Finite Group Databasebeta

← #582902

Subgroups — #582902

Order 512.

2938 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2#14669C212yesmaximal, complemented
3#14638C214yesmaximal, complemented
4#14635C212yesmaximal, complemented
5#14596C212yesmaximal, complemented
6#14514C212yesmaximal, complemented
7#14427C214yesmaximal
8#14261C212yesmaximal, complemented
9#14139C212yesmaximal, complemented
10#14021C212yesmaximal, complemented
11#13997C212yesmaximal, complemented
12#13994C212yesmaximal, complemented
13#13922C212yesmaximal, complemented
14#13872C211yesmaximal, characteristic
15#13847C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2409C4116yescomplemented
21C2×#395C2212yes—
22C2×#393C2212yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#385C2212yes—
33C2×#380C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×#376C2211yescharacteristic
41#1446C2214yes—
42#1446C2214yes—
43#1445C2212yes—
44#1445C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2214yes—
54#1441C2214yes—
55#1440C2214yes—
56#1439C2211yescharacteristic
57#1439C2211yescharacteristic
58#1439C2211yescharacteristic
59#1439C2211yescharacteristic
60#1439C2214yes—
61#1439C2212yes—
62#1438C2212yes—
63#1437C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C23×Q8—42no—
77C23×Q8—42no—
78C23×Q8—44no—
79C23×Q8—44no—
80C23×Q8C2×C418yescomplemented
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2312yes—
88C2×C22⋊Q8—44no—
89C2×C22⋊Q8—44no—
90C2×C22⋊Q8C2×C4116yes—
91C2×C4×D4—42no—
92C2×C4×D4—42no—
93C2×C4×D4C2×C418yes—
94C22×C4⋊C4—44no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119#395C2×C4116yescomplemented
120#393C2×C4116yes—
121#391—44no—
122#391—44no—
123#391—44no—
124#391—44no—
125#391C2×C4116yescomplemented
126#389—44no—
127#388—48no—
128#386—44no—
129#386C2×C4116yes—
130#385C2×C4116yescomplemented
131#380—44no—
132#380—42no—
133#380C2×C418yescomplemented
134C4×C4⋊C4—42no—
135C4×C4⋊C4—44no—
136C4×C22⋊C4—44no—
137C4×C22⋊C4—44no—
138#376—42no—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8—416no—
181C22×Q8—432no—
182C22×Q8—416no—
183C22×Q8—432no—
184C22×Q8—416no—
185C22×Q8C22×C418yes—
186C22×Q8C22×C418yes—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—41no—
192C22×D4—41no—
193C22×D4C22×C414yes—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4C22×C414yes—
212C23×C4C22×C414yes—
213C23×C4C22×C418yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C22⋊Q8—432no—
228C4×D4—416no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C22×C418yes—
252C2×C4⋊C4C22×C414yes—
253C2×C4⋊C4C22×C4116yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4C22×C418yes—
281C2×C22⋊C4C22×C414yes—
282C2×C22⋊C4C22×C414yes—
283C2×C22⋊C4C22×C4116yes—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42C22×C414yes—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—41no—
300#95—41no—
301#95—42no—
302#95—42no—
303#95—42no—
304#95—42no—
305#95—42no—
306#95—42no—
307#95—44no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C22×C418yes—
323#95C22×C4116yes—
324#95C22×C4116yes—
325C24—22no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24C23×C412yes—
342C24C23×C412yes—
343C24H(5,𝔽2)12yes—
344C242−1+412yes—
345C242−1+412yes—
346C242−1+412yes—
347C24C2511yescharacteristic
348C2×Q8—216no—
349C2×Q8—216no—
350C2×Q8—216no—
351C2×Q8—48no—
352C2×Q8—416no—
353C2×Q8—4128no—
354C2×Q8—48no—
355C2×Q8—416no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—84no—
456C22×C4—88no—
457C22×C4—84no—
458C22×C4—88no—
459C22×C4C23×C412yes—
460C22×C4C23×C412yes—
461C22×C4C23×C418yes—
462C22×C4C23×C412yes—
463C22×C4C23×C418yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C4H(5,𝔽2)14yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C42−1+414yes—
471C4⋊C4—216no—
472C4⋊C4—44no—
473C4⋊C4—416no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—88no—
480C22⋊C4—216no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—416no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—84no—
489C22⋊C4—88no—
490C42—28no—
491C42—48no—
492C42—48no—
493C42—88no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—41no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23#51914yes—
539C23#52014yes—
540C23C24×C411yescharacteristic
541C23C2×H(5,𝔽2)12yes—
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×H(5,𝔽2)11yescharacteristic
544C23C2×H(5,𝔽2)12yes—
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×2−1+411yescharacteristic
547C23C2×2−1+411yescharacteristic
548C23C2×2−1+411yescharacteristic
549C23C2×2−1+412yes—
550C23#58512yes—
551Q8—432no—
552Q8—432no—
553D4—416no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—432no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—816no—
636C2×C4#51814yes—
637C2×C4#51818yes—
638C2×C4#56014yes—
639C2×C4#56018yes—
640C2×C4H(3,𝔽4)14yes—
641C22—24no—
642C22—24no—
643C22—28no—
644C22—28no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—42no—
662C22—42no—
663C22C2×#51811yescharacteristic
664C22C2×#51812yes—
665C22C2×#51911yescharacteristic
666C22C2×#52011yescharacteristic
667C22C4×2−1+412yes—
668C22C2×#56011yescharacteristic
669C22C2×#56012yes—
670C22C2×H(3,𝔽4)11yescharacteristic
671C22#353611yescharacteristic
672C22#353612yes—
673C22#353611yescharacteristic
674C22#353812yes—
675C22#353811yescharacteristic
676C22#353811yescharacteristic
677C22#353811yescharacteristic
678C22#354011yescharacteristic
679C22#354112yes—
680C22#354112yes—
681C22#354712yes—
682C22#355212yes—
683C22#355212yes—
684C22#355812yes—
685C22#356112yes—
686C4—44no—
687C4—42no—
688C4—42no—
689C4—48no—
690C4—42no—
691C4—42no—
692C4—48no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—84no—
697C4—84no—
698C4#237518yes—
699C2—44no—
700C2—44no—
701C2C2×#237511yescharacteristic
702C2#3438011yescharacteristic
703C2#3441211yescharacteristic
704C2#3447212yes—
705C2#3447512yes—
706C2#5296212yes—
707C2#5299011yescharacteristic
708C2#5299311yescharacteristic
709C2#5301112yes—
710C2#5303211yescharacteristic
711C2#5313711yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14669C2semidirect
3. #14638C2semidirect
4. #14635C2semidirect
5. #14596C2semidirect
6. #14514C2semidirect
8. #14261C2semidirect
9. #14139C2semidirect
10. #14021C2semidirect
11. #13997C2semidirect
12. #13994C2semidirect
13. #13922C2semidirect
20. #2409C4semidirect
80. C23×Q8C2×C4semidirect
119. #395C2×C4semidirect
125. #391C2×C4semidirect
130. #385C2×C4semidirect
133. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
#146692
#146382
#146352
#145962
#145142
#144272
#142612
#141392
#140212
#139972
#139942
#139222
#138722
#138472