Finite Group Databasebeta

← #6851924

Subgroups — #6851924

Order 512.

1529 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#32143C214yesmaximal, complemented
5#30959C214yesmaximal, complemented
6#29973C212yesmaximal, complemented
7#28811C212yesmaximal, complemented
8#27948C211yesmaximal, characteristic, complemented
9#27814C212yesmaximal, complemented
10#27677C211yesmaximal, characteristic
11#26445C212yesmaximal, complemented
12#26045C212yesmaximal, complemented
13#25815C212yesmaximal, complemented
14#24305C212yesmaximal
15#22625C211yesmaximal, characteristic, complemented
16#22076C212yesmaximal
17C4×#481C211yesmaximal, characteristic
18C22×#125C2212yes—
19C2×#499C2214yes—
20C2×#481C2211yescharacteristic
21C2×#478C2214yes—
22#2857C2211yescharacteristic
23#2851C2212yes—
24#2589—22no—
25#2563C2211yescharacteristic
26#2552—22no—
27#2552—21no—
28#2552—21no—
29#2552C2212yes—
30#2552C2212yes—
31#2552C2214yes—
32#2552C2211yescharacteristic
33#2509C2212yes—
34#2497—21no—
35#2487—21no—
36#2454C2212yes—
37#2450—22no—
38C2×#398—22no—
39#2369C2211yescharacteristic
40D4×C4⋊C4—21no—
41D4×C4⋊C4C2211yescharacteristic
42#2345—21no—
43C4×#121C2211yescharacteristic
44C4×#121C2212yes—
45C2×#389—21no—
46C2×#389C2212yes—
47C2×#384—21no—
48C2×#384C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2101C2214yes—
52#2099C2212yes—
53#2093C2212yes—
54#2088C2214yes—
55#2086C2214yes—
56#2084C2214yes—
57#2059C2214yes—
58#2059C2214yes—
59#2059C2214yes—
60#2059C2214yes—
61#2025C2214yes—
62#1991C2214yes—
63#1990C2212yes—
64#1989C2214yes—
65#1987C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1941C2214yes—
69#1939C2212yes—
70#1934C2214yes—
71#1926C2212yes—
72#1921C2214yes—
73#1920C2214yes—
74#1919C2212yes—
75#1916C2214yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1864C2214yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2211yescharacteristic
82#1852C2212yes—
83#1852C2212yes—
84#1844C2212yes—
85#1824C2211yescharacteristic
86#1824C2212yes—
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#106C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#125—22no—
97C2×#125—22no—
98C2×#125—48no—
99C2×#125C2314yes—
100C2×#125C2314yes—
101C2×#125C2312yes—
102C2×C22⋊Q8—21no—
103C2×#121C2311yescharacteristic
104C2×C4×D4—21no—
105C2×C4×D4—22no—
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2312yes—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#499—48no—
123#481Q818yescomplemented
124#478—48no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2312yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×#102C2312yes—
134C2×#102C2314yes—
135C2×#102C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2311yescharacteristic
138C2×C22⋊C8C2311yescharacteristic
139C2×C4×C8C2312yes—
140C2×C4×C8C2314yes—
141#400—22no—
142#400—22no—
143#399—22no—
144#398—22no—
145#398—24no—
146#398—24no—
147#398—24no—
148#398—24no—
149#396—22no—
150#393—22no—
151#391—21no—
152#391—22no—
153#390C2312yes—
154#390C2311yescharacteristic
155#389—22no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389—24no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#389C2312yes—
169#389C2314yes—
170#389C2311yescharacteristic
171#388—21no—
172#388—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#386—21no—
177#386—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#384—21no—
183#384—22no—
184#384—22no—
185#384—24no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2312yes—
190#384C2311yescharacteristic
191#382C2312yes—
192#380—21no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380—22no—
197#380C2311yescharacteristic
198#380C2312yes—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—21no—
214C2×#95—21no—
215C2×#95—22no—
216C2×#95D412yes—
217#340C2314yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2314yes—
231#336C2312yes—
232C22×Q8—21no—
233C22×D4—21no—
