Subgroups — #101993

Order 512.

442 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#500	C₂	1	1	yes	maximal, characteristic
2	#9673	C₂	1	2	yes	maximal
3	C₄×#335	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#7570	C₂	1	2	yes	maximal
6	C₂×#500	C₄	1	1	yes	characteristic
7	C₄×C₄⋊Q₈	C₂²	1	2	yes	—
8	#1990	C₄	1	2	yes	—
9	#1942	—	2	1	no	—
10	#1942	—	2	1	no	—
11	#1845	C₄	1	1	yes	characteristic
12	C₄×#107	—	2	1	no	—
13	C₄×#107	—	2	1	no	—
14	C₄×C₄⋊C₈	C₂²	1	2	yes	—
15	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	C₂×#335	—	2	1	no	—
19	C₂×Q₈⋊C₈	—	2	4	no	—
20	C₂×C₈²	—	2	1	no	—
21	#1277	—	2	1	no	—
22	#1273	—	2	1	no	—
23	#1271	—	2	4	no	—
24	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×Q₈	—	2	2	no	—
26	#500	—	2	2	no	—
27	#500	C₈	1	4	yes	complemented
28	C₂×#107	—	2	1	no	—
29	C₂×#107	—	2	1	no	—
30	C₂×#107	—	2	1	no	—
31	C₂×#107	—	2	1	no	—
32	C₂×C₄⋊C₈	—	2	1	no	—
33	C₂×C₄⋊C₈	—	2	2	no	—
34	C₂×C₄⋊C₈	D₄	1	2	yes	—
35	C₂×C₄×C₈	—	2	1	no	—
36	C₂×C₄×C₈	—	2	1	no	—
37	C₂×C₄×C₈	—	2	2	no	—
38	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	#391	—	2	2	no	—
45	#389	—	2	2	no	—
46	#389	—	2	2	no	—
47	#384	C₂×C₄	1	2	yes	—
48	C₄×C₄⋊C₄	—	2	2	no	—
49	C₄×C₄⋊C₄	—	2	2	no	—
50	C₄³	C₂³	1	1	yes	characteristic
51	#340	—	2	2	no	—
52	#340	—	2	2	no	—
53	#336	—	2	2	no	—
54	#336	—	2	1	no	—
55	#336	D₄	1	2	yes	—
56	#335	—	2	4	no	—
57	Q₈⋊C₈	—	4	8	no	—
58	C₈²	—	4	2	no	—
59	C₂²×Q₈	—	2	2	no	—
60	C₂²×C₈	—	2	2	no	—
61	C₂²×C₈	—	2	2	no	—
62	C₂²×C₈	—	4	2	no	—
63	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₄⋊Q₈	—	2	2	no	—
68	C₄⋊Q₈	—	2	2	no	—
69	C₄⋊Q₈	—	2	2	no	—
70	C₄⋊Q₈	C₂×C₈	1	4	yes	—
71	C₄×Q₈	—	4	4	no	—
72	C₄×Q₈	—	4	4	no	—
73	C₂×C₄⋊C₄	—	2	2	no	—
74	C₂×C₄⋊C₄	—	2	2	no	—
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	4	2	no	—
79	C₂×C₄²	—	4	2	no	—
80	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄²	#41	1	1	yes	characteristic
86	C₂×C₄²	#41	1	1	yes	characteristic
87	#107	—	2	4	no	—
88	#107	—	2	4	no	—
89	#107	—	4	2	no	—
90	#107	—	4	2	no	—
91	C₄⋊C₈	—	2	4	no	—
92	C₄⋊C₈	—	2	4	no	—
93	C₄⋊C₈	—	2	2	no	—
94	C₄⋊C₈	—	4	2	no	—
95	C₄⋊C₈	D₈	1	4	yes	—
96	C₄×C₈	—	2	2	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	—	4	1	no	—
102	C₄×C₈	—	4	4	no	—
103	C₄×C₈	—	4	1	no	—
104	C₄×C₈	—	4	1	no	—
105	C₄×C₈	—	4	1	no	—
106	C₄×C₈	—	4	1	no	—
107	C₄×C₈	C₂×C₈	1	2	yes	—
108	C₄×C₈	M₁₆	1	2	yes	—
109	C₄×C₈	SD₁₆	1	2	yes	—
110	C₄×C₈	Q₁₆	1	2	yes	—
111	#95	—	4	2	no	—
112	#95	—	4	2	no	—
113	C₂×Q₈	—	2	4	no	—
114	C₂×Q₈	—	4	2	no	—
115	C₂×Q₈	—	4	4	no	—
116	C₂×Q₈	—	4	4	no	—
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	—	4	2	no	—
119	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
122	C₂²×C₄	#121	1	1	yes	characteristic
123	C₂²×C₄	#121	1	1	yes	characteristic
124	C₂²×C₄	#124	1	1	yes	characteristic
125	C₂²×C₄	#127	1	1	yes	characteristic
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	8	2	no	—
