Finite Group Databasebeta

← #6986543

Subgroups — #6986543

Order 512.

1443 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36481C211yesmaximal, characteristic
2#35320C211yesmaximal, characteristic
3#32842C212yesmaximal, complemented
4#31951C214yesmaximal
5#31857C214yesmaximal
6#30344C212yesmaximal, complemented
7#30042C212yesmaximal, complemented
8#29664C211yesmaximal, characteristic
9#29147C212yesmaximal, complemented
10#28417C212yesmaximal, complemented
11#27737C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic
13#25323C212yesmaximal, complemented
14#24818C212yesmaximal, complemented
15#24301C211yesmaximal, characteristic
16#23571C211yesmaximal, characteristic
17#22643C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#487C2212yes—
21C2×#475C2212yes—
22C2×#462C2212yes—
23C2×C4×Q16C2212yes—
24#2863C2212yes—
25#2862C2212yes—
26#2862C2212yes—
27#2857C2211yescharacteristic
28#2642—22no—
29#2606—22no—
30#2568—21no—
31#2568—21no—
32#2568C2212yes—
33#2568C2212yes—
34#2552C2211yescharacteristic
35#2546C2211yescharacteristic
36#2522—22no—
37#2507—21no—
38#2503—22no—
39#2503C2214yes—
40#2441—21no—
41#2429C2212yes—
42#2423C2212yes—
43#2370—21no—
44#2365C2211yescharacteristic
45Q8×C4⋊C4C2211yescharacteristic
46#2347—21no—
47#2310—21no—
48#2310—21no—
49C4×C4⋊Q8C2212yes—
50C4×C22⋊Q8C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2097C2214yes—
54#2085C2214yes—
55#2074C2214yes—
56#2072C2214yes—
57#2066C2214yes—
58#2058C2212yes—
59#2057C2211yescharacteristic
60#2056C2214yes—
61#2043C2211yescharacteristic
62#2036C2212yes—
63#2024C2211yescharacteristic
64#2024C2212yes—
65#2024C2212yes—
66#2024C2212yes—
67#2024C2212yes—
68#2016C2214yes—
69#2016C2211yescharacteristic
70#2016C2212yes—
71#2001C2212yes—
72#2001C2214yes—
73#1990C2212yes—
74#1988C2212yes—
75#1971C2214yes—
76#1960C2214yes—
77#1939C2212yes—
78#1935C2214yes—
79#1924C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1872C2214yes—
83#1872C2212yes—
84#1853C2212yes—
85#1839C2212yes—
86#1836C2214yes—
87#1805C2214yes—
88#1805C2214yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91#1770C2212yes—
92#1767C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#125—21no—
109C2×#125—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2311yescharacteristic
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C4×Q8—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×#117—21no—
120C2×#117D411yescharacteristic
121C2×#117D411yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125#487—44no—
126#475—44no—
127#462—44no—
128C4×Q16—44no—
129C2×#107C2312yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146#400—24no—
147#400—24no—
148#400—24no—
149#400—24no—
150#399—22no—
151#398—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#396—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391C2311yescharacteristic
167#391C2312yes—
168#391C2312yes—
169#391C2314yes—
170#391C2314yes—
171#391C2312yes—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389C2311yescharacteristic
183#389C2312yes—
184#389C2312yes—
185#388—22no—
186#388—22no—
187#387—21no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#385—21no—
194#384—21no—
195#384C2312yes—
196#384C2311yescharacteristic
197#383—22no—
198#382—22no—
199#382—22no—
200#382C2314yes—
201#382C2314yes—
202#381—24no—
203#381—24no—
204#380—21no—
205#380—21no—
206#380C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233#336C2311yescharacteristic
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—44no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×Q8C2412yes—
241C22×Q8C2412yes—
242C22×Q8C2411yescharacteristic
243C23×C4—22no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×Q16—24no—
248C2×Q16—42no—
249C2×Q16—42no—
250C2×Q16—44no—
251C2×Q16—44no—
252C2×Q16—44no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—42no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266#125—44no—
267#125—44no—
268C22⋊Q8—22no—
269C22⋊Q8—41no—
270C22⋊Q8—42no—
271C22⋊Q8—42no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280#117—24no—
281#117—42no—
282#117—44no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341#107—42no—
342#106—44no—
343C4⋊C8—24no—
344C4⋊C8—44no—
345#103—42no—
346#103—42no—
347#103—44no—
348#103—42no—
349#103—42no—
350#103—44no—
351#103—44no—
352#103—44no—
353#103—44no—
354C22⋊C8—42no—
355C22⋊C8—42no—
356#97—44no—
357C4×C8—24no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24C22×D411yescharacteristic
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—816no—
407C2×Q82−1+412yes—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4#13714yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2511yescharacteristic
440Q16—48no—
441Q16—88no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C22⋊C4—24no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—81no—
493C22⋊C4—82no—
494C42—22no—
495C42—22no—
496C42—24no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42—42no—
503C42—42no—
504C42—44no—
505C42—42no—
506C42—42no—
507C42—82no—
508C42—82no—
509C42H(5,𝔽2)14yes—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23C2×#13712yes—
517C23C23×D411yescharacteristic
518Q8—24no—
519Q8—42no—
520Q8—42no—
521Q8—44no—
522Q8—42no—
523Q8—42no—
524Q8—44no—
525Q8—42no—
526Q8—84no—
527Q8—88no—
528Q8—88no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—88no—
582C2×C4#53512yes—
583C2×C4#53512yes—
584C2×C4C2×#13712yes—
585C2×C4C2×#13712yes—
586C2×C4#57512yes—
587C2×C4#57812yes—
588C2×C4C2×H(5,𝔽2)12yes—
589C2×C4C2×2−1+411yescharacteristic
590C2×C4#58512yes—
591C2×C4#58511yescharacteristic
592C8—42no—
593C8—42no—
594C8—44no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22C2×#53511yescharacteristic
603C22#348911yescharacteristic
604C22#350011yescharacteristic
605C22C22×#13711yescharacteristic
606C22C2×#57511yescharacteristic
607C22C2×#57811yescharacteristic
608C22#358611yescharacteristic
609C4—22no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—42no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—84no—
619C4—81no—
620C4—82no—
621C4—81no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—81no—
626C4#321514yes—
627C4#354112yes—
628C4#355112yes—
629C2—42no—
630C2#4291011yescharacteristic
631C2C2×#321511yescharacteristic
632C2#6081411yescharacteristic
633C2#6211311yescharacteristic
634C2#6214611yescharacteristic
635C2#6253611yescharacteristic
636C2#6258711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32842C2semidirect
6. #30344C2semidirect
7. #30042C2semidirect
9. #29147C2semidirect
10. #28417C2semidirect
11. #27737C2semidirect
13. #25323C2semidirect
14. #24818C2semidirect

Maximal subgroups

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

subgroupindex
#364812
#353202
#328422
#319512
#318572
#303442
#300422
#296642
#291472
#284172
#277372
#259022
#253232
#248182
#243012
#235712
#226432