77	#358	C₂×C₄	1	2	yes	—
78	#357	C₂×C₄	1	2	yes	—
79	#357	C₂×C₄	1	2	yes	—
80	C₂²⋊C₁₆	—	2	1	no	—
81	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
82	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
83	C₄×C₁₆	C₂³	1	1	yes	characteristic
84	C₄×C₁₆	C₂³	1	1	yes	characteristic
85	C₄×C₁₆	C₂³	1	1	yes	characteristic
86	C₄×C₁₆	C₂³	1	1	yes	characteristic
87	#335	—	2	2	no	—
88	#335	C₂³	1	2	yes	—
89	#335	C₂³	1	2	yes	—
90	Q₈⋊C₈	—	2	2	no	—
91	Q₈⋊C₈	—	2	2	no	—
92	D₄⋊C₈	—	2	2	no	—
93	D₄⋊C₈	—	2	2	no	—
94	#322	D₄	1	1	yes	characteristic
95	#322	D₄	1	1	yes	characteristic
96	C₈²	C₂³	1	1	yes	characteristic
97	C₈²	C₂³	1	1	yes	characteristic
98	C₂×Q₁₆	C₂²×C₄	1	2	yes	—
99	C₂×SD₁₆	—	2	2	no	—
100	C₂×D₈	C₂²×C₄	1	2	yes	—
101	C₂×M₁₆	—	2	1	no	—
102	C₂×M₁₆	—	4	2	no	—
103	C₂²×C₈	—	4	2	no	—
104	C₂²×C₈	—	4	2	no	—
105	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₈	#41	1	1	yes	characteristic
107	#126	—	4	2	no	—
108	#125	—	2	2	no	—
109	#124	—	2	2	no	—
110	#123	—	4	2	no	—
111	C₂²⋊Q₈	—	2	2	no	—
112	#121	—	2	2	no	—
113	C₄×Q₈	—	2	2	no	—
114	C₄×D₄	—	2	2	no	—
115	C₄×D₄	—	4	2	no	—
116	#117	—	4	2	no	—
117	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
118	Q₃₂	—	2	2	no	—
119	SD₃₂	—	4	2	no	—
120	D₁₆	—	2	2	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	—	4	1	no	—
125	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
129	#107	C₂²×C₄	1	2	yes	—
130	#107	C₂²×C₄	1	2	yes	—
131	#106	—	2	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	4	2	no	—
135	C₄⋊C₈	—	4	2	no	—
136	C₄⋊C₈	—	4	2	no	—
137	C₄⋊C₈	D₈	1	2	yes	—
138	#103	—	2	2	no	—
139	#103	—	2	2	no	—
140	#102	—	2	2	no	—
141	#102	—	2	2	no	—
142	C₂²⋊C₈	—	4	2	no	—
143	C₂²⋊C₈	—	4	2	no	—
144	C₂²⋊C₈	D₈	1	2	yes	—
145	#97	—	2	1	no	—
146	#97	—	4	2	no	—
147	C₄×C₈	—	4	2	no	—
148	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
149	C₄×C₈	C₂×D₄	1	1	yes	characteristic
150	C₄×C₈	C₂×D₄	1	1	yes	characteristic
151	C₄×C₈	C₂×D₄	1	1	yes	characteristic
152	C₄×C₈	C₂×D₄	1	1	yes	characteristic
153	C₄×C₈	C₂×D₄	1	1	yes	characteristic
154	C₄×C₈	#41	1	1	yes	characteristic
155	C₄×C₈	C₂⁴	1	1	yes	characteristic
156	C₂×Q₈	—	2	2	no	—
157	C₂×D₄	—	2	2	no	—
158	C₂×D₄	—	4	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
162	C₂²×C₄	#135	1	1	yes	characteristic
163	Q₁₆	—	2	2	no	—
164	Q₁₆	—	2	2	no	—
165	SD₁₆	—	8	2	no	—
166	D₈	—	2	2	no	—
167	D₈	—	2	2	no	—
168	M₁₆	—	4	2	no	—
169	M₁₆	—	16	2	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	16	2	no	—
178	C₂×C₈	—	16	2	no	—
179	C₂×C₈	C₄×D₄	1	1	yes	characteristic
180	C₂×C₈	C₄×D₄	1	1	yes	characteristic
181	C₂×C₈	C₂×D₈	1	2	yes	—
182	C₂×C₈	C₂×D₈	1	2	yes	—
183	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₈	C₂×#41	1	1	yes	characteristic
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₂²⋊C₄	—	4	2	no	—
192	C₂²⋊C₄	—	4	2	no	—
193	C₂²⋊C₄	—	4	2	no	—
194	C₂²⋊C₄	—	4	2	no	—
195	C₂²⋊C₄	—	4	2	no	—
196	C₄²	—	4	2	no	—
197	C₄²	C₄×D₄	1	1	yes	characteristic
198	C₄²	C₂×D₈	1	1	yes	characteristic
199	C₄²	#135	1	1	yes	characteristic
200	C₄²	C₂²×D₄	1	1	yes	characteristic
201	C₁₆	—	2	1	no	—
202	C₁₆	—	2	1	no	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	4	1	no	—
205	C₁₆	—	4	1	no	—
206	C₁₆	#131	1	2	yes	—
207	C₂³	—	4	2	no	—
208	C₂³	C₄×D₈	1	1	yes	characteristic
209	Q₈	—	4	2	no	—
210	Q₈	—	4	2	no	—
211	D₄	—	4	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	8	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	C₄×D₈	1	1	yes	characteristic
224	C₂×C₄	C₄×D₈	1	1	yes	characteristic
225	C₂×C₄	C₄×D₈	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
227	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
228	C₂×C₄	C₂×#135	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	4	1	no	—
235	C₈	—	4	1	no	—
236	C₈	—	8	2	no	—
237	C₈	—	8	2	no	—
238	C₈	C₂×#131	1	1	yes	characteristic
239	C₈	#568	1	1	yes	characteristic
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	8	2	no	—
244	C₂²	—	8	2	no	—
245	C₂²	#2179	1	2	yes	—
246	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	8	2	no	—
252	C₄	—	8	2	no	—
253	C₄	—	8	2	no	—
254	C₄	#2935	1	1	yes	characteristic
255	C₄	#2957	1	1	yes	characteristic
256	C₄	#3416	1	2	yes	—
257	C₂	—	2	2	no	—
258	C₂	—	16	2	no	—
259	C₂	#18066	1	1	yes	characteristic
260	C₂	#33460	1	1	yes	characteristic
261	C₂	C₂×#2179	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#2215	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #18087	C₂	semidirect
3. #12086	C₂	semidirect
4. #12085	C₂	semidirect
5. #12071	C₂	semidirect
6. #12070	C₂	semidirect
7. #12045	C₂	semidirect
8. #12044	C₂	semidirect
9. C₂×#1372	C₂	semidirect
10. #11648	C₂	semidirect
11. #11408	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33735	2
#18087	2
#12086	2
#12085	2
#12071	2
#12070	2
#12045	2
#12044	2
C₂×#1372	2
#11648	2
#11408	2