Finite Group Databasebeta

← #125550

Subgroups — #125550

Order 512.

550 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 76 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#15015C211yesmaximal, characteristic
2#8698C211yesmaximal, characteristic
3#8698C211yesmaximal, characteristic
4#7132C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C2×#380C411yescharacteristic
7C22×#95C2211yescharacteristic
8#1779—21no—
9#1779—21no—
10#1445C411yescharacteristic
11#1445C2211yescharacteristic
12#1443—21no—
13#1443—21no—
14#1442C411yescharacteristic
15#1442C2211yescharacteristic
16C22⋊#95—21no—
17C22⋊#95—21no—
18C22⋊#95C411yescharacteristic
19#1280—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C24×C4D411yescharacteristic
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C2×C22⋊C8—22no—
31C2×C22⋊C8—22no—
32C2×C22⋊C8—41no—
33C2×C22⋊C8—41no—
34C2×C22⋊C8—41no—
35C2×C22⋊C8—41no—
36#380—41no—
37#380—41no—
38C2×#95—21no—
39C2×#95—21no—
40C2×#95—21no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—41no—
49C2×#95—41no—
50C2×#95C2×C411yescharacteristic
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95D411yescharacteristic
54C2×#95D411yescharacteristic
55C2×#95C2311yescharacteristic
56C25C22⋊C411yescharacteristic
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—41no—
71C23×C4—41no—
72C23×C4—41no—
73C23×C4—42no—
74C23×C4—42no—
75C23×C4—81no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4—81no—
79C23×C4C22⋊C411yescharacteristic
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22×C411yescharacteristic
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C22×C8—42no—
86C22×C8—42no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—81no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C22⋊C8—42no—
100C22⋊C8—42no—
101C22⋊C8—42no—
102C22⋊C8—42no—
103C22⋊C8—82no—
104C22⋊C8—82no—
105C22⋊C8—82no—
106C22⋊C8—82no—
107#95—22no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—42no—
114#95—42no—
115#95—42no—
116#95—81no—
117#95—81no—
118#95—81no—
119#95—81no—
120#95—81no—
121#95—81no—
122#95—81no—
123C24—22no—
124C24—22no—
125C24—21no—
126C24—21no—
127C24—21no—
128C24—21no—
129C24—21no—
130C24—21no—
131C24—41no—
132C24—41no—
133C24#9911yescharacteristic
134C24#9911yescharacteristic
135C24C2×C22⋊C411yescharacteristic
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—84no—
210C22×C4—84no—
211C22×C4—84no—
212C22×C4—84no—
213C22×C4—84no—
214C22×C4—84no—
215C22×C4#13112yes—
216C22×C4#13512yes—
217C22×C4#13712yes—
218C2×C8—82no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C2×C8—82no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C22⋊C4—84no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—24no—
234C23—24no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#35212yes—
262C23#35512yes—
263C23#40811yescharacteristic
264C23C2×#9911yescharacteristic
265C23#41311yescharacteristic
266C23#41611yescharacteristic
267C23#41911yescharacteristic
268C23#42111yescharacteristic
269C23#42111yescharacteristic
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—84no—
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—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—81no—
334C8—162no—
335C8—162no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—44no—
355C22—44no—
356C22—44no—
357C22—44no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22#146811yescharacteristic
367C22#148211yescharacteristic
368C22#149711yescharacteristic
369C22#151111yescharacteristic
370C22#151611yescharacteristic
371C22#153511yescharacteristic
372C22#155411yescharacteristic
373C22C2×#35211yescharacteristic
374C22#212111yescharacteristic
375C22#212711yescharacteristic
376C22C2×#35511yescharacteristic
377C4—42no—
378C4—42no—
379C4—42no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—161no—
384C4—161no—
385C4—161no—
386C4—161no—
387C2—21no—
388C2—21no—
389C2—21no—
390C2—21no—
391C2—42no—
392C2—42no—
393C2#772111yescharacteristic
394C2#1154311yescharacteristic
395C2#1155311yescharacteristic
396C2#1181111yescharacteristic
397C2#1181411yescharacteristic
398C2#1184111yescharacteristic
399C2#1185211yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C22×#95 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7132C2semidirect
5. #7122C2semidirect

Maximal subgroups

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

subgroupindex
#150152
#86982
#71322
#71222