Finite Group Databasebeta

← #6528511

Subgroups — #6528511

Order 512.

1652 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35239C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic
3#29110C212yesmaximal
4#29103C212yesmaximal
5#27969C212yesmaximal, complemented
6#27964C212yesmaximal, complemented
7#27826C214yesmaximal, complemented
8#27821C212yesmaximal, complemented
9#27814C212yesmaximal, complemented
10#27740C214yesmaximal, complemented
11#27735C212yesmaximal, complemented
12#27728C212yesmaximal, complemented
13#21840C211yesmaximal, characteristic, complemented
14#21833C211yesmaximal, characteristic, complemented
15#20738C211yesmaximal, characteristic
16#20733C211yesmaximal, characteristic
17#20535C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2857C2212yes—
23#2851C2212yes—
24#2666—22no—
25#2571C2211yescharacteristic, complemented
26#2570C2212yescomplemented
27#2563—21no—
28#2563—21no—
29#2563—22no—
30#2563C2212yes—
31#2563C2212yes—
32#2563C2214yes—
33#2552C2211yescharacteristic, complemented
34#2552C2211yescharacteristic, complemented
35#2551C2211yescharacteristic, complemented
36#2550C2212yescomplemented
37#2442—21no—
38#2440—21no—
39#2438—22no—
40C2×#393—22no—
41#2331—21no—
42#2330—21no—
43C2×#390C2212yes—
44#2285C2211yescharacteristic
45#2283C2211yescharacteristic
46C2×#382C2212yes—
47C2×#380—21no—
48C2×#380—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yescomplemented
51#2101C2214yescomplemented
52#2101C2214yescomplemented
53#2101C2214yescomplemented
54#2073C2214yes—
55#2071C2214yes—
56#2025C2214yes—
57#2023C2214yes—
58#1993C2214yescomplemented
59#1993C2212yescomplemented
60#1993C2212yescomplemented
61#1991C2214yescomplemented
62#1990C2212yescomplemented
63#1990C2212yescomplemented
64#1988C2214yescomplemented
65#1987C2212yescomplemented
66#1987C2212yescomplemented
67#1936C2214yes—
68#1936C2214yes—
69#1934C2214yes—
70#1934C2214yes—
71#1836C2214yescomplemented
72#1835C2214yescomplemented
73#1824C2212yes—
74#1824C2211yescharacteristic
75#1824C2211yescharacteristic
76#1823C2212yes—
77#1823C2212yes—
78#1823C2212yes—
79#1823C2212yes—
80#1804C2212yes—
81#1804C2212yes—
82#1802C2212yes—
83#1802C2212yes—
84#1786C2212yes—
85#1786C2211yescharacteristic
86#1786C2211yescharacteristic
87#1771C2214yescomplemented
88#1770C2212yescomplemented
89C8×C4⋊C4C2212yescomplemented
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2212yes—
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×#125—21no—
107C2×#125C2312yes—
108C2×C22⋊Q8—22no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—24no—
112C2×C4×D4—24no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×#117—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#432—24no—
127#431—24no—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×#102C2314yes—
131C2×#102C2312yes—
132C2×#102C2312yes—
133C2×#102C2314yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×C22⋊C8C2311yescharacteristic
137C2×C22⋊C8C2311yescharacteristic
138C2×C22⋊C8C2312yes—
139C2×#97C2312yes—
140C2×#97C2314yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—22no—
144#400—22no—
145#398—22no—
146#398—22no—
147#398—22no—
148#398—22no—
149#395—22no—
150#395—22no—
151#393—24no—
152#393—24no—
153#393—24no—
154#393—24no—
155#391—22no—
156#391—22no—
157#390—22no—
158#390—22no—
159#390C2314yes—
160#390C2314yes—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#385—22no—
176#385—22no—
177#384—21no—
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2312yes—
181#383—21no—
182#382—21no—
183#382—22no—
184#382—22no—
185#382—21no—
186#382C2314yes—
187#382C2314yes—
188#382C2311yescharacteristic
189#382C2312yes—
190#382C2311yescharacteristic
191#381—21no—
192#380—22no—
193#380—21no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—24no—
199#380—24no—
200#380C2311yescharacteristic
201#380C2312yes—
202#380C2311yescharacteristic
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214#376—21no—
215C2×#95—22no—
216C2×#95—22no—
217C2×#95D412yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234C25—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—44no—
