Finite Group Databasebeta

← #508482

Subgroups — #508482

Order 512.

738 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32931C211yesmaximal, characteristic
2C2×#1808C211yesmaximal, characteristic
3#13494C212yesmaximal, complemented
4#13492C212yesmaximal, complemented
5#13383C212yesmaximal, complemented
6#13382C212yesmaximal, complemented
7#12993C211yesmaximal, characteristic, complemented
8#12993C211yesmaximal, characteristic, complemented
9#12920C211yesmaximal, characteristic, complemented
10#12919C211yesmaximal, characteristic, complemented
11#12679C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C2×#467C2211yescharacteristic
14C2×#467C2211yescharacteristic
15C2×#467C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18#2848—21no—
19C2×#380—21no—
20C2×#367C2212yescomplemented
21C2×#366C2212yescomplemented
22C2×#358C2212yescomplemented
23C2×#358C2212yescomplemented
24C2×C22⋊C16C2211yescharacteristic
25C2×C22⋊C16C2211yescharacteristic
26C2×C22⋊C16C2212yes—
27C2×C22⋊C16C2212yes—
28#1892—21no—
29#1892—21no—
30#1892—21no—
31#1808—21no—
32#1808—21no—
33#1808C2212yes—
34#1808C2212yes—
35#1808C2211yescharacteristic
36#1808C2211yescharacteristic
37#1808C2211yescharacteristic
38C2×#340—21no—
39#1380C2212yescomplemented
40#1380C2212yescomplemented
41#1380C2212yescomplemented
42#1380C2212yescomplemented
43C24×C4D411yescharacteristic
44C22×Q16C2311yescharacteristic
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—42no—
48C23×C8—42no—
49C23×C8D411yescharacteristic
50C23×C8D411yescharacteristic
51C23×C8D411yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C22⋊C4—41no—
61C22×C16C2312yes—
62C22×C16C2312yes—
63#467—22no—
64#467—22no—
65#467—42no—
66#467—42no—
67C2×#107—21no—
68C2×#107—21no—
69C2×#107—42no—
70C2×#107—42no—
71C2×#107C2312yes—
72C2×#107C2312yes—
73C2×#107C2311yescharacteristic
74C2×#107C2311yescharacteristic
75C2×#107C2311yescharacteristic
76C2×#103—21no—
77C2×#103—21no—
78C2×#103—21no—
79#385—41no—
80#385—41no—
81#385—41no—
82#380—22no—
83#380—22no—
84#380—21no—
85#380—21no—
86#380—21no—
87#380—41no—
88#380—41no—
89#379—41no—
90C2×#95—41no—
91#367—42no—
92#366—42no—
93#358—42no—
94#358—42no—
95C22⋊C16—22no—
96C22⋊C16—22no—
97C22⋊C16—42no—
98C22⋊C16—42no—
99#340—22no—
100#340—22no—
101#340—21no—
102#340—21no—
103#340—21no—
104C25D811yescharacteristic
105C22×Q8—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4—41no—
111C23×C4—41no—
112C23×C4—41no—
113C23×C4—41no—
114C23×C4D811yescharacteristic
115C23×C4D811yescharacteristic
116C23×C4D811yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×Q16—21no—
121C2×Q16—21no—
122C2×Q16—22no—
123C2×Q16—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—41no—
141C22×C8—41no—
142C22×C8—41no—
143C22×C8—48no—
144C22×C8—48no—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8#4112yes—
149C22×C8#4112yes—
150C22×C8C2411yescharacteristic
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153C22⋊Q8—42no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—84no—
180C2×C16—22no—
181C2×C16—22no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16—22no—
185C2×C16—22no—
186#107—21no—
187#107—21no—
188#107—22no—
189#107—22no—
190#107—22no—
191#107—22no—
192#107—22no—
193#107—22no—
194#107—42no—
195#107—42no—
196#103—42no—
197#103—42no—
198#103—42no—
199#95—42no—
200#95—42no—
201#95—41no—
202#95—41no—
203#95—41no—
204C24—22no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—22no—
209C24—41no—
210C24—41no—
211C24C2×D811yescharacteristic
212C24C2×D811yescharacteristic
213C24C2×D811yescharacteristic
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—41no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4#12312yes—
262C22×C4C2×D811yescharacteristic
263C22×C4C2×D811yescharacteristic
264C22×C4C2×D811yescharacteristic
265C22×C4#13712yes—
266C22×C4C22×D411yescharacteristic
267Q16—42no—
268Q16—42no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—44no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8#12312yes—
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8H(5,𝔽2)12yes—
290C2×C8H(5,𝔽2)12yes—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4—82no—
306C22⋊C4—84no—
307C22⋊C4—82no—
308C22⋊C4—82no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—84no—
313C22⋊C4—82no—
314C16—42no—
315C16—42no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—28no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—44no—
334C23—44no—
335C23—44no—
336C23—44no—
337C23—44no—
338C23—44no—
339C23#48012yes—
340C23#50812yes—
341C23#50912yes—
342C23#51012yes—
343C23#51012yes—
344C23C22×D811yescharacteristic
345Q8—81no—
346Q8—81no—
347Q8—81no—
348Q8—81no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—164no—
391C2×C4—164no—
392C2×C4#48012yes—
393C2×C4C2×#12311yescharacteristic
394C2×C4#53411yescharacteristic
395C2×C4#53411yescharacteristic
396C2×C4C2×#13711yescharacteristic
397C2×C4#57511yescharacteristic
398C2×C4#57511yescharacteristic
399C8—22no—
400C8—41no—
401C8—41no—
402C8#53812yes—
403C8#53812yes—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—416no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22#223714yes—
426C22C2×#48011yescharacteristic
427C22#318611yescharacteristic
428C22#318611yescharacteristic
429C22C2×#50811yescharacteristic
430C22C2×#50911yescharacteristic
431C22C2×#51011yescharacteristic
432C22C2×#51011yescharacteristic
433C4—22no—
434C4—41no—
435C4—41no—
436C4—161no—
437C4—161no—
438C4—161no—
439C4—162no—
440C4—161no—
441C4—162no—
442C4#262211yescharacteristic
443C4#309311yescharacteristic
444C4#319611yescharacteristic
445C4#319611yescharacteristic
446C2—24no—
447C2—42no—
448C2—42no—
449C2#2989011yescharacteristic
450C2C2×#223711yescharacteristic
451C2#3364611yescharacteristic
452C2#3375411yescharacteristic
453C2#3375611yescharacteristic
454C2#3375911yescharacteristic
455C2#3376111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13494C2semidirect
4. #13492C2semidirect
5. #13383C2semidirect
6. #13382C2semidirect
7. #12993C2semidirect
8. #12993C2semidirect
9. #12920C2semidirect
10. #12919C2semidirect
20. C2×#367C22semidirect
21. C2×#366C22semidirect
22. C2×#358C22semidirect
23. C2×#358C22semidirect
39. #1380C22semidirect
40. #1380C22semidirect
41. #1380C22semidirect
42. #1380C22semidirect

Maximal subgroups

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

subgroupindex
#329312
C2×#18082
#134942
#134922
#133832
#133822
#129932
#129202
#129192
#126792