Finite Group Databasebeta

← #110563

Subgroups — #110563

Order 512.

988 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 170 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×#1508C212yesmaximal
2#14159C211yesmaximal, characteristic
3#13796C211yesmaximal, characteristic
4C2×#1297C212yesmaximal
5#7909C211yesmaximal, characteristic
6C22×C22⋊Q8C2211yescharacteristic
7C2×#407C2212yes—
8C2×C4×C22⋊C4C2211yescharacteristic
9C2×C4×C22⋊C4C2212yes—
10C22×#95C2211yescharacteristic
11#1508—24no—
12#1508C418yescomplemented
13C2×#328—24no—
14#1446C412yescomplemented
15#1445C412yescomplemented
16#1436—22no—
17#1434—24no—
18C22⋊#95—22no—
19C22⋊#95—22no—
20#1297—24no—
21#1280—22no—
22C23×Q8C2×C411yescharacteristic
23C24×C4D412yes—
24C24×C4C2311yescharacteristic
25C22×M16—22no—
26C2×C22⋊Q8—22no—
27C2×C22⋊Q8—22no—
28C2×C22⋊Q8—22no—
29C2×C22⋊Q8—22no—
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8C2×C414yes—
32C22×C4⋊C4—21no—
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—22no—
37C22×C42—22no—
38C22×C42—21no—
39#407—22no—
40#407—22no—
41#407—22no—
42#407C2×C414yes—
43C2×C22⋊C8—22no—
44C2×C22⋊C8—22no—
45C4×C22⋊C4—22no—
46C4×C22⋊C4—22no—
47C4×C22⋊C4—22no—
48C4×C22⋊C4—22no—
49C4×C22⋊C4C2×C414yes—
50C4×C22⋊C4D414yes—
51C2×#95—22no—
52C2×#95—22no—
53C2×#95—24no—
54C2×#95—24no—
55C2×#95—24no—
56C2×#95—24no—
57C2×#95C2×C412yes—
58C2×#95C2×C412yes—
59C2×#95D412yes—
60C2×#95D414yes—
61#328—216no—
62C25C2×D411yescharacteristic
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—82no—
67C22×Q8C4212yes—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—22no—
71C23×C4—24no—
72C23×C4—22no—
73C23×C4—22no—
74C23×C4—44no—
75C23×C4—44no—
76C23×C4—82no—
77C23×C4—82no—
78C23×C4C22⋊C414yes—
79C23×C4C22×C411yescharacteristic
80C23×C4C22×C412yes—
81C23×C4C2×D412yes—
82C23×C4C2×D411yescharacteristic
83C23×C4C2×D412yes—
84C23×C4#4112yes—
85C23×C4#4112yes—
86C23×C4#4112yes—
87C2×M16—24no—
88C2×M16—42no—
89C2×M16—84no—
90C22×C8—42no—
91C22⋊Q8—24no—
92C22⋊Q8—24no—
93C22⋊Q8—24no—
94C22⋊Q8—24no—
95C22⋊Q8—42no—
96C22⋊Q8—42no—
97C22⋊Q8—42no—
98C22⋊Q8—42no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C4212yes—
107C2×C22⋊C4—24no—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—24no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—48no—
112C2×C22⋊C4—42no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4—42no—
115C2×C22⋊C4—42no—
116C2×C22⋊C4—41no—
117C2×C42—24no—
118C2×C42—24no—
119C2×C42—41no—
120C2×C42—41no—
121C2×C42—84no—
122C2×C42—82no—
123C22⋊C8—24no—
124C22⋊C8—24no—
125C22⋊C8—42no—
126C22⋊C8—42no—
127#95—24no—
128#95—24no—
129#95—24no—
130#95—24no—
131#95—48no—
132#95—84no—
133#95—84no—
134#95—84no—
135#95C22⋊C418yes—
136C24—24no—
137C24—22no—
138C24—22no—
139C24—42no—
140C24C2×C22⋊C411yescharacteristic
141C24C4×D412yes—
142C24#12011yescharacteristic
143C24#12112yes—
144C24#12311yescharacteristic
145C2×Q8—24no—
146C2×Q8—42no—
147C2×Q8—84no—
148C2×Q8—82no—
149C2×Q8—82no—
150C2×Q8—84no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—28no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—42no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—44no—
163C22×C4—44no—
164C22×C4—48no—
165C22×C4—44no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—42no—
169C22×C4—48no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—82no—
183C22×C4—82no—
184C22×C4—82no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4—816no—
189C22×C4C2×C4211yescharacteristic
190C22×C4C2×C22⋊C412yes—
191C22×C4#11712yes—
192C22×C4#11711yescharacteristic
193C22×C4C4×D412yes—
194C22×C4C4×D412yes—
195C22×C4C4×D412yes—
196C22×C4C4×D412yes—
197C22×C4C4×D412yes—
198C22×C4#12112yes—
199C22×C4#12112yes—
200C22×C4#12312yes—
201C22×C4#12411yescharacteristic
202C22×C4#12412yes—
203C22×C4#12612yes—
204C22×C4#12711yescharacteristic
205M16—84no—
206M16—84no—
207C2×C8—44no—
208C2×C8—84no—
209C4⋊C4—24no—
210C4⋊C4—42no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C22⋊C4—48no—
214C22⋊C4—48no—
215C22⋊C4—48no—
216C22⋊C4—44no—
217C22⋊C4—44no—
218C42—84no—
219C42—84no—
220C42—82no—
221C42—82no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—21no—
230C23—21no—
231C23—22no—
232C23—22no—
233C23—24no—
234C23—24no—
235C23—42no—
236C23—42no—
237C23—48no—
238C23—48no—
239C23C4×C22⋊C411yescharacteristic
240C23#38612yes—
241C23#38611yescharacteristic
242C23#38812yes—
243C23#39011yescharacteristic
244C23#42114yes—
245C23#45512yes—
246C23#45612yes—
247Q8—84no—
248Q8—84no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—48no—
266C2×C4—48no—
267C2×C4—44no—
268C2×C4—48no—
269C2×C4—48no—
270C2×C4—44no—
271C2×C4—48no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—82no—
279C2×C4—84no—
280C2×C4—88no—
281C2×C4—88no—
282C2×C4—88no—
283C2×C4—82no—
284C2×C4—88no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—84no—
290C2×C4C4×C22⋊C412yes—
291C2×C4#38812yes—
292C2×C4#38812yes—
293C2×C4#39012yes—
294C8—84no—
295C22—22no—
296C22—21no—
297C22—22no—
298C22—21no—
299C22—24no—
300C22—24no—
301C22—24no—
302C22—24no—
303C22—21no—
304C22—21no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—48no—
309C22—48no—
310C22—42no—
311C22—42no—
312C22#143711yescharacteristic
313C22#175912yes—
314C22#188214yes—
315C22#188511yescharacteristic
316C22#188611yescharacteristic
317C22#189712yes—
318C22#191312yes—
319C22#196312yes—
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—88no—
325C4—84no—
326C4—82no—
327C4—82no—
328C4—84no—
329C2—24no—
330C2—22no—
331C2—22no—
332C2—42no—
333C2#941112yes—
334C2#941212yes—
335C2#1092411yescharacteristic
336C2#1093811yescharacteristic
337C2#1095711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #1508C4semidirect
14. #1446C4semidirect
15. #1445C4semidirect

Maximal subgroups

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

subgroupindex
C2×#15082
#141592
#137962
C2×#12972
#79092