Finite Group Databasebeta

← #125395

Subgroups — #125395

Order 512.

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

normal subgroups only

This group has 604 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#15019C211yesmaximal, characteristic
2#8698C211yesmaximal, characteristic
3#7909C211yesmaximal, characteristic
4#7122C212yesmaximal, complemented
5#7104C212yesmaximal, complemented
6C2×#407—21no—
7C2×#381—21no—
8C2×#380C411yescharacteristic
9C22×#95C2211yescharacteristic
10#1779—21no—
11#1445C411yescharacteristic
12#1445C2211yescharacteristic
13#1443—21no—
14#1443—21no—
15#1442C411yescharacteristic
16#1442C2211yescharacteristic
17C22⋊#95—21no—
18C22⋊#95C411yescharacteristic
19#1280—21no—
20#1280—21no—
21#1280C2212yes—
22#1280C2212yes—
23C24×C4C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C24×C4D411yescharacteristic
26C22×M16—41no—
27C22×C4⋊C4C2×C411yescharacteristic
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C22⋊C4—21no—
30C22×C22⋊C4—21no—
31#407—24no—
32#407—42no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—22no—
35C2×C22⋊C8—41no—
36C2×C22⋊C8—41no—
37C2×C22⋊C8—41no—
38#381—44no—
39#380—41no—
40#380—41no—
41C2×#95—21no—
42C2×#95—21no—
43C2×#95—21no—
44C2×#95—21no—
45C2×#95—21no—
46C2×#95—21no—
47C2×#95—21no—
48C2×#95—21no—
49C2×#95—21no—
50C2×#95—21no—
51C2×#95—41no—
52C2×#95—41no—
53C2×#95C2×C411yescharacteristic
54C2×#95C2×C411yescharacteristic
55C2×#95C2×C411yescharacteristic
56C2×#95D411yescharacteristic
57C2×#95D411yescharacteristic
58C2×#95C2311yescharacteristic
59C25C22⋊C411yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—22no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—41no—
76C23×C4—41no—
77C23×C4—42no—
78C23×C4—42no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—81no—
82C23×C4—81no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22×C411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C2×M16—44no—
90C2×M16—82no—
91C2×M16—82no—
92C22×C8—42no—
93C22×C8—42no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—81no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—42no—
103C2×C22⋊C4—42no—
104C2×C22⋊C4—41no—
105C2×C22⋊C4—41no—
106C22⋊C8—24no—
107C22⋊C8—44no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110C22⋊C8—42no—
111C22⋊C8—82no—
112C22⋊C8—82no—
113#95—22no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—42no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—41no—
124#95—42no—
125#95—42no—
126#95—41no—
127#95—81no—
128#95—81no—
129#95—81no—
130C24—22no—
131C24—22no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—21no—
137C24—21no—
138C24—41no—
139C24—41no—
140C24#9911yescharacteristic
141C24#9911yescharacteristic
142C24C2×C22⋊C411yescharacteristic
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
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—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
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—42no—
178C22×C4—48no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
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—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—82no—
213C22×C4—84no—
214C22×C4—82no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—84no—
222C22×C4—84no—
223C22×C4—84no—
224C22×C4#10412yes—
225C22×C4#13112yes—
226C22×C4AGL(1,ℤ/8)11yescharacteristic
227C22×C4AGL(1,ℤ/8)12yes—
228C22×C4#13711yescharacteristic
229M16—88no—
230C2×C8—44no—
231C2×C8—82no—
232C2×C8—82no—
233C2×C8—82no—
234C2×C8—82no—
235C2×C8—82no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C22⋊C4—84no—
239C22⋊C4—82no—
240C22⋊C4—82no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—24no—
246C23—24no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23—42no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23#35212yes—
274C23#35412yes—
275C23#40811yescharacteristic
276C23C2×#9911yescharacteristic
277C23#41311yescharacteristic
278C23#41711yescharacteristic
279C23#41811yescharacteristic
280C23C2×#10411yescharacteristic
281C23#42111yescharacteristic
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—84no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—84no—
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×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C8—84no—
351C8—162no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—44no—
371C22—44no—
372C22—44no—
373C22—44no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22#146811yescharacteristic
383C22#148411yescharacteristic
384C22#149511yescharacteristic
385C22#150411yescharacteristic
386C22#151011yescharacteristic
387C22#152911yescharacteristic
388C22#155711yescharacteristic
389C22C2×#35211yescharacteristic
390C22#212111yescharacteristic
391C22C2×#35411yescharacteristic
392C22#213311yescharacteristic
393C4—42no—
394C4—42no—
395C4—41no—
396C4—42no—
397C4—41no—
398C4—81no—
399C4—81no—
400C4—161no—
401C4—161no—
402C4—161no—
403C4—161no—
404C2—21no—
405C2—21no—
406C2—21no—
407C2—21no—
408C2—42no—
409C2—42no—
410C2#769411yescharacteristic
411C2#1154311yescharacteristic
412C2#1155111yescharacteristic
413C2#1175911yescharacteristic
414C2#1176411yescharacteristic
415C2#1179711yescharacteristic
416C2#1180511yescharacteristic
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. #7122C2semidirect
5. #7104C2semidirect

Maximal subgroups

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

subgroupindex
#150192
#86982
#79092
#71222
#71042