Finite Group Databasebeta

← #6458466

Subgroups — #6458466

Order 512.

1730 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic, complemented
2C2×#2566C211yesmaximal, characteristic
3#27743C214yesmaximal, complemented
4#27739C214yesmaximal, complemented
5#27733C214yesmaximal, complemented
6#27731C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23745C211yesmaximal, characteristic, complemented
9#23628C211yesmaximal, characteristic
10#23628C211yesmaximal, characteristic
11#21325C211yesmaximal, characteristic, complemented
12#21314C211yesmaximal, characteristic, complemented
13#21095C211yesmaximal, characteristic
14#21095C211yesmaximal, characteristic
15#20765C211yesmaximal, characteristic, complemented
16#20764C211yesmaximal, characteristic, complemented
17#20646C211yesmaximal, characteristic
18C4⋊C4×M16C211yesmaximal, characteristic, complemented
19C4×#418C211yesmaximal, characteristic
20C22×#125C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#418C2211yescharacteristic
24C2×#418C2211yescharacteristic
25C2×#418C2211yescharacteristic
26C2×#418C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30#2571C2212yescomplemented
31#2570C2212yescomplemented
32#2570C2212yescomplemented
33#2569—22no—
34#2566—22no—
35#2566—22no—
36#2566C2212yes—
37#2566C2212yes—
38#2566C2212yes—
39#2566C2212yes—
40#2552C2212yescomplemented
41#2442—22no—
42#2402—22no—
43C2×#396—22no—
44C4×#125—22no—
45#2296C2211yescharacteristic
46#2296C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#385—22no—
50#2285C2211yescharacteristic
51#2285C2211yescharacteristic
52#2285C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C4×#117C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57#2101C2214yescomplemented
58#2101C2214yescomplemented
59#2100C2214yescomplemented
60#2099C2214yescomplemented
61#2073C2214yes—
62#2073C2214yes—
63#2071C2214yes—
64#2071C2214yes—
65#1992C2214yescomplemented
66#1991C2214yescomplemented
67#1989C2214yescomplemented
68#1989C2214yescomplemented
69#1936C2214yes—
70#1936C2214yes—
71#1934C2214yes—
72#1934C2214yes—
73#1836C2214yescomplemented
74#1836C2214yescomplemented
75#1835C2214yescomplemented
76#1834C2214yescomplemented
77#1804C2214yes—
78#1802C2214yes—
79#1771C2212yescomplemented
80#1771C2212yescomplemented
81#1770C2212yescomplemented
82C8×C4⋊C4C2212yescomplemented
83#1762C2214yes—
84C4×#102C2214yes—
85#1736C2211yescharacteristic
86#1736C2211yescharacteristic
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1728C2211yescharacteristic
98#1728C2211yescharacteristic
99C23×D4C2311yescharacteristic
100C24×C4—21no—
101C22×M16C2311yescharacteristic
102C22×M16C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C2×#125—22no—
106C2×#125—21no—
107C2×#125—21no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125C2312yes—
111C2×#125C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4—24no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4D412yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—22no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#423—24no—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138#418—24no—
139#418—24no—
140#418—24no—
141#418—24no—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C4×M16—24no—
147C4×M16—24no—
148C4×M16—24no—
149C2×#97C2312yes—
150C2×#97C2312yes—
151C2×#97C2312yes—
152C2×C4×C8C2312yes—
153C2×C4×C8C2312yes—
154C2×C4×C8C2312yes—
155#400—22no—
156#400—22no—
157#400—22no—
158#400—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#396—24no—
164#396—24no—
165#396—24no—
166#396—24no—
167#390—22no—
168#390—22no—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#390C2312yes—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#385—22no—
185#385—24no—
186#385—24no—
187#385—24no—
188#385—24no—
189#384—22no—
190#384C2312yes—
191#384C2312yes—
192#384C2312yes—
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196#382—22no—
197#382C2312yes—
198#382C2312yes—
199#382C2312yes—
200#382C2312yes—
201#382C2312yes—
202#380—21no—
203#380—22no—
204#380—22no—
205#380—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222#376—21no—
223C2×#95—22no—
224C2×#95D412yes—
225C43—21no—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#336C2312yes—
239#336C2312yes—
240#336C2312yes—
241#336C2312yes—
242C25—21no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4#4114yes—
