Finite Group Databasebeta

← #6993550

Subgroups — #6993550

Order 512.

1473 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36287C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#32821C212yesmaximal, complemented
4#32284C214yesmaximal
5#31936C214yesmaximal
6#30304C212yesmaximal, complemented
7#29984C212yesmaximal, complemented
8#29691C211yesmaximal, characteristic
9#29553C212yesmaximal, complemented
10#28531C212yesmaximal, complemented
11#27729C212yesmaximal, complemented
12#25813C211yesmaximal, characteristic
13#25455C212yesmaximal, complemented
14#24817C212yesmaximal, complemented
15#24302C211yesmaximal, characteristic
16#23571C211yesmaximal, characteristic
17#22632C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#500C2212yes—
21C2×#477C2212yes—
22C2×#477C2212yes—
23C2×C4×Q16C2212yes—
24#2868C2212yes—
25#2862C2212yes—
26#2859C2212yes—
27#2857C2211yescharacteristic
28#2660—22no—
29#2612—22no—
30#2567C2212yes—
31#2567C2212yes—
32#2567C2211yescharacteristic
33#2556—22no—
34#2552C2211yescharacteristic
35#2546—21no—
36#2546—21no—
37#2506—21no—
38#2502C2214yes—
39#2497C2212yes—
40#2441—21no—
41#2434—22no—
42#2423C2212yes—
43#2370—21no—
44#2369C2211yescharacteristic
45Q8×C4⋊C4C2211yescharacteristic
46#2346—21no—
47#2317—21no—
48#2317—21no—
49C4×C22⋊Q8C2211yescharacteristic
50#2295C2212yes—
51C4×#117C2211yescharacteristic
52#2100C2214yes—
53#2095C2214yes—
54#2095C2214yes—
55#2083C2214yes—
56#2083C2214yes—
57#2072C2214yes—
58#2068C2214yes—
59#2068C2214yes—
60#2057C2211yescharacteristic
61#2057C2212yes—
62#2057C2212yes—
63#2057C2212yes—
64#2057C2212yes—
65#2043C2211yescharacteristic
66#2043C2212yes—
67#2036C2212yes—
68#2036C2214yes—
69#2024C2211yescharacteristic
70#2024C2212yes—
71#2016C2211yescharacteristic
72#2001C2212yes—
73#1990C2212yes—
74#1987C2212yes—
75#1969C2214yes—
76#1947C2214yes—
77#1939C2212yes—
78#1933C2214yes—
79#1928C2214yes—
80#1923C2214yes—
81#1878C2214yes—
82#1872C2212yes—
83#1852C2212yes—
84#1852C2214yes—
85#1845C2214yes—
86#1834C2214yes—
87#1831C2212yes—
88#1803C2214yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C23×Q8—21no—
96C22×Q16C2312yes—
97#533—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125—21no—
109C2×#125—21no—
110C2×#125—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2311yescharacteristic
114C2×C22⋊Q8C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×#117—21no—
120C2×#117D411yescharacteristic
121C2×#117D411yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125#500—44no—
126#477—44no—
127#477—44no—
128C4×Q16—44no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C4×C8C2314yes—
146C2×C4×C8C2312yes—
147#400—22no—
148#399—22no—
149#398—22no—
150#398—22no—
151#398—24no—
152#398—24no—
153#396—22no—
154#395—22no—
155#395—24no—
156#395—24no—
157#391—21no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391C2314yes—
162#391C2314yes—
163#391C2312yes—
164#391C2311yescharacteristic
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389C2312yes—
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#387—21no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#385—21no—
190#385—24no—
191#385—24no—
192#384—21no—
193#384—22no—
194#384—22no—
195#384C2314yes—
196#384C2314yes—
197#384C2312yes—
198#384C2311yescharacteristic
199#384C2312yes—
200#382—22no—
201#382—22no—
202#382—22no—
203#380—21no—
204#380C2312yes—
205#380C2311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376—22no—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—44no—
239C22×Q8—82no—
240C22×Q8—82no—
241C22×Q8C2412yes—
242C22×Q8C2411yescharacteristic
243C23×C4—22no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×Q16—24no—
248C2×Q16—24no—
249C2×Q16—42no—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—44no—
257C4⋊Q8—42no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—84no—
262C4⋊Q8—84no—
263#125—44no—
264#125—44no—
265#125—84no—
266C22⋊Q8—22no—
267C22⋊Q8—41no—
268C22⋊Q8—42no—
269C22⋊Q8—42no—
270C22⋊Q8—44no—
271C22⋊Q8—82no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—42no—
277C4×Q8—44no—
278#117—24no—
279#117—44no—
280#117Q1618yescomplemented
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343#107—42no—
344#107—44no—
345#107—44no—
346#107—44no—
347#107—44no—
348C4⋊C8—24no—
349C4⋊C8—44no—
350#103—24no—
351#103—44no—
352#103—42no—
353#103—42no—
354#103—42no—
355#103—44no—
356#103—44no—
357C22⋊C8—24no—
358C22⋊C8—42no—
359C4×C8—28no—
360C4×C8—24no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24C22×D411yescharacteristic
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×Q8—816no—
405C2×Q82−1+412yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4—44no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4C2×Q1614yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2511yescharacteristic
438Q16—48no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C82−1+414yes—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—82no—
478C4⋊C4—81no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4C2×Q1614yes—
488C4⋊C4C2×Q1614yes—
489C4⋊C4#13714yes—
490C22⋊C4—24no—
491C22⋊C4—42no—
492C22⋊C4—44no—
493C22⋊C4—81no—
494C22⋊C4—82no—
495C22⋊C4C2×Q1618yes—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—24no—
500C42—22no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—42no—
505C42—44no—
506C42—44no—
507C42—42no—
508C42—42no—
509C42—82no—
510C42C2×Q1618yes—
511C42H(5,𝔽2)14yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23C22×Q1612yes—
519C23C23×D411yescharacteristic
520Q8—24no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—84no—
527Q8—88no—
528Q8—88no—
529Q8—162no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—88no—
580C2×C4#53512yes—
581C2×C4#53512yes—
582C2×C4C22×Q1612yes—
583C2×C4C22×Q1614yes—
584C2×C4C22×Q1614yes—
585C2×C4C22×Q1612yes—
586C2×C4C2×#13714yes—
587C2×C4C2×#13712yes—
588C2×C4#57612yes—
589C2×C4C2×H(5,𝔽2)12yes—
590C2×C4C2×2−1+411yescharacteristic
591C2×C4C2×2−1+412yes—
592C2×C4#58511yescharacteristic
593C8—24no—
594C8—42no—
595C8—44no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22C2×#53511yescharacteristic
604C22#348911yescharacteristic
605C22#349911yescharacteristic
606C22C23×Q1611yescharacteristic
607C22C22×#13711yescharacteristic
608C22C2×#57611yescharacteristic
609C22#358511yescharacteristic
610C4—24no—
611C4—24no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—24no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—81no—
620C4—81no—
621C4—84no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—81no—
626C4—82no—
627C4#320914yes—
628C4#353812yes—
629C4#354712yes—
630C2—42no—
631C2#4299411yescharacteristic
632C2C2×#320911yescharacteristic
633C2#6081111yescharacteristic
634C2#6210011yescharacteristic
635C2#6214911yescharacteristic
636C2#6247211yescharacteristic
637C2#6250411yescharacteristic
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. #32821C2semidirect
6. #30304C2semidirect
7. #29984C2semidirect
9. #29553C2semidirect
10. #28531C2semidirect
11. #27729C2semidirect
13. #25455C2semidirect
14. #24817C2semidirect
280. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#362872
#352812
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