141	C₂×C₈	—	8	2	no	—
142	C₂×C₈	C₂²⋊C₈	1	2	yes	—
143	C₂×C₈	C₂²⋊C₈	1	2	yes	—
144	C₂×C₈	#103	1	2	yes	—
145	C₂×C₈	#103	1	2	yes	—
146	C₂×C₈	#104	1	2	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	2	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	2	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	—	8	2	no	—
155	C₄⋊C₄	—	8	2	no	—
156	C₄⋊C₄	—	8	2	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	4	1	no	—
162	C₄²	—	8	4	no	—
163	C₄²	C₂²×C₈	1	1	yes	characteristic
164	C₄²	C₂×M₁₆	1	1	yes	characteristic
165	C₄²	#131	1	1	yes	characteristic
166	C₄²	#131	1	1	yes	characteristic
167	C₄²	C₂×D₈	1	2	yes	—
168	C₄²	C₂×SD₁₆	1	1	yes	characteristic
169	C₄²	C₂×Q₁₆	1	1	yes	characteristic
170	C₄²	#135	1	2	yes	—
171	C₄²	#137	1	1	yes	characteristic
172	C₄²	#137	1	1	yes	characteristic
173	C₂³	#390	1	1	yes	characteristic
174	Q₈	—	4	8	no	—
175	Q₈	—	8	4	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	2	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
192	C₂×C₄	#408	1	1	yes	characteristic
193	C₂×C₄	C₂×#103	1	1	yes	characteristic
194	C₂×C₄	#419	1	1	yes	characteristic
195	C₂×C₄	C₂×#104	1	1	yes	characteristic
196	C₂×C₄	#421	1	1	yes	characteristic
197	C₂×C₄	C₈×D₄	1	1	yes	characteristic
198	C₂×C₄	C₈×D₄	1	1	yes	characteristic
199	C₂×C₄	#436	1	1	yes	characteristic
200	C₂×C₄	#436	1	1	yes	characteristic
201	C₂×C₄	C₄×D₈	1	2	yes	—
202	C₂×C₄	#443	1	2	yes	—
203	C₂×C₄	#461	1	1	yes	characteristic
204	C₂×C₄	#461	1	1	yes	characteristic
205	C₂×C₄	#462	1	1	yes	characteristic
206	C₂×C₄	#462	1	1	yes	characteristic
207	C₂×C₄	#466	1	2	yes	—
208	C₂×C₄	#473	1	2	yes	—
209	C₂×C₄	#493	1	1	yes	characteristic
210	C₂×C₄	#495	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₈	—	4	4	no	—
214	C₈	—	4	4	no	—
215	C₈	—	4	1	no	—
216	C₈	—	8	4	no	—
217	C₈	Q₈⋊C₈	1	4	yes	complemented
218	C₂²	#1954	1	1	yes	characteristic
219	C₂²	#1970	1	1	yes	characteristic
220	C₂²	#1974	1	1	yes	characteristic
221	C₂²	#1976	1	1	yes	characteristic
222	C₂²	#1979	1	1	yes	characteristic
223	C₂²	#1980	1	1	yes	characteristic
224	C₂²	#1981	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	8	4	no	—
233	C₄	—	16	2	no	—
234	C₄	C₂×Q₈⋊C₈	1	1	yes	characteristic
235	C₄	#1479	1	1	yes	characteristic
236	C₄	#1481	1	1	yes	characteristic
237	C₄	#1485	1	1	yes	characteristic
238	C₄	C₈×D₈	1	2	yes	—
239	C₄	#1589	1	2	yes	—
240	C₄	#1673	1	2	yes	—
241	C₄	#1674	1	2	yes	—
242	C₂	#9657	1	1	yes	characteristic
243	C₂	#9669	1	1	yes	characteristic
244	C₂	#9673	1	1	yes	characteristic
245	C₂	#9719	1	1	yes	characteristic
246	C₂	#9725	1	1	yes	characteristic
247	C₂	#9731	1	1	yes	characteristic
248	C₂	#9749	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#390
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
27. #500	C₈	semidirect
217. C₈	Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#500	2
#9673	2
C₄×#335	2
C₈×C₄⋊C₈	2
#7570	2