Finite Group Databasebeta

← #6617011

Subgroups — #6617011

Order 512.

1592 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#36176C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#33096C212yesmaximal
4#32957C211yesmaximal, characteristic, complemented
5#27948C211yesmaximal, characteristic, complemented
6#27677C211yesmaximal, characteristic
7#27589C212yesmaximal, complemented
8#27489C214yesmaximal, complemented
9#27359C214yesmaximal, complemented
10#26089C212yesmaximal, complemented
11#24679C212yesmaximal, complemented
12#24503C212yesmaximal
13#23063C212yesmaximal, complemented
14#22805C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16C4×#448C211yesmaximal, characteristic
17#21361C212yesmaximal, complemented
18C22×C4×Q8C2211yescharacteristic
19C2×#477C2212yes—
20C2×#448C2211yescharacteristic
21C2×C4×SD16C2212yes—
22C2×#431C2212yes—
23#2857C2211yescharacteristic
24#2854C2212yes—
25#2571C2211yescharacteristic
26#2563C2211yescharacteristic
27#2532—22no—
28#2513C2212yes—
29#2508C2212yes—
30#2495—21no—
31#2488—22no—
32#2486—21no—
33#2369C2211yescharacteristic
34#2355—21no—
35#2355—21no—
36#2355—22no—
37#2355C2212yes—
38#2355C2214yes—
39#2355C2212yes—
40#2354C2211yescharacteristic
41#2352—21no—
42#2346—21no—
43C4×#121C2211yescharacteristic
44C2×#391C2212yes—
45C2×#387—22no—
46C2×#384C2211yescharacteristic
47#2283C2212yes—
48C2×#382—21no—
49C2×C4×C4⋊C4—21no—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2099C2212yes—
53#2097C2214yes—
54#2093C2212yes—
55#2084C2214yes—
56#2074C2214yes—
57#2066C2214yes—
58#2055C2214yes—
59#1990C2212yes—
60#1968C2214yes—
61#1949C2214yes—
62#1949C2214yes—
63#1947C2212yes—
64#1939C2212yes—
65#1933C2214yes—
66#1928C2214yes—
67#1927C2214yes—
68#1926C2214yes—
69#1925C2214yes—
70#1921C2214yes—
71#1920C2214yes—
72#1916C2214yes—
73#1880C2212yes—
74#1874C2212yes—
75#1874C2211yescharacteristic
76#1865C2212yes—
77#1863C2214yes—
78#1853C2212yes—
79#1853C2212yes—
80#1853C2211yescharacteristic
81#1852C2212yes—
82#1852C2212yes—
83#1844C2212yes—
84#1817C2212yes—
85#1803C2212yes—
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1787C2212yes—
89#1787C2211yescharacteristic
90C4×#107C2214yes—
91C4×#106C2214yes—
92C8×C4⋊C4C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×SD16C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#123—22no—
99C2×C22⋊Q8—21no—
100C2×#121C2311yescharacteristic
101C2×C4×Q8—28no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—21no—
107C2×C4×D4C2311yescharacteristic
108C2×#117—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C42—21no—
117C22×C42—22no—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120#477—28no—
121#448—24no—
122C4×SD16—28no—
123#431—28no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2312yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#102C2312yes—
137C2×#102C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2311yescharacteristic
140C2×C22⋊C8C2311yescharacteristic
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—22no—
144#400—22no—
145#398—22no—
146#397—22no—
147#397—22no—
148#393—22no—
149#391—21no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391C2314yes—
154#391C2314yes—
155#390C2312yes—
156#390C2312yes—
157#390C2311yescharacteristic
158#389—21no—
159#389—21no—
160#389—21no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2311yescharacteristic
165#388—22no—
166#388—22no—
167#388—22no—
168#388—21no—
169#388—22no—
170#388—22no—
171#387—22no—
172#387—24no—
173#387—24no—
174#387—24no—
175#387—24no—
176#386—22no—
177#386—21no—
178#384—21no—
179#384—21no—
180#384C2312yes—
181#384C2311yescharacteristic
182#384C2312yes—
183#383—22no—
184#382—21no—
185#382—22no—
186#382—22no—
187#382—24no—
188#382—22no—
189#382C2312yes—
190#382C2312yes—
191#381—22no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380—21no—
196#380C2312yes—
197#380C2311yescharacteristic
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—24no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2311yescharacteristic