250C22×D4C2412yes—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—44no—
257C23×C4—84no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16#4114yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279#125—24no—
280#125—24no—
281#125—24no—
282#125—24no—
283#125—48no—
284#125—44no—
285#125—44no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—44no—
291C4×D4—44no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—84no—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—48no—
351C2×C22⋊C4—44no—
352C2×C22⋊C4—84no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42#4114yes—
368C2×C42#4114yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C4⋊C8—24no—
376C4⋊C8—24no—
377#102—28no—
378#102—28no—
379#102—28no—
380#102—28no—
381#97—24no—
382#97—24no—
383#97—24no—
384C4×C8—24no—
385C4×C8—24no—
386C4×C8—24no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405C24—24no—
406C24—22no—
407C24—44no—
408C24—44no—
409C24—42no—
410C24—42no—
411C24C22×D411yescharacteristic
412C2×D4—28no—
413C2×D4—24no—
414C2×D4—28no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—44no—
424C2×D4C2×#4114yes—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4—816no—
451C22×C4#12414yes—
452C22×C4#12414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C42−1+412yes—
467C22×C42−1+412yes—
468C22×C4C2511yescharacteristic
469M16—28no—
470M16—44no—
471M16—44no—
472M16—44no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8C2×#4114yes—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C22⋊C4—48no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C42—24no—
533C42—24no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—44no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—42no—
547C42—82no—
548C42#12414yes—
549C42#12414yes—
550C42C2×#4112yes—
551C42C2×#4112yes—
552C42C2×#4112yes—
553C42C2×#4112yes—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—416no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—48no—
562C23—44no—
563C23—84no—
564C23—88no—
565C23—84no—
566C23C2×#12412yes—
567C23#54712yes—
568C23#54812yes—
569C23C23×D411yescharacteristic
570D4—24no—
571D4—24no—
572D4—42no—
573D4—42no—
574D4—48no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—88no—
612C2×C4—88no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4C2×#12412yes—
618C2×C4C2×#12412yes—
619C2×C4C2×#12412yes—
620C2×C4C2×#12412yes—
621C2×C4C2×#12412yes—
622C2×C4#52914yes—
623C2×C4#54114yes—
624C2×C4#54712yes—
625C2×C4#54812yes—
626C2×C4#55214yes—
627C2×C4#55214yes—
628C2×C4#57514yes—
629C2×C4C22×#4111yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C2×C4C2×2−1+411yescharacteristic
635C8—24no—
636C8—42no—
637C8—42no—
638C8—42no—
639C22—24no—
640C22—24no—
641C22—22no—
642C22—22no—
643C22—48no—
644C22—44no—
645C22—816no—
646C22—84no—
647C22—82no—
648C22—82no—
649C22#248914yes—
650C22C22×#12411yescharacteristic
651C22C2×#54711yescharacteristic
652C22C2×#54811yescharacteristic
653C22C2×#57512yes—
654C22#358512yes—
655C4—24no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#248914yes—
671C4C2×#52911yescharacteristic
672C4C2×#54111yescharacteristic
673C4C2×#55211yescharacteristic
674C4C2×#55211yescharacteristic
675C2—24no—
676C2—84no—
677C2C2×#248911yescharacteristic
678C2#6061811yescharacteristic
679C2#6061911yescharacteristic
680C2#6125512yes—
681C2#6131812yes—
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. #34948C2semidirect
3. #27743C2semidirect
4. #27739C2semidirect
5. #27733C2semidirect
6. #27731C2semidirect
7. #23746C2semidirect
8. #23745C2semidirect
11. #21325C2semidirect
12. #21314C2semidirect
15. #20765C2semidirect
16. #20764C2semidirect
18. C4⋊C4×M16C2semidirect
30. #2571C22semidirect
31. #2570C22semidirect
32. #2570C22semidirect
40. #2552C22semidirect
57. #2101C22semidirect
58. #2101C22semidirect
59. #2100C22semidirect
60. #2099C22semidirect
65. #1992C22semidirect
66. #1991C22semidirect
67. #1989C22semidirect
68. #1989C22semidirect
73. #1836C22semidirect
74. #1836C22semidirect
75. #1835C22semidirect
76. #1834C22semidirect
79. #1771C22semidirect
80. #1771C22semidirect
81. #1770C22semidirect
82. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#349482
C2×#25662
#277432
#277392
#277332
#277312
#237462
#237452
#236282
#213252
#213142
#210952
#207652
#207642
#206462
C4⋊C4×M162
C4×#4182