Finite Group Databasebeta

← #125375

Subgroups — #125375

Order 512.

568 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 84 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#15016C211yesmaximal, characteristic
2#8152C211yesmaximal, characteristic
3#7907C211yesmaximal, characteristic
4#7132C212yesmaximal, complemented
5#7104C212yesmaximal, complemented
6C22×C22⋊C8—21no—
7C2×#399C411yescharacteristic
8C2×#397C411yescharacteristic
9C2×#382C411yescharacteristic
10C2×C4×C22⋊C4C2211yescharacteristic
11#1749—21no—
12C2×#323—21no—
13#1445C2211yescharacteristic
14#1445C2211yescharacteristic
15#1442—21no—
16#1440—21no—
17#1436—21no—
18#1434C411yescharacteristic
19C22⋊#95—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4D411yescharacteristic
24C23×C8—41no—
25C22×C4⋊C4—21no—
26C22×C4⋊C4C2×C411yescharacteristic
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C22⋊C4—21no—
29C22×C22⋊C4—21no—
30C22×C22⋊C4C2×C411yescharacteristic
31C22×C22⋊C4D411yescharacteristic
32C22×C42C2×C411yescharacteristic
33C22×C42D411yescharacteristic
34C2×C22⋊C8—24no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—41no—
38C2×C22⋊C8—42no—
39C2×#97—41no—
40#399—42no—
41#397—42no—
42#382—41no—
43#382—41no—
44C4×C22⋊C4—41no—
45C4×C22⋊C4—41no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—21no—
52C2×#95—21no—
53C2×#95—21no—
54C2×#95—21no—
55C2×#95C2×C411yescharacteristic
56C2×#95C2×C411yescharacteristic
57C2×#95D411yescharacteristic
58C2×#95C2311yescharacteristic
59#336—41no—
60#323—24no—
61C25C22⋊C411yescharacteristic
62C23×C4—22no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—42no—
72C23×C4—81no—
73C23×C4C22⋊C411yescharacteristic
74C23×C4C22⋊C411yescharacteristic
75C23×C4C22⋊C411yescharacteristic
76C23×C4C22×C411yescharacteristic
77C23×C4C2×D411yescharacteristic
78C23×C4C2×D411yescharacteristic
79C22×C8—44no—
80C22×C8—42no—
81C22×C8—42no—
82C22×C8—82no—
83C22×C8—82no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—41no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—81no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—81no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—21no—
108C2×C42—21no—
109C2×C42—42no—
110C2×C42—81no—
111C22⋊C8—24no—
112C22⋊C8—24no—
113C22⋊C8—44no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116#97—82no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—41no—
126#95—41no—
127#95—42no—
128#95—41no—
129#95—41no—
130#95—81no—
131#95—81no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—41no—
137C24—41no—
138C24—41no—
139C24—41no—
140C24#10012yes—
141C24#10412yes—
142C24#10412yes—
143C24C2×C22⋊C411yescharacteristic
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
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—41no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4—84no—
209C22×C4#9911yescharacteristic
210C22×C4#9911yescharacteristic
211C22×C4#13112yes—
212C22×C4#13512yes—
213C22×C4AGL(1,ℤ/8)11yescharacteristic
214C22×C4#13711yescharacteristic
215C2×C8—44no—
216C2×C8—44no—
217C2×C8—82no—
218C2×C8—82no—
219C2×C8—88no—
220C2×C8—82no—
221C2×C8—82no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C4⋊C4—84no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—84no—
228C22⋊C4—84no—
229C42—42no—
230C42—42no—
231C42—42no—
232C42—81no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—81no—
254C23—81no—
255C23—81no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23#32914yes—
260C23#40811yescharacteristic
261C23C2×#9911yescharacteristic
262C23C2×#10011yescharacteristic
263C23#41611yescharacteristic
264C23#41711yescharacteristic
265C23C2×#10411yescharacteristic
266C23C2×#10411yescharacteristic
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
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#35212yes—
334C8—84no—
335C8—84no—
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—24no—
349C22—44no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—82no—
357C22—82no—
358C22—82no—
359C22#147111yescharacteristic
360C22#148311yescharacteristic
361C22#149411yescharacteristic
362C22#150111yescharacteristic
363C22#150411yescharacteristic
364C22C2×#32911yescharacteristic
365C22#155111yescharacteristic
366C22C2×#35211yescharacteristic
367C22#212311yescharacteristic
368C22#212711yescharacteristic
369C22#213211yescharacteristic
370C4—22no—
371C4—42no—
372C4—42no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—82no—
377C4—81no—
378C4—81no—
379C4—161no—
380C4—161no—
381C4—161no—
382C2—21no—
383C2—21no—
384C2—21no—
385C2—21no—
386C2—44no—
387C2#769011yescharacteristic
388C2#1155811yescharacteristic
389C2#1156611yescharacteristic
390C2#1173211yescharacteristic
391C2#1173511yescharacteristic
392C2#1175511yescharacteristic
393C2#1176711yescharacteristic
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. #7132C2semidirect
5. #7104C2semidirect

Maximal subgroups

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

subgroupindex
#150162
#81522
#79072
#71322
#71042