Finite Group Databasebeta

← #382692

Subgroups — #382692

Order 512.

2670 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2670 conjugacy classes of subgroups — too many to draw. Only its 314 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
1C22×#380C211yesmaximal, characteristic
2#20579C212yesmaximal
3#11198C218yesmaximal, complemented
4#10727C214yesmaximal, complemented
5C25×C4C2211yescharacteristic
6C23×C4⋊C4C2212yes—
7C23×C22⋊C4C412yes—
8C2×#407C2214yes—
9C2×#407C2218yes—
10C2×#380—24no—
11C2×#380—22no—
12C2×#380—22no—
13C2×#380C414yes—
14C2×#380C2214yes—
15C22×#95C412yes—
16#1779—24no—
17#1779—22no—
18#1779—22no—
19C2×#343C22116yescomplemented
20C26C2×C411yescharacteristic
21C24×C4—28no—
22C24×C4—48no—
23C24×C4C2×C411yescharacteristic
24C24×C4C2×C412yes—
25C24×C4C2×C414yes—
26C24×C4D414yes—
27C24×C4D412yes—
28C24×C4D412yes—
29C24×C4C2312yes—
30C24×C4C2311yescharacteristic
31C22×M16C2318yes—
32C22×C4⋊C4—24no—
33C22×C4⋊C4—24no—
34C22×C4⋊C4—48no—
35C22×C4⋊C4C2×C414yes—
36C22×C4⋊C4C2314yes—
37C22×C22⋊C4—24no—
38C22×C22⋊C4—22no—
39C22×C22⋊C4—22no—
40C22×C22⋊C4—48no—
41C22×C22⋊C4C2×C418yes—
42#407—216no—
43#407—416no—
44C2×C22⋊C8—28no—
45C2×C22⋊C8—28no—
46C2×C22⋊C8—24no—
47C2×C22⋊C8—24no—
48#380—28no—
49#380—28no—
50#380—28no—
51#380—28no—
52C2×#95—24no—
53C2×#95—24no—
54C2×#95C2×C418yes—
55#343—416no—
56C25—24no—
57C25—24no—
58C25—24no—
59C25—44no—
60C25—44no—
61C25C22⋊C412yes—
62C25C22⋊C411yescharacteristic
63C25C22⋊C411yescharacteristic
64C25C22×C411yescharacteristic
65C25C22×C412yes—
66C23×C4—28no—
67C23×C4—24no—
68C23×C4—28no—
69C23×C4—28no—
70C23×C4—28no—
71C23×C4—216no—
72C23×C4—28no—
73C23×C4—216no—
74C23×C4—24no—
75C23×C4—24no—
76C23×C4—48no—
77C23×C4—48no—
78C23×C4—48no—
79C23×C4—48no—
80C23×C4—48no—
81C23×C4—44no—
82C23×C4—48no—
83C23×C4—464no—
84C23×C4—48no—
85C23×C4—48no—
86C23×C4C22⋊C412yes—
87C23×C4C22⋊C412yes—
88C23×C4C22⋊C414yes—
89C23×C4C22⋊C414yes—
90C23×C4C22×C412yes—
91C23×C4C22×C411yescharacteristic
92C23×C4C22×C418yes—
93C23×C4C2×D414yes—
94C23×C4C2×D414yes—
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4#4118yes—
98C23×C4C2411yescharacteristic
99C2×M16—216no—
100C2×M16—232no—
101C22×C8—28no—
102C2×C4⋊C4—28no—
103C2×C4⋊C4—28no—
104C2×C4⋊C4—216no—
105C2×C4⋊C4—48no—
106C2×C4⋊C4—416no—
107C2×C4⋊C4—416no—
108C2×C22⋊C4—28no—
109C2×C22⋊C4—28no—
110C2×C22⋊C4—48no—
111C2×C22⋊C4—416no—
112C2×C22⋊C4—48no—
113C2×C22⋊C4—416no—
114C22⋊C8—416no—
115C22⋊C8—416no—
116C22⋊C8—48no—
117C22⋊C8—48no—
118#95—216no—
119C24—24no—
120C24—22no—
121C24—22no—
122C24—24no—
123C24—232no—
124C24—24no—
125C24—24no—
126C24—22no—
127C24—22no—
128C24—24no—
129C24—22no—
130C24—24no—
131C24—24no—
132C24—22no—
133C24—24no—
134C24—28no—
135C24—28no—
136C24—42no—
137C24—42no—
138C24—44no—
139C24—42no—
140C24—42no—
141C24—44no—
142C24—416no—
143C24—432no—
144C24—432no—
145C24—416no—
146C24#10014yes—
147C24#10114yes—
148C24C2×C22⋊C412yes—
149C24C2×C22⋊C412yes—
150C24C2×C22⋊C411yescharacteristic
151C24C2×C22⋊C411yescharacteristic
152C24#11714yes—
