Finite Group Databasebeta

← #6497356

Subgroups — #6497356

Order 512.

1528 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35322C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#33289C211yesmaximal, characteristic, complemented
5#33287C211yesmaximal, characteristic, complemented
6#33278C211yesmaximal, characteristic, complemented
7#30229C211yesmaximal, characteristic, complemented
8#30228C211yesmaximal, characteristic, complemented
9#30205C211yesmaximal, characteristic, complemented
10#30205C211yesmaximal, characteristic, complemented
11#27827C214yesmaximal, complemented
12#27741C214yesmaximal, complemented
13#27722C214yesmaximal
14#27706C214yesmaximal
15#21860C211yesmaximal, characteristic
16#21835C211yesmaximal, characteristic
17#21409C211yesmaximal, characteristic
18#21322C211yesmaximal, characteristic
19#20472C211yesmaximal, characteristic
20C2×#544—21no—
21C2×#533C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2868C2211yescharacteristic
28#2862C2211yescharacteristic
29#2862C2211yescharacteristic
30#2862C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2660—21no—
34#2660—21no—
35#2621—21no—
36#2621—21no—
37#2565—21no—
38#2565—21no—
39#2565C2212yes—
40#2565C2212yes—
41#2552—21no—
42#2552C2212yes—
43#2551—21no—
44#2551C2212yes—
45#2499—21no—
46#2499—21no—
47#2499C2212yes—
48#2499C2212yes—
49#2440—21no—
50#2440—21no—
51#2440C2212yes—
52#2440C2212yes—
53#2327—21no—
54#2327—21no—
55#2317—21no—
56#2285C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2101C2214yes—
60#2101C2214yes—
61#2099C2214yes—
62#2099C2214yes—
63#2072C2214yes—
64#2072C2214yes—
65#2024C2214yes—
66#2024C2214yes—
67#1993C2214yes—
68#1992C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2214yes—
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1990C2211yescharacteristic
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1989C2214yes—
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1935C2214yes—
88#1935C2214yes—
89#1933C2214yes—
90#1933C2214yes—
91#1894C2214yes—
92#1872C2214yes—
93#1836C2214yes—
94#1834C2214yes—
95#1825C2214yes—
96#1803C2214yes—
97#1771C2212yes—
98#1771C2212yes—
99#1763C2214yes—
100#1753C2212yes—
101#1753C2212yes—
102#1744C2211yescharacteristic
103#1744C2211yescharacteristic
104#1736C2214yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×M16C2312yes—
108#544—48no—
109#533—24no—
110#533—24no—
111#533—24no—
112#533—24no—
113C2×C4⋊Q8—22no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2312yes—
125C2×C4⋊Q8C2312yes—
126C2×#125—22no—
127C2×#125—21no—
128C2×#125—21no—
129C2×#125—21no—
130C2×#125C2312yes—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—21no—
134C2×C22⋊Q8—22no—
135C2×C22⋊Q8—22no—
136C2×C22⋊Q8C2312yes—
137C2×C4×Q8—21no—
138C2×C4×Q8—21no—
139C2×C4×Q8—21no—
140C2×C4×Q8C2312yes—
141C2×#117—21no—
142C2×#117—21no—
143C2×#117D411yescharacteristic
144C2×#117D411yescharacteristic
145C2×#117C2312yes—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4C2312yes—
149C22×C42C2311yescharacteristic
150#431—24no—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×C22⋊C8C2312yes—
157C4×M16—24no—
158C2×#97C2312yes—
159C2×#97C2312yes—
160C2×#97C2312yes—
161C2×C4×C8C2312yes—
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2312yes—
167#398—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#398—22no—
172#398—22no—
173#398—22no—
174#398—22no—
175#397—22no—
176#397—22no—
177#397—22no—
178#397—22no—
179#395—22no—
180#395—22no—
181#395—22no—
182#395—22no—
183#391—21no—
184#391—21no—
185#391—21no—
186#391—21no—
187#391—21no—
188#391—21no—
189#391—21no—
190#391—21no—
191#391C2312yes—
192#391C2312yes—
193#391C2312yes—
194#391C2312yes—
195#389—22no—
196#389—22no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#385—22no—
214#385—22no—
215#384—21no—
216#384—21no—
217#384—21no—
218#384—21no—
219#384—21no—
220#384—21no—
221#384—21no—
222#384—21no—
223#384C2312yes—
224#384C2312yes—
225#384C2312yes—
226#384C2312yes—
227#384C2312yes—
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#382—21no—
233#382—21no—
234#382—21no—
235#382—21no—
236#382—21no—
237#382—21no—
238#382—21no—
239#382C2312yes—
240#382C2312yes—
241#380—21no—
242#380C2312yes—
243C4×C4⋊C4—21no—
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2311yescharacteristic
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4C2311yescharacteristic
252C4×C22⋊C4C2311yescharacteristic
