Finite Group Databasebeta

← #285771

Subgroups — #285771

Order 512.

619 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27877C211yesmaximal, characteristic
2#27649C211yesmaximal, characteristic
3#24760C211yesmaximal, characteristic
4#22952C211yesmaximal, characteristic
5#10651C211yesmaximal, characteristic, complemented
6#10623C211yesmaximal, characteristic, complemented
7#10594C211yesmaximal, characteristic, complemented
8#10478C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9565C211yesmaximal, characteristic, complemented
11#9218C211yesmaximal, characteristic, complemented
12#9129C211yesmaximal, characteristic
13#8832C211yesmaximal, characteristic
14#8645C211yesmaximal, characteristic, complemented
15#8045C211yesmaximal, characteristic
16C2×#485—21no—
17C2×#470—21no—
18C2×#462C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22#2857C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24#2285C2211yescharacteristic
25#2085—21no—
26#2075—21no—
27#2075—21no—
28#2074—21no—
29#2066—21no—
30#2056—21no—
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1984C2211yescharacteristic
34#1981C2211yescharacteristic
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1947C2211yescharacteristic
38#1939—21no—
39#1939C2211yescharacteristic
40#1918—21no—
41#1918—21no—
42#1916—21no—
43#1894—21no—
44#1850—21no—
45#1845C2211yescharacteristic
46#1834C2211yescharacteristic
47#1825—21no—
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C8×C22⋊C4—21no—
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×Q16—21no—
68C22×M16—21no—
69C23×C8—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8D411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×#117—21no—
75C22×C42C2311yescharacteristic
76#485—44no—
77#470—44no—
78#462—22no—
79#462—41no—
80C8×Q8—22no—
81C8×Q8—41no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#106—21no—
86C2×#106—21no—
87#423—22no—
88#423—41no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—41no—
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8D411yescharacteristic
109C2×C22⋊C8D411yescharacteristic
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#391D411yescharacteristic
116#389—21no—
117#389—21no—
118#389D411yescharacteristic
119#387—21no—
120#385—21no—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#382—21no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380D411yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336D411yescharacteristic
153#336D411yescharacteristic
154#335—22no—
155#335—41no—
156#332—22no—
157#332—41no—
158C22×Q8—21no—
159C22×Q8C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C2×Q16—42no—
164C2×Q16—42no—
165C2×Q16—81no—
166C2×M16—42no—
167C2×M16—42no—
168C2×M16—81no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8—81no—
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C22⋊Q8—22no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189C22⋊Q8D814yes—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194#117—42no—
195#117—42no—
196C2×C4⋊C4—21no—
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—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—41no—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42C2411yescharacteristic
229#107—42no—
230#107—42no—
231#107—42no—
232#106—42no—
233#106—42no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—81no—
248C4⋊C8#4112yes—
249C4⋊C8#4112yes—
250#103—42no—
251#103—42no—
252#103—42no—
253#103—42no—
254#103—42no—
255#103—81no—
256C22⋊C8—22no—
257C22⋊C8—41no—
258C22⋊C8—42no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—42no—
264C4×C8—81no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274C24#12111yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8C2×D812yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4C2×D812yes—
313C22×C4#13512yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4111yescharacteristic
317Q16—84no—
318M16—84no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—84no—
349C2×C8#13712yes—
350C4⋊C4—21no—
351C4⋊C4—24no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—81no—
375C4⋊C4—81no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4—82no—
379C4⋊C4C2×D812yes—
380C22⋊C4—24no—
381C22⋊C4—42no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—82no—
395C42C2×#4111yescharacteristic
396C42C2×#4111yescharacteristic
397C42H(5,𝔽2)11yescharacteristic
398C422−1+411yescharacteristic
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23#46612yes—
406C23C2×#12111yescharacteristic
407Q8—22no—
408Q8—22no—
409Q8—41no—
410Q8—41no—
411Q8—81no—
412Q8—81no—
413Q8—81no—
414Q8—81no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4#46612yes—
460C2×C4#52911yescharacteristic
461C2×C4#53911yescharacteristic
462C2×C4#54611yescharacteristic
463C2×C4#54611yescharacteristic
464C2×C4#54711yescharacteristic
465C2×C4#54711yescharacteristic
466C2×C4C22×D811yescharacteristic
467C2×C4C2×#13511yescharacteristic
468C2×C4C2×#13711yescharacteristic
469C2×C4#57511yescharacteristic
470C2×C4#57711yescharacteristic
471C2×C4#57711yescharacteristic
472C2×C4#57811yescharacteristic
473C2×C4#57811yescharacteristic
474C8—22no—
475C8—41no—
476C8—41no—
477C8—42no—
478C8—41no—
479C8—42no—
480C8—81no—
481C8—81no—
482C8—81no—
483C8—81no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22#242511yescharacteristic
492C22#303711yescharacteristic
493C22#304611yescharacteristic
494C22C2×#46611yescharacteristic
495C22#306311yescharacteristic
496C22#306711yescharacteristic
497C22#306911yescharacteristic
498C4—22no—
499C4—21no—
500C4—21no—
501C4—41no—
502C4—42no—
503C4—41no—
504C4—42no—
505C4—81no—
506C4—81no—
507C4—161no—
508C4—161no—
509C4—161no—
510C4—161no—
511C4#329411yescharacteristic
512C4#330011yescharacteristic
513C4#331211yescharacteristic
514C4#332311yescharacteristic
515C4#333411yescharacteristic
516C4#333711yescharacteristic
517C4#334911yescharacteristic
518C4#335611yescharacteristic
519C2—42no—
520C2#1926111yescharacteristic
521C2#1937511yescharacteristic
522C2#1941611yescharacteristic
523C2#1958511yescharacteristic
524C2#2522011yescharacteristic
525C2#2523211yescharacteristic
526C2#2526511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10651C2semidirect
6. #10623C2semidirect
7. #10594C2semidirect
8. #10478C2semidirect
9. #9616C2semidirect
10. #9565C2semidirect
11. #9218C2semidirect
14. #8645C2semidirect

Maximal subgroups

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

subgroupindex
#278772
#276492
#247602
#229522
#106512
#106232
#105942
#104782
#96162
#95652
#92182
#91292
#88322
#86452
#80452