153C24C23×C411yescharacteristic
154C22×C4—24no—
155C22×C4—28no—
156C22×C4—28no—
157C22×C4—232no—
158C22×C4—24no—
159C22×C4—28no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—28no—
163C22×C4—28no—
164C22×C4—28no—
165C22×C4—28no—
166C22×C4—28no—
167C22×C4—28no—
168C22×C4—28no—
169C22×C4—28no—
170C22×C4—28no—
171C22×C4—216no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—432no—
180C22×C4—432no—
181C22×C4—432no—
182C22×C4—464no—
183C22×C4—416no—
184C22×C4—432no—
185C22×C4—48no—
186C22×C4—48no—
187C22×C4—48no—
188C22×C4—48no—
189C22×C4—48no—
190C22×C4—48no—
191C22×C4—48no—
192C22×C4—48no—
193C22×C4—832no—
194C22×C4C2×C22⋊C414yes—
195C22×C4C2×C22⋊C412yes—
196C22×C4#11714yes—
197C22×C4#12318yes—
198C22×C4#12318yes—
199C22×C4C22×D412yes—
200C22×C4C2×#4114yes—
201C22×C4H(5,𝔽2)18yes—
202M16—432no—
203C2×C8—48no—
204C2×C8—416no—
205C4⋊C4—416no—
206C4⋊C4—416no—
207C22⋊C4—416no—
208C22⋊C4—416no—
209C23—216no—
210C23—216no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—28no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—22no—
221C23—22no—
222C23—216no—
223C23—216no—
224C23—24no—
225C23—24no—
226C23—28no—
227C23—28no—
228C23—22no—
229C23—22no—
230C23—24no—
231C23—24no—
232C23—22no—
233C23—22no—
234C23—48no—
235C23—48no—
236C23—416no—
237C23—416no—
238C23—416no—
239C23—416no—
240C23—44no—
241C23—44no—
242C23—48no—
243C23—4128no—
244C23—48no—
245C23—44no—
246C23—44no—
247C23—48no—
248C23—48no—
249C23—48no—
250C23—48no—
251C23#38112yes—
252C23#38112yes—
253C23C2×#10014yes—
254C23C2×#10012yes—
255C23C2×#10114yes—
256C23C2×#10112yes—
257C23#41314yes—
258C23C22×C22⋊C411yescharacteristic
259C23C2×#11712yes—
260C23#51814yes—
261C2×C4—24no—
262C2×C4—28no—
263C2×C4—216no—
264C2×C4—216no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—416no—
269C2×C4—416no—
270C2×C4—48no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—416no—
274C2×C4—416no—
275C2×C4—48no—
276C2×C4—48no—
277C2×C4—48no—
278C2×C4—816no—
279C2×C4—816no—
280C2×C4—816no—
281C2×C4#38114yes—
282C2×C4C2×#12314yes—
283C2×C4#53418yes—
284C2×C4#538116yes—
285C8—88no—
286C22—232no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—24no—
294C22—24no—
295C22—24no—
296C22—24no—
297C22—24no—
298C22—24no—
299C22—24no—
300C22—24no—
301C22—28no—
302C22—28no—
303C22—432no—
304C22—416no—
305C22—416no—
306C22—432no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—44no—
312C22—44no—
313C22#182114yes—
314C22#182214yes—
315C22C2×#38111yescharacteristic
316C22#230812yes—
317C22#231214yes—
318C22C22×#10011yescharacteristic
319C22C22×#10111yescharacteristic
320C22C2×#41312yes—
321C4—24no—
322C4—44no—
323C4—88no—
324C4#262218yes—
325C2—24no—
326C2—24no—
327C2—24no—
328C2—44no—
329C2—44no—
330C2#1434111yescharacteristic
331C2C2×#182111yescharacteristic
332C2C2×#182211yescharacteristic
333C2#2175212yes—
334C2#2175312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11198C2semidirect
4. #10727C2semidirect
19. C2×#343C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3802
#205792
#111982
#107272