Finite Group Databasebeta

← #875060

Subgroups — #875060

Order 512.

3270 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×#2591C212yesmaximal, complemented
2#17076C211yesmaximal, characteristic, complemented
3#17068C211yesmaximal, characteristic, complemented
4#16932C212yesmaximal, complemented
5#15742C214yesmaximal, complemented
6#15655C214yesmaximal, complemented
7#14757C214yesmaximal, complemented
8#14719C212yesmaximal, complemented
9#14698C212yesmaximal, complemented
10#14589C212yesmaximal, complemented
11#14445C212yesmaximal, complemented
12#13999C214yesmaximal, complemented
13C2×C4⋊#95C211yesmaximal, characteristic, complemented
14C22×#121C2212yes—
15C22×#121C2212yes—
16C22×#121C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2591—216no—
21C2×#400C2212yes—
22C2×#400C2212yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25C2×#394C2212yes—
26C2×#394C2212yes—
27C2×#392C2214yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#380C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C22×#95C2214yes—
40#1444C2214yes—
41#1443C2214yes—
42#1442C2214yes—
43#1441C2212yes—
44#1441C2212yes—
45#1440C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2211yescharacteristic
66#1434C2214yes—
67C22⋊#95C2212yes—
68C23×D4C2314yes—
69C23×D4C2312yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2312yes—
76C2×#121—28no—
77C2×#121—28no—
78C2×#121—44no—
79C2×#121—44no—
80C2×#121—44no—
81C2×#121—44no—
82C2×#121—44no—
83C2×#121—48no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4—42no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4D418yescomplemented
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C42—24no—
114C22×C42—42no—
115C22×C42—42no—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120#400—28no—
121#400—28no—
122#394—28no—
123#394—28no—
124#394—28no—
125#394—44no—
126#392—48no—
127#390—28no—
128#390—44no—
129#388—216no—
130#388—216no—
131#388—48no—
132#386—216no—
133#386—216no—
134#386—216no—
135#386—48no—
136#380—44no—
137C4×C22⋊C4—28no—
138C2×#95—48no—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C25C2414yes—
158C25C2411yescharacteristic
159C22×D4—28no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—44no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—42no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C2×D414yes—
205C23×C4C2×D418yes—
206C23×C4#4118yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218#121—432no—
219#121—432no—
220#121—432no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—416no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4#4114yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—41no—
296C2×C42—41no—
297C2×C42—42no—
298C2×C42—41no—
299C2×C42—41no—
300C2×C42—416no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42#4114yes—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—42no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—21no—
327C24—22no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24C22×D412yes—
344C24C2×#4114yes—
345C24H(5,𝔽2)12yes—
346C24H(5,𝔽2)12yes—
347C24H(5,𝔽2)14yes—
348C24C2511yescharacteristic
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—416no—
354C2×D4—416no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—416no—
359C2×D4—416no—
360C2×D4—48no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—88no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—88no—
370C2×D4—84no—
371C2×D4—84no—
372C2×D4—84no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—216no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4114yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C42−1+412yes—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C42—44no—
490C42—48no—
491C42—44no—
492C42—88no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23#53414yes—
547C23#53818yes—
548C23#55018yes—
549C23#55514yes—
550C23H(3,𝔽4)14yes—
551C23C23×D411yescharacteristic
552C23C22×#4111yescharacteristic
553C23C22×#4111yescharacteristic
554C23C22×#4112yes—
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×H(5,𝔽2)11yescharacteristic
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×H(5,𝔽2)12yes—
559C23C2×H(5,𝔽2)12yes—
560C23C2×2−1+411yescharacteristic
561C23#58512yes—
562D4—816no—
563D4—816no—
564D4—816no—
565D4—832no—
566C2×C4—232no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—432no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—816no—
635C2×C4—816no—
636C2×C4#53214yes—
637C2×C4#53714yes—
638C2×C4#55318yes—
639C22—28no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—28no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22#273318yes—
665C22C2×#53211yescharacteristic
666C22C2×#53411yescharacteristic
667C22C2×#53711yescharacteristic
668C22C2×#53812yes—
669C22C2×#55012yes—
670C22C2×#55312yes—
671C22C2×#55511yescharacteristic
672C22#348012yes—
673C22#349011yescharacteristic
674C22#349011yescharacteristic
675C22#349211yescharacteristic
676C22#349211yescharacteristic
677C22#349412yes—
678C22#349612yes—
679C22#350512yes—
680C22#351112yes—
681C22#351912yes—
682C22C2×H(3,𝔽4)11yescharacteristic
683C22#353412yes—
684C22#353712yes—
685C22#354311yescharacteristic
686C22#354511yescharacteristic
687C22#355912yes—
688C4—28no—
689C4—24no—
690C4—24no—
691C4—44no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—82no—
696C4—84no—
697C4—84no—
698C4—84no—
699C2—28no—
700C2—164no—
701C2#3505111yescharacteristic
702C2#3659612yes—
703C2#3712911yescharacteristic
704C2#3789112yes—
705C2#3848712yes—
706C2#3974212yes—
707C2#4599111yescharacteristic
708C2#4608911yescharacteristic
709C2C2×#273311yescharacteristic
710C2#5930311yescharacteristic
711C2#5988511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2591C2semidirect
2. #17076C2semidirect
3. #17068C2semidirect
4. #16932C2semidirect
5. #15742C2semidirect
6. #15655C2semidirect
7. #14757C2semidirect
8. #14719C2semidirect
9. #14698C2semidirect
10. #14589C2semidirect
11. #14445C2semidirect
12. #13999C2semidirect
13. C2×C4⋊#95C2semidirect
89. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
#170762
#170682
#169322
#157422
#156552
#147572
#147192
#146982
#145892
#144452
#139992
C2×C4⋊#952