Finite Group Databasebeta

← #642001

Subgroups — #642001

Order 512.

2780 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C2×#2290C211yesmaximal, characteristic
3#16897C214yesmaximal, complemented
4#16896C214yesmaximal, complemented
5#15235C211yesmaximal, characteristic, complemented
6#15223C211yesmaximal, characteristic, complemented
7#15218C211yesmaximal, characteristic, complemented
8#15216C211yesmaximal, characteristic, complemented
9#15014C214yesmaximal, complemented
10#14921C214yesmaximal
11#14266C212yesmaximal, complemented
12#14264C212yesmaximal, complemented
13#13907C212yesmaximal, complemented
14#13872C212yesmaximal
15C2×C4⋊#95C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C42C2211yescharacteristic
21#2851—28no—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25#2290—28no—
26C2×#388C2212yes—
27C2×#387C2214yes—
28C2×#387C2212yes—
29C2×#386C2212yes—
30C2×#386C2214yes—
31C2×#385C2214yes—
32C2×#385C2212yes—
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2214yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×#376C2211yescharacteristic
42C22×#95C2214yes—
43#1445C2214yes—
44#1445C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C23×D4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2314yes—
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C2×#125—24no—
79C2×#125—44no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4Q818yes—
83C2×#117—24no—
84C2×#117—44no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C42—44no—
109C22×C42D418yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116#389—28no—
117#389—28no—
118#389—44no—
119#388—28no—
120#387—216no—
121#387—28no—
122#386—28no—
123#386—48no—
124#385—216no—
125#385—28no—
126#382—24no—
127#380—28no—
128#380—28no—
129#380—48no—
130#380—44no—
131C4×C4⋊C4—24no—
132C4×C22⋊C4—24no—
133C4×C22⋊C4—24no—
134#376—24no—
135C2×#95—48no—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C25C2411yescharacteristic
161C25C2411yescharacteristic
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4C2×Q814yes—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4C2×D414yes—
187C23×C4C2×D418yes—
188C23×C4C2×Q814yes—
189C23×C4C2×Q814yes—
190C23×C4#4114yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204#125—416no—
205C4×D4—232no—
206#117—416no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—432no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
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—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×Q814yes—
243C2×C4⋊C4#4118yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4C2×Q814yes—
267C2×C22⋊C4C2×Q814yes—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—416no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42C2×D414yes—
280C2×C42C2×D414yes—
281C2×C42C2×D418yes—
282C2×C42C2×Q814yes—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—21no—
312C24—22no—
313C24—21no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24C22×D412yes—
320C24C22×Q812yes—
321C24C22×Q812yes—
322C24C2×#4112yes—
323C24H(5,𝔽2)14yes—
324C242−1+414yes—
325C24C2511yescharacteristic
326C2×D4—28no—
327C2×D4—216no—
328C2×D4—48no—
329C2×D4—44no—
330C2×D4—44no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—216no—
336C22×C4—216no—
337C22×C4—216no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
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—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—28no—
369C22×C4—216no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—416no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4—88no—
421C22×C4C22⋊Q8116yes—
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×Q812yes—
427C22×C4C22×Q812yes—
428C22×C4C22×Q812yes—
429C22×C4C22×Q812yes—
430C22×C4C22×Q812yes—
431C22×C4C2×#4118yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4114yes—
435C22×C4H(5,𝔽2)14yes—
436C4⋊C4—28no—
437C4⋊C4—216no—
438C4⋊C4—416no—
439C4⋊C4—48no—
440C4⋊C4—416no—
441C4⋊C4—48no—
442C4⋊C4—416no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—88no—
447C4⋊C4—88no—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—416no—
454C22⋊C4—88no—
455C22⋊C4—88no—
456C42—28no—
457C42—28no—
458C42—28no—
459C42—48no—
460C42—44no—
461C42—44no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23C2×C22⋊Q814yes—
500C23#53718yes—
501C23#54014yes—
502C23#54114yes—
503C23#54418yes—
504C23C23×D411yescharacteristic
505C23C23×Q811yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4112yes—
508C23C2×H(5,𝔽2)12yes—
509C23C2×H(5,𝔽2)12yes—
510C23C2×2−1+412yes—
511C23#58512yes—
512C23#58512yes—
513D4—416no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—232no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—432no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—416no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—832no—
593C2×C4C2×C22⋊Q814yes—
594C2×C4C2×C22⋊Q814yes—
595C2×C4#53214yes—
596C2×C4#53414yes—
597C2×C4#55118yes—
598C2×C4#55418yes—
599C22—24no—
600C22—24no—
601C22—28no—
602C22—28no—
603C22—28no—
604C22—28no—
605C22—28no—
606C22—28no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—44no—
611C22—44no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22#262918yes—
617C22C22×C22⋊Q811yescharacteristic
618C22C2×#53211yescharacteristic
619C22C2×#53411yescharacteristic
620C22C2×#53712yes—
621C22#344912yes—
622C22C2×#54011yescharacteristic
623C22C2×#54111yescharacteristic
624C22C2×#54412yes—
625C22C2×#55112yes—
626C22C2×#55412yes—
627C22#350411yescharacteristic
628C22#350411yescharacteristic
629C22#351012yes—
630C22#351312yes—
631C22#351312yes—
632C22#351612yes—
633C22#351612yes—
634C22#354112yes—
635C22#354112yes—
636C22#354212yes—
637C22#354212yes—
638C4—24no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—44no—
643C4—44no—
644C4—48no—
645C4—82no—
646C4—84no—
647C4—82no—
648C4—84no—
649C4—84no—
650C4#256418yes—
651C2—28no—
652C2—82no—
653C2C2×#262911yescharacteristic
654C2C2×#256411yescharacteristic
655C2#3523412yes—
656C2#3533411yescharacteristic
657C2#3744412yes—
658C2#3810012yes—
659C2#3810112yes—
660C2#3811212yes—
661C2#3811312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2851C2semidirect
3. #16897C2semidirect
4. #16896C2semidirect
5. #15235C2semidirect
6. #15223C2semidirect
7. #15218C2semidirect
8. #15216C2semidirect
9. #15014C2semidirect
11. #14266C2semidirect
12. #14264C2semidirect
13. #13907C2semidirect
15. C2×C4⋊#95C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#22902
#168972
#168962
#152352
#152232
#152182
#152162
#150142
#149212
#142662
#142642
#139072
#138722
C2×C4⋊#952