239C22×Q8—82no—
240C22×Q8—82no—
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—42no—
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—44no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2411yescharacteristic
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270#125—44no—
271#125—44no—
272C22⋊Q8—84no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—44no—
278C4×D4—44no—
279#117—82no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—84no—
329C2×C22⋊C4—84no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—44no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42#4114yes—
339C2×C42#4114yes—
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C4⋊C8—24no—
345C4⋊C8—24no—
346#102—28no—
347#102—24no—
348#102—24no—
349#102—42no—
350#102—44no—
351#102—42no—
352C22⋊C8—22no—
353C22⋊C8—22no—
354C22⋊C8—24no—
355#97—24no—
356#97—44no—
357C4×C8—24no—
358C4×C8—44no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—24no—
384C24—22no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24C22×D411yescharacteristic
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—816no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D42−1+414yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—816no—
446C22×C4#12414yes—
447C22×C4#12414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2511yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C42—24no—
512C42—24no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—82no—
518C42#12414yes—
519C42#12414yes—
520C42C2×#4112yes—
521C42C2×#4112yes—
522C42C2×#4112yes—
523C42C2×#4112yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—24no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23C2×#12412yes—
543C23#57612yes—
544C23#57712yes—
545C23C23×D411yescharacteristic
546Q8—88no—
547Q8—88no—
548Q8—164no—
549D4—24no—
550D4—24no—
551D4—48no—
552D4—42no—
553D4—42no—
554C2×C4—28no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—88no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—88no—
607C2×C4—88no—
608C2×C4C2×#12412yes—
609C2×C4C2×#12412yes—
610C2×C4C2×#12412yes—
611C2×C4C2×#12412yes—
612C2×C4C2×#12412yes—
613C2×C4#57512yes—
614C2×C4#57512yes—
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C2×2−1+412yes—
618C2×C4#58512yes—
619C8—42no—
620C8—42no—
621C8—44no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—816no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22#313614yes—
637C22C22×#12411yescharacteristic
638C22#348511yescharacteristic
639C22#348611yescharacteristic
640C22C2×#57511yescharacteristic
641C22C2×#57511yescharacteristic
642C22C2×#57611yescharacteristic
643C22C2×#57711yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—81no—
653C4—82no—
654C4—81no—
655C4—81no—
656C4—82no—
657C4—81no—
658C4—84no—
659C4—82no—
660C4#314014yes—
661C4#349512yes—
662C4#349512yes—
663C2—24no—
664C2—84no—
665C2#3564411yescharacteristic
666C2C2×#313611yescharacteristic
667C2C2×#314011yescharacteristic
668C2#6173111yescharacteristic
669C2#6174011yescharacteristic
670C2#6183511yescharacteristic
671C2#6184111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
1. #35239C2semidirect
5. #27969C2semidirect
6. #27964C2semidirect
7. #27826C2semidirect
8. #27821C2semidirect
9. #27814C2semidirect
10. #27740C2semidirect
11. #27735C2semidirect
12. #27728C2semidirect
13. #21840C2semidirect
14. #21833C2semidirect
17. #20535C2semidirect
25. #2571C22semidirect
26. #2570C22semidirect
33. #2552C22semidirect
34. #2552C22semidirect
35. #2551C22semidirect
36. #2550C22semidirect
50. #2101C22semidirect
51. #2101C22semidirect
52. #2101C22semidirect
53. #2101C22semidirect
58. #1993C22semidirect
59. #1993C22semidirect
60. #1993C22semidirect
61. #1991C22semidirect
62. #1990C22semidirect
63. #1990C22semidirect
64. #1988C22semidirect
65. #1987C22semidirect
66. #1987C22semidirect
71. #1836C22semidirect
72. #1835C22semidirect
87. #1771C22semidirect
88. #1770C22semidirect
89. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#352392
C2×#25632
#291102
#291032
#279692
#279642
#278262
#278212
#278142
#277402
#277352
#277282
#218402
#218332
#207382
#207332
#205352