Finite Group Databasebeta

← #6565049

Subgroups — #6565049

Order 512.

1499 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34321C211yesmaximal, characteristic
3#30106C212yesmaximal, complemented
4#29981C211yesmaximal, characteristic, complemented
5#29338C212yesmaximal, complemented
6#29218C211yesmaximal, characteristic, complemented
7#28206C211yesmaximal, characteristic, complemented
8#27612C212yesmaximal, complemented
9#27356C214yesmaximal, complemented
10#26746C214yesmaximal, complemented
11#24972C212yesmaximal, complemented
12#24121C212yesmaximal
13#23760C212yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23069C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22762C212yesmaximal, complemented
18C2×#555C2211yescharacteristic
19C2×#477C2212yes—
20C2×C4×SD16C2212yes—
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2571—21no—
24#2563—21no—
25#2538C2212yes—
26#2497—22no—
27#2492C2214yes—
28#2439—22no—
29#2369—21no—
30#2369C2211yescharacteristic
31#2367—22no—
32#2363C2212yes—
33#2361—21no—
34#2355C2212yes—
35#2354—21no—
36#2354C2211yescharacteristic
37D4×C4⋊C4C2211yescharacteristic
38#2347—21no—
39#2347C2212yes—
40#2337—22no—
41#2324—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#121C2212yes—
45#2296C2212yes—
46#2293C2211yescharacteristic
47#2285C2212yes—
48#2274—21no—
49C42×D4C2211yescharacteristic
50#2100C2212yes—
51#2100C2212yes—
52#2100C2212yes—
53#2094C2212yes—
54#2084C2211yescharacteristic
55#2083C2212yes—
56#2077C2214yes—
57#2071C2211yescharacteristic
58#2066C2212yes—
59#2058C2212yes—
60#2055C2212yes—
61#2043C2212yes—
62#2043C2211yescharacteristic
63#2035C2214yes—
64#2033C2214yes—
65#2023C2212yes—
66#2013C2211yescharacteristic
67#2000C2214yes—
68#1990C2212yes—
69#1987C2214yes—
70#1954C2212yes—
71#1948C2212yes—
72#1935C2214yes—
73#1935C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1924C2214yes—
77#1920C2214yes—
78#1918C2214yes—
79#1877C2214yes—
80#1873C2214yes—
81#1873C2214yes—
82#1854C2214yes—
83#1834C2212yes—
84#1831C2211yescharacteristic
85#1825C2214yes—
86#1817C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1762C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C22×SD16C2312yes—
96#555—24no—
97#555—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#124—22no—
104C2×C22⋊Q8C2314yes—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8—21no—
108C2×C4×Q8C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2312yes—
114C2×C4×D4C2311yescharacteristic
115C2×#117—22no—
116C2×#117—21no—
117C2×#117—21no—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C22×C4⋊C4—21no—
121C22×C42C2312yes—
122#477—28no—
123C4×SD16—28no—
124C2×#107C2314yes—
125C2×#106C2312yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#102C2312yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2314yes—
139C2×C4×C8C2312yes—
140#400—24no—
141#396—24no—
142#393—24no—
143#391—21no—
144#391—22no—
145#391—24no—
146#391C2312yes—
147#391C2314yes—
148#390—22no—
149#390—21no—
150#390—22no—
151#390C2311yescharacteristic
152#390C2311yescharacteristic
153#389—22no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389C2312yes—
161#389C2311yescharacteristic
162#389C2311yescharacteristic
163#389C2312yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—24no—
169#388—21no—
170#388—22no—
171#388—21no—
172#388—22no—
173#387—24no—
174#386—22no—
175#386—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#384—21no—
180#384—21no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2312yes—
185#383—22no—
186#382—22no—
187#382—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380—21no—
192#380C2312yes—
193#380C2314yes—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C43C2311yescharacteristic
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8C2412yes—
235C22×D4—22no—
236C22×D4C2×D411yescharacteristic
237C22×D4C2411yescharacteristic
238C23×C4—21no—
239C23×C4—22no—
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2412yes—
