Finite Group Databasebeta

← #6796040

Subgroups — #6796040

Order 512.

1429 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34943C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#32418C214yesmaximal, complemented
5#31618C214yesmaximal, complemented
6#29942C212yesmaximal, complemented
7#29290C211yesmaximal, characteristic, complemented
8#28711C212yesmaximal, complemented
9#28689C211yesmaximal, characteristic, complemented
10#27624C212yesmaximal, complemented
11#26511C212yesmaximal, complemented
12#25450C212yesmaximal, complemented
13#25346C212yesmaximal
14#24843C212yesmaximal, complemented
15#22650C211yesmaximal, characteristic, complemented
16#21963C212yesmaximal, complemented
17C4×#476C211yesmaximal, characteristic, complemented
18C2×#500C2212yes—
19C2×#476C2211yescharacteristic
20C2×#476C2212yes—
21C2×#436C2212yes—
22C2×#423C2212yes—
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2572—22no—
26#2571—21no—
27#2571C2211yescharacteristic
28#2563C2211yescharacteristic
29#2563C2212yes—
30#2563C2211yescharacteristic
31#2538—21no—
32#2537C2211yescharacteristic
33#2497C2214yes—
34#2495—21no—
35#2495C2212yes—
36#2492—22no—
37#2439—22no—
38#2420—21no—
39#2420C2212yes—
40#2398—22no—
41#2369C2212yes—
42#2355—21no—
43#2354—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#121—21no—
47C4×#121C2212yes—
48#2285C2212yes—
49#2283—21no—
50#2283C2212yes—
51C42×D4C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2101C2214yes—
54#2088C2214yes—
55#2087C2214yes—
56#2085C2214yes—
57#2083C2214yes—
58#2077C2214yes—
59#2075C2214yes—
60#2075C2214yes—
61#2067C2214yes—
62#2033C2214yes—
63#1990C2212yes—
64#1987C2212yes—
65#1987C2212yes—
66#1987C2212yes—
67#1987C2211yescharacteristic
68#1949C2214yes—
69#1949C2214yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1947C2212yes—
77#1939C2212yes—
78#1925C2214yes—
79#1920C2212yes—
80#1918C2214yes—
81#1895C2214yes—
82#1864C2212yes—
83#1851C2214yes—
84#1842C2211yescharacteristic
85#1809C2214yes—
86#1809C2214yes—
87#1809C2214yes—
88#1802C2212yes—
89#1802C2211yescharacteristic
90#1802C2212yes—
91#1797C2212yes—
92C4×#106C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×M16C2314yes—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C22⋊Q8—22no—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×D4—22no—
107C2×C4×D4D411yescharacteristic
108C2×C4×D4D411yescharacteristic
109C2×C4×D4C2311yescharacteristic
110C2×C4×D4C2311yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×#117—22no—
114C22×C42—21no—
115C22×C42C2312yes—
116#500—44no—
117#476—24no—
118#476—44no—
119#436—44no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106C2312yes—
124C2×#106C2312yes—
125C2×#106C2314yes—
126#423—28no—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2314yes—
137C2×C4×C8C2312yes—
138#400—24no—
139#400—22no—
140#400—22no—
141#397—22no—
142#397—22no—
143#396—24no—
144#394—22no—
145#394—22no—
146#393—22no—
147#393—24no—
148#393—22no—
149#391—21no—
150#391—21no—
151#391—22no—
152#391—24no—
153#390—22no—
154#390—21no—
155#390C2312yes—
156#389—22no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2314yes—
165#388—22no—
166#388—24no—
167#388—22no—
168#387—24no—
169#386—22no—
170#386—22no—
171#384—21no—
172#384—21no—
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2312yes—
176#384C2312yes—
177#383—22no—
178#383—22no—
179#382—22no—
180#382—21no—
181#382—21no—
182#382—22no—
183#382C2312yes—
184#382C2312yes—
185#381—22no—
186#381—22no—
187#380—22no—
188#380—22no—
189#380—21no—
190#380—21no—
191#380—21no—
192#380C2312yes—
193#380C2312yes—
194#380C2314yes—
195#380C2314yes—
196#380C2312yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218#376—22no—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×D4—22no—
237C22×D4C2×D411yescharacteristic
238C22×D4C2411yescharacteristic
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2412yes—
245C2×M16—28no—
246C2×M16—44no—
247C2×M16—44no—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C22×C8C2412yes—
251C4⋊Q8—22no—
