Finite Group Databasebeta

← #729465

Subgroups — #729465

Order 512.

2136 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2323C211yesmaximal, characteristic
3#17009C212yesmaximal
4#16963C212yesmaximal, complemented
5#15903C214yesmaximal, complemented
6#15841C214yesmaximal
7#15236C212yesmaximal
8#15232C212yesmaximal, complemented
9#15207C212yesmaximal, complemented
10#15025C212yesmaximal
11#14340C211yesmaximal, characteristic
12#14142C212yesmaximal, complemented
13#13993C212yesmaximal
14#13942C211yesmaximal, characteristic, complemented
15#13926C211yesmaximal, characteristic, complemented
16#13852C211yesmaximal, characteristic
17#13804C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#2571—28no—
22C2×#400C2214yes—
23#2323—28no—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#388C2212yes—
32C2×#388C2212yes—
33C2×#387C2212yes—
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#382C2212yes—
37C2×#382C2212yes—
38C2×#380C2212yes—
39C2×#380C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45#1445C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2212yes—
55#1441C2212yes—
56#1441C2212yes—
57#1441C2211yescharacteristic
58#1441C2211yescharacteristic
59#1441C2212yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2212yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2211yescharacteristic
68#1439C2211yescharacteristic
69#1439C2212yes—
70#1439C2214yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2214yes—
81#1434C2214yes—
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117—24no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#400—216no—
127#391—28no—
128#391—24no—
129#391—44no—
130#389—28no—
131#389—24no—
132#389—28no—
133#389—28no—
134#388—28no—
135#388—28no—
136#387—28no—
137#384—24no—
138#382—28no—
139#382—28no—
140#382—24no—
141#380—28no—
142#380—24no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—44no—
146C4×C4⋊C4D418yes—
147C4×C22⋊C4—44no—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C25C2411yescharacteristic
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8#4114yes—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4#4114yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C22⋊Q8—232no—
213C4×Q8—416no—
214#117—416no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C2×D418yes—
295C2×C42C2×D414yes—
296C2×C42#4114yes—
297C2×C42#4114yes—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—22no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—44no—
316#95—42no—
317#95—44no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—42no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95C2×D418yes—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24C2×#4112yes—
342C24C2511yescharacteristic
343C2×Q8—28no—
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—44no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C42−1+414yes—
454C22×C42−1+412yes—
455C4⋊C4—216no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—88no—
471C22⋊C4—216no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—84no—
477C22⋊C4—88no—
478C42—48no—
479C42—44no—
480C42—48no—
481C42—48no—
482C42—48no—
483C42—88no—
484C42#124116yes—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23C23×D411yescharacteristic
509C23C22×#4111yescharacteristic
510C23C22×#4111yescharacteristic
511C23C22×#4111yescharacteristic
512C23C22×#4111yescharacteristic
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×2−1+411yescharacteristic
515C23C2×2−1+412yes—
516C23#58512yes—
517C23#58512yes—
518C23#58512yes—
519Q8—416no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—416no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4#52514yes—
611C2×C4C2×#12414yes—
612C2×C4C2×#12414yes—
613C2×C4C2×#12414yes—
614C2×C4#53714yes—
615C2×C4#55214yes—
616C2×C4#55214yes—
617C2×C4#55314yes—
618C2×C4#55314yes—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—42no—
624C22—81no—
625C22—81no—
626C22—82no—
627C22—81no—
628C22—81no—
629C22—82no—
630C22C2×#52511yescharacteristic
631C22C22×#12411yescharacteristic
632C22C2×#53711yescharacteristic
633C22#346112yes—
634C22C2×#55211yescharacteristic
635C22C2×#55211yescharacteristic
636C22C2×#55311yescharacteristic
637C22C2×#55311yescharacteristic
638C22#348511yescharacteristic
639C22#348711yescharacteristic
640C22#349512yes—
641C22#349911yescharacteristic
642C22#350011yescharacteristic
643C22#350712yes—
644C22#351012yes—
645C22#351212yes—
646C22#351212yes—
647C22#351612yes—
648C22#352412yes—
649C22#354012yes—
650C22#354112yes—
651C22#355512yes—
652C22#356512yes—
653C4—24no—
654C4—24no—
655C4—44no—
656C4—42no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—81no—
661C4—82no—
662C4—82no—
663C4—84no—
664C4—82no—
665C4—84no—
666C4—81no—
667C4—82no—
668C4#256218yes—
669C2—82no—
670C2C2×#256211yescharacteristic
671C2#3531211yescharacteristic
672C2#3564911yescharacteristic
673C2#3584611yescharacteristic
674C2#3604111yescharacteristic
675C2#3827912yes—
676C2#3982012yes—
677C2#4265011yescharacteristic
678C2#4287811yescharacteristic
679C2#4514512yes—
680C2#4529712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2571C2semidirect
4. #16963C2semidirect
5. #15903C2semidirect
8. #15232C2semidirect
9. #15207C2semidirect
12. #14142C2semidirect
14. #13942C2semidirect
15. #13926C2semidirect
17. #13804C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#23232
#170092
#169632
#159032
#158412
#152362
#152322
#152072
#150252
#143402
#141422
#139932
#139422
#139262
#138522
#138042