Finite Group Databasebeta

← #125285

Subgroups — #125285

Order 512.

662 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 96 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#7909C211yesmaximal, characteristic
3C2×#1280C211yesmaximal, characteristic
4#7122C212yesmaximal, complemented
5#7104C212yesmaximal, complemented
6C2×#407—21no—
7C22×C22⋊C8—21no—
8C2×#381—21no—
9C2×#380C411yescharacteristic
10C22×#95C2211yescharacteristic
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—
27C23×C8—41no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C22⋊C4—21no—
31C22×C22⋊C4—21no—
32#407—24no—
33#407—42no—
34C2×C22⋊C8—24no—
35C2×C22⋊C8—22no—
36C2×C22⋊C8—22no—
37C2×C22⋊C8—41no—
38C2×C22⋊C8—42no—
39C2×C22⋊C8—41no—
40#381—42no—
41#381—42no—
42#380—41no—
43#380—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×#95—21no—
53C2×#95—21no—
54C2×#95—41no—
55C2×#95—41no—
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95D411yescharacteristic
60C2×#95D411yescharacteristic
61C2×#95C2311yescharacteristic
62C25C22⋊C411yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—22no—
68C23×C4—22no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—22no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—41no—
79C23×C4—42no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—81no—
84C23×C4—81no—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4D812yes—
89C23×C4SD1612yes—
90C23×C4C22×C411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D411yescharacteristic
93C2×M16—44no—
94C2×M16—82no—
95C2×M16—82no—
96C22×C8—44no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8—82no—
100C22×C8—82no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—81no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4—42no—
113C22⋊C8—24no—
114C22⋊C8—24no—
115C22⋊C8—44no—
116C22⋊C8—44no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119#95—22no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124#95—41no—
125#95—42no—
126#95—42no—
127#95—42no—
128#95—42no—
129#95—41no—
130#95—41no—
131#95—41no—
132#95—41no—
133#95—41no—
134#95—41no—
135#95—81no—
136#95—81no—
137#95—81no—
138C24—22no—
139C24—22no—
140C24—21no—
141C24—21no—
142C24—21no—
143C24—21no—
144C24—41no—
145C24—41no—
146C24#9911yescharacteristic
147C24#9911yescharacteristic
148C24#10212yes—
149C24#10412yes—
150C24C2×C22⋊C411yescharacteristic
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
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—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
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—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—82no—
223C22×C4—84no—
224C22×C4—82no—
225C22×C4—84no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4—81no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—84no—
233C22×C4—84no—
234C22×C4#10212yes—
235C22×C4#10412yes—
236C22×C4#13112yes—
237C22×C4C2×D811yescharacteristic
238C22×C4C2×SD1611yescharacteristic
239C22×C4AGL(1,ℤ/8)11yescharacteristic
240C22×C4#13711yescharacteristic
241M16—88no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—88no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C2×C8—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C22⋊C4—84no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—24no—
264C23—24no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—22no—
272C23—21no—
273C23—22no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23#32712yes—
285C23#32712yes—
286C23#35212yes—
287C23#35412yes—
288C23#40811yescharacteristic
289C23C2×#9911yescharacteristic
290C23#41311yescharacteristic
291C23C2×#10211yescharacteristic
292C23#41711yescharacteristic
293C23C2×#10411yescharacteristic
294C23C2×#10411yescharacteristic
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—84no—
336C2×C4—84no—
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—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C8—84no—
368C8—84no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22#134414yes—
399C22#146811yescharacteristic
400C22#148311yescharacteristic
401C22#149211yescharacteristic
402C22C2×#32711yescharacteristic
403C22#150411yescharacteristic
404C22#152411yescharacteristic
405C22#154711yescharacteristic
406C22C2×#35211yescharacteristic
407C22#212111yescharacteristic
408C22C2×#35411yescharacteristic
409C22#213311yescharacteristic
410C4—42no—
411C4—41no—
412C4—42no—
413C4—41no—
414C4—41no—
415C4—42no—
416C4—41no—
417C4—81no—
418C4—161no—
419C4—161no—
420C4—161no—
421C4—161no—
422C2—21no—
423C2—21no—
424C2—24no—
425C2—21no—
426C2—21no—
427C2—42no—
428C2#767811yescharacteristic
429C2#1154711yescharacteristic
430C2#1155111yescharacteristic
431C2C2×#134411yescharacteristic
432C2#1169811yescharacteristic
433C2#1175911yescharacteristic
434C2#1176411yescharacteristic
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
#79092
C2×#12802
#71222
#71042