Finite Group Databasebeta

← #6628103

Subgroups — #6628103

Order 512.

1635 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36587C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#33111C212yesmaximal, complemented
4#33106C211yesmaximal, characteristic
5#29330C212yesmaximal, complemented
6#29238C212yesmaximal, complemented
7#27133C214yesmaximal, complemented
8#26693C214yesmaximal, complemented
9#26511C212yesmaximal, complemented
10#25903C211yesmaximal, characteristic
11#25277C212yesmaximal, complemented
12#24302C211yesmaximal, characteristic, complemented
13#23215C212yesmaximal, complemented
14#23063C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16C4×#483C211yesmaximal, characteristic, complemented
17C4×#480C212yesmaximal, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#486C2212yes—
20C2×#483C2211yescharacteristic
21C2×#480C2212yes—
22C2×#477C2212yes—
23#2856C2211yescharacteristic
24#2850C2212yes—
25#2597—22no—
26#2571C2211yescharacteristic
27#2548C2211yescharacteristic
28#2513—21no—
29#2513C2212yes—
30#2504—21no—
31#2447C2212yes—
32#2444—22no—
33#2369—22no—
34#2369—21no—
35#2369—21no—
36#2369C2212yes—
37#2369C2214yes—
38#2369C2212yes—
39#2366C2211yescharacteristic
40#2361—21no—
41#2357—21no—
42#2355C2211yescharacteristic
43C4×C22⋊Q8C2211yescharacteristic
44C4×#121C2212yes—
45C2×#391—22no—
46C2×#389—21no—
47C2×#389C2212yes—
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2101C2214yes—
53#2088C2214yes—
54#2067C2214yes—
55#2057C2212yes—
56#2042C2214yes—
57#2024C2212yes—
58#2023C2214yes—
59#1990C2212yes—
60#1987C2212yes—
61#1983C2214yes—
62#1982C2212yes—
63#1949C2214yes—
64#1947C2212yes—
65#1939C2212yes—
66#1934C2214yes—
67#1933C2214yes—
68#1928C2214yes—
69#1927C2212yes—
70#1926C2214yes—
71#1925C2214yes—
72#1925C2214yes—
73#1921C2214yes—
74#1920C2212yes—
75#1918C2214yes—
76#1916C2214yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1872C2211yescharacteristic
81#1872C2212yes—
82#1864C2214yes—
83#1853C2211yescharacteristic
84#1852C2212yes—
85#1823C2211yescharacteristic
86#1797C2212yes—
87#1797C2211yescharacteristic
88C4×#107C2214yes—
89C4×#106C2212yes—
90C8×C4⋊C4C2212yes—
91C4×#103C2212yes—
92C4×#102C2214yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#124C2312yes—
103C2×C22⋊Q8C2311yescharacteristic
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C42—21no—
119C22×C42—22no—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#486—28no—
123#483—24no—
124#480—28no—
125#477—28no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2312yes—
129C2×#106C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2311yescharacteristic
141C2×C22⋊C8C2311yescharacteristic
142C2×C4×C8C2312yes—
143C2×C4×C8C2314yes—
144#400—22no—
145#397—22no—
146#397—22no—
147#397—22no—
148#397—22no—
149#394—22no—
150#391—24no—
151#391—24no—
152#391—24no—
153#391—24no—
154#391C2312yes—
155#391C2311yescharacteristic
156#390—21no—
157#390—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390C2312yes—
162#390C2312yes—
163#390C2312yes—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—24no—
168#389—22no—
169#389—22no—
170#389C2311yescharacteristic
171#389C2312yes—
172#389C2314yes—
173#389C2314yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—21no—
180#387—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#385—21no—
186#385—22no—
187#384C2311yescharacteristic
188#382—21no—
189#382—21no—
190#382C2312yes—
191#381—22no—
192#380C2311yescharacteristic
193C4×C4⋊C4—21no—
194C4×C4⋊C4—24no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C2×#95—22no—
217C2×#95—22no—
218C2×#95—21no—
219C2×#95D412yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—28no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—82no—
237C22×Q8—82no—
238C22×Q8C2411yescharacteristic
