Finite Group Databasebeta

← #125528

Subgroups — #125528

Order 512.

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

normal subgroups only

This group has 586 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#15014C211yesmaximal, characteristic
2#8698C211yesmaximal, characteristic
3#8698C211yesmaximal, characteristic
4#7123C212yesmaximal, complemented
5#7104C212yesmaximal, complemented
6C2×#385—21no—
7C2×#380C411yescharacteristic
8C22×#95C2211yescharacteristic
9#1779—21no—
10#1779—21no—
11#1445C411yescharacteristic
12#1445C2211yescharacteristic
13#1444C411yescharacteristic
14#1444C2211yescharacteristic
15#1442—21no—
16#1442—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⋊C4—21no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32C2×C22⋊C8—22no—
33C2×C22⋊C8—22no—
34C2×C22⋊C8—41no—
35C2×C22⋊C8—41no—
36C2×C22⋊C8—41no—
37C2×C22⋊C8—41no—
38#385—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—41no—
50C2×#95—41no—
51C2×#95C2×C411yescharacteristic
52C2×#95C2×C411yescharacteristic
53C2×#95C2×C411yescharacteristic
54C2×#95D411yescharacteristic
55C2×#95D411yescharacteristic
56C2×#95C2311yescharacteristic
57C25C22⋊C411yescharacteristic
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—22no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—41no—
72C23×C4—41no—
73C23×C4—41no—
74C23×C4—42no—
75C23×C4—42no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4—81no—
79C23×C4—81no—
80C23×C4C22⋊C411yescharacteristic
81C23×C4C22⋊C411yescharacteristic
82C23×C4C22⋊C411yescharacteristic
83C23×C4C22×C411yescharacteristic
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D411yescharacteristic
86C22×C8—42no—
87C22×C8—42no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—81no—
95C2×C4⋊C4—81no—
96C2×C4⋊C4—81no—
97C2×C4⋊C4—81no—
98C2×C4⋊C4—81no—
99C2×C22⋊C4—24no—
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—42no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109C22⋊C8—42no—
110C22⋊C8—82no—
111C22⋊C8—82no—
112C22⋊C8—82no—
113C22⋊C8—82no—
114#95—22no—
115#95—41no—
116#95—41no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—42no—
121#95—42no—
122#95—42no—
123#95—42no—
124#95—42no—
125#95—81no—
126#95—81no—
127#95—81no—
128C24—22no—
129C24—22no—
130C24—21no—
131C24—21no—
132C24—21no—
133C24—21no—
134C24—21no—
135C24—21no—
136C24—41no—
137C24—41no—
138C24#9911yescharacteristic
139C24#9911yescharacteristic
140C24C2×C22⋊C411yescharacteristic
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
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—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
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—81no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—84no—
215C22×C4—84no—
216C22×C4—84no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4—84no—
220C22×C4#13112yes—
221C22×C4#13512yes—
222C22×C4AGL(1,ℤ/8)12yes—
223C2×C8—82no—
224C2×C8—82no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8—82no—
228C2×C8—82no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C22⋊C4—48no—
234C22⋊C4—82no—
235C22⋊C4—82no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—24no—
241C23—24no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23#35212yes—
269C23#35612yes—
270C23#40811yescharacteristic
271C23C2×#9911yescharacteristic
272C23#41311yescharacteristic
273C23#41611yescharacteristic
274C23#41811yescharacteristic
275C23#42111yescharacteristic
276C23#42111yescharacteristic
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—84no—
310C2×C4—84no—
311C2×C4—84no—
312C2×C4—84no—
313C2×C4—84no—
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×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C8—162no—
342C8—162no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—44no—
362C22—44no—
363C22—44no—
364C22—44no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22#146811yescharacteristic
374C22#148211yescharacteristic
375C22#149611yescharacteristic
376C22#151011yescharacteristic
377C22#151611yescharacteristic
378C22#153511yescharacteristic
379C22#155311yescharacteristic
380C22C2×#35211yescharacteristic
381C22#212111yescharacteristic
382C22#212711yescharacteristic
383C22C2×#35611yescharacteristic
384C4—42no—
385C4—42no—
386C4—42no—
387C4—82no—
388C4—82no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—161no—
393C4—161no—
394C2—21no—
395C2—21no—
396C2—21no—
397C2—21no—
398C2—42no—
399C2—42no—
400C2#771811yescharacteristic
401C2#1154311yescharacteristic
402C2#1155511yescharacteristic
403C2#1179911yescharacteristic
404C2#1180311yescharacteristic
405C2#1184111yescharacteristic
406C2#1185411yescharacteristic
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. #7123C2semidirect
5. #7104C2semidirect

Maximal subgroups

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

subgroupindex
#150142
#86982
#71232
#71042