Finite Group Databasebeta

← #358924

Subgroups — #358924

Order 512.

2798 conjugacy classes of subgroups; only the 326 normal subgroups are shown.

normal subgroups only

This group has 2798 conjugacy classes of subgroups — too many to draw. Only its 326 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×#2280C211yesmaximal, characteristic
2C4×#379C212yesmaximal
3#11353C214yesmaximal, complemented
4#11196C214yesmaximal
5C4×#342C214yesmaximal, complemented
6C25×C4C2211yescharacteristic
7C22×#117C412yescomplemented
8#2280—24no—
9#2280—24no—
10C2×#381C412yescomplemented
11C2×#380C412yescomplemented
12C2×#380C2212yes—
13C2×#379C2214yes—
14C2×#379C2212yes—
15C2×C4×C22⋊C4—24no—
16C2×C4×C22⋊C4C412yescomplemented
17C2×C4×C22⋊C4C2214yes—
18C2×C4×C22⋊C4C2212yes—
19C2×#342C22116yes—
20C22⋊#95—24no—
21C22⋊#95C2214yes—
22C26C2311yescharacteristic
23C24×C4—28no—
24C24×C4—48no—
25C24×C4C2×C411yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C24×C4D414yes—
30C24×C4D414yes—
31C24×C4C2311yescharacteristic
32C24×C4C2311yescharacteristic
33C2×#117—22no—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—48no—
38C22×C4⋊C4C2×C414yes—
39C22×C22⋊C4—28no—
40C22×C22⋊C4C2×C414yes—
41C22×C22⋊C4C2×C414yes—
42C22×C22⋊C4C2314yes—
43C22×C22⋊C4C2318yes—
44C22×C42—24no—
45C22×C42C2×C414yes—
46#381—28no—
47#380—24no—
48#380—24no—
49#380—24no—
50#380—24no—
51#379—28no—
52#379—216no—
53C4×C22⋊C4—24no—
54C4×C22⋊C4—24no—
55C4×C22⋊C4—24no—
56C4×C22⋊C4—24no—
57C4×C22⋊C4—216no—
58C4×C22⋊C4—416no—
59C2×#95—24no—
60C2×#95—28no—
61C2×#95C2×C414yes—
62C2×#95C2×C414yes—
63#342—232no—
64C25—24no—
65C25—28no—
66C25—44no—
67C25—44no—
68C25C22×C411yescharacteristic
69C25C22×C411yescharacteristic
70C25C2×D414yes—
71C25C2411yescharacteristic
72C23×C4—28no—
73C23×C4—24no—
74C23×C4—24no—
75C23×C4—216no—
76C23×C4—24no—
77C23×C4—24no—
78C23×C4—24no—
79C23×C4—216no—
80C23×C4—216no—
81C23×C4—28no—
82C23×C4—44no—
83C23×C4—44no—
84C23×C4—48no—
85C23×C4—48no—
86C23×C4—44no—
87C23×C4—44no—
88C23×C4—44no—
89C23×C4—48no—
90C23×C4—464no—
91C23×C4C22⋊C412yes—
92C23×C4C22⋊C412yes—
93C23×C4C22⋊C414yes—
94C23×C4C22⋊C414yes—
95C23×C4C22⋊C412yes—
96C23×C4C22⋊C412yes—
97C23×C4C22×C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C418yes—
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C411yescharacteristic
102C23×C4C2×D414yes—
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4#4118yes—
106#117—48no—
107#117—48no—
108#117—48no—
109#117—48no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—48no—
113C2×C22⋊C4—216no—
114C2×C22⋊C4—216no—
115C2×C22⋊C4—232no—
116C2×C22⋊C4—24no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—216no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—416no—
125C2×C22⋊C4—416no—
126C2×C22⋊C4—48no—
127C2×C22⋊C4—48no—
128C2×C22⋊C4#41116yes—
129C2×C42—24no—
130C2×C42—24no—
131C2×C42—48no—
132C2×C42—416no—
133C2×C42—48no—
134#95—28no—
135#95—28no—
136#95—416no—
137#95—48no—
138C24—22no—
139C24—22no—
140C24—28no—
141C24—24no—
142C24—28no—
143C24—24no—
144C24—28no—
145C24—24no—
146C24—24no—
147C24—28no—
148C24—28no—
149C24—42no—
150C24—44no—
151C24—42no—
152C24—42no—
153C24—44no—
154C24—42no—
155C24—416no—
156C24—464no—
157C24—416no—
158C24—416no—
159C24C2×C22⋊C412yes—
160C24C2×C22⋊C412yes—
161C24C23×C411yescharacteristic