252C4⋊Q8—24no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—82no—
258C4⋊Q8—82no—
259C22⋊Q8—84no—
260#121—24no—
261#121—42no—
262#121—44no—
263C4×Q8—44no—
264C4×Q8—82no—
265C4×D4—22no—
266C4×D4—24no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—42no—
274C4×D4—41no—
275C4×D4—84no—
276#117—84no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
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—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—24no—
325C2×C42—44no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—44no—
337#107—44no—
338#107—44no—
339#106—24no—
340#106—42no—
341#106—44no—
342C4⋊C8—28no—
343C4⋊C8—24no—
344C4⋊C8—24no—
345C4⋊C8—22no—
346C4⋊C8—42no—
347C4⋊C8#4114yes—
348#102—24no—
349#102—24no—
350#102—44no—
351C22⋊C8—44no—
352C4×C8—24no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—21no—
376C24C22×D412yes—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—44no—
381C2×Q8—42no—
382C2×Q8—81no—
383C2×Q8—81no—
384C2×Q8—82no—
385C2×Q8—82no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—82no—
397C2×D42−1+412yes—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—81no—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2511yescharacteristic
437M16—88no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8C2×#4114yes—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—24no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—82no—
476C4⋊C4—81no—
477C4⋊C4—82no—
478C4⋊C4—84no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—84no—
493C22⋊C4—82no—
494C22⋊C4—84no—
495C42—24no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—21no—
502C42—21no—
503C42—41no—
504C42—42no—
505C42—42no—
506C42—42no—
507C42—44no—
508C42—44no—
509C42—84no—
510C42#13714yes—
511C42C2×#4112yes—
512C42H(5,𝔽2)12yes—
513C422−1+412yes—
514C422−1+412yes—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23C23×D411yescharacteristic
530Q8—84no—
531Q8—84no—
532Q8—164no—
533D4—22no—
534D4—22no—
535D4—44no—
536D4—41no—
537D4—41no—
538D4—88no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
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—21no—
556C2×C4—22no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4#53712yes—
591C2×C4#53712yes—
592C2×C4#54712yes—
593C2×C4#54712yes—
594C2×C4C2×#13712yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4#57512yes—
598C2×C4#57612yes—
599C2×C4C22×#4111yescharacteristic
600C2×C4C2×H(5,𝔽2)11yescharacteristic
601C2×C4C2×2−1+411yescharacteristic
602C2×C4C2×2−1+411yescharacteristic
603C2×C4C2×2−1+411yescharacteristic
604C2×C4#58511yescharacteristic
605C8—24no—
606C8—44no—
607C8—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22C2×#53711yescharacteristic
623C22C2×#54711yescharacteristic
624C22#348911yescharacteristic
625C22C22×#13711yescharacteristic
626C22C2×#57511yescharacteristic
627C22C2×#57611yescharacteristic
628C22#358511yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—41no—
636C4—81no—
637C4—82no—
638C4—81no—
639C4—82no—
640C4—82no—
641C4—81no—
642C4—82no—
643C4—84no—
644C4#325214yes—
645C4#335714yes—
646C4#351411yescharacteristic
647C4#352311yescharacteristic
648C4#355211yescharacteristic
649C4#355511yescharacteristic
650C2—82no—
651C2—82no—
652C2#3892311yescharacteristic
653C2C2×#325211yescharacteristic
654C2#6092211yescharacteristic
655C2C2×#335711yescharacteristic
656C2#6125611yescharacteristic
657C2#6210811yescharacteristic
658C2#6214311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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. #33197C2semidirect
4. #32418C2semidirect
5. #31618C2semidirect
6. #29942C2semidirect
7. #29290C2semidirect
8. #28711C2semidirect
9. #28689C2semidirect
10. #27624C2semidirect
11. #26511C2semidirect
12. #25450C2semidirect
14. #24843C2semidirect
15. #22650C2semidirect
16. #21963C2semidirect
17. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349432
#331972
#324182
#316182
#299422
#292902
#287112
#286892
#276242
#265112
#254502
#253462
#248432
#226502
#219632
C4×#4762