Finite Group Databasebeta

← #6854498

Subgroups — #6854498

Order 512.

1444 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36560C211yesmaximal, characteristic, complemented
2#36555C211yesmaximal, characteristic
3#33294C211yesmaximal, characteristic, complemented
4#31541C214yesmaximal, complemented
5#31539C214yesmaximal
6#30311C212yesmaximal, complemented
7#29196C212yesmaximal, complemented
8#29150C211yesmaximal, characteristic, complemented
9#29146C212yesmaximal
10#28520C211yesmaximal, characteristic
11#24818C212yesmaximal, complemented
12#24816C212yesmaximal
13#24332C212yesmaximal, complemented
14#24328C212yesmaximal
15#22273C212yesmaximal
16C4×#497C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic, complemented
18C2×#557C2212yes—
19C2×#497C2211yescharacteristic
20C2×#475C2212yes—
21C2×#474C2212yes—
22C2×C4×Q16C2212yes—
23C2×C4×SD16C2212yes—
24#2868C2211yescharacteristic
25#2864C2212yes—
26#2859C2211yescharacteristic
27#2859C2212yes—
28#2587—22no—
29#2582C2211yescharacteristic
30#2568C2212yes—
31#2565C2214yes—
32#2546—21no—
33#2546—21no—
34#2538C2211yescharacteristic
35#2522—22no—
36#2506—22no—
37#2506—22no—
38#2497—21no—
39#2454C2212yes—
40#2453C2212yes—
41#2423—21no—
42#2365—21no—
43#2365—21no—
44#2360—21no—
45Q8×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4C2211yescharacteristic
47#2345—21no—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×#124C2212yes—
51C4×#124C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53#2095C2212yes—
54#2094C2212yes—
55#2085C2214yes—
56#2085C2214yes—
57#2068C2214yes—
58#2066C2214yes—
59#2059C2214yes—
60#2059C2214yes—
61#2026C2214yes—
62#2026C2214yes—
63#2025C2214yes—
64#1992C2214yes—
65#1990C2212yes—
66#1990C2212yes—
67#1984C2212yes—
68#1973C2212yes—
69#1971C2212yes—
70#1960C2212yes—
71#1960C2211yescharacteristic
72#1959C2212yes—
73#1959C2211yescharacteristic
74#1939C2212yes—
75#1917C2214yes—
76#1880C2214yes—
77#1878C2214yes—
78#1878C2214yes—
79#1877C2214yes—
80#1865C2214yes—
81#1863C2214yes—
82#1847C2212yes—
83#1847C2211yescharacteristic
84#1836C2212yes—
85#1836C2212yes—
86#1836C2212yes—
87#1805C2214yes—
88#1804C2214yes—
89#1803C2212yes—
90#1803C2212yes—
91#1767C2212yes—
92#1767C2211yescharacteristic
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C22×Q16C2312yes—
96C22×SD16C2312yes—
97#557—44no—
98#557—44no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#125—24no—
105C2×#125C2314yes—
106C2×#125C2312yes—
107C2×#124C2311yescharacteristic
108C2×C22⋊Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119C22×C4⋊C4—21no—
120#497Q818yescomplemented
121#475—44no—
122#474—44no—
123C4×Q16—44no—
124C4×SD16—44no—
125C2×#107C2312yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#97C2311yescharacteristic
140C2×#97C2312yes—
141C2×#97C2312yes—
142C2×#97C2311yescharacteristic
143C2×C4×C8C2312yes—
144#400—24no—
145#400—22no—
146#399—22no—
147#398—24no—
148#398—24no—
149#398—22no—
150#398—22no—
151#396—22no—
152#396—22no—
153#391—21no—
154#391—22no—
155#391—22no—
156#391C2312yes—
157#391C2314yes—
158#391C2312yes—
159#391C2311yescharacteristic
160#390C2311yescharacteristic
161#390C2312yes—
162#389—21no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—21no—
176#389—22no—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—21no—
182#388—22no—
183#388—22no—
184#388—21no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—24no—
190#387—24no—
191#387—22no—
192#385—22no—
193#385—22no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#382C2314yes—
203#382C2312yes—
204#382C2311yescharacteristic
205#380—21no—
206#380—21no—
