Subgroups — #483379

Order 512.

426 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33503	C₂	1	2	yes	maximal
2	#17328	C₂	1	1	yes	maximal, characteristic
3	#12018	C₂	1	2	yes	maximal, complemented
4	#12017	C₂	1	2	yes	maximal, complemented
5	#12016	C₂	1	2	yes	maximal, complemented
6	#12014	C₂	1	2	yes	maximal, complemented
7	#12013	C₂	1	1	yes	maximal, characteristic, complemented
8	#11997	C₂	1	2	yes	maximal, complemented
9	#11996	C₂	1	1	yes	maximal, characteristic, complemented
10	#2965	C₄	1	1	yes	characteristic
11	C₈×#41	C₂²	1	1	yes	characteristic
12	#2922	C₄	1	1	yes	characteristic
13	#2183	—	2	2	no	—
14	#2169	—	2	2	no	—
15	#2168	—	2	2	no	—
16	#2168	—	2	2	no	—
17	#2163	C₂²	1	2	yes	—
18	#2163	C₂²	1	2	yes	—
19	#2162	C₂²	1	2	yes	—
20	#2150	C₂²	1	2	yes	—
21	#2150	C₂²	1	2	yes	—
22	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
23	#1454	C₄	1	2	yes	—
24	#1453	C₄	1	2	yes	—
25	#1452	C₄	1	1	yes	characteristic
26	#1451	C₄	1	1	yes	characteristic
27	#1450	C₂²	1	2	yes	—
28	C₈×M₁₆	C₂²	1	2	yes	—
29	C₈×M₁₆	C₂²	1	1	yes	characteristic
30	C₂×C₈²	C₂²	1	1	yes	characteristic
31	#1370	C₂²	1	2	yes	—
32	#1370	C₂²	1	2	yes	—
33	#1370	C₂²	1	2	yes	—
34	#1367	C₂²	1	2	yes	—
35	#1367	C₂²	1	1	yes	characteristic
36	#1367	C₂²	1	1	yes	characteristic
37	#1367	C₂²	1	1	yes	characteristic
38	#1367	C₂²	1	2	yes	—
39	#1367	C₂²	1	2	yes	—
40	#1367	C₂²	1	1	yes	characteristic
41	C₄×#41	C₂×C₄	1	1	yes	characteristic
42	C₂×M₃₂	—	2	2	no	—
43	C₂×M₃₂	—	2	2	no	—
44	C₂²×C₁₆	—	2	2	no	—
45	#446	—	2	1	no	—
46	C₈×Q₈	D₄	1	1	yes	characteristic
47	C₈×Q₈	Q₈	1	1	yes	characteristic
48	#436	—	2	1	no	—
49	#435	—	2	2	no	—
50	C₈×D₄	D₄	1	1	yes	characteristic
51	C₈×D₄	D₄	1	2	yes	—
52	C₈×D₄	Q₈	1	2	yes	—
53	C₈×D₄	Q₈	1	1	yes	characteristic
54	#433	C₈	1	4	yes	—
55	#431	C₂×C₄	1	1	yes	characteristic
56	#431	C₂×C₄	1	2	yes	—
57	#431	C₂×C₄	1	1	yes	characteristic
58	#431	C₂×C₄	1	2	yes	—
59	#431	C₂×C₄	1	2	yes	—
60	#431	C₂×C₄	1	2	yes	—
61	#424	C₈	1	2	yes	—
62	#408	C₈	1	2	yes	—
63	C₄×M₁₆	C₂×C₄	1	2	yes	—
64	C₄×M₁₆	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	2	yes	—
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	C₄⋊C₁₆	—	2	2	no	—
70	C₄⋊C₁₆	—	2	2	no	—
71	C₄⋊C₁₆	—	2	2	no	—
72	C₄⋊C₁₆	C₂³	1	2	yes	—
73	C₄⋊C₁₆	C₂³	1	2	yes	—
74	C₄⋊C₁₆	C₂³	1	2	yes	—
75	C₄⋊C₁₆	C₂³	1	2	yes	—
76	C₂²⋊C₁₆	—	2	2	no	—
77	C₂²⋊C₁₆	—	2	2	no	—
78	C₂²⋊C₁₆	—	2	2	no	—
79	#346	—	2	2	no	—
80	#346	—	2	2	no	—
81	C₄×C₁₆	—	2	2	no	—
82	#322	C₂×C₄	1	2	yes	—
83	#322	C₂×C₄	1	2	yes	—
84	#322	C₂×C₄	1	2	yes	—
85	#322	C₂×C₄	1	2	yes	—
86	#322	C₂×C₄	1	2	yes	—
87	C₈²	C₂×C₄	1	1	yes	characteristic
88	C₈²	C₂×C₄	1	1	yes	characteristic
89	C₈²	C₂³	1	1	yes	characteristic
90	C₈²	C₂³	1	1	yes	characteristic
91	C₈²	C₂³	1	2	yes	—
92	C₂×#41	C₂×C₈	1	1	yes	characteristic
93	C₂×M₁₆	—	2	2	no	—
94	C₂×M₁₆	C₂×C₈	1	2	yes	—
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	C₂×C₈	1	2	yes	—
97	C₂²×C₈	C₂×D₄	1	2	yes	—
98	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₈	C₂×Q₈	1	2	yes	—
100	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
101	C₄×Q₈	C₄⋊C₄	1	1	yes	characteristic
102	C₄×Q₈	M₁₆	1	1	yes	characteristic
103	C₄×D₄	C₄⋊C₄	1	1	yes	characteristic
104	C₄×D₄	C₄⋊C₄	1	2	yes	—
105	C₄×D₄	M₁₆	1	2	yes	—
106	C₄×D₄	M₁₆	1	1	yes	characteristic
107	#117	C₂×C₈	1	1	yes	characteristic
108	#117	C₂×C₈	1	2	yes	—
109	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄²	C₂²×C₄	1	2	yes	—
111	M₃₂	—	8	2	no	—
112	M₃₂	—	8	2	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	2	2	no	—
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	8	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	2	1	no	—
120	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
121	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
122	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
