Finite Group Databasebeta

← #6563535

Subgroups — #6563535

Order 512.

1617 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36275C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#29938C211yesmaximal, characteristic, complemented
4#29911C212yesmaximal, complemented
5#27950C211yesmaximal, characteristic, complemented
6#27861C212yesmaximal, complemented
7#27648C211yesmaximal, characteristic, complemented
8#27610C212yesmaximal, complemented
9#26279C212yesmaximal, complemented
10#26145C214yesmaximal, complemented
11#25063C212yesmaximal, complemented
12#24952C214yesmaximal, complemented
13#23756C211yesmaximal, characteristic, complemented
14#23657C212yesmaximal, complemented
15#22665C212yesmaximal, complemented
16#22502C212yesmaximal, complemented
17#22473C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#484C2212yes—
21C2×#436C2212yes—
22C2×#428C2214yes—
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2582C2211yescharacteristic
26#2567C2211yescharacteristic
27#2487C2212yes—
28#2453—21no—
29#2449—22no—
30#2441—21no—
31#2438—22no—
32#2417C2212yes—
33#2355—21no—
34#2355C2212yes—
35#2354—21no—
36#2354C2212yes—
37#2352—22no—
38#2352C2214yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2211yescharacteristic
41#2339—21no—
42#2314—22no—
43#2311—21no—
44C4×C22⋊Q8C2211yescharacteristic
45C2×#389C2212yes—
46#2285C2211yescharacteristic
47#2285C2212yes—
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50#2274—21no—
51#2101C2214yes—
52#2099C2211yescharacteristic
53#2099C2212yes—
54#2097C2214yes—
55#2096C2211yescharacteristic
56#2085C2214yes—
57#2085C2214yes—
58#2084C2211yescharacteristic
59#2084C2212yes—
60#2083C2212yes—
61#2083C2212yes—
62#2083C2212yes—
63#2077C2214yes—
64#2077C2212yes—
65#2072C2212yes—
66#2072C2211yescharacteristic
67#2035C2212yes—
68#1990C2212yes—
69#1987C2212yes—
70#1984C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1928C2214yes—
74#1919C2214yes—
75#1918C2214yes—
76#1918C2214yes—
77#1894C2214yes—
78#1873C2214yes—
79#1870C2214yes—
80#1854C2214yes—
81#1852C2214yes—
82#1852C2214yes—
83#1845C2212yes—
84#1824C2212yes—
85#1823C2214yes—
86#1817C2212yes—
87#1809C2214yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90C4×#106C2212yes—
91C4×#103C2212yes—
92C4×C22⋊C8C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C22×M16C2314yes—
97#554—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2311yescharacteristic
104C4×#41—28no—
105C2×C4×Q8D411yescharacteristic
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D412yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2311yescharacteristic
114C2×#117—21no—
115C2×#117—22no—
116C2×#117D412yes—
117C2×#117D411yescharacteristic
118C2×#117C2314yes—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#484—44no—
128#436—28no—
129#428—48no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2312yes—
134C2×#106C2312yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C4×C8C2312yes—
144#400—22no—
145#399—22no—
146#398—22no—
147#398—22no—
148#396—22no—
149#395—22no—
150#391—22no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391C2312yes—
155#391C2311yescharacteristic
156#390—22no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2311yescharacteristic
167#388—24no—
168#388—24no—
169#387—24no—
170#387—24no—
171#387—22no—
172#387—22no—
173#387—21no—
174#386—24no—
175#386—22no—
176#386—22no—
177#385—22no—
178#385—21no—
179#385—22no—
180#385—22no—
181#384—21no—
182#384—21no—
183#384—21no—
184#384C2312yes—
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382—21no—
190#382—22no—
191#382C2314yes—
192#381—22no—
193#380—22no—
194#380—22no—
195#380—21no—
196#380—21no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200#380C2314yes—
201#380C2312yes—
202#380C2311yescharacteristic
203#380C2312yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—24no—
216C2×#95—22no—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C2×#41—24no—
234C2×#41—24no—
235C2×#41—42no—