234C22×D4C2411yescharacteristic
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2411yescharacteristic
245C22×C8C2412yes—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C4⋊Q8—24no—
249#125—28no—
250#125—28no—
251#125—44no—
252#125—44no—
253#125—44no—
254#125—48no—
255#125—48no—
256C22⋊Q8—82no—
257#121—22no—
258#121C2×Q814yes—
259C4×D4—24no—
260C4×D4—44no—
261C4×D4—44no—
262C4×D4—84no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—48no—
293C2×C4⋊C4—48no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4—84no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×Q814yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C42—24no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2412yes—
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—28no—
337#107—44no—
338#106—24no—
339#106—44no—
340#106—44no—
341#106C2×Q818yes—
342C4⋊C8—24no—
343C4⋊C8—44no—
344#102—24no—
345#102—44no—
346#102C2×Q818yes—
347C22⋊C8—24no—
348C22⋊C8—22no—
349C22⋊C8C2×Q814yes—
350C4×C8—24no—
351C4×C8—44no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—21no—
357#95—22no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371C24—21no—
372C24C22×D411yescharacteristic
373C2×Q8—42no—
374C2×Q8—81no—
375C2×Q8—81no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4C22×Q812yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4#13514yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×Q812yes—
430C22×C42−1+412yes—
431C22×C4C2511yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8C22×Q814yes—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—42no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C4⋊C4—82no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4—81no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4C22×Q814yes—
480C4⋊C4C22×Q812yes—
481C4⋊C4H(5,𝔽2)14yes—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—81no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—24no—
495C42—22no—
496C42—42no—
497C42—42no—
498C42—44no—
499C42—42no—
500C42—44no—
501C42—82no—
502C422−1+414yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23D4×Q812yes—
515C23C2×#13512yes—
516C23#57712yes—
517C23C23×D411yescharacteristic
518Q8—162no—
519D4—24no—
520D4—48no—
521D4—42no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4—88no—
576C2×C4—84no—
577C2×C4—164no—
578C2×C4#53612yes—
579C2×C4#53612yes—
580C2×C4D4×Q812yes—
581C2×C4C2×#13512yes—
582C2×C4C2×#13512yes—
583C2×C4#57512yes—
584C2×C4C23×Q811yescharacteristic
585C2×C4C2×H(5,𝔽2)12yes—
586C2×C4C2×2−1+412yes—
587C2×C4C2×2−1+411yescharacteristic
588C8—24no—
589C8—44no—
590C8—42no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22#338114yes—
605C22C2×#53611yescharacteristic
606C22C2×D4×Q811yescharacteristic
607C22#348411yescharacteristic
608C22C22×#13511yescharacteristic
609C22C2×#57511yescharacteristic
610C22C2×#57711yescharacteristic
611C22#358511yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—44no—
620C4—42no—
621C4—44no—
622C4—41no—
623C4—81no—
624C4—84no—
625C4—82no—
626C4—81no—
627C4—82no—
628C4#323214yes—
629C4#350312yes—
630C4#355212yes—
631C2—24no—
632C2—82no—
633C2#3959211yescharacteristic
634C2C2×#323211yescharacteristic
635C2#6087311yescharacteristic
636C2C2×#338111yescharacteristic
637C2#6136711yescharacteristic
638C2#6156611yescharacteristic
639C2#6160411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2552C2semidirect
3. #33149C2semidirect
4. #32143C2semidirect
5. #30959C2semidirect
6. #29973C2semidirect
7. #28811C2semidirect
8. #27948C2semidirect
9. #27814C2semidirect
11. #26445C2semidirect
12. #26045C2semidirect
13. #25815C2semidirect
15. #22625C2semidirect
123. #481Q8semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×#25522
#331492
#321432
#309592
#299732
#288112
#279482
#278142
#276772
#264452
#260452
#258152
#243052
#226252
#220762
C4×#4812