Finite Group Databasebeta

← #135318

Subgroups — #135318

Order 512.

646 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 110 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#20658C211yesmaximal, characteristic
2#12679C211yesmaximal, characteristic
3#10721C211yesmaximal, characteristic
4#7258C212yesmaximal, complemented
5#7258C212yesmaximal, complemented
6C24×C8C2211yescharacteristic
7C2×#427C411yescharacteristic, complemented
8C2×#427C411yescharacteristic, complemented
9#2280—21no—
10C2×#380—21no—
11C2×C22⋊C16—21no—
12C2×C22⋊C16—21no—
13C2×C22⋊C16C2212yes—
14#1808C411yescharacteristic, complemented
15#1808C411yescharacteristic, complemented
16#1808C2211yescharacteristic
17#1808C2211yescharacteristic
18#1807—21no—
19#1807—21no—
20#1735—21no—
21C2×#340—21no—
22C2×#340C2212yes—
23C24×C4D411yescharacteristic
24C23×C8—24no—
25C23×C8—42no—
26C23×C8—42no—
27C23×C8C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C23×C8D411yescharacteristic
30C23×C8D411yescharacteristic
31C23×C8D411yescharacteristic
32C23×C8C2311yescharacteristic
33C2×#117—21no—
34C2×#117—21no—
35C22×C4⋊C4—22no—
36C22×C22⋊C4—41no—
37C22×C16—22no—
38#427—42no—
39#427—42no—
40C2×#107C2×C411yescharacteristic
41C2×#107C2×C411yescharacteristic
42C2×#107C2×C411yescharacteristic
43C2×#107C2×C411yescharacteristic
44C2×#106—21no—
45C2×#106—21no—
46#381—41no—
47#380—24no—
48#380—21no—
49#380—21no—
50#380—21no—
51#380—21no—
52#380—42no—
53#379—41no—
54C4×C22⋊C4—41no—
55C4×C22⋊C4—41no—
56C2×#95—41no—
57C22⋊C16—24no—
58C22⋊C16—42no—
59C22⋊C16—42no—
60#340—21no—
61#340—21no—
62#340—21no—
63#340—21no—
64#340—24no—
65#340—42no—
66C25D811yescharacteristic
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—41no—
72C23×C4—41no—
73C23×C4—42no—
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4D811yescharacteristic
77C23×C4SD1611yescharacteristic
78C23×C4SD1611yescharacteristic
79C23×C4C2×D411yescharacteristic
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8—41no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8—416no—
98C22×C8C22⋊C411yescharacteristic
99C22×C8C22⋊C411yescharacteristic
100C22×C8C22×C411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C22×C8#4114yes—
103#117—42no—
104#117—42no—
105#117—42no—
106#117—42no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C22⋊C4—44no—
118C2×C22⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—82no—
125C2×C22⋊C4—82no—
126C2×C42—41no—
127C2×C42—41no—
128C2×C16—42no—
129C2×C16—42no—
130C2×C16—42no—
131#107—41no—
132#107—41no—
133#107—41no—
134#107—41no—
135#106—42no—
136#106—42no—
137#95—44no—
138#95—41no—
139#95—41no—
140#95—41no—
141#95—41no—
142C24—22no—
143C24—22no—
144C24—21no—
145C24—21no—
146C24—22no—
147C24—22no—
148C24—41no—
149C24—41no—
150C24#10211yescharacteristic
151C24#10211yescharacteristic
152C24C2×D811yescharacteristic
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4—82no—
189C22×C4—82no—
190C22×C4—82no—
191C22×C4#9912yes—
192C22×C4#10211yescharacteristic
193C22×C4#10211yescharacteristic
194C22×C4#10412yes—
195C22×C4C2×C22⋊C411yescharacteristic
196C22×C4#12312yes—
197C22×C4C2×SD1611yescharacteristic
198C22×C4#13712yes—
199C2×C8—28no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—44no—
210C2×C8—44no—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8—44no—
214C2×C8#9912yes—
215C2×C8#11712yes—
216C2×C8#12312yes—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—82no—
228C4⋊C4—82no—
229C22⋊C4—82no—
230C22⋊C4—88no—
231C22⋊C4—82no—
232C22⋊C4—82no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C42—82no—
238C42—82no—
239C16—82no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—24no—
245C23—24no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—44no—
259C23—44no—
260C23—44no—
261C23—44no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23#32712yes—
267C23#36012yes—
268C23C2×#10211yescharacteristic
269C23#48012yes—
270C23#50812yes—
271C23#50911yescharacteristic
272C23#51011yescharacteristic
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—84no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—162no—
311C2×C4—162no—
312C2×C4—162no—
313C2×C4—162no—
314C2×C4#32712yes—
315C2×C4#38111yescharacteristic
316C2×C4C2×#9911yescharacteristic
317C2×C4C2×#9911yescharacteristic
318C2×C4#41911yescharacteristic
319C2×C4C2×#10411yescharacteristic
320C2×C4#42111yescharacteristic
321C2×C4#48312yes—
322C8—22no—
323C8—22no—
324C8—22no—
325C8—41no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—21no—
333C22—21no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—48no—
341C22—48no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22C2×#32711yescharacteristic
351C22#151911yescharacteristic
352C22#182311yescharacteristic
353C22#213811yescharacteristic
354C22#213911yescharacteristic
355C22C2×#36011yescharacteristic
356C22#214511yescharacteristic
357C22#223412yes—
358C22#223712yes—
359C4—22no—
360C4—22no—
361C4—41no—
362C4—164no—
363C4—161no—
364C4—161no—
365C4—161no—
366C4—161no—
367C4#150811yescharacteristic
368C4#151111yescharacteristic
369C4#181911yescharacteristic
370C4#182611yescharacteristic
371C2—22no—
372C2—22no—
373C2—42no—
374C2—42no—
375C2#871211yescharacteristic
376C2#1170311yescharacteristic
377C2#1170411yescharacteristic
378C2#1188511yescharacteristic
379C2#1189011yescharacteristic
380C2#1286811yescharacteristic
381C2#1287011yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7258C2semidirect
5. #7258C2semidirect
7. C2×#427C4semidirect
8. C2×#427C4semidirect
14. #1808C4semidirect
15. #1808C4semidirect

Maximal subgroups

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

subgroupindex
#206582
#126792
#107212
#72582