207#380—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C43C2312yes—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2314yes—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×Q8C2412yes—
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—22no—
242C2×Q16—42no—
243C2×Q16—42no—
244C2×Q16C2×Q818yes—
245C2×SD16—42no—
246C2×SD16—42no—
247C2×SD16C2×Q818yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C4⋊Q8—48no—
257C4⋊Q8C2×Q814yes—
258#125—44no—
259#125—44no—
260#125—44no—
261#125—416no—
262#124—22no—
263#124C2×Q814yes—
264C22⋊Q8—44no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—42no—
270C4×Q8—42no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—48no—
275C4×D4—24no—
276C4×D4—44no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
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—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2412yes—
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—42no—
341#107—44no—
342#106—42no—
343#106—44no—
344C4⋊C8—24no—
345C4⋊C8—44no—
346#103—44no—
347#103—44no—
348#103—42no—
349#103—42no—
350#103—42no—
351#103—44no—
352#102—42no—
353#102—44no—
354#97—22no—
355#97—24no—
356#97—42no—
357#97C2×Q814yes—
358C4×C8—24no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8C22×Q812yes—
396C2×Q8C22×Q814yes—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4C22×Q812yes—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2511yescharacteristic
426Q16—48no—
427SD16—48no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8C22×Q814yes—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—44no—
479C22⋊C4—42no—
480C22⋊C4—82no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—82no—
484C42—22no—
485C42—22no—
486C42—24no—
487C42—22no—
488C42—24no—
489C42—41no—
490C42—41no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—48no—
496C42—42no—
497C42—44no—
498C42—82no—
499C42C22×Q812yes—
500C42H(5,𝔽2)14yes—
501C422−1+412yes—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C23×D411yescharacteristic
510Q8—24no—
511Q8—24no—
512Q8—24no—
513Q8—44no—
514Q8—42no—
515Q8—42no—
516Q8—42no—
517Q8—88no—
518D4—24no—
519D4—42no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—84no—
569C2×C4—164no—
570C2×C4#53512yes—
571C2×C4#53512yes—
572C2×C4D4×Q812yes—
573C2×C4D4×Q812yes—
574C2×C4#57512yes—
575C2×C4#57512yes—
576C2×C4C23×Q811yescharacteristic
577C2×C4C2×H(5,𝔽2)12yes—
578C2×C4C2×2−1+411yescharacteristic
579C2×C4#58512yes—
580C8—24no—
581C8—42no—
582C8—44no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22C2×#53511yescharacteristic
591C22C2×D4×Q811yescharacteristic
592C22#350011yescharacteristic
593C22C2×#57511yescharacteristic
594C22C2×#57511yescharacteristic
595C22#358611yescharacteristic
596C22#358611yescharacteristic
597C4—22no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—44no—
608C4—81no—
609C4—84no—
610C4—82no—
611C4—82no—
612C4—81no—
613C4—82no—
614C4#321914yes—
615C4#350112yes—
616C4#354712yes—
617C2—82no—
618C2#3961811yescharacteristic
619C2C2×#321911yescharacteristic
620C2#6083111yescharacteristic
621C2#6136911yescharacteristic
622C2#6136911yescharacteristic
623C2#6258711yescharacteristic
624C2#6259311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36560C2semidirect
3. #33294C2semidirect
4. #31541C2semidirect
6. #30311C2semidirect
7. #29196C2semidirect
8. #29150C2semidirect
11. #24818C2semidirect
13. #24332C2semidirect
17. #21345C2semidirect
120. #497Q8semidirect

Maximal subgroups

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

subgroupindex
#365602
#365552
#332942
#315412
#315392
#303112
#291962
#291502
#291462
#285202
#248182
#248162
#243322
#243282
#222732
C4×#4972
#213452