Finite Group Databasebeta

← #6628095

Subgroups — #6628095

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33110C212yesmaximal
4#33109C211yesmaximal, characteristic, complemented
5#29330C212yesmaximal
6#29243C212yesmaximal, complemented
7#27127C214yesmaximal, complemented
8#26691C214yesmaximal, complemented
9#26509C212yesmaximal, complemented
10#25902C211yesmaximal, characteristic, complemented
11#25275C212yesmaximal, complemented
12#24298C211yesmaximal, characteristic
13#23214C212yesmaximal, complemented
14#23060C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16C4×#484C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8—21no—
20C2×#487C2212yes—
21C2×#484C2212yes—
22C2×#481C2211yescharacteristic
23C2×#474C2212yes—
24#2857C2211yescharacteristic
25#2850C2212yes—
26#2596—22no—
27#2571C2211yescharacteristic
28#2547C2211yescharacteristic
29#2514C2212yes—
30#2510—22no—
31#2494—21no—
32#2487—21no—
33#2448C2212yes—
34#2365—22no—
35#2365—21no—
36#2365—21no—
37#2365C2212yes—
38#2365C2214yes—
39#2365C2212yes—
40#2365C2211yescharacteristic
41#2354C2211yescharacteristic
42#2352—21no—
43#2347—21no—
44C4×C22⋊Q8C2212yes—
45C4×#121C2211yescharacteristic
46C2×#391—21no—
47C2×#389C2212yes—
48C2×#387—22no—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2101C2214yes—
53#2088C2214yes—
54#2068C2214yes—
55#2057C2212yes—
56#2042C2214yes—
57#2024C2214yes—
58#2023C2212yes—
59#1990C2212yes—
60#1988C2212yes—
61#1983C2214yes—
62#1981C2212yes—
63#1949C2214yes—
64#1947C2212yes—
65#1939C2212yes—
66#1934C2214yes—
67#1933C2214yes—
68#1928C2214yes—
69#1928C2214yes—
70#1927C2214yes—
71#1926C2212yes—
72#1925C2214yes—
73#1921C2214yes—
74#1919C2212yes—
75#1918C2214yes—
76#1916C2214yes—
77#1874C2211yescharacteristic
78#1874C2212yes—
79#1872C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1865C2214yes—
83#1853C2211yescharacteristic
84#1852C2212yes—
85#1824C2211yescharacteristic
86#1797C2212yes—
87#1797C2211yescharacteristic
88C4×#107C2214yes—
89C4×#106C2212yes—
90C8×C4⋊C4C2212yes—
91C4×#103C2214yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×#124C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×#121C2311yescharacteristic
105C2×C4×Q8—22no—
106C2×C4×Q8—24no—
107C2×C4×Q8—22no—
108C2×C4×Q8—48no—
109C2×C4×Q8C2312yes—
110C2×C4×D4—21no—
111C2×C4×D4C2311yescharacteristic
112C2×#117—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C42—22no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#487—28no—
125#484—28no—
126#481—24no—
127#474—28no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2312yes—
131C2×#106C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2312yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×C22⋊C8C2311yescharacteristic
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2311yescharacteristic
144C2×C4×C8C2312yes—
145C2×C4×C8C2314yes—
146#401—22no—
147#400—22no—
148#400—22no—
149#399—22no—
150#397—22no—
151#397—22no—
152#391—22no—
153#391—22no—
154#391—24no—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#390C2312yes—
159#390C2311yescharacteristic
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389C2311yescharacteristic
167#389C2312yes—
168#389C2314yes—
169#389C2314yes—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—21no—
174#388—22no—
175#388—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—24no—
180#387—24no—
181#387—24no—
182#387—24no—
183#386—22no—
184#386—22no—
185#386—21no—
186#385—22no—
187#384C2311yescharacteristic
188#383—22no—
189#382—21no—
190#382C2312yes—
191#380—21no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380C2311yescharacteristic
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—28no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—44no—
238C22×Q8—44no—
239C22×Q8C2412yes—
240C22×D4—21no—
241C22×D4C2411yescharacteristic
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2411yescharacteristic
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C4⋊Q8—28no—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8#4118yes—
260#124—44no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263#121—22no—
264#121—22no—
265C4×Q8—44no—
266C4×Q8—48no—
267C4×Q8—416no—
268C4×Q8—48no—
269C4×D4—24no—
270C4×D4—44no—
271#117—44no—
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—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—48no—
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4#4114yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42—48no—
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—28no—
345#103—28no—
346#103—44no—
347#102—28no—
348#102—24no—
349#102—42no—
350C22⋊C8—22no—
351C22⋊C8—24no—
352C22⋊C8—22no—
353C4×C8—24no—
354C4×C8—44no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—21no—
379C24C22×D411yescharacteristic
380C2×Q8—28no—
381C2×Q8—28no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—44no—
387C2×Q8—432no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—42no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4#13514yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4112yes—
442C22×C42−1+412yes—
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—42no—
473C4⋊C4—48no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4AGL(1,ℤ/8)14yes—
484C4⋊C4C2×#4114yes—
485C4⋊C4C2×#4118yes—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—81no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—24no—
500C42—24no—
501C42—42no—
502C42—42no—
503C42—44no—
504C42—48no—
505C42—42no—
506C42—48no—
507C42—44no—
508C42—44no—
509C42—82no—
510C42C2×#4114yes—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23#54812yes—
523C23C2×#13512yes—
524C23#57812yes—
525C23C23×D411yescharacteristic
526Q8—416no—
527Q8—416no—
528Q8—44no—
529Q8—44no—
530D4—42no—
531D4—42no—
532D4—84no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—416no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4#52512yes—
588C2×C4#52512yes—
589C2×C4#54812yes—
590C2×C4C2×#13512yes—
591C2×C4C2×#13512yes—
592C2×C4C2×AGL(1,ℤ/8)14yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C22×#4111yescharacteristic
595C2×C4C22×#4112yes—
596C2×C4C2×2−1+411yescharacteristic
597C2×C4#58512yes—
598C8—42no—
599C8—44no—
600C8—42no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22#336414yes—
615C22C2×#52511yescharacteristic
616C22C2×#54811yescharacteristic
617C22#348511yescharacteristic
618C22C22×#13511yescharacteristic
619C22C22×AGL(1,ℤ/8)11yescharacteristic
620C22C2×#57811yescharacteristic
621C22#358611yescharacteristic
622C4—24no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—48no—
630C4—44no—
631C4—41no—
632C4—81no—
633C4—82no—
634C4—81no—
635C4—81no—
636C4—82no—
637C4—82no—
638C4#310514yes—
639C4#348012yes—
640C4#351612yes—
641C2—24no—
642C2—161no—
643C2#3658211yescharacteristic
644C2C2×#310511yescharacteristic
645C2#6058311yescharacteristic
646C2C2×#336411yescharacteristic
647C2#6131711yescharacteristic
648C2#6170111yescharacteristic
649C2#6173711yescharacteristic
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×#2365C2semidirect
4. #33109C2semidirect
6. #29243C2semidirect
7. #27127C2semidirect
8. #26691C2semidirect
9. #26509C2semidirect
10. #25902C2semidirect
11. #25275C2semidirect
13. #23214C2semidirect
14. #23060C2semidirect
15. #22625C2semidirect
16. C4×#484C2semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#23652
#331102
#331092
#293302
#292432
#271272
#266912
#265092
#259022
#252752
#242982
#232142
#230602
#226252
C4×#4842
C4×#4812