Finite Group Databasebeta

← #108239

Subgroups — #108239

Order 512.

1066 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 190 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
1#34178C211yesmaximal, characteristic
2C2×#1821C211yesmaximal, characteristic
3#14342C211yesmaximal, characteristic
4#7943C212yesmaximal
5#7943C212yesmaximal
6C23×C4⋊C4C2211yescharacteristic
7C2×#407C2212yes—
8#2345—24no—
9#2344—24no—
10#2316C412yescomplemented
11#2313—22no—
12#2280—22no—
13#2280C2212yes—
14C2×#381C2211yescharacteristic
15C2×#381C2211yescharacteristic
16#2274C412yescomplemented
17#1821—22no—
18#1821C414yescomplemented
19C2×#343—22no—
20#1445—22no—
21#1445C414yescomplemented
22C22⋊#95—22no—
23#1284—24no—
24#1284—24no—
25C24×C4D412yes—
26C24×C4C2311yescharacteristic
27C22×M16—22no—
28C2×#117—24no—
29C2×#117C2×C414yes—
30C22×C4⋊C4—22no—
31C22×C4⋊C4—22no—
32C22×C4⋊C4—22no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4D412yes—
39C22×C4⋊C4D412yes—
40C22×C4⋊C4Q812yes—
41C22×C22⋊C4—21no—
42C22×C22⋊C4—21no—
43#407—22no—
44#407—22no—
45#407—22no—
46#407C2×C414yes—
47C2×C22⋊C8—22no—
48C2×C22⋊C8—22no—
49#389—28no—
50#387—216no—
51#381—24no—
52#381—22no—
53#381—22no—
54#381—22no—
55#381—22no—
56#381—22no—
57#380—24no—
58#380—24no—
59#380—22no—
60#380—22no—
61#379—22no—
62C4×C4⋊C4—28no—
63C4×C22⋊C4—24no—
64C4×C22⋊C4—24no—
65C4×C22⋊C4D414yes—
66C4×C22⋊C4D414yes—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—28no—
70C2×#95C2×C412yes—
71C2×#95C2×C412yes—
72#343—28no—
73C25C2×D411yescharacteristic
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—44no—
81C23×C4—84no—
82C23×C4C22⋊C414yes—
83C23×C4C22⋊C412yes—
84C23×C4C22⋊C412yes—
85C23×C4C4⋊C414yes—
86C23×C4C22×C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C22×C411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×Q811yescharacteristic
94C23×C4#4112yes—
95C23×C4#4112yes—
96C2×M16—24no—
97C2×M16—42no—
98C2×M16—48no—
99C22×C8—42no—
100#117—48no—
101#117—88no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—28no—
118C2×C4⋊C4C4212yes—
119C2×C4⋊C4C4212yes—
120C2×C4⋊C4C22⋊C414yes—
121C2×C4⋊C4C4⋊C412yes—
122C2×C4⋊C4C4⋊C412yes—
123C2×C4⋊C4C4⋊C414yes—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—44no—
128C2×C22⋊C4—44no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C42—28no—
134C2×C42C22⋊C418yes—
135C22⋊C8—24no—
136C22⋊C8—24no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139#95—28no—
140#95—28no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—42no—
146#95—42no—
147#95—88no—
148C24—24no—
149C24—42no—
150C24—42no—
151C24C2×C22⋊C411yescharacteristic
152C24C2×C22⋊C411yescharacteristic
153C24C2×C4⋊C411yescharacteristic
154C24#12011yescharacteristic
155C24#12011yescharacteristic
156C24#12311yescharacteristic
157C24#12311yescharacteristic
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—28no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—44no—
176C22×C4—44no—
177C22×C4—48no—
178C22×C4—48no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4—84no—
192C22×C4—84no—
193C22×C4#9514yes—
194C22×C4#9514yes—
195C22×C4C2×C4211yescharacteristic
196C22×C4C2×C22⋊C412yes—
197C22×C4C2×C22⋊C411yescharacteristic
198C22×C4C2×C4⋊C411yescharacteristic
199C22×C4C2×C4⋊C411yescharacteristic
200C22×C4#11712yes—
201C22×C4C4×D412yes—
202C22×C4C4×D412yes—
203C22×C4C4×D412yes—
204C22×C4C4×D412yes—
205C22×C4#12012yes—
206C22×C4#12112yes—
207C22×C4#12112yes—
208C22×C4C22⋊Q812yes—
209C22×C4C22⋊Q812yes—
210C22×C4#12312yes—
211M16—48no—
212M16—84no—
213C2×C8—44no—
214C2×C8—84no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—28no—
220C4⋊C4—28no—
221C4⋊C4—48no—
222C4⋊C4—88no—
223C22⋊C4—48no—
224C22⋊C4—48no—
225C22⋊C4—48no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C42—48no—
231C42—88no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—416no—
253C23C2×#9511yescharacteristic
254C23#37911yescharacteristic
255C23#38111yescharacteristic
256C23#38511yescharacteristic
257C23#38511yescharacteristic
258C23#38611yescharacteristic
259C23#38611yescharacteristic
260C23#41014yes—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—48no—
282C2×C4—48no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—48no—
286C2×C4—48no—
287C2×C4—48no—
288C2×C4—84no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—84no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—88no—
295C2×C4—88no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4C4×C22⋊C412yes—
301C2×C4#38012yes—
302C2×C4#38512yes—
303C2×C4#38612yes—
304C8—84no—
305C22—28no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—44no—
327C22—44no—
328C22C22⋊#9511yescharacteristic
329C22#174012yes—
330C22#178212yes—
331C22#182012yes—
332C22#191112yes—
333C22#191212yes—
334C4—22no—
335C4—22no—
336C4—22no—
337C4—22no—
338C4—44no—
339C4—44no—
340C4—88no—
341C4—162no—
342C4—162no—
343C2—22no—
344C2—22no—
345C2—22no—
346C2—22no—
347C2—22no—
348C2—41no—
349C2—41no—
350C2#1072611yescharacteristic
351C2#1074011yescharacteristic
352C2#1074111yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2316C4semidirect
16. #2274C4semidirect
18. #1821C4semidirect
21. #1445C4semidirect

Maximal subgroups

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

subgroupindex
#341782
C2×#18212
#143422
#79432