Finite Group Databasebeta

← #6628094

Subgroups — #6628094

Order 512.

1592 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3#33111C212yesmaximal
4#33108C211yesmaximal, characteristic, complemented
5#29330C212yesmaximal
6#29244C212yesmaximal, complemented
7#27131C214yesmaximal, complemented
8#26695C214yesmaximal, complemented
9#26507C212yesmaximal, complemented
10#25902C211yesmaximal, characteristic, complemented
11#25273C212yesmaximal, complemented
12#24299C211yesmaximal, characteristic
13#23217C212yesmaximal, complemented
14#23061C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16C4×#483C212yesmaximal, complemented
17C4×#480C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#487C2212yes—
20C2×#483C2212yes—
21C2×#480C2211yescharacteristic
22C2×#476C2212yes—
23#2857C2211yescharacteristic
24#2850C2212yes—
25#2596—22no—
26#2571C2211yescharacteristic
27#2547C2211yescharacteristic
28#2514C2212yes—
29#2510—22no—
30#2494—21no—
31#2487—21no—
32#2448C2212yes—
33#2369—22no—
34#2369—21no—
35#2369—21no—
36#2369C2212yes—
37#2369C2214yes—
38#2369C2212yes—
39#2365C2211yescharacteristic
40#2354C2211yescharacteristic
41#2352—21no—
42#2347—21no—
43C4×C22⋊Q8C2212yes—
44C4×#121C2211yescharacteristic
45C2×#391—22no—
46C2×#389—21no—
47C2×#389C2212yes—
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2101C2214yes—
53#2086C2214yes—
54#2066C2214yes—
55#2058C2212yes—
56#2042C2214yes—
57#2024C2214yes—
58#2023C2212yes—
59#1990C2212yes—
60#1987C2212yes—
61#1983C2214yes—
62#1981C2212yes—
63#1949C2214yes—
64#1948C2212yes—
65#1939C2212yes—
66#1934C2214yes—
67#1933C2214yes—
68#1928C2214yes—
69#1927C2214yes—
70#1927C2214yes—
71#1926C2214yes—
72#1925C2212yes—
73#1920C2214yes—
74#1920C2214yes—
75#1918C2212yes—
76#1916C2214yes—
77#1874C2211yescharacteristic
78#1874C2212yes—
79#1872C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1863C2214yes—
83#1853C2212yes—
84#1852C2211yescharacteristic
85#1823C2211yescharacteristic
86#1797C2212yes—
87#1797C2211yescharacteristic
88C4×#107C2212yes—
89C4×#106C2214yes—
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×Q8C2312yes—
106C2×C4×D4—21no—
107C2×C4×D4C2311yescharacteristic
108C2×#117—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C42—21no—
117C22×C42—22no—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120#487—28no—
121#483—28no—
122#480—24no—
123#476—28no—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2312yes—
135C2×#102C2314yes—
136C2×#102C2312yes—
137C2×C22⋊C8C2311yescharacteristic
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2311yescharacteristic
140C2×C4×C8C2312yes—
141C2×C4×C8C2314yes—
142#401—22no—
143#400—22no—
144#400—22no—
145#399—22no—
146#397—22no—
147#397—22no—
148#391—24no—
149#391—24no—
150#391—24no—
151#391—24no—
152#391C2312yes—
153#391C2312yes—
154#391C2312yes—
155#390C2312yes—
156#390C2311yescharacteristic
157#389—21no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389—24no—
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#386—22no—
180#386—22no—
181#386—21no—
182#385—22no—
183#384C2311yescharacteristic
184#383—22no—
185#382—21no—
186#382C2312yes—
187#380—21no—
188#380—21no—
189#380—21no—
190#380—21no—
191#380—21no—
192#380C2311yescharacteristic
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—24no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
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⋊C4C2312yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—21no—
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×Q8C2412yes—
239C22×D4—21no—
240C22×D4C2411yescharacteristic
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2411yescharacteristic
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—28no—
254C4⋊Q8—28no—
255C4⋊Q8—28no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8#4118yes—
259#124—44no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262#121—22no—
263#121—22no—
264C4×Q8—44no—
265C4×D4—24no—
266C4×D4—44no—
267#117—44no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—84no—
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4#4114yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42—84no—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2412yes—
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107—24no—
334#107—42no—
335#106—28no—
336#106—28no—
337C4⋊C8—24no—
338C4⋊C8—44no—
339#103—28no—
340#103—28no—
341#103—28no—
342#103—44no—
343#102—28no—
344#102—24no—
345#102—42no—
346C22⋊C8—22no—
347C22⋊C8—24no—
348C22⋊C8—22no—
349C4×C8—24no—
350C4×C8—44no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95—21no—
358#95—22no—
359#95—22no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—21no—
375C24C22×D411yescharacteristic
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—22no—
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—82no—
383C2×Q8—82no—
384C2×Q8—82no—
385C2×Q8—82no—
386C2×Q8—816no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4#13514yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4112yes—
438C22×C42−1+412yes—
439C22×C4C2511yescharacteristic
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—28no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—81no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—82no—
479C4⋊C4#13714yes—
480C4⋊C4C2×#4114yes—
481C4⋊C4C2×#4118yes—
482C22⋊C4—44no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—82no—
489C22⋊C4—81no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—24no—
497C42—42no—
498C42—42no—
499C42—44no—
500C42—44no—
501C42—42no—
502C42—44no—
503C42—84no—
504C42—84no—
505C42—82no—
506C42C2×#4114yes—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23#54812yes—
519C23C2×#13512yes—
520C23#57712yes—
521C23C23×D411yescharacteristic
522Q8—44no—
523Q8—44no—
524Q8—88no—
525Q8—88no—
526D4—42no—
527D4—42no—
528D4—84no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—416no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4#52512yes—
584C2×C4#52512yes—
585C2×C4#54812yes—
586C2×C4C2×#13512yes—
587C2×C4C2×#13512yes—
588C2×C4C2×#13714yes—
589C2×C4C2×#13712yes—
590C2×C4C22×#4111yescharacteristic
591C2×C4C22×#4112yes—
592C2×C4C2×2−1+411yescharacteristic
593C2×C4#58512yes—
594C8—42no—
595C8—44no—
596C8—42no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22#336014yes—
611C22C2×#52511yescharacteristic
612C22C2×#54811yescharacteristic
613C22#348511yescharacteristic
614C22C22×#13511yescharacteristic
615C22C22×#13711yescharacteristic
616C22C2×#57711yescharacteristic
617C22#358511yescharacteristic
618C4—24no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—44no—
626C4—41no—
627C4—48no—
628C4—81no—
629C4—82no—
630C4—81no—
631C4—81no—
632C4—82no—
633C4—82no—
634C4#310814yes—
635C4#348012yes—
636C4#351612yes—
637C2—24no—
638C2—161no—
639C2#3658211yescharacteristic
640C2C2×#310811yescharacteristic
641C2#6057911yescharacteristic
642C2C2×#336011yescharacteristic
643C2#6131911yescharacteristic
644C2#6170011yescharacteristic
645C2#6173511yescharacteristic
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×#2369C2semidirect
4. #33108C2semidirect
6. #29244C2semidirect
7. #27131C2semidirect
8. #26695C2semidirect
9. #26507C2semidirect
10. #25902C2semidirect
11. #25273C2semidirect
13. #23217C2semidirect
14. #23061C2semidirect
15. #22625C2semidirect
16. C4×#483C2semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#23692
#331112
#331082
#293302
#292442
#271312
#266952
#265072
#259022
#252732
#242992
#232172
#230612
#226252
C4×#4832
C4×#4802