253#376D411yescharacteristic
254#376D411yescharacteristic
255C43C2311yescharacteristic
256C43C2311yescharacteristic
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×Q8—22no—
278C22×Q8C2412yes—
279C23×C4—22no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283C2×M16—24no—
284C2×M16—24no—
285C2×M16—42no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300C4⋊Q8—44no—
301C4⋊Q8—48no—
302C4⋊Q8—44no—
303C4⋊Q8—44no—
304C4⋊Q8—44no—
305C4⋊Q8—82no—
306C4⋊Q8—82no—
307#125—24no—
308#125—24no—
309#125—44no—
310#125—48no—
311#125—44no—
312#125—44no—
313C22⋊Q8—24no—
314C22⋊Q8—24no—
315C22⋊Q8—48no—
316C22⋊Q8—48no—
317C22⋊Q8—44no—
318C22⋊Q8—44no—
319C22⋊Q8—44no—
320C4×Q8—24no—
321C4×Q8—24no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8—44no—
325#117—24no—
326#117—24no—
327#117—24no—
328#117—24no—
329#117—44no—
330#117—44no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—44no—
366C2×C4⋊C4—44no—
367C2×C4⋊C4—82no—
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2×D411yescharacteristic
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C4⋊C4C2412yes—
377C2×C4⋊C4C2412yes—
378C2×C4⋊C4C2412yes—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C22⋊C4C2×D411yescharacteristic
385C2×C22⋊C4C2×D411yescharacteristic
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42#4114yes—
400C2×C42#4114yes—
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C4⋊C8—24no—
408#103—28no—
409#103—44no—
410#103—44no—
411#103—44no—
412C22⋊C8—24no—
413#97—24no—
414#97—24no—
415#97—42no—
416C4×C8—24no—
417C4×C8—22no—
418C4×C8—22no—
419C4×C8—22no—
420C4×C8—22no—
421C4×C8—42no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448C24C22×D411yescharacteristic
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—24no—
452C2×Q8—24no—
453C2×Q8—44no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—44no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—82no—
465C2×Q8—82no—
466C2×Q8—82no—
467C2×Q8—82no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—44no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4#12414yes—
486C22×C4#12414yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4C2511yescharacteristic
503M16—48no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C82−1+414yes—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—44no—
582C42—44no—
583C42—44no—
584C42#12414yes—
585C42#12414yes—
586C42AGL(1,ℤ/8)14yes—
587C42#13714yes—
588C42C2×#4112yes—
589C42C2×#4112yes—
590C42C2×#4112yes—
591C42C2×#4112yes—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23C2×#12412yes—
599C23C23×D411yescharacteristic
600Q8—44no—
601Q8—44no—
602Q8—84no—
603Q8—84no—
604Q8—84no—
605Q8—164no—
606Q8—164no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—28no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—164no—
662C2×C4C2×#12412yes—
663C2×C4C2×#12412yes—
664C2×C4C2×#12412yes—
665C2×C4C2×#12412yes—
666C2×C4C2×#12412yes—
667C2×C4C2×AGL(1,ℤ/8)12yes—
668C2×C4C2×AGL(1,ℤ/8)12yes—
669C2×C4C2×AGL(1,ℤ/8)12yes—
670C2×C4C2×#13712yes—
671C2×C4C2×#13712yes—
672C2×C4C2×#13712yes—
673C2×C4#57712yes—
674C2×C4#57812yes—
675C2×C4C22×#4111yescharacteristic
676C2×C4C22×#4111yescharacteristic
677C2×C4C2×2−1+412yes—
678C2×C4#58512yes—
679C8—24no—
680C8—42no—
681C8—42no—
682C8—42no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22C22×#12411yescharacteristic
691C22#349911yescharacteristic
692C22#350011yescharacteristic
693C22C22×AGL(1,ℤ/8)11yescharacteristic
694C22C22×#13711yescharacteristic
695C22C2×#57711yescharacteristic
696C22C2×#57811yescharacteristic
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4#313714yes—
717C4#314114yes—
718C4#346112yes—
719C4#349512yes—
720C2—42no—
721C2#3531211yescharacteristic
722C2C2×#313711yescharacteristic
723C2C2×#314111yescharacteristic
724C2#6247811yescharacteristic
725C2#6247911yescharacteristic
726C2#6253611yescharacteristic
727C2#6253911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #33293C2semidirect
4. #33289C2semidirect
5. #33287C2semidirect
6. #33278C2semidirect
7. #30229C2semidirect
8. #30228C2semidirect
9. #30205C2semidirect
10. #30205C2semidirect
11. #27827C2semidirect
12. #27741C2semidirect

Maximal subgroups

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

subgroupindex
#353222
#352752
#332932
#332892
#332872
#332782
#302292
#302282
#302052
#278272
#277412
#277222
#277062
#218602
#218352
#214092
#213222
#204722