Finite Group Databasebeta

← #213619

Subgroups — #213619

Order 512.

1122 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1122 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#9126C218yesmaximal, complemented
4#8719C212yesmaximal, complemented
5#8335C212yesmaximal, complemented
6C22×#125C2211yescharacteristic
7C23×C42C2211yescharacteristic
8C2×#432C2214yes—
9#2851—22no—
10#2851—22no—
11#2851C414yes—
12#2851C2214yes—
13C2×#389C412yes—
14C2×#387—22no—
15C2×#385—22no—
16C2×#382C2211yescharacteristic
17C2×#380C412yes—
18#1817—22no—
19#1797—22no—
20#1797—22no—
21#1779—22no—
22#1749C2214yes—
23#1749C2214yes—
24C2×#330C2218yescomplemented
25#1295C2218yescomplemented
26C24×C4C2×C411yescharacteristic
27C24×C4D412yes—
28C2×#125—22no—
29C2×#125—44no—
30C2×#125C2×C412yes—
31C2×#125C2312yes—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C412yes—
36C22×C22⋊C4—22no—
37C22×C42—44no—
38C22×C42C2×C412yes—
39C22×C42C2×C411yescharacteristic
40C22×C42D412yes—
41C22×C42D412yes—
42C22×C42D412yes—
43C22×C42C2312yes—
44C22×C42C2311yescharacteristic
45#432—48no—
46C2×C4⋊C8—24no—
47C2×C22⋊C8—24no—
48C2×C22⋊C8—24no—
49C2×C22⋊C8—24no—
50C2×#97C2318yes—
51#389—22no—
52#389—22no—
53#389C2×C414yes—
54#389C2×C414yes—
55#387—24no—
56#387—24no—
57#387—24no—
58#387—24no—
59#385—24no—
60#385—24no—
61#385—24no—
62#385—24no—
63#382—22no—
64#382C2×C412yes—
65#382C2312yes—
66#380—22no—
67#380—22no—
68#380C2×C414yes—
69#380C2×C414yes—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73#336—24no—
74#336—24no—
75#330—48no—
76C25C22⋊C411yescharacteristic
77C23×C4—24no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—44no—
83C23×C4—44no—
84C23×C4—44no—
85C23×C4C22⋊C412yes—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C412yes—
89C23×C4C22⋊C412yes—
90C23×C4C22×C411yescharacteristic
91C23×C4C22×C412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C22×C8—28no—
96#125—28no—
97#125—44no—
98#125—48no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—44no—
106C2×C4⋊C4—48no—
107C2×C4⋊C4—48no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4C22×C414yes—
110C2×C4⋊C4C22×C414yes—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—48no—
114C2×C42—24no—
115C2×C42—28no—
116C2×C42—42no—
117C2×C42—44no—
118C2×C42—44no—
119C2×C42—44no—
120C2×C42—44no—
121C2×C42C22⋊C412yes—
122C2×C42C22⋊C412yes—
123C2×C42C22⋊C411yescharacteristic
124C2×C42C22⋊C411yescharacteristic
125C2×C42C22×C411yescharacteristic
126C2×C42C22×C412yes—
127C2×C42C2×D412yes—
128C2×C42C2×D412yes—
129C2×C42C2×D412yes—
130C2×C42C2411yescharacteristic
131C4⋊C8—48no—
132C22⋊C8—48no—
133C22⋊C8—48no—
134C22⋊C8—48no—
135#97—216no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—24no—
140#95—24no—
141#95—24no—
142C24—24no—
143C24—41no—
144C24—41no—
145C24—42no—
146C24#10114yes—
147C24C2×C22⋊C412yes—
148C24C2×C22⋊C411yescharacteristic
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—22no—
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—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—416no—
182C22×C4—432no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4C2×C22⋊C412yes—
190C22×C4C2×C22⋊C412yes—
191C22×C4C2×C22⋊C412yes—
192C22×C4C2×C22⋊C411yescharacteristic
193C22×C4C2×C22⋊C412yes—
194C22×C4C23×C411yescharacteristic
195C22×C4C22×D412yes—
196C2×C8—48no—
197C2×C8—48no—
198C2×C8—48no—
199C4⋊C4—28no—
200C4⋊C4—44no—
201C4⋊C4—48no—
202C4⋊C4—44no—
203C4⋊C4—48no—
204C4⋊C4—48no—
205C4⋊C4—44no—
206C4⋊C4—48no—
207C22⋊C4—48no—
208C22⋊C4—48no—
209C42—24no—
210C42—22no—
211C42—24no—
212C42—24no—
213C42—24no—
214C42—41no—
215C42—41no—
216C42—42no—
217C42—42no—
218C42H(5,𝔽2)14yes—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—28no—
229C23—44no—
230C23—44no—
231C23—48no—
232C23—48no—
233C23C2×#10114yes—
234C23C2×#10112yes—
235C23#42114yes—
236C23C22×C22⋊C411yescharacteristic
237C2×C4—28no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
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—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—48no—
256C2×C4—48no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—48no—
260C2×C4—48no—
261C2×C4—48no—
262C2×C4—44no—
263C2×C4—48no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—88no—
271C2×C4—88no—
272C2×C4#41312yes—
273C2×C4#42114yes—
274C2×C4#51812yes—
275C2×C4#53414yes—
276C8—88no—
277C22—24no—
278C22—22no—
279C22—24no—
280C22—22no—
281C22—24no—
282C22—24no—
283C22—24no—
284C22—416no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22#153414yes—
290C22#230811yescharacteristic
291C22C22×#10111yescharacteristic
292C22C2×#41311yescharacteristic
293C22C2×#42112yes—
294C22C2×#42112yes—
295C4—24no—
296C4—24no—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—42no—
301C4—42no—
302C4—84no—
303C4—84no—
304C2—24no—
305C2—44no—
306C2C2×#153411yescharacteristic
307C2#1767012yes—
308C2#1767611yescharacteristic
309C2#2175311yescharacteristic
310C2#2177412yes—
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 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9126C2semidirect
4. #8719C2semidirect
5. #8335C2semidirect
24. C2×#330C22semidirect
25. #1295C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
#207152
#91262
#87192
#83352