Finite Group Databasebeta

← #497544

Subgroups — #497544

Order 512.

838 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3336C211yesmaximal, characteristic
2C2×#2169C211yesmaximal, characteristic
3#22656C211yesmaximal, characteristic
4#13281C211yesmaximal, characteristic, complemented
5#13277C211yesmaximal, characteristic, complemented
6#13062C211yesmaximal, characteristic, complemented
7#13062C211yesmaximal, characteristic, complemented
8#12994C211yesmaximal, characteristic, complemented
9#12993C211yesmaximal, characteristic, complemented
10#12868C211yesmaximal, characteristic, complemented
11#12867C211yesmaximal, characteristic, complemented
12#12814C211yesmaximal, characteristic, complemented
13C4×#358C211yesmaximal, characteristic, complemented
14#12557C211yesmaximal, characteristic, complemented
15#12557C211yesmaximal, characteristic, complemented
16C2×#552—21no—
17C22×M32C2211yescharacteristic
18C22×M32C2211yescharacteristic
19#3336—24no—
20#3336C418yescomplemented
21C2×#499C2211yescharacteristic
22C2×#485—21no—
23C2×#483—21no—
24C2×#479—21no—
25C2×#467C2211yescharacteristic
26C2×#467C2211yescharacteristic
27C2×C4×Q16C2211yescharacteristic
28C2×C8×D4C2211yescharacteristic
29C2×#427C2211yescharacteristic
30C2×#427C2211yescharacteristic
31C22×#107C2211yescharacteristic
32C22×#107C2211yescharacteristic
33#2352—21no—
34#2169—24no—
35C2×C4⋊C16C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#358C2211yescharacteristic
38C2×#358C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×C22⋊C16C2211yescharacteristic
42C2×C4×C16C2211yescharacteristic
43#1949—21no—
44#1854—21no—
45#1853—21no—
46#1846C2211yescharacteristic
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49C4×#107C2211yescharacteristic
50#1380C2211yescharacteristic
51#1380C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62C22×Q16C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65#552—28no—
66#552—44no—
67C2×#126—41no—
68C2×#125—21no—
69C2×#125—41no—
70C2×#123—41no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×D4D411yescharacteristic
75C2×#117—21no—
76C2×#117—21no—
77C2×#117—41no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C2×M32—22no—
81C2×M32—22no—
82C2×M32—22no—
83C2×M32—22no—
84C2×M32—41no—
85C2×M32—41no—
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90#499—22no—
91#499C2×C414yescomplemented
92#485—28no—
93#483—28no—
94#479—28no—
95#467—22no—
96#467—22no—
97#467C2×C414yes—
98#467C2×C414yescomplemented
99C4×Q16—22no—
100C4×Q16C2×C414yescomplemented
101C8×D4C2×C414yes—
102C8×D4D414yes—
103#427—22no—
104#427—22no—
105#427—41no—
106#427C2×C414yes—
107C2×#107—21no—
108C2×#107—22no—
109C2×#107—22no—
110C2×#107—21no—
111C2×#107—41no—
112C2×#107C2×C414yescomplemented
113C2×#107C2311yescharacteristic
114C2×#107C2311yescharacteristic
115C2×#107C2311yescharacteristic
116C2×#107C2311yescharacteristic
117C2×#107C2311yescharacteristic
118C2×#107C2311yescharacteristic
119C2×#107C2311yescharacteristic
120C2×#106—21no—
121C2×#106—21no—
122C2×#106—21no—
123C2×#106—21no—
124C2×C4⋊C8D411yescharacteristic
125C2×#103—21no—
126C2×#103—21no—
127C2×#103—21no—
128C2×C22⋊C8D411yescharacteristic
129C2×C22⋊C8D411yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#388—41no—
132#387—41no—
133#386—41no—
134#382—21no—
135#380—21no—
136#380—21no—
137C4×C4⋊C4—21no—
138C4×C22⋊C4—41no—
139C4⋊C16—22no—
140#358—22no—
141#358—22no—
142#358—41no—
143#358—41no—
144C22⋊C16—22no—
145C22⋊C16—22no—
146C4×C16—22no—
147#340—21no—
148#340—21no—
149#340—21no—
150C22×Q8—21no—
151C22×D4D811yescharacteristic
152C23×C4—41no—
153C23×C4—41no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C2×Q16—21no—
157C2×Q16—21no—
158C2×Q16—21no—
159C2×Q16—21no—
160C2×Q16—21no—
161C2×Q16C22×C412yes—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C22×C412yes—
171C22×C8C22×C412yes—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D412yes—
176C22×C8C2×D412yes—
177C22×C8C2411yescharacteristic
178#126—48no—
179#125—24no—
180#125—48no—
181#125—42no—
182#123—48no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C4×Q8—24no—
188C4×Q8—42no—
189C4×D4C22⋊C414yes—
190C4×D4SD1614yes—
191#117—24no—
192#117—42no—
193#117—48no—
194#117—42no—
