Finite Group Databasebeta

← #552732

Subgroups — #552732

Order 512.

3026 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2311C211yesmaximal, characteristic
2#14666C212yesmaximal, complemented
3#14489C212yesmaximal, complemented
4#14471C212yesmaximal, complemented
5#14444C214yesmaximal, complemented
6#14337C211yesmaximal, characteristic
7#14267C214yesmaximal, complemented
8#14145C212yesmaximal, complemented
9#13994C212yesmaximal, complemented
10#13993C212yesmaximal, complemented
11#13974C212yesmaximal, complemented
12#13973C212yesmaximal, complemented
13#13951C212yesmaximal, complemented
14#13851C211yesmaximal, characteristic
15C22⋊C4×C4⋊C4C212yesmaximal, complemented
16C23×#41C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2311C4116yescomplemented
20C2×#391C2212yes—
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#388C2214yes—
25C2×#388C2214yes—
26C2×#386C2212yes—
27C2×#386C2212yes—
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2214yes—
41#1446C2212yes—
42#1446C2211yescharacteristic
43#1446C2211yescharacteristic
44#1443C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2212yes—
55#1438C2212yes—
56#1438C2212yes—
57#1438C2211yescharacteristic
58#1438C2211yescharacteristic
59#1438C2212yes—
60#1438C2212yes—
61#1437C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73#1434C2214yes—
74#1434C2214yes—
75C22⋊#95C2214yes—
76C22×#41—28no—
77C22×#41—42no—
78C22×#41—42no—
79C22×#41—44no—
80C22×#41C2×C418yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×#117—44no—
90C2×#117C2×C418yes—
91C22×C4⋊C4—42no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4C2×C418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2314yes—
117C22×C42C2314yes—
118C22×C42C2312yes—
119#391C2×C4116yes—
120#390C2×C4116yes—
121#389—44no—
122#389—44no—
123#388—48no—
124#388—48no—
125#386C2×C4116yes—
126#386C2×C4116yes—
127#380—28no—
128#380—42no—
129#380—42no—
130#380—44no—
131#380C2×C4116yes—
132#380C2×C4116yes—
133C4×C4⋊C4—44no—
134C4×C4⋊C4—44no—
135C4×C22⋊C4—28no—
136C4×C22⋊C4—48no—
137C4×C22⋊C4—42no—
138C4×C22⋊C4—42no—
139C4×C22⋊C4—44no—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2311yescharacteristic
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2314yes—
166C25C2411yescharacteristic
167C25C2411yescharacteristic
168C25C2411yescharacteristic
169C2×#41—232no—
170C2×#41—416no—
171C2×#41—416no—
172C2×#41—432no—
173C2×#41—432no—
174C22×Q8—24no—
175C22×Q8—42no—
176C22×Q8—41no—
177C22×Q8—41no—
178C22×Q8C22×C414yes—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—42no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4C22×C414yes—
192C22×D4C22×C414yes—
193C22×D4C22×C414yes—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C22×C414yes—
210C23×C4C22×C414yes—
211C23×C4C22×C414yes—
212C23×C4C22×C418yes—
213C23×C4#4118yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226#117—416no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
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—28no—
238C2×C4⋊C4—416no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C22×C418yes—
247C2×C4⋊C4C22×C418yes—
248C2×C4⋊C4C22×C418yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C22×C418yes—
283C2×C22⋊C4C22×C4116yes—
284C2×C22⋊C4C22×C4116yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42C22×C418yes—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—41no—
311#95—41no—
312#95—42no—
313#95—41no—
314#95—41no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C22×C4116yes—
324#95C22×C4116yes—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24C23×C412yes—
349C24C23×C412yes—
350C24C23×C412yes—
351C24C2×#4114yes—
352C24C2511yescharacteristic
353#41—4128no—
354C2×Q8—28no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—88no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4#117116yes—
457C22×C4C23×C412yes—
458C22×C4C23×C412yes—
459C22×C4C23×C418yes—
460C22×C4C23×C418yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4114yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C42−1+414yes—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—88no—
476C4⋊C4—88no—
477C22⋊C4—216no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—88no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C42—48no—
491C42—48no—
492C42—48no—
493C42—88no—
494C42—88no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23C2×#11714yes—
558C23C24×C411yescharacteristic
559C23C22×#4112yes—
560C23C2×H(5,𝔽2)12yes—
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)11yescharacteristic
563C23C2×H(5,𝔽2)12yes—
564C23C2×2−1+412yes—
565C23#58512yes—
566C23#58512yes—
567Q8—416no—
568D4—416no—
569D4—416no—
570D4—416no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—232no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—416no—
615C2×C4—416no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—816no—
645C2×C4C2×#11714yes—
646C2×C4C2×#11714yes—
647C2×C4#51814yes—
648C2×C4#51814yes—
649C2×C4#53818yes—
650C2×C4#53818yes—
651C22—28no—
652C22—24no—
653C22—44no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—44no—
668C22—42no—
669C22—42no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22C22×#11711yescharacteristic
679C22C2×#51811yescharacteristic
680C22C2×#51811yescharacteristic
681C22C4×H(5,𝔽2)12yes—
682C22C4×2−1+412yes—
683C22C2×#53812yes—
684C22C2×#53812yes—
685C22#349212yes—
686C22#349312yes—
687C22#350612yes—
688C22#350712yes—
689C22#353612yes—
690C22#353912yes—
691C22#354112yes—
692C22#354212yes—
693C22#354612yes—
694C22#354712yes—
695C22#354911yescharacteristic
696C22#355011yescharacteristic
697C22#355111yescharacteristic
698C22#355111yescharacteristic
699C4—24no—
700C4—48no—
701C4—48no—
702C4—44no—
703C4—44no—
704C4—42no—
705C4—42no—
706C4—82no—
707C4—84no—
708C4—82no—
709C4—84no—
710C4#231218yes—
711C2—44no—
712C2—82no—
713C2—82no—
714C2C2×#231211yescharacteristic
715C2#3425212yes—
716C2#3441012yes—
717C2#3441311yescharacteristic
718C2#3441511yescharacteristic
719C2#4244612yes—
720C2#4244712yes—
721C2#4246212yes—
722C2#4265312yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14666C2semidirect
3. #14489C2semidirect
4. #14471C2semidirect
5. #14444C2semidirect
7. #14267C2semidirect
8. #14145C2semidirect
9. #13994C2semidirect
10. #13993C2semidirect
11. #13974C2semidirect
12. #13973C2semidirect
13. #13951C2semidirect
15. C22⋊C4×C4⋊C4C2semidirect
19. #2311C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23112
#146662
#144892
#144712
#144442
#143372
#142672
#141452
#139942
#139932
#139742
#139732
#139512
#138512
C22⋊C4×C4⋊C42