162C24C22×D412yes—
163C24C2×#4114yes—
164C24C2×#4118yes—
165C22×C4—232no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—28no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—22no—
176C22×C4—28no—
177C22×C4—28no—
178C22×C4—28no—
179C22×C4—28no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—28no—
183C22×C4—28no—
184C22×C4—28no—
185C22×C4—28no—
186C22×C4—216no—
187C22×C4—216no—
188C22×C4—416no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—44no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—464no—
197C22×C4—416no—
198C22×C4—464no—
199C22×C4—416no—
200C22×C4—416no—
201C22×C4—416no—
202C22×C4—416no—
203C22×C4—416no—
204C22×C4—48no—
205C22×C4—48no—
206C22×C4—48no—
207C22×C4—48no—
208C22×C4#9918yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4#11714yes—
215C22×C4#11714yes—
216C22×C4#12318yes—
217C22×C4#12318yes—
218C4⋊C4—48no—
219C4⋊C4—48no—
220C22⋊C4—232no—
221C22⋊C4—464no—
222C22⋊C4—48no—
223C22⋊C4—48no—
224C22⋊C4—48no—
225C22⋊C4—416no—
226C22⋊C4—48no—
227C22⋊C4—48no—
228C22⋊C4—48no—
229C42—48no—
230C42—48no—
231C42—816no—
232C23—22no—
233C23—24no—
234C23—22no—
235C23—24no—
236C23—232no—
237C23—28no—
238C23—28no—
239C23—22no—
240C23—24no—
241C23—22no—
242C23—24no—
243C23—22no—
244C23—22no—
245C23—24no—
246C23—24no—
247C23—44no—
248C23—416no—
249C23—44no—
250C23—416no—
251C23—416no—
252C23—416no—
253C23—416no—
254C23—416no—
255C23—416no—
256C23—44no—
257C23—416no—
258C23—44no—
259C23—4128no—
260C23—416no—
261C23—44no—
262C23—44no—
263C23C2×#9914yes—
264C23#41014yes—
265C23C22×C22⋊C411yescharacteristic
266C23C2×#11712yes—
267C23C4×#4114yes—
268C23C2×#12314yes—
269C23#52514yes—
270C23#529116yes—
271C23#53214yes—
272C2×C4—216no—
273C2×C4—28no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—216no—
279C2×C4—216no—
280C2×C4—28no—
281C2×C4—24no—
282C2×C4—22no—
283C2×C4—44no—
284C2×C4—48no—
285C2×C4—416no—
286C2×C4—416no—
287C2×C4—416no—
288C2×C4—48no—
289C2×C4—44no—
290C2×C4—48no—
291C2×C4—432no—
292C2×C4—48no—
293C2×C4—48no—
294C2×C4—416no—
295C2×C4—48no—
296C2×C4—88no—
297C2×C4—816no—
298C2×C4—88no—
299C2×C4—816no—
300C2×C4#38112yes—
301C2×C4#38114yes—
302C2×C4#38112yes—
303C2×C4C2×#9914yes—
304C2×C4C2×#9914yes—
305C22—28no—
306C22—28no—
307C22—28no—
308C22—22no—
309C22—24no—
310C22—24no—
311C22—22no—
312C22—28no—
313C22—28no—
314C22—24no—
315C22—24no—
316C22—416no—
317C22—464no—
318C22—416no—
319C22—416no—
320C22—42no—
321C22—42no—
322C22—44no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22C2×#38111yescharacteristic
327C22#229712yes—
328C22#229814yes—
329C22#239518yes—
330C22C22×#9912yes—
331C22C2×#41012yes—
332C4—22no—
333C4—24no—
334C4—24no—
335C4—42no—
336C4—816no—
337C4—88no—
338C4#181914yes—
339C2—28no—
340C2—24no—
341C2—44no—
342C2—44no—
343C2#1385311yescharacteristic
344C2C2×#181911yescharacteristic
345C2#2068811yescharacteristic
346C2#2142014yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
socleC23C22×C22⋊C4
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C25×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11353C2semidirect
5. C4×#342C2semidirect
7. C22×#117C4semidirect
10. C2×#381C4semidirect
11. C2×#380C4semidirect
16. C2×C4×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22802
C4×#3792
#113532
#111962
C4×#3422