Finite Group Databasebeta

← #1040081

Subgroups — #1040081

Order 512.

2470 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17273C212yesmaximal, complemented
2#16896C214yesmaximal, complemented
3#16707C212yesmaximal, complemented
4#16240C212yesmaximal, complemented
5#15334C214yesmaximal, complemented
6#15195C212yesmaximal, complemented
7#15181C212yesmaximal, complemented
8#15135C211yesmaximal, characteristic, complemented
9#14521C212yesmaximal
10#14029C212yesmaximal, complemented
11#13924C211yesmaximal, characteristic, complemented
12#13906C212yesmaximal, complemented
13#13876C212yesmaximal, complemented
14#13872C212yesmaximal, complemented
15C4×#392C211yesmaximal, characteristic
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#399C2212yes—
20C2×#392C2211yescharacteristic
21C2×#390C2214yes—
22C2×#389C2212yes—
23C2×#388C2214yes—
24C2×#387C2214yes—
25C2×#386C2214yes—
26C2×#384C2212yes—
27C2×#384C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1445C2212yes—
38#1445C2212yes—
39#1445C2214yes—
40#1445C2212yes—
41#1444C2212yes—
42#1444C2211yescharacteristic
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2212yes—
56#1439C2211yescharacteristic
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2211yescharacteristic
61#1438C2212yes—
62#1438C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C4×#95C2211yescharacteristic
75C23×D4C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4×D4—28no—
81C2×C4×D4—28no—
82C2×C4×D4—44no—
83C2×C4×D4—42no—
84C2×C4×D4—44no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2314yes—
117#399—44no—
118#392Q818yes—
119#390—48no—
120#389—44no—
121#388—48no—
122#387—48no—
123#386—48no—
124#384—28no—
125#384—42no—
126#380—44no—
127#380—44no—
128#380—44no—
129#380—42no—
130#380—44no—
131#380—44no—
132C4×C4⋊C4—44no—
133C4×C22⋊C4—28no—
134C4×C22⋊C4—42no—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2311yescharacteristic
157C2×#95C2314yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C25C2412yes—
161C25C2411yescharacteristic
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—41no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—41no—
172C22×D4C2×Q818yes—
173C22×D4C2×Q814yes—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4#4114yes—
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C4×D4—432no—
205C4×D4—416no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—416no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C2×Q818yes—
279C2×C22⋊C4C2×Q814yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—28no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42#4118yes—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—44no—
299#95—41no—
300#95—41no—
301#95—41no—
302#95—44no—
303#95—44no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—42no—
314#95C2×Q814yes—
315C24—22no—
316C24—22no—
317C24—21no—
318C24—21no—
319C24—22no—
320C24—21no—
321C24—21no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24C22×Q814yes—
338C24C22×Q812yes—
339C24C2×#4112yes—
340C24H(5,𝔽2)12yes—
341C24C2511yescharacteristic
342C2×D4—216no—
343C2×D4—216no—
344C2×D4—48no—
345C2×D4—44no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—44no—
352C2×D4—48no—
353C2×D4—44no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—88no—
434C22×C4—84no—
435C22×C4—88no—
436C22×C4—84no—
437C22×C4C22×Q812yes—
438C22×C4C22×Q812yes—
439C22×C4C22×Q814yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4114yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C42−1+412yes—
450C4⋊C4—48no—
451C4⋊C4—416no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—88no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—88no—
462C22⋊C4—216no—
463C22⋊C4—44no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—88no—
470C22⋊C4—84no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C42—216no—
476C42—44no—
477C42—44no—
478C42—88no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23#55414yes—
537C23C23×Q811yescharacteristic
538C23C22×#4111yescharacteristic
539C23C22×#4112yes—
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×H(5,𝔽2)12yes—
543C23C2×H(5,𝔽2)11yescharacteristic
544C23C2×H(5,𝔽2)12yes—
545C23C2×2−1+411yescharacteristic
546C23#58512yes—
547C23#58511yescharacteristic
548D4—432no—
549D4—416no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—416no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—816no—
625C2×C4—816no—
626C2×C4#53814yes—
627C2×C4#54018yes—
628C2×C4#55118yes—
629C2×C4#55418yes—
630C2×C4#55914yes—
631C2×C4#56014yes—
632C22—28no—
633C22—24no—
634C22—44no—
635C22—42no—
636C22—42no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22C2×#53811yescharacteristic
659C22C2×#54012yes—
660C22C2×#55112yes—
661C22C2×#55411yescharacteristic
662C22C2×#55412yes—
663C22C2×#55911yescharacteristic
664C22#349412yes—
665C22#349412yes—
666C22#349512yes—
667C22#350112yes—
668C22#350111yescharacteristic
669C22#350311yescharacteristic
670C22#350612yes—
671C22#350711yescharacteristic
672C22C2×#56011yescharacteristic
673C22#353611yescharacteristic
674C22#353712yes—
675C22#353912yes—
676C22#354011yescharacteristic
677C22#354612yes—
678C22#354812yes—
679C22#354911yescharacteristic
680C22#355811yescharacteristic
681C4—28no—
682C4—44no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—82no—
688C4—81no—
689C4—84no—
690C4—84no—
691C4—84no—
692C4—84no—
693C4#275218yes—
694C2—44no—
695C2—84no—
696C2#3960612yes—
697C2#3963211yescharacteristic
698C2#4247112yes—
699C2#4297511yescharacteristic
700C2#4490012yes—
701C2#4543311yescharacteristic
702C2#4549311yescharacteristic
703C2C2×#275211yescharacteristic
704C2#5295711yescharacteristic
705C2#5919912yes—
706C2#5986411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17273C2semidirect
2. #16896C2semidirect
3. #16707C2semidirect
4. #16240C2semidirect
5. #15334C2semidirect
6. #15195C2semidirect
7. #15181C2semidirect
8. #15135C2semidirect
10. #14029C2semidirect
11. #13924C2semidirect
12. #13906C2semidirect
13. #13876C2semidirect
14. #13872C2semidirect

Maximal subgroups

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

subgroupindex
#172732
#168962
#167072
#162402
#153342
#151952
#151812
#151352
#145212
#140292
#139242
#139062
#138762
#138722
C4×#3922