236C22×Q8—21no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2411yescharacteristic
239C22×D4C2×D412yes—
240C22×D4C2×D411yescharacteristic
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C2×M16—28no—
251C2×M16—28no—
252C2×M16—44no—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257#125—44no—
258C22⋊Q8—28no—
259C22⋊Q8—22no—
260C22⋊Q8—41no—
261C22⋊Q8—48no—
262C22⋊Q8—82no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×D4—24no—
267C4×D4—28no—
268C4×D4—28no—
269C4×D4—44no—
270C4×D4—44no—
271C4×D4#4118yes—
272#117—24no—
273#117—28no—
274#117—44no—
275#117—44no—
276#117—44no—
277#117—84no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—84no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—28no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—28no—
346#107—44no—
347#107—44no—
348#106—24no—
349#106—24no—
350#106—44no—
351C4⋊C8—24no—
352#103—24no—
353#103—42no—
354#103—44no—
355C22⋊C8—28no—
356C22⋊C8—24no—
357C22⋊C8—42no—
358C4×C8—24no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—21no—
366#95—24no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—24no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378C24C22×D412yes—
379C24C22×D411yescharacteristic
380#41—416no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—42no—
389C2×Q8—81no—
390C2×Q8—81no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4#12314yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4#13714yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4118yes—
443C22×C4C2511yescharacteristic
444M16—416no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C82−1+414yes—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—81no—
491C4⋊C4#12314yes—
492C4⋊C4H(5,𝔽2)12yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—81no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—82no—
503C22⋊C4#12318yes—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—22no—
509C42—24no—
510C42—42no—
511C42—44no—
512C42—42no—
513C42—84no—
514C42—84no—
515C42C2×#4114yes—
516C23—21no—
517C23—22no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23C2×#12314yes—
529C23C2×#13712yes—
530C23C23×D411yescharacteristic
531Q8—44no—
532Q8—42no—
533Q8—42no—
534Q8—162no—
535D4—28no—
536D4—48no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—88no—
590C2×C4—164no—
591C2×C4—164no—
592C2×C4C2×#12314yes—
593C2×C4C2×#12312yes—
594C2×C4C2×#12312yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4#57512yes—
598C2×C4#57612yes—
599C2×C4C22×#4112yes—
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4C2×2−1+412yes—
602C2×C4#58511yescharacteristic
603C8—24no—
604C8—42no—
605C8—44no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—41no—
619C22—41no—
620C22C22×#12311yescharacteristic
621C22#348811yescharacteristic
622C22#349911yescharacteristic
623C22C22×#13711yescharacteristic
624C22C2×#57511yescharacteristic
625C22C2×#57611yescharacteristic
626C22#358611yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—24no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—41no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—84no—
639C4—81no—
640C4—81no—
641C4—82no—
642C4—81no—
643C4#309514yes—
644C4#351512yes—
645C4#352012yes—
646C2—42no—
647C2—44no—
648C2#3599811yescharacteristic
649C2C2×#309511yescharacteristic
650C2#6052111yescharacteristic
651C2#6196611yescharacteristic
652C2#6199511yescharacteristic
653C2#6248011yescharacteristic
654C2#6251111yescharacteristic
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. #29938C2semidirect
4. #29911C2semidirect
5. #27950C2semidirect
6. #27861C2semidirect
7. #27648C2semidirect
8. #27610C2semidirect
9. #26279C2semidirect
10. #26145C2semidirect
11. #25063C2semidirect
12. #24952C2semidirect
13. #23756C2semidirect
14. #23657C2semidirect
15. #22665C2semidirect
16. #22502C2semidirect

Maximal subgroups

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

subgroupindex
#362752
#341882
#299382
#299112
#279502
#278612
#276482
#276102
#262792
#261452
#250632
#249522
#237562
#236572
#226652
#225022
#224732