Subgroups — #498524

Order 512.

486 conjugacy classes of subgroups.

The subgroup lattice has 265 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
2	#23160	C₂	1	2	yes	maximal
3	#13277	C₂	1	2	yes	maximal
4	#13165	C₂	1	1	yes	maximal, characteristic
5	#13165	C₂	1	1	yes	maximal, characteristic
6	#13111	C₂	1	2	yes	maximal
7	#13110	C₂	1	2	yes	maximal
8	C₄×#366	C₂	1	1	yes	maximal, characteristic
9	C₄×#366	C₂	1	1	yes	maximal, characteristic
10	C₄×#366	C₂	1	1	yes	maximal, characteristic
11	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂×#499	C₂²	1	2	yes	—
13	#2370	—	2	2	no	—
14	#2253	C₄	1	8	yes	complemented
15	C₂×#367	C₂²	1	2	yes	—
16	C₂×#366	C₂²	1	1	yes	characteristic
17	C₂×#366	C₂²	1	1	yes	characteristic
18	C₂×#366	C₂²	1	1	yes	characteristic
19	C₂×#366	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
24	#1989	—	2	2	no	—
25	#1942	C₂²	1	2	yes	—
26	#1942	C₂²	1	2	yes	—
27	#1920	—	2	2	no	—
28	#1919	—	2	2	no	—
29	C₄×#107	C₂²	1	2	yes	—
30	C₄×#107	C₂²	1	2	yes	—
31	C₄×#107	C₂²	1	2	yes	—
32	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	#1380	C₂²	1	2	yes	—
34	#1380	C₂²	1	2	yes	—
35	#1380	C₂²	1	2	yes	—
36	#1380	C₂²	1	2	yes	—
37	#1380	C₂²	1	2	yes	—
38	#1374	C₂²	1	1	yes	characteristic
39	#1374	C₂²	1	1	yes	characteristic
40	C₂×#125	—	2	2	no	—
41	C₂²×C₁₆	C₂³	1	1	yes	characteristic
42	C₂²×C₁₆	C₂³	1	1	yes	characteristic
43	C₂²×C₁₆	C₂³	1	1	yes	characteristic
44	C₂²×C₁₆	C₂³	1	1	yes	characteristic
45	#499	—	2	4	no	—
46	#499	C₂×C₄	1	8	yes	—
47	C₂×#107	C₂³	1	2	yes	—
48	C₂×#107	C₂³	1	2	yes	—
49	C₂×#107	C₂³	1	2	yes	—
50	C₂×#107	C₂³	1	2	yes	—
51	C₂×#106	—	2	2	no	—
52	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
56	#389	—	2	2	no	—
57	#389	—	2	2	no	—
58	#389	—	4	2	no	—
59	#387	—	4	2	no	—
60	#387	—	4	2	no	—
61	C₄×C₄⋊C₄	—	2	2	no	—
62	C₄×C₄⋊C₄	—	2	2	no	—
63	C₄×C₄⋊C₄	—	2	2	no	—
64	C₄×C₄⋊C₄	—	4	2	no	—
65	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
66	#367	C₂×C₄	1	8	yes	—
67	#366	—	2	2	no	—
68	#366	C₂×C₄	1	4	yes	—
69	#366	C₂×C₄	1	4	yes	—
70	#366	Q₈	1	4	yes	—
71	C₄⋊C₁₆	—	2	2	no	—
72	C₄×C₁₆	—	2	2	no	—
73	C₄×C₁₆	—	2	2	no	—
74	C₄×C₁₆	C₂×C₄	1	4	yes	—
75	#340	—	2	2	no	—
76	#340	—	2	2	no	—
77	#340	—	2	2	no	—
78	#340	—	2	2	no	—
79	#340	—	2	2	no	—
80	#336	D₄	1	1	yes	characteristic
81	#336	D₄	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₈	C₂⁴	1	1	yes	characteristic
86	#125	—	2	8	no	—
87	#125	—	4	4	no	—
88	C₂×C₄⋊C₄	—	2	2	no	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	4	2	no	—
93	C₂×C₄⋊C₄	—	4	2	no	—
94	C₂×C₄⋊C₄	—	4	2	no	—
95	C₂×C₄⋊C₄	—	4	2	no	—
96	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
97	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
98	C₂×C₄²	—	4	2	no	—
99	C₂×C₄²	—	4	2	no	—
100	C₂×C₄²	—	4	2	no	—
101	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
104	C₂×C₁₆	—	2	1	no	—
105	C₂×C₁₆	—	2	1	no	—
106	C₂×C₁₆	—	2	1	no	—
107	C₂×C₁₆	—	2	1	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	—	2	1	no	—
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	C₂²×C₄	1	2	yes	—