243C2×SD16—24no—
244C2×SD16—42no—
245C2×SD16—42no—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C22×C8C2412yes—
249C4⋊Q8—22no—
250C4⋊Q8—44no—
251C4⋊Q8—42no—
252C4⋊Q8—44no—
253C4⋊Q8—82no—
254C4⋊Q8#4114yes—
255#124—216no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258#121—28no—
259#121—24no—
260#121—24no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×Q8—82no—
264C4×D4—24no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—42no—
269C4×D4—44no—
270C4×D4—84no—
271C4×D4#4114yes—
272#117—28no—
273#117—24no—
274#117—24no—
275#117—44no—
276#117—44no—
277#117—84no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—44no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—28no—
342#106—24no—
343#106—44no—
344C4⋊C8—24no—
345C4⋊C8—42no—
346#103—24no—
347#103—28no—
348#103—42no—
349#103—42no—
350#103—44no—
351#102—24no—
352#102—44no—
353C22⋊C8—44no—
354C22⋊C8—44no—
355#97—44no—
356C4×C8—24no—
357#95—22no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—24no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—21no—
376C24C22×D412yes—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—44no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×Q8—82no—
388C2×Q8—82no—
389C2×D4—28no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—82no—
397C2×D4—82no—
398C2×D4C2×#4112yes—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C2511yescharacteristic
433SD16—48no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—82no—
470C4⋊C4—81no—
471C4⋊C4—84no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4C2×#4112yes—
475C4⋊C4C2×#4114yes—
476C22⋊C4—28no—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—81no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—84no—
489C22⋊C4—82no—
490C22⋊C4—84no—
491C42—28no—
492C42—22no—
493C42—22no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—42no—
498C42—42no—
499C42—41no—
500C42—44no—
501C42—42no—
502C42—82no—
503C42—84no—
504C42—82no—
505C42—82no—
506C42#13714yes—
507C42C2×#4112yes—
508C42C2×#4112yes—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C23×D411yescharacteristic
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—84no—
529Q8—164no—
530D4—24no—
531D4—44no—
532D4—42no—
533D4—88no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
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—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—84no—
583C2×C4—164no—
584C2×C4#52512yes—
585C2×C4#52512yes—
586C2×C4C2×#13712yes—
587C2×C4C2×#13712yes—
588C2×C4C2×#13712yes—
589C2×C4#57512yes—
590C2×C4#57712yes—
591C2×C4C22×#4111yescharacteristic
592C2×C4C22×#4111yescharacteristic
593C2×C4#58512yes—
594C2×C4#58512yes—
595C8—42no—
596C8—42no—
597C8—44no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22C2×#52511yescharacteristic
613C22#348511yescharacteristic
614C22#350011yescharacteristic
615C22C22×#13711yescharacteristic
616C22C2×#57511yescharacteristic
617C22C2×#57711yescharacteristic
618C22#358511yescharacteristic
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—42no—
628C4—44no—
629C4—81no—
630C4—82no—
631C4—81no—
632C4—84no—
633C4—82no—
634C4—82no—
635C4#311614yes—
636C4#349312yes—
637C4#352412yes—
638C2—82no—
639C2—82no—
640C2#3601111yescharacteristic
641C2C2×#311611yescharacteristic
642C2#6059711yescharacteristic
643C2#6171911yescharacteristic
644C2#6173911yescharacteristic
645C2#6254011yescharacteristic
646C2#6258011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #30106C2semidirect
4. #29981C2semidirect
5. #29338C2semidirect
6. #29218C2semidirect
7. #28206C2semidirect
8. #27612C2semidirect
9. #27356C2semidirect
10. #26746C2semidirect
11. #24972C2semidirect
13. #23760C2semidirect
14. #23677C2semidirect
15. #23069C2semidirect
16. #22839C2semidirect
17. #22762C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#343212
#301062
#299812
#293382
#292182
#282062
#276122
#273562
#267462
#249722
#241212
#237602
#236772
#230692
#228392
#227622