Finite Group Databasebeta

← #895132

Subgroups — #895132

Order 512.

3130 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic
3#17077C212yesmaximal, complemented
4#16570C212yesmaximal, complemented
5#15358C214yesmaximal, complemented
6#15331C214yesmaximal, complemented
7#14865C214yesmaximal, complemented
8#14825C212yesmaximal, complemented
9#14703C212yesmaximal, complemented
10#14440C211yesmaximal, characteristic, complemented
11#14419C212yesmaximal
12#14016C212yesmaximal, complemented
13#13899C214yesmaximal, complemented
14C22×#127C2211yescharacteristic
15C22×C4×D4C2214yes—
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#2804—28no—
19C2×#400C2212yes—
20C2×#399C2212yes—
21C2×#393C2214yes—
22C2×#393C2214yes—
23C4×#127—28no—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2214yes—
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#385C2212yes—
38C2×#384C2211yescharacteristic
39C2×#380C2212yes—
40C22×#95C2212yes—
41C2×C43C2211yescharacteristic
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2212yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2212yes—
54#1439C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62#1434C2214yes—
63#1434C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×D4C2314yes—
71C23×D4C2311yescharacteristic
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2312yes—
76C24×C4C2312yes—
77C2×#127—24no—
78C2×#127—24no—
79C2×#127Q818yescomplemented
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4—216no—
83C2×C4×D4—216no—
84C2×C4×D4—48no—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115#400—28no—
116#399—44no—
117#393—216no—
118#393—48no—
119#391—28no—
120#391—28no—
121#391—44no—
122#390—216no—
123#390—24no—
124#390D418yes—
125#389—44no—
126#386—28no—
127#386—28no—
128#386—48no—
129#386—48no—
130#385—216no—
131#385—48no—
132#385—44no—
133#384—24no—
134#380—44no—
135C2×#95—44no—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C43—24no—
156C25C2412yes—
157C25C2412yes—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—22no—
168C22×D4—28no—
169C22×D4—22no—
170C22×D4—28no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—44no—
174C22×D4—41no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—44no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4C2×D414yes—
182C22×D4C2×Q8116yes—
183C22×D4C2×Q814yes—
184C22×D4C2×Q814yes—
185C22×D4#4114yes—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—44no—
195C23×C4—42no—
196C23×C4—44no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217#127—232no—
218C4×D4—464no—
219C4×D4—416no—
220C4×D4—416no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4C2×D414yes—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
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⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C2×D4116yes—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—41no—
293C2×C42—41no—
294C2×C42—42no—
295C2×C42C2×D414yes—
296C2×C42C2×Q814yes—
297C2×C42#4118yes—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—42no—
317C24—24no—
318C24—22no—
319C24—24no—
320C24—24no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—24no—
325C24—22no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24C22×D414yes—
336C24C22×Q814yes—
337C24C22×Q814yes—
338C24C2×#4114yes—
339C24C2511yescharacteristic
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—84no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×D4—232no—
348C2×D4—28no—
349C2×D4—232no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—416no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—416no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4C22⋊Q8116yes—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
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—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
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—24no—
423C22×C4—24no—
424C22×C4—416no—
425C22×C4—48no—
426C22×C4—416no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—88no—
438C22×C4—88no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D414yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q814yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4H(5,𝔽2)12yes—
457C4⋊C4—44no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—88no—
465C4⋊C4—84no—
466C22⋊C4—232no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C22⋊C4—84no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C42—28no—
479C42—28no—
480C42—216no—
481C42—44no—
482C42—48no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—44no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23C2×C22⋊Q818yes—
529C23C23×D411yescharacteristic
530C23C23×Q811yescharacteristic
531C23C22×#4111yescharacteristic
532C23C22×#4112yes—
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×H(5,𝔽2)12yes—
536C23C2×H(5,𝔽2)12yes—
537C23#58511yescharacteristic
538C23#58512yes—
539C23#58511yescharacteristic
540Q8—816no—
541D4—416no—
542D4—464no—
543D4—416no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—816no—
610C2×C4—816no—
611C2×C4—816no—
612C2×C4—816no—
613C2×C4C2×C22⋊Q814yes—
614C2×C4#54618yes—
615C2×C4#55418yes—
616C2×C4#55914yes—
617C22—216no—
618C22—28no—
619C22—44no—
620C22—44no—
621C22—48no—
622C22—44no—
623C22—44no—
624C22—48no—
625C22—44no—
626C22—48no—
627C22—44no—
628C22—48no—
629C22—44no—
630C22—44no—
631C22—48no—
632C22—44no—
633C22—82no—
634C22—82no—
635C22—84no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22C22×C22⋊Q811yescharacteristic
640C22#345512yes—
641C22#345911yescharacteristic
642C22#345911yescharacteristic
643C22C2×#54612yes—
644C22C2×#55412yes—
645C22#347512yes—
646C22C2×#55911yescharacteristic
647C22#348411yescharacteristic
648C22#348511yescharacteristic
649C22#348812yes—
650C22#350112yes—
651C22#350111yescharacteristic
652C22#350111yescharacteristic
653C22#350612yes—
654C22#350812yes—
655C22#351912yes—
656C22#352911yescharacteristic
657C22#354311yescharacteristic
658C22#354911yescharacteristic
659C22#355812yes—
660C22#355812yes—
661C22#356012yes—
662C4—24no—
663C4—28no—
664C4—28no—
665C4—24no—
666C4—44no—
667C4—42no—
668C4—82no—
669C4—84no—
670C4—84no—
671C4—84no—
672C4—84no—
673C2—48no—
674C2—84no—
675C2#3498511yescharacteristic
676C2#3615412yes—
677C2#3724911yescharacteristic
678C2#3769712yes—
679C2#3957212yes—
680C2#3962812yes—
681C2#4622511yescharacteristic
682C2#4628211yescharacteristic
683C2#5227311yescharacteristic
684C2#5914311yescharacteristic
685C2#5986411yescharacteristic
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×#2804C2semidirect
3. #17077C2semidirect
4. #16570C2semidirect
5. #15358C2semidirect
6. #15331C2semidirect
7. #14865C2semidirect
8. #14825C2semidirect
9. #14703C2semidirect
10. #14440C2semidirect
12. #14016C2semidirect
13. #13899C2semidirect
79. C2×#127Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×C4×#1272
#170772
#165702
#153582
#153312
#148652
#148252
#147032
#144402
#144192
#140162
#138992