195#117—81no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—81no—
214C2×C4⋊C4D811yescharacteristic
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4D811yescharacteristic
222C2×C22⋊C4D811yescharacteristic
223C2×C42—41no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42C2×D411yescharacteristic
228M32—44no—
229M32—44no—
230C2×C16—21no—
231C2×C16—21no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16#4112yes—
242#107—24no—
243#107—21no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107—42no—
254#107C22×C412yes—
255#107C22×C412yes—
256#107C22×C412yes—
257#107#4112yes—
258#106—24no—
259#106—24no—
260#106—42no—
261#106—81no—
262C4⋊C8C22⋊C414yes—
263#103—24no—
264#103—24no—
265#103—24no—
266C22⋊C8C22⋊C414yes—
267C22⋊C8C22⋊C414yes—
268C4×C8C22×C412yes—
269C4×C8C2×D412yes—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275C24C2×D811yescharacteristic
276C24C2×D811yescharacteristic
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4#10214yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—44no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—161no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×D811yescharacteristic
311C22×C4C2×D811yescharacteristic
312C22×C4C2×D811yescharacteristic
313C22×C4C2×D811yescharacteristic
314C22×C4C2×SD1612yes—
315C22×C4C2×SD1612yes—
316C22×C4C22×D411yescharacteristic
317Q16—22no—
318Q16—22no—
319Q16—41no—
320Q16—41no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8C2×C22⋊C412yes—
336C2×C8C2×C22⋊C412yes—
337C2×C8C2×C22⋊C412yes—
338C2×C8C23×C411yescharacteristic
339C2×C8C22×D411yescharacteristic
340C2×C8C2×#4111yescharacteristic
341C2×C8C2×#4111yescharacteristic
342C2×C8H(5,𝔽2)11yescharacteristic
343C2×C82−1+411yescharacteristic
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—82no—
365C4⋊C4#10214yes—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C22⋊C4—161no—
372C22⋊C4#10214yes—
373C22⋊C4#10214yes—
374C42—44no—
375C42—44no—
376C42—82no—
377C42—161no—
378C42C2×C22⋊C412yes—
379C42C2×SD1612yes—
380C16—22no—
381C16—41no—
382C16—41no—
383C16—41no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23C2×#10212yes—
397C23C2×#10212yes—
398C23C22×D811yescharacteristic
399Q8—42no—
400Q8—42no—
401Q8—81no—
402Q8—81no—
403D4—44no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—88no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—164no—
446C2×C4C2×#10212yes—
447C2×C4C2×#10212yes—
448C2×C4C2×#10212yes—
449C2×C4C2×#10212yes—
450C2×C4C22×C22⋊C411yescharacteristic
451C2×C4#54611yescharacteristic
452C2×C4#54711yescharacteristic
453C2×C4C22×SD1611yescharacteristic
454C2×C4#57711yescharacteristic
455C2×C4#57811yescharacteristic
456C8—22no—
457C8—22no—
458C8—22no—
459C8C4×#4111yescharacteristic
460C8#52011yescharacteristic
461C8#54611yescharacteristic
462C8#54711yescharacteristic
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22C22×#10211yescharacteristic
478C22#329411yescharacteristic
479C22#333411yescharacteristic
480C22#341411yescharacteristic
481C22#341411yescharacteristic
482C22#341611yescharacteristic
483C22#341711yescharacteristic
484C4—22no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—82no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—82no—
493C4—161no—
494C4—161no—
495C4—161no—
496C4#233911yescharacteristic
497C4#289811yescharacteristic
498C4#329811yescharacteristic
499C4#333511yescharacteristic
500C2—42no—
501C2—42no—
502C2#2248511yescharacteristic
503C2#3338011yescharacteristic
504C2#3338211yescharacteristic
505C2#3389011yescharacteristic
506C2#3389411yescharacteristic
507C2#3391811yescharacteristic
508C2#3391911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×D4D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13281C2semidirect
5. #13277C2semidirect
6. #13062C2semidirect
7. #13062C2semidirect
8. #12994C2semidirect
9. #12993C2semidirect
10. #12868C2semidirect
11. #12867C2semidirect
12. #12814C2semidirect
13. C4×#358C2semidirect
14. #12557C2semidirect
15. #12557C2semidirect
20. #3336C4semidirect
91. #499C2×C4semidirect
98. #467C2×C4semidirect
100. C4×Q16C2×C4semidirect
112. C2×#107C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#33362
C2×#21692
#226562
#132812
#132772
#130622
#129942
#129932
#128682
#128672
#128142
C4×#3582
#125572