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C23×C4—22no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—24no—
259C4⋊Q8#4118yes—
260#127—24no—
261#124—44no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264#121—24no—
265#121—24no—
266#121—44no—
267#121—82no—
268C4×Q8—24no—
269C4×D4—44no—
270C4×D4—44no—
271C4×D4—84no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—84no—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4#4114yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C42—24no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42—84no—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—28no—
338#107—28no—
339#106—24no—
340#106—42no—
341C4⋊C8—24no—
342C4⋊C8—44no—
343#103—28no—
344#103—24no—
345#103—42no—
346#102—28no—
347#102—28no—
348#102—28no—
349#102—44no—
350C22⋊C8—24no—
351C22⋊C8—22no—
352C22⋊C8—22no—
353C4×C8—24no—
354C4×C8—44no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—22no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24—22no—
377C24C22×D411yescharacteristic
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—82no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8—82no—
388C2×Q8—816no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—81no—
401C2×D4—81no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4#13514yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C2×#4112yes—
446C22×C42−1+412yes—
447C22×C4C2511yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—84no—
482C4⋊C4—84no—
483C4⋊C4—82no—
484C4⋊C4—81no—
485C4⋊C4#13714yes—
486C4⋊C4C2×#4114yes—
487C4⋊C4C2×#4118yes—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—24no—
504C42—24no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42—44no—
509C42—44no—
510C42—42no—
511C42—84no—
512C42—84no—
513C42—82no—
514C42C2×#4114yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23#54812yes—
527C23C2×#13512yes—
528C23#57612yes—
529C23C23×D411yescharacteristic
530Q8—42no—
531Q8—42no—
532Q8—88no—
533Q8—88no—
534D4—44no—
535D4—44no—
536D4—88no—
537D4—84no—
538D4—162no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—48no—
565C2×C4—44no—
566C2×C4—416no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—164no—
593C2×C4#52512yes—
594C2×C4#52512yes—
595C2×C4#54812yes—
596C2×C4C2×#13512yes—
597C2×C4C2×#13512yes—
598C2×C4C2×#13714yes—
599C2×C4C2×#13712yes—
600C2×C4C22×#4111yescharacteristic
601C2×C4C22×#4112yes—
602C2×C4C2×2−1+411yescharacteristic
603C2×C4#58512yes—
604C8—42no—
605C8—44no—
606C8—42no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22#336114yes—
621C22C2×#52511yescharacteristic
622C22C2×#54811yescharacteristic
623C22#348511yescharacteristic
624C22C22×#13511yescharacteristic
625C22C22×#13711yescharacteristic
626C22C2×#57611yescharacteristic
627C22#358511yescharacteristic
628C4—24no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—41no—
637C4—48no—
638C4—82no—
639C4—81no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4#310814yes—
645C4#348012yes—
646C4#351612yes—
647C2—24no—
648C2—162no—
649C2#3658211yescharacteristic
650C2C2×#310811yescharacteristic
651C2#6057811yescharacteristic
652C2C2×#336111yescharacteristic
653C2#6131911yescharacteristic
654C2#6170011yescharacteristic
655C2#6172611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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×#2369C2semidirect
3. #33111C2semidirect
5. #29330C2semidirect
6. #29238C2semidirect
7. #27133C2semidirect
8. #26693C2semidirect
9. #26511C2semidirect
11. #25277C2semidirect
12. #24302C2semidirect
13. #23215C2semidirect
14. #23063C2semidirect
15. #22625C2semidirect
16. C4×#483C2semidirect
17. C4×#480C2semidirect

Maximal subgroups

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

subgroupindex
#365872
C2×#23692
#331112
#331062
#293302
#292382
#271332
#266932
#265112
#259032
#252772
#243022
#232152
#230632
#226252
C4×#4832
C4×#4802