123	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
124	C₄⋊C₈	C₂×C₈	1	4	yes	—
125	C₄⋊C₈	C₂×C₈	1	4	yes	—
126	C₂²⋊C₈	—	2	1	no	—
127	C₂²⋊C₈	—	2	2	no	—
128	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
129	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
130	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
131	C₂²⋊C₈	C₂×C₈	1	4	yes	—
132	C₂²⋊C₈	C₂×C₈	1	4	yes	—
133	#97	—	2	2	no	—
134	#97	C₂×C₈	1	2	yes	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	C₂×C₈	1	2	yes	—
137	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
138	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
139	C₄×C₈	C₂²×C₄	1	2	yes	—
140	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
141	C₄×C₈	C₂²×C₄	1	2	yes	—
142	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
143	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
144	C₄×C₈	C₂²×C₄	1	2	yes	—
145	C₄×C₈	C₂×D₄	1	2	yes	—
146	C₄×C₈	C₂×D₄	1	1	yes	characteristic
147	C₄×C₈	C₂×Q₈	1	2	yes	—
148	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
149	C₄×C₈	C₂⁴	1	1	yes	characteristic
150	#41	—	4	2	no	—
151	C₂×Q₈	C₄⋊C₈	1	1	yes	characteristic
152	C₂×D₄	C₄⋊C₈	1	2	yes	—
153	C₂×D₄	C₄⋊C₈	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
156	C₂²×C₄	C₂²×C₈	1	2	yes	—
157	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
158	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
159	C₂²×C₄	C₂×M₁₆	1	2	yes	—
160	M₁₆	—	4	2	no	—
161	M₁₆	—	4	4	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	1	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
174	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
176	C₂×C₈	C₂²×C₈	1	2	yes	—
177	C₂×C₈	C₂²×C₈	1	2	yes	—
178	C₂×C₈	C₂²×C₈	1	4	yes	—
179	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
181	C₄⋊C₄	C₄⋊C₈	1	1	yes	characteristic
182	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
183	C₄⋊C₄	C₄⋊C₈	1	1	yes	characteristic
184	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
185	C₂²⋊C₄	C₄⋊C₈	1	2	yes	—
186	C₂²⋊C₄	C₄⋊C₈	1	2	yes	—
187	C₂²⋊C₄	C₄⋊C₈	1	2	yes	—
188	C₄²	C₂×C₄⋊C₄	1	2	yes	—
189	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
190	C₄²	C₂²×C₈	1	1	yes	characteristic
191	C₄²	C₂²×C₈	1	2	yes	—
192	C₄²	C₂×M₁₆	1	1	yes	characteristic
193	C₄²	C₂×M₁₆	1	2	yes	—
194	C₄²	C₂³×C₄	1	1	yes	characteristic
195	C₁₆	—	4	2	no	—
196	C₁₆	—	4	2	no	—
197	C₁₆	—	4	2	no	—
198	C₁₆	—	4	2	no	—
199	C₂³	C₂×C₄⋊C₈	1	2	yes	—
200	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
201	Q₈	—	4	1	no	—
202	D₄	—	4	1	no	—
203	D₄	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
214	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
216	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
218	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
219	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
220	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	2	no	—
224	C₈	—	2	4	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	1	no	—
227	C₈	—	2	1	no	—
228	C₈	—	2	1	no	—
229	C₈	—	2	2	no	—
230	C₈	—	4	1	no	—
231	C₈	—	4	2	no	—
232	C₈	#577	1	2	yes	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	1	no	—
239	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	1	no	—
247	C₄	#2912	1	1	yes	characteristic
248	C₄	#2916	1	1	yes	characteristic
249	C₄	#3407	1	2	yes	—
250	C₂	—	4	1	no	—
251	C₂	—	4	2	no	—
252	C₂	#17788	1	1	yes	characteristic
253	C₂	#33397	1	1	yes	characteristic
254	C₂	#33401	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41 ◁ 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
3. #12018	C₂	semidirect
4. #12017	C₂	semidirect
5. #12016	C₂	semidirect
6. #12014	C₂	semidirect
7. #12013	C₂	semidirect
8. #11997	C₂	semidirect
9. #11996	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33503	2
#17328	2
#12018	2
#12017	2
#12016	2
#12014	2
#12013	2
#11997	2
#11996	2