Finite Group Databasebeta

← #6563280

Subgroups — #6563280

Order 512.

1581 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36456C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#29939C211yesmaximal, characteristic, complemented
4#29937C212yesmaximal, complemented
5#27955C212yesmaximal, complemented
6#27948C211yesmaximal, characteristic, complemented
7#27652C212yesmaximal, complemented
8#27647C211yesmaximal, characteristic, complemented
9#26313C214yesmaximal, complemented
10#26297C212yesmaximal, complemented
11#25029C214yesmaximal, complemented
12#25015C212yesmaximal, complemented
13#23762C212yesmaximal, complemented
14#23757C211yesmaximal, characteristic, complemented
15#22729C212yesmaximal, complemented
16#22532C211yesmaximal, characteristic
17#22525C212yesmaximal, complemented
18C2×Q82C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#485C2214yes—
21C2×#483C2212yes—
22C2×#446C2212yes—
23#2857C2211yescharacteristic
24#2857C2212yes—
25#2563C2211yescharacteristic
26#2563C2212yes—
27#2538C2211yescharacteristic
28#2538C2212yes—
29#2452—22no—
30#2452—21no—
31#2439—22no—
32#2439—21no—
33#2355—21no—
34#2355C2212yes—
35#2354—21no—
36#2354C2212yes—
37#2352—22no—
38#2352C2214yes—
39#2351—21no—
40Q8×C4⋊C4C2211yescharacteristic
41#2341—21no—
42#2314—22no—
43#2311—21no—
44C4×C22⋊Q8C2212yes—
45C4×C22⋊Q8C2211yescharacteristic
46#2285C2211yescharacteristic
47#2285C2212yes—
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51#2100C2211yescharacteristic
52#2100C2212yes—
53#2095C2211yescharacteristic
54#2084C2212yes—
55#2084C2212yes—
56#2084C2212yes—
57#2083C2212yes—
58#2083C2211yescharacteristic
59#2077C2214yes—
60#2077C2214yes—
61#2076C2214yes—
62#2076C2212yes—
63#2075C2214yes—
64#2072C2211yescharacteristic
65#2072C2212yes—
66#2036C2212yes—
67#2035C2214yes—
68#1993C2214yes—
69#1990C2212yes—
70#1988C2214yes—
71#1987C2212yes—
72#1955C2214yes—
73#1954C2212yes—
74#1949C2214yes—
75#1948C2212yes—
76#1948C2214yes—
77#1947C2212yes—
78#1854C2214yes—
79#1854C2214yes—
80#1853C2214yes—
81#1847C2214yes—
82#1844C2212yes—
83#1825C2214yes—
84#1823C2212yes—
85#1805C2214yes—
86#1803C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1776C2214yes—
90C4×#107C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41D411yescharacteristic
96Q82—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8C2311yescharacteristic
104C2×C22⋊Q8C2312yes—
105C4×#41—28no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4D412yes—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2312yes—
115C2×C4×D4C2311yescharacteristic
116C2×#117—21no—
117C2×#117—22no—
118C2×#117D412yes—
119C2×#117D411yescharacteristic
120C2×#117C2314yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#485—48no—
127#483—44no—
128#446—28no—
129C2×#107C2312yes—
130C2×#107C2314yes—
131C2×#107C2312yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2314yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146#396—22no—
147#396—22no—
148#396—22no—
149#393—22no—
150#393—22no—
151#393—22no—
152#391—22no—
153#391—22no—
154#391—21no—
155#391C2311yescharacteristic
156#390—21no—
157#390—22no—
158#390—22no—
159#390—21no—
160#390—22no—
161#389—21no—
162#389—22no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—21no—
167#389C2312yes—
168#389C2311yescharacteristic
169#388—24no—
170#388—24no—
171#387—24no—
172#387—24no—
173#387—22no—
174#387—21no—
175#386—24no—
176#385—21no—
177#385—22no—
178#384—21no—
179#384—21no—
180#384C2312yes—
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2311yescharacteristic
186#382—21no—
187#382—22no—
188#382—21no—
189#382—22no—
190#382C2314yes—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—21no—
195#380—21no—
196#380—22no—
197#380C2312yes—
198#380C2311yescharacteristic
199#380C2312yes—
200#380C2312yes—
201#380C2312yes—
202#380C2314yes—
203#380C2312yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—24no—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—42no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2411yescharacteristic