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—
217#376—22no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95D412yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—24no—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8C2412yes—
240C22×D4—21no—
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×SD16—24no—
252C2×SD16—24no—
253C2×SD16—42no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258#123—84no—
259C22⋊Q8—82no—
260#121—22no—
261#121—22no—
262C4×Q8—28no—
263C4×Q8—28no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8#4118yes—
268C4×D4—24no—
269C4×D4—44no—
270#117—44no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2414yes—
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C42—28no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—84no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2412yes—
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336#107—28no—
337#107—44no—
338#106—24no—
339#106—44no—
340#106—44no—
341C4⋊C8—28no—
342C4⋊C8—24no—
343#103—24no—
344#103—28no—
345#103—24no—
346#103—44no—
347#103—42no—
348#102—24no—
349#102—44no—
350C22⋊C8—24no—
351C22⋊C8—22no—
352C22⋊C8#4114yes—
353C4×C8—28no—
354C4×C8—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—21no—
379C24C22×D411yescharacteristic
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×Q8C2×#4114yes—
394C2×D4—22no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—42no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4#13514yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C2×#4112yes—
445C22×C42−1+412yes—
446C22×C4C2511yescharacteristic
447SD16—48no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8C2×#4114yes—
457C4⋊C4—24no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4#13714yes—
487C4⋊C4C2×#4114yes—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—82no—
495C22⋊C4—81no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C42—28no—
499C42—28no—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—42no—
510C42—44no—
511C42—82no—
512C42—84no—
513C42—84no—
514C42C2×#4114yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23#54712yes—
527C23C2×#13512yes—
528C23#57712yes—
529C23C23×D411yescharacteristic
530Q8—24no—
531Q8—24no—
532Q8—48no—
533Q8—42no—
534Q8—42no—
535Q8—162no—
536D4—42no—
537D4—42no—
538D4—84no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
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—48no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—88no—
592C2×C4—84no—
593C2×C4#52512yes—
594C2×C4#52512yes—
595C2×C4#54712yes—
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—24no—
605C8—44no—
606C8—42no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22#334014yes—
621C22C2×#52511yescharacteristic
622C22C2×#54711yescharacteristic
623C22#348711yescharacteristic
624C22C22×#13511yescharacteristic
625C22C22×#13711yescharacteristic
626C22C2×#57711yescharacteristic
627C22#358511yescharacteristic
628C4—24no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—41no—
637C4—44no—
638C4—81no—
639C4—81no—
640C4—82no—
641C4—81no—
642C4—84no—
643C4—82no—
644C4#310814yes—
645C4#347512yes—
646C4#349512yes—
647C2—24no—
648C2—161no—
649C2#3642711yescharacteristic
650C2C2×#310811yescharacteristic
651C2#6058811yescharacteristic
652C2C2×#334011yescharacteristic
653C2#6125411yescharacteristic
654C2#6188511yescharacteristic
655C2#6190611yescharacteristic
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×#2355C2semidirect
4. #32957C2semidirect
5. #27948C2semidirect
7. #27589C2semidirect
8. #27489C2semidirect
9. #27359C2semidirect
10. #26089C2semidirect
11. #24679C2semidirect
13. #23063C2semidirect
14. #22805C2semidirect
15. #22599C2semidirect
17. #21361C2semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×#23552
#330962
#329572
#279482
#276772
#275892
#274892
#273592
#260892
#246792
#245032
#230632
#228052
#225992
C4×#4482
#213612