Finite Group Databasebeta

← #797654

Subgroups — #797654

Order 512.

3078 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2592C212yesmaximal, complemented
2C2×#2545C212yesmaximal, complemented
3#16556C212yesmaximal, complemented
4#15733C214yesmaximal, complemented
5#15569C212yesmaximal, complemented
6#15417C212yesmaximal, complemented
7#15227C212yesmaximal
8#15180C211yesmaximal, characteristic, complemented
9#14976C214yesmaximal, complemented
10#14157C212yesmaximal, complemented
11#14154C211yesmaximal, characteristic
12#14026C212yesmaximal, complemented
13#14022C212yesmaximal, complemented
14C4×#395C211yesmaximal, characteristic, complemented
15#13817C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#124C2212yes—
19C22×#124C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22#2592—216no—
23#2545—216no—
24C2×#397C2214yes—
25C2×#396C2214yes—
26C2×#396C2214yes—
27C2×#395C2211yescharacteristic
28C2×#393C2214yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#388C2214yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#388C2214yes—
42C2×#388C2214yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47#1446C2211yescharacteristic
48#1444C2211yescharacteristic
49#1444C2212yes—
50#1442C2212yes—
51#1442C2212yes—
52#1441C2211yescharacteristic
53#1441C2212yes—
54#1441C2212yes—
55#1441C2212yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2212yes—
61#1438C2214yes—
62#1438C2212yes—
63#1438C2214yes—
64#1438C2214yes—
65#1438C2214yes—
66#1437C2214yes—
67#1437C2214yes—
68#1437C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73#1434C2214yes—
74C4×#95C2212yes—
75C4×#95C2211yescharacteristic
76C4×#95C2214yes—
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2312yes—
81C23×D4C2312yes—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—42no—
84C2×#127—24no—
85C2×#127—42no—
86C2×#127—42no—
87C2×#124—28no—
88C2×#124—28no—
89C2×#124—28no—
90C2×#124—28no—
91C2×#124—44no—
92C2×#124—44no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—42no—
97C2×C4×Q8—44no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2314yes—
129#397—216no—
130#396—216no—
131#396—216no—
132#395—24no—
133#393—216no—
134#391—28no—
135#391—28no—
136#391D4116yescomplemented
137#390—28no—
138#390—28no—
139#390—28no—
140#390—28no—
141#390—28no—
142#390—44no—
143#388—216no—
144#388—216no—
145#388—216no—
146#388—216no—
147#388—48no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—42no—
151C4×C4⋊C4—42no—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2314yes—
175C25C2414yes—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—41no—
185C22×Q8—42no—
186C22×Q8—41no—
187C22×Q8C2×D418yes—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—44no—
207C23×C4C2411yescharacteristic
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C4⋊Q8—416no—
222#127—416no—
223#124—432no—
224#124—432no—
225C4×Q8—416no—
226C4×Q8—432no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4C2×D418yes—
256C2×C4⋊C4#4114yes—
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—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
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×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—28no—
306C2×C42—24no—
307C2×C42—44no—
308C2×C42—42no—
309C2×C42—44no—
310C2×C42—42no—
311C2×C42—44no—
312C2×C42C2×D418yes—
313C2×C42#4118yes—
314C2×C42#4114yes—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—44no—
327#95—42no—
328#95—42no—
329#95—42no—
330#95—42no—
331#95—41no—
332#95—41no—
333#95—41no—
334#95—42no—
335#95—44no—
336#95C2×D4116yes—
337#95C2×D4116yes—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24C2511yescharacteristic
354C2×Q8—216no—
355C2×Q8—216no—
356C2×Q8—216no—
357C2×Q8—44no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—84no—
381C2×D4—88no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4—88no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4C22×D418yes—
454C22×C4C22×D418yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4114yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C42−1+412yes—
471C4⋊C4—216no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—416no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—416no—
489C22⋊C4—416no—
490C22⋊C4—416no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C42—28no—
495C42—216no—
496C42—216no—
497C42—416no—
498C42—44no—
499C42—44no—
500C42—84no—
501C42—88no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
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—
537C23C23×D412yes—
538C23C22×#4111yescharacteristic
539C23C22×#4112yes—
540C23C22×#4111yescharacteristic
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×H(5,𝔽2)11yescharacteristic
544C23C2×2−1+411yescharacteristic
545C23#58512yes—
546C23#58512yes—
547C23#58511yescharacteristic
548Q8—416no—
549Q8—432no—
550D4—816no—
551D4—816no—
552D4—816no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4#53218yes—
634C2×C4#53814yes—
635C2×C4D4214yes—
636C2×C4#54618yes—
637C2×C4#55514yes—
638C2×C4#55614yes—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22C2×#53212yes—
655C22C2×#53811yescharacteristic
656C22C2×D4211yescharacteristic
657C22C2×#54612yes—
658C22D4×#4112yes—
659C22C2×#55511yescharacteristic
660C22C2×#55611yescharacteristic
661C22#348012yes—
662C22#348712yes—
663C22#348812yes—
664C22#349712yes—
665C22#350511yescharacteristic
666C22#350612yes—
667C22#350611yescharacteristic
668C22#350712yes—
669C22#350811yescharacteristic
670C22#351012yes—
671C22#351012yes—
672C22#353611yescharacteristic
673C22#354011yescharacteristic
674C22#355111yescharacteristic
675C22#356111yescharacteristic
676C22#356212yes—
677C4—24no—
678C4—48no—
679C4—44no—
680C4—48no—
681C4—42no—
682C4—44no—
683C4—42no—
684C4—44no—
685C4—44no—
686C4—42no—
687C4—42no—
688C4—84no—
689C4—81no—
690C4#255518yes—
691C2—164no—
692C2C2×#255511yescharacteristic
693C2#3595612yes—
694C2#3627112yes—
695C2#3951111yescharacteristic
696C2#4298311yescharacteristic
697C2#4524012yes—
698C2#4524411yescharacteristic
699C2#4536911yescharacteristic
700C2#5900111yescharacteristic
701C2#5914512yes—
702C2#5990811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2592C2semidirect
2. C2×#2545C2semidirect
3. #16556C2semidirect
4. #15733C2semidirect
5. #15569C2semidirect
6. #15417C2semidirect
8. #15180C2semidirect
9. #14976C2semidirect
10. #14157C2semidirect
12. #14026C2semidirect
13. #14022C2semidirect
14. C4×#395C2semidirect
15. #13817C2semidirect
136. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25922
C2×#25452
#165562
#157332
#155692
#154172
#152272
#151802
#149762
#141572
#141542
#140262
#140222
C4×#3952
#138172