Finite Group Databasebeta

← #163263

Subgroups — #163263

Order 512.

1102 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2#20715C212yesmaximal
3#7731C218yesmaximal, complemented
4#7627C214yesmaximal, complemented
5C22×#125C411yescharacteristic
6C23×C42C2211yescharacteristic
7C2×#432—22no—
8#2851—22no—
9#2851—22no—
10#2851C414yes—
11#2851C2214yes—
12C2×#389C2212yes—
13C2×#387—22no—
14C2×#385—22no—
15C2×#382C411yescharacteristic
16C2×#380C412yes—
17#1817C2214yes—
18#1797—22no—
19#1797C2214yes—
20#1779—22no—
21#1749—22no—
22#1749C2214yes—
23#1271C22116yescomplemented
24C24×C4C2×C411yescharacteristic
25C24×C4D412yes—
26C2×#125—21no—
27C2×#125—21no—
28C2×#125—44no—
29C2×#125C2×C414yes—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4C2×C412yes—
33C22×C4⋊C4C2×C412yes—
34C22×C22⋊C4—22no—
35C22×C42—44no—
36C22×C42C2×C411yescharacteristic
37C22×C42C2×C412yes—
38C22×C42D412yes—
39C22×C42D412yes—
40C22×C42D412yes—
41C22×C42C2312yes—
42C22×C42C2311yescharacteristic
43#432—48no—
44C2×C4⋊C8—24no—
45C2×C22⋊C8—24no—
46C2×C22⋊C8—24no—
47C2×C22⋊C8—24no—
48C2×#97—24no—
49#389—22no—
50#389—22no—
51#389C2×C414yes—
52#389C2314yes—
53#387—24no—
54#387—24no—
55#387—24no—
56#387—24no—
57#385—24no—
58#385—24no—
59#385—24no—
60#385—24no—
61#382—21no—
62#382—21no—
63#382C2×C414yes—
64#380—22no—
65#380—22no—
66#380C2×C414yes—
67#380C2×C414yes—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71#336—24no—
72#336C2318yes—
73C25C22⋊C411yescharacteristic
74C23×C4—24no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—44no—
80C23×C4—44no—
81C23×C4—44no—
82C23×C4C22⋊C412yes—
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C412yes—
86C23×C4C22⋊C412yes—
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C412yes—
89C23×C4C2×D412yes—
90C23×C4C2×D412yes—
91C23×C4C2×D412yes—
92C22×C8—28no—
93#125—48no—
94#125—44no—
95#125—44no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—44no—
103C2×C4⋊C4—48no—
104C2×C4⋊C4—48no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4C22×C414yes—
107C2×C4⋊C4C22×C414yes—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—48no—
111C2×C42—24no—
112C2×C42—28no—
113C2×C42—44no—
114C2×C42—44no—
115C2×C42—44no—
116C2×C42—42no—
117C2×C42—44no—
118C2×C42C22⋊C411yescharacteristic
119C2×C42C22⋊C412yes—
120C2×C42C22⋊C412yes—
121C2×C42C22⋊C411yescharacteristic
122C2×C42C22×C412yes—
123C2×C42C22×C411yescharacteristic
124C2×C42C2×D412yes—
125C2×C42C2×D412yes—
126C2×C42C2×D412yes—
127C2×C42C2411yescharacteristic
128C4⋊C8—48no—
129C22⋊C8—48no—
130C22⋊C8—48no—
131C22⋊C8—48no—
132#97—48no—
133#95—24no—
134#95—24no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95—24no—
139C24—24no—
140C24—41no—
141C24—41no—
142C24—42no—
143C24#10114yes—
144C24C2×C22⋊C412yes—
145C24C2×C22⋊C411yescharacteristic
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—416no—
179C22×C4—432no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4C2×C22⋊C412yes—
187C22×C4C2×C22⋊C412yes—
188C22×C4C2×C22⋊C412yes—
189C22×C4C2×C22⋊C412yes—
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4C23×C411yescharacteristic
192C22×C4C22×D412yes—
193C2×C8—48no—
194C2×C8—48no—
195C2×C8—48no—
196C4⋊C4—28no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—48no—
200C4⋊C4—48no—
201C4⋊C4—48no—
202C4⋊C4—48no—
203C4⋊C4—44no—
204C22⋊C4—48no—
205C22⋊C4—48no—
206C42—24no—
207C42—24no—
208C42—24no—
209C42—22no—
210C42—24no—
211C42—42no—
212C42—41no—
213C42—42no—
214C42—41no—
215C42#9914yes—
216C23—21no—
217C23—21no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—28no—
226C23—44no—
227C23—44no—
228C23—48no—
229C23—48no—
230C23C2×#10114yes—
231C23C2×#10112yes—
232C23#42114yes—
233C23C22×C22⋊C411yescharacteristic
234C2×C4—28no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—48no—
253C2×C4—48no—
254C2×C4—48no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—44no—
260C2×C4—48no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—88no—
268C2×C4—88no—
269C2×C4C2×#9914yes—
270C2×C4C2×#9912yes—
271C2×C4#56812yes—
272C2×C4#57514yes—
273C8—88no—
274C22—24no—
275C22—22no—
276C22—24no—
277C22—22no—
278C22—24no—
279C22—24no—
280C22—24no—
281C22—416no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22#153414yes—
287C22#287911yescharacteristic
288C22C22×#9911yescharacteristic
289C22C22×#10111yescharacteristic
290C22#289612yes—
291C22C2×#42112yes—
292C4—24no—
293C4—24no—
294C4—22no—
295C4—22no—
296C4—42no—
297C4—42no—
298C4—42no—
299C4—84no—
300C4—84no—
301C2—24no—
302C2—44no—
303C2#1739711yescharacteristic
304C2#1749712yes—
305C2#1758212yes—
306C2C2×#153411yescharacteristic
307C2#1775511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7731C2semidirect
4. #7627C2semidirect
23. #1271C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
#207152
#77312
#76272