Finite Group Databasebeta

← #125265

Subgroups — #125265

Order 512.

602 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 78 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#15001C211yesmaximal, characteristic
2#8152C211yesmaximal, characteristic
3C4×#323C211yesmaximal, characteristic
4#7131C212yesmaximal, complemented
5#7122C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C2×#387—21no—
8C2×C4×C22⋊C4C2211yescharacteristic
9#1749—21no—
10C4×C22⋊C8—21no—
11C2×#323—21no—
12C2×#323—21no—
13#1446C411yescharacteristic
14#1446C2211yescharacteristic
15#1444C411yescharacteristic
16#1442C2211yescharacteristic
17#1440—21no—
18#1439—21no—
19#1436—21no—
20C22⋊#95C411yescharacteristic
21#1280C2212yes—
22#1280C2212yes—
23C23×Q8C2×C411yescharacteristic
24C24×C4C2×C411yescharacteristic
25C2×C22⋊Q8—44no—
26C2×C22⋊Q8—41no—
27C2×C22⋊Q8—41no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4—21no—
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4D411yescharacteristic
34C22×C22⋊C4D411yescharacteristic
35C22×C42D411yescharacteristic
36C22×C42D411yescharacteristic
37C2×C22⋊C8—22no—
38C2×C22⋊C8—22no—
39C2×#97—41no—
40C2×C4×C8—41no—
41#387—44no—
42C4×C22⋊C4—41no—
43C4×C22⋊C4—41no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95C2311yescharacteristic
56#336—41no—
57#336—41no—
58#323—24no—
59#323—24no—
60C25C22⋊C411yescharacteristic
61C22×Q8—22no—
62C22×Q8—42no—
63C22×Q8—41no—
64C22×Q8—41no—
65C22×Q8—81no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—22no—
72C23×C4—21no—
73C23×C4—41no—
74C23×C4—42no—
75C23×C4—81no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C22×C8—42no—
83C22×C8—42no—
84C22⋊Q8—44no—
85C22⋊Q8—44no—
86C22⋊Q8—44no—
87C22⋊Q8—44no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4—22no—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—42no—
105C2×C22⋊C4—42no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—81no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—42no—
114C2×C42—81no—
115C22⋊C8—24no—
116C22⋊C8—24no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119#97—82no—
120C4×C8—82no—
121#95—22no—
122#95—41no—
123#95—41no—
124#95—41no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—42no—
130#95—42no—
131#95—81no—
132#95—81no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—21no—
137C24—21no—
138C24—21no—
139C24—41no—
140C24—41no—
141C24—41no—
142C24—41no—
143C24#10412yes—
144C24C2×C22⋊C411yescharacteristic
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—82no—
148C2×Q8—84no—
149C2×Q8—84no—
150C2×Q8—82no—
151C22×C4—21no—
152C22×C4—21no—
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—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—41no—
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—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—84no—
211C22×C4—84no—
212C22×C4#9911yescharacteristic
213C22×C4#9911yescharacteristic
214C22×C4#10412yes—
215C22×C4#13112yes—
216C22×C4#13512yes—
217C22×C4#13712yes—
218C2×C8—44no—
219C2×C8—44no—
220C2×C8—82no—
221C2×C8—82no—
222C2×C8—82no—
223C2×C8—82no—
224C4⋊C4—44no—
225C4⋊C4—42no—
226C4⋊C4—48no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—48no—
232C22⋊C4—84no—
233C42—42no—
234C42—42no—
235C42—81no—
236C42—81no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23#40811yescharacteristic
265C23C2×#9911yescharacteristic
266C23#41311yescharacteristic
267C23#41611yescharacteristic
268C23#41911yescharacteristic
269C23C2×#10411yescharacteristic
270C23C2×#10411yescharacteristic
271Q8—84no—
272Q8—84no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
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—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—84no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—84no—
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—
334C2×C4—81no—
335C2×C4AΣL(1,𝔽16)12yes—
336C8—84no—
337C8—84no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—82no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22#146811yescharacteristic
362C22#148511yescharacteristic
363C22#149411yescharacteristic
364C22#150111yescharacteristic
365C22#150811yescharacteristic
366C22#152411yescharacteristic
367C22#155411yescharacteristic
368C22C2×AΣL(1,𝔽16)11yescharacteristic
369C22#212111yescharacteristic
370C22#212911yescharacteristic
371C22#213211yescharacteristic
372C4—22no—
373C4—42no—
374C4—41no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—82no—
379C4—81no—
380C4—82no—
381C4—82no—
382C4—161no—
383C2—21no—
384C2—21no—
385C2—21no—
386C2—21no—
387C2—82no—
388C2#767511yescharacteristic
389C2#1154111yescharacteristic
390C2#1155611yescharacteristic
391C2#1173111yescharacteristic
392C2#1173311yescharacteristic
393C2#1178511yescharacteristic
394C2#1179311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC2×#95C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7131C2semidirect
5. #7122C2semidirect

Maximal subgroups

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

subgroupindex
#150012
#81522
C4×#3232
#71312
#71222