240C22×D4C2×D411yescharacteristic
241C22×D4C2×D412yes—
242C23×C4—22no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C4⋊Q8—24no—
253C4⋊Q8—28no—
254C4⋊Q8—48no—
255C4⋊Q8—44no—
256C22⋊Q8—22no—
257C22⋊Q8—24no—
258C22⋊Q8—42no—
259C22⋊Q8—41no—
260C22⋊Q8—82no—
261C22⋊Q8—84no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—28no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×Q8#4118yes—
268C4×D4—28no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—28no—
272#117—24no—
273#117—28no—
274#117—44no—
275#117—44no—
276#117—44no—
277#117—84no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C42—24no—
329C2×C42—28no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344#107—24no—
345#107—42no—
346#107—44no—
347#106—44no—
348#106—44no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—28no—
352#103—28no—
353#103—24no—
354#103—24no—
355C22⋊C8—28no—
356C22⋊C8—24no—
357C22⋊C8—42no—
358C22⋊C8—44no—
359#97—28no—
360C4×C8—24no—
361#95—21no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—21no—
366#95—21no—
367#95—24no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378C24C22×D412yes—
379C24C22×D411yescharacteristic
380#41—416no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—82no—
389C2×Q8—81no—
390C2×Q8—82no—
391C2×Q8—81no—
392C2×Q8#12314yes—
393C2×Q8H(5,𝔽2)12yes—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—42no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4#13714yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2511yescharacteristic
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C82−1+414yes—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—81no—
486C4⋊C4#12318yes—
487C4⋊C4#12314yes—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—42no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—81no—
495C22⋊C4—82no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C42—24no—
502C42—22no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—44no—
507C42—44no—
508C42—42no—
509C42—44no—
510C42—42no—
511C42—84no—
512C42—84no—
513C42C2×#4114yes—
514C42C2×#4118yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23C2×#13712yes—
529C23C23×D411yescharacteristic
530Q8—28no—
531Q8—24no—
532Q8—42no—
533Q8—164no—
534Q8—162no—
535D4—48no—
536D4—44no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—48no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—164no—
589C2×C4C2×#12314yes—
590C2×C4C2×#12312yes—
591C2×C4C2×#12314yes—
592C2×C4C2×#12312yes—
593C2×C4C2×#13712yes—
594C2×C4C2×#13712yes—
595C2×C4#57512yes—
596C2×C4#57712yes—
597C2×C4C22×#4112yes—
598C2×C4C2×H(5,𝔽2)11yescharacteristic
599C2×C4C2×2−1+412yes—
600C2×C4#58511yescharacteristic
601C8—24no—
602C8—42no—
603C8—44no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22C22×#12311yescharacteristic
619C22#348811yescharacteristic
620C22#349911yescharacteristic
621C22C22×#13711yescharacteristic
622C22C2×#57511yescharacteristic
623C22C2×#57711yescharacteristic
624C22#358511yescharacteristic
625C4—22no—
626C4—24no—
627C4—22no—
628C4—24no—
629C4—22no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—84no—
634C4—82no—
635C4—81no—
636C4—82no—
637C4—82no—
638C4—81no—
639C4—81no—
640C4—82no—
641C4#309514yes—
642C4#351612yes—
643C4#352212yes—
644C2—42no—
645C2—82no—
646C2#3599611yescharacteristic
647C2C2×#309511yescharacteristic
648C2#6052511yescharacteristic
649C2#6196511yescharacteristic
650C2#6198711yescharacteristic
651C2#6247811yescharacteristic
652C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29939C2semidirect
4. #29937C2semidirect
5. #27955C2semidirect
6. #27948C2semidirect
7. #27652C2semidirect
8. #27647C2semidirect
9. #26313C2semidirect
10. #26297C2semidirect
11. #25029C2semidirect
12. #25015C2semidirect
13. #23762C2semidirect
14. #23757C2semidirect
15. #22729C2semidirect
17. #22525C2semidirect

Maximal subgroups

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

subgroupindex
#364562
#341882
#299392
#299372
#279552
#279482
#276522
#276472
#263132
#262972
#250292
#250152
#237622
#237572
#227292
#225322
#225252