Subgroups — #483380

Order 512.

477 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33504	C₂	1	2	yes	maximal
2	C₈×#131	C₂	1	1	yes	maximal, characteristic
3	#12018	C₂	1	2	yes	maximal, complemented
4	#12016	C₂	1	4	yes	maximal, complemented
5	#12013	C₂	1	2	yes	maximal, complemented
6	#12012	C₂	1	1	yes	maximal, characteristic, complemented
7	#11996	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#1367	C₂	1	2	yes	maximal, complemented
9	C₈×#41	C₄	1	1	yes	characteristic
10	C₈×#41	C₂²	1	1	yes	characteristic
11	C₄×#131	C₄	1	1	yes	characteristic
12	#2183	—	2	2	no	—
13	#2168	—	2	4	no	—
14	D₄×C₁₆	—	2	2	no	—
15	#2163	C₂²	1	4	yes	—
16	#2162	C₂²	1	2	yes	—
17	#2150	C₂²	1	2	yes	—
18	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
19	#1450	C₄	1	1	yes	characteristic
20	#1450	C₄	1	2	yes	—
21	C₈×M₁₆	C₄	1	1	yes	characteristic
22	C₈×M₁₆	C₄	1	2	yes	—
23	C₈×M₁₆	C₂²	1	2	yes	—
24	C₈×M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₈²	C₂²	1	2	yes	—
26	C₂×C₈²	C₂²	1	1	yes	characteristic
27	#1370	C₂²	1	2	yes	—
28	#1370	C₂²	1	4	yes	—
29	#1367	C₂²	1	1	yes	characteristic
30	#1367	C₂²	1	2	yes	—
31	#1367	C₂²	1	4	yes	—
32	#1367	C₂²	1	2	yes	—
33	#1367	C₂²	1	1	yes	characteristic
34	C₂×#131	—	2	1	no	—
35	C₄×#41	C₂×C₄	1	1	yes	characteristic
36	C₂×M₃₂	—	2	2	no	—
37	C₂²×C₁₆	—	2	4	no	—
38	C₈×Q₈	C₈	1	1	yes	characteristic
39	C₈×Q₈	C₈	1	1	yes	characteristic
40	C₈×Q₈	D₄	1	1	yes	characteristic
41	C₈×Q₈	Q₈	1	1	yes	characteristic
42	C₈×D₄	C₈	1	2	yes	—
43	C₈×D₄	C₈	1	1	yes	characteristic
44	C₈×D₄	C₈	1	2	yes	—
45	C₈×D₄	C₈	1	1	yes	characteristic
46	C₈×D₄	D₄	1	1	yes	characteristic
47	C₈×D₄	D₄	1	2	yes	—
48	C₈×D₄	Q₈	1	2	yes	—
49	C₈×D₄	Q₈	1	1	yes	characteristic
50	#431	C₂×C₄	1	1	yes	characteristic
51	#431	C₂×C₄	1	2	yes	—
52	#431	C₂×C₄	1	2	yes	—
53	#431	C₂×C₄	1	1	yes	characteristic
54	#405	—	2	1	no	—
55	#405	—	2	2	no	—
56	C₄×M₁₆	C₂×C₄	1	2	yes	—
57	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
59	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
61	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂³	1	2	yes	—
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₄⋊C₁₆	—	2	4	no	—
65	C₄⋊C₁₆	—	2	2	no	—
66	C₄⋊C₁₆	C₂³	1	2	yes	—
67	C₄⋊C₁₆	C₂³	1	4	yes	—
68	C₄⋊C₁₆	C₂³	1	2	yes	—
69	C₂²⋊C₁₆	—	2	2	no	—
70	C₂²⋊C₁₆	—	2	4	no	—
71	#346	—	2	4	no	—
72	C₄×C₁₆	—	2	2	no	—
73	#322	C₂×C₄	1	2	yes	—
74	#322	C₂×C₄	1	2	yes	—
75	#322	C₂×C₄	1	2	yes	—
76	C₈²	C₂×C₄	1	2	yes	—
77	C₈²	C₂×C₄	1	1	yes	characteristic
78	C₈²	C₂×C₄	1	2	yes	—
79	C₈²	C₂×C₄	1	1	yes	characteristic
80	C₈²	C₂³	1	1	yes	characteristic
81	C₈²	C₂³	1	2	yes	—
82	C₈²	C₂³	1	1	yes	characteristic
83	C₂×#41	M₁₆	1	1	yes	characteristic
84	#131	—	2	8	no	—
85	C₂×M₁₆	—	2	2	no	—
86	C₂×M₁₆	—	2	1	no	—
87	C₂²×C₈	—	2	2	no	—
88	C₂²×C₈	—	2	1	no	—
89	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
90	C₂²×C₈	C₂×C₈	1	2	yes	—
91	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
92	C₂²×C₈	C₂×C₈	1	2	yes	—
93	C₂²×C₈	C₂×D₄	1	2	yes	—
94	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₈	C₂×Q₈	1	2	yes	—
96	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
97	C₄×Q₈	C₄⋊C₄	1	1	yes	characteristic
98	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
99	C₄×D₄	C₄⋊C₄	1	2	yes	—
100	C₄×D₄	C₄⋊C₄	1	1	yes	characteristic
101	C₄×D₄	C₂×C₈	1	2	yes	—
102	C₄×D₄	C₂×C₈	1	1	yes	characteristic
103	#117	M₁₆	1	1	yes	characteristic
104	#117	M₁₆	1	2	yes	—
105	C₂×C₄²	C₂²×C₄	1	2	yes	—
106	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
107	M₃₂	—	8	2	no	—
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	2	4	no	—
111	C₂×C₁₆	—	8	4	no	—
112	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
113	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
115	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
116	C₄⋊C₈	C₂×C₈	1	2	yes	—
117	C₄⋊C₈	C₂×C₈	1	2	yes	—
118	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
119	C₄⋊C₈	C₂×C₈	1	1	yes	characteristic
120	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