113	C₂×C₁₆	C₂²×C₄	1	2	yes	—
114	C₂×C₁₆	C₂×Q₈	1	2	yes	—
115	C₂×C₁₆	#41	1	2	yes	—
116	#107	—	2	2	no	—
117	#107	—	2	2	no	—
118	#107	—	2	2	no	—
119	#107	—	2	2	no	—
120	#107	—	2	2	no	—
121	#107	C₂²×C₄	1	4	yes	—
122	#107	C₂²×C₄	1	4	yes	—
123	#107	C₂×Q₈	1	4	yes	—
124	#106	—	2	8	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄×C₈	—	2	1	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	C₂²×C₄	1	2	yes	—
131	C₄×C₈	#41	1	2	yes	—
132	#95	—	4	2	no	—
133	#95	—	4	2	no	—
134	#95	—	4	2	no	—
135	#95	—	4	2	no	—
136	#95	—	4	2	no	—
137	#95	D₈	1	1	yes	characteristic
138	#95	D₈	1	1	yes	characteristic
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
146	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	C₄×D₄	1	2	yes	—
159	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₈	C₂×#41	1	1	yes	characteristic
162	C₂×C₈	C₂×#41	1	1	yes	characteristic
163	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
164	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	8	no	—
174	C₄⋊C₄	—	4	8	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	8	4	no	—
178	C₄⋊C₄	—	16	2	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	8	4	no	—
184	C₄²	—	16	2	no	—
185	C₄²	—	16	2	no	—
186	C₄²	C₄×D₄	1	2	yes	—
187	C₄²	#135	1	2	yes	—
188	C₁₆	—	2	2	no	—
189	C₁₆	—	4	1	no	—
190	C₁₆	—	4	1	no	—
191	C₁₆	C₄×Q₈	1	4	yes	—
192	C₂³	C₂²×D₈	1	1	yes	characteristic
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	2	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	C₄×D₈	1	2	yes	—
220	C₂×C₄	C₄×D₈	1	2	yes	—
221	C₂×C₄	#508	1	2	yes	—
222	C₂×C₄	#508	1	2	yes	—
223	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
224	C₂×C₄	#547	1	1	yes	characteristic
225	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
226	C₂×C₄	C₂×#135	1	1	yes	characteristic
227	C₂×C₄	#578	1	1	yes	characteristic
228	C₂×C₄	#578	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	—	4	1	no	—
231	C₈	—	4	1	no	—
232	C₈	C₂×C₄×Q₈	1	1	yes	characteristic
233	C₈	#519	1	1	yes	characteristic
234	C₈	#556	1	1	yes	characteristic
235	C₈	#557	1	1	yes	characteristic
236	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
237	C₂²	#3334	1	1	yes	characteristic
238	C₂²	Q₈×D₈	1	1	yes	characteristic
239	C₂²	C₂×#508	1	1	yes	characteristic
240	C₂²	C₂×#508	1	1	yes	characteristic
241	C₂²	#3415	1	1	yes	characteristic
242	C₂²	#3417	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	8	4	no	—
250	C₄	—	8	4	no	—
251	C₄	—	8	4	no	—
252	C₄	—	16	2	no	—
253	C₄	#2365	1	1	yes	characteristic
254	C₄	#2950	1	1	yes	characteristic
255	C₄	#3342	1	1	yes	characteristic
256	C₄	#3383	1	1	yes	characteristic
257	C₂	#23157	1	1	yes	characteristic
258	C₂	C₄×#508	1	1	yes	characteristic
259	C₂	#33459	1	1	yes	characteristic
260	C₂	#33903	1	1	yes	characteristic
261	C₂	#33906	1	1	yes	characteristic
262	C₂	#33947	1	1	yes	characteristic
263	C₂	#33950	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
14. #2253	C₄	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
#23160	2
#13277	2
#13165	2
#13111	2
#13110	2
C₄×#366	2
C₁₆×C₄⋊C₄	2