Finite Group Databasebeta

← #375735

Subgroups — #375735

Order 512.

1518 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1518 conjugacy classes of subgroups — too many to draw. Only its 274 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#13753C212yesmaximal
3#11353C214yesmaximal
4#10912C218yesmaximal
5C22×#125C411yescharacteristic, complemented
6C23×C42C2211yescharacteristic
7#2851—22no—
8#2851—22no—
9#2851—22no—
10#2851—22no—
11C2×#389C412yescomplemented
12C2×#387C414yescomplemented
13C2×#385C2214yes—
14C2×#382C411yescharacteristic, complemented
15C2×#380C2212yes—
16C2×C4×C22⋊C4C2214yes—
17C2×#376—22no—
18C2×#342C22116yes—
19#1434—22no—
20#1434—22no—
21C22⋊#95C2214yes—
22C22⋊#95C2214yes—
23C4×#95—22no—
24C24×C4C2311yescharacteristic
25C24×C4C2312yes—
26C2×#125—21no—
27C2×#125—22no—
28C2×#125—21no—
29C2×#125—44no—
30C22×C4⋊C4C2×C412yes—
31C22×C4⋊C4C2×C414yes—
32C22×C4⋊C4C2×C412yes—
33C22×C4⋊C4C2×C414yes—
34C22×C22⋊C4C2314yes—
35C22×C22⋊C4C2318yes—
36C22×C42—24no—
37C22×C42—44no—
38C22×C42C2×C411yescharacteristic
39C22×C42C2×C412yes—
40C22×C42C2×C411yescharacteristic
41C22×C42D412yes—
42C22×C42D412yes—
43C22×C42D412yes—
44C22×C42D412yes—
45#389—22no—
46#389—22no—
47#389—22no—
48#389—22no—
49#387—24no—
50#387—24no—
51#387—24no—
52#387—24no—
53#385—24no—
54#385—24no—
55#385—24no—
56#385—24no—
57#382—21no—
58#382—22no—
59#382—21no—
60#380—22no—
61#380—22no—
62#380—22no—
63#380—22no—
64C4×C22⋊C4—216no—
65#376—48no—
66C2×#95—24no—
67C2×#95—24no—
68C2×#95—24no—
69C2×#95—24no—
70C2×#95—24no—
71C2×#95C2×C414yes—
72C2×#95C2×C414yes—
73C2×#95C2×C414yes—
74#342—232no—
75C25C2411yescharacteristic
76C23×C4—28no—
77C23×C4—24no—
78C23×C4—28no—
79C23×C4—44no—
80C23×C4—42no—
81C23×C4—44no—
82C23×C4—42no—
83C23×C4C22×C411yescharacteristic
84C23×C4C22×C412yes—
85C23×C4C22×C412yes—
86C23×C4C22×C414yes—
87C23×C4C22×C414yes—
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D412yes—
90C23×C4C2×D412yes—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95#125—48no—
96#125—44no—
97#125—44no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—48no—
108C2×C4⋊C4—44no—
109C2×C4⋊C4—48no—
110C2×C22⋊C4—216no—
111C2×C22⋊C4—216no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—216no—
115C2×C22⋊C4—28no—
116C2×C22⋊C4#41116yes—
117C2×C42—24no—
118C2×C42—28no—
119C2×C42—216no—
120C2×C42—42no—
121C2×C42—44no—
122C2×C42—44no—
123C2×C42—44no—
124C2×C42—44no—
125C2×C42—48no—
126C2×C42—48no—
127C2×C42C22⋊C411yescharacteristic
128C2×C42C22⋊C412yes—
129C2×C42C22⋊C412yes—
130C2×C42C22⋊C412yes—
131C2×C42C22⋊C412yes—
132C2×C42C22⋊C412yes—
133C2×C42C22⋊C412yes—
134C2×C42C22⋊C411yescharacteristic
135C2×C42C22⋊C411yescharacteristic
136C2×C42C22⋊C411yescharacteristic
137#95—216no—
138#95—24no—
139#95—24no—
140#95—24no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—48no—
145#95—48no—
146#95—48no—
147#95—48no—
148C24—41no—
149C24—42no—
150C24—41no—
151C24C23×C411yescharacteristic
152C24C22×D412yes—
153C24C2×#4118yes—
154C242−1+414yes—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—216no—
178C22×C4—28no—
179C22×C4—28no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—432no—
191C22×C4—416no—
192C22×C4—48no—
193C22×C4—48no—
194C22×C4—48no—
195C22×C4—48no—
196C22×C4—48no—
197C22×C4—48no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4C2×C22⋊C412yes—
203C22×C4C2×C22⋊C412yes—
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C22⋊C412yes—
206C22×C4C2×C22⋊C411yescharacteristic
207C22×C4C2×C22⋊C412yes—
208C22×C4C2×C22⋊C411yescharacteristic
209C4⋊C4—44no—
210C4⋊C4—48no—
211C4⋊C4—44no—
212C4⋊C4—48no—
213C4⋊C4—44no—
214C4⋊C4—48no—
215C4⋊C4—44no—
216C4⋊C4—48no—
217C22⋊C4—232no—
218C22⋊C4—48no—
219C22⋊C4—48no—
220C22⋊C4—416no—
221C42—24no—
222C42—24no—
223C42—24no—
224C42—24no—
225C42—22no—
226C42—41no—
227C42—41no—
228C42—42no—
229C42—42no—
230C42—432no—
231C42#9914yes—
232C23—21no—
233C23—22no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—48no—
240C23—44no—
241C23—48no—
242C23—44no—
243C23C22×C22⋊C411yescharacteristic
244C23C4×#4114yes—
245C23#51912yes—
246C23#52514yes—
247C23#53214yes—
248C23#53614yes—
249C23#552116yes—
250C2×C4—216no—
251C2×C4—28no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—48no—
270C2×C4—48no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—48no—
274C2×C4—48no—
275C2×C4—44no—
276C2×C4—48no—
277C2×C4—416no—
278C2×C4—416no—
279C2×C4—416no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—88no—
287C2×C4—88no—
288C2×C4—88no—
289C2×C4—88no—
290C2×C4—88no—
291C2×C4—88no—
292C2×C4C2×#9912yes—
293C2×C4C2×#9914yes—
294C2×C4#41012yes—
295C2×C4#41014yes—
296C22—22no—
297C22—24no—
298C22—24no—
299C22—22no—
300C22—24no—
301C22—24no—
302C22—24no—
303C22—24no—
304C22—24no—
305C22—24no—
306C22—416no—
307C22#229712yes—
308C22#231011yescharacteristic
309C22#237114yes—
310C22#261118yes—
311C22C22×#9911yescharacteristic
312C22C2×#41011yescharacteristic
313C22C2×#41012yes—
314C4—22no—
315C4—22no—
316C4—24no—
317C4—24no—
318C4—42no—
319C4—42no—
320C4—42no—
321C4—816no—
322C4—84no—
323C4—84no—
324C2—28no—
325C2—24no—
326C2#1417011yescharacteristic
327C2#2142012yes—
328C2#2142112yes—
329C2#2182511yescharacteristic
330C2#2182611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. C22×#125C4semidirect
11. C2×#389C4semidirect
12. C2×#387C4semidirect
14. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
#137532
#113532
#109122