121	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
122	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
123	C₂²⋊C₈	C₂×C₈	1	2	yes	—
124	C₂²⋊C₈	C₂×C₈	1	2	yes	—
125	C₂²⋊C₈	C₂×C₈	1	2	yes	—
126	#97	—	2	2	no	—
127	#97	—	2	1	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	C₂×C₈	1	2	yes	—
131	C₄×C₈	C₂×C₈	1	1	yes	characteristic
132	C₄×C₈	C₂×C₈	1	2	yes	—
133	C₄×C₈	C₂×C₈	1	1	yes	characteristic
134	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
135	C₄×C₈	C₂²×C₄	1	2	yes	—
136	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
137	C₄×C₈	C₂²×C₄	1	2	yes	—
138	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
139	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
140	C₄×C₈	C₂²×C₄	1	2	yes	—
141	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
142	C₄×C₈	C₂×D₄	1	1	yes	characteristic
143	C₄×C₈	C₂×D₄	1	2	yes	—
144	C₄×C₈	C₂×Q₈	1	2	yes	—
145	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
146	C₄×C₈	C₂⁴	1	1	yes	characteristic
147	#41	—	2	4	no	—
148	C₂×Q₈	C₄⋊C₈	1	1	yes	characteristic
149	C₂×D₄	C₄⋊C₈	1	2	yes	—
150	C₂×D₄	C₄⋊C₈	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
153	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
154	C₂²×C₄	C₂²×C₈	1	2	yes	—
155	C₂²×C₄	C₂×M₁₆	1	2	yes	—
156	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
157	M₁₆	—	2	8	no	—
158	M₁₆	—	2	4	no	—
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	—	4	1	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	1	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	2	yes	—
175	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
177	C₂×C₈	C₂²×C₈	1	2	yes	—
178	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
179	C₂×C₈	C₂²×C₈	1	2	yes	—
180	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
181	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
182	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
184	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
185	C₄⋊C₄	C₄⋊C₈	1	1	yes	characteristic
186	C₄⋊C₄	C₄⋊C₈	1	2	yes	—
187	C₄⋊C₄	C₄⋊C₈	1	1	yes	characteristic
188	C₂²⋊C₄	C₄⋊C₈	1	2	yes	—
189	C₂²⋊C₄	C₄⋊C₈	1	2	yes	—
190	C₂²⋊C₄	C₄⋊C₈	1	2	yes	—
191	C₄²	C₂×C₄⋊C₄	1	2	yes	—
192	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₄²	C₂²×C₈	1	2	yes	—
194	C₄²	C₂²×C₈	1	1	yes	characteristic
195	C₄²	C₂×M₁₆	1	2	yes	—
196	C₄²	C₂×M₁₆	1	1	yes	characteristic
197	C₄²	C₂³×C₄	1	1	yes	characteristic
198	C₁₆	—	4	2	no	—
199	C₁₆	—	4	2	no	—
200	C₁₆	—	4	4	no	—
201	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
202	C₂³	C₂×C₄⋊C₈	1	2	yes	—
203	Q₈	—	2	2	no	—
204	D₄	—	2	4	no	—
205	D₄	#364	1	4	yes	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
217	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
219	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
220	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
224	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	2	4	no	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	2	1	no	—
230	C₈	—	2	1	no	—
231	C₈	—	2	2	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	2	no	—
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	#504	1	2	yes	—
239	C₈	#577	1	2	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	C₂×#364	1	4	yes	—
246	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	1	no	—
253	C₄	C₂×#364	1	2	yes	—
254	C₄	#2912	1	1	yes	characteristic
255	C₄	#2916	1	1	yes	characteristic
256	C₄	C₂×#504	1	1	yes	characteristic
257	C₄	#3407	1	1	yes	characteristic
258	C₂	—	2	4	no	—
259	C₂	—	4	1	no	—
260	C₂	#17788	1	1	yes	characteristic
261	C₂	#33398	1	1	yes	characteristic
262	C₂	C₂²×#364	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	M₁₆
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. #12016	C₂	semidirect
5. #12013	C₂	semidirect
6. #12012	C₂	semidirect
7. #11996	C₂	semidirect
8. C₂×#1367	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33504	2
C₈×#131	2
#12018	2
#12016	2
#12013	2
#12012	2
#11996	2
C₂×#1367	2