Finite Group Databasebeta

← #796934

Subgroups — #796934

Order 512.

2614 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2447C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3#16699C212yesmaximal, complemented
4#15557C214yesmaximal, complemented
5#15546C214yesmaximal, complemented
6#15240C211yesmaximal, characteristic, complemented
7#15192C211yesmaximal, characteristic, complemented
8#15152C211yesmaximal, characteristic, complemented
9#14906C212yesmaximal, complemented
10#14826C212yesmaximal, complemented
11#14030C212yesmaximal, complemented
12#14027C212yesmaximal
13#14021C212yesmaximal, complemented
14#13958C212yesmaximal, complemented
15#13941C212yesmaximal
16C4×#398C211yesmaximal, characteristic, complemented
17D4×#95C211yesmaximal, characteristic
18C22×#124C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2212yes—
22#2447—28no—
23C2×#400C2212yes—
24C2×#398C2211yescharacteristic
25C2×#397C2212yes—
26C2×#397C2212yes—
27#2354—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#390C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#388C2214yes—
36C2×#385C2212yes—
37C2×#384C2211yescharacteristic
38C2×#382C2212yes—
39C2×#381C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C22×#95C2212yes—
46#1445C2211yescharacteristic
47#1445C2212yes—
48#1445C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1445C2214yes—
52#1444C2211yescharacteristic
53#1443C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1442C2212yes—
57#1442C2212yes—
58#1442C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2212yes—
62#1441C2211yescharacteristic
63#1441C2212yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76#1434C2214yes—
77C22⋊#95C2212yes—
78C22⋊#95C2214yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2314yes—
89C2×#124—24no—
90C2×#124—28no—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—28no—
94C2×C4×D4—44no—
95C2×C4×D4D418yes—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117—44no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#400—44no—
127#398—24no—
128#397—28no—
129#397—28no—
130#391—24no—
131#391—24no—
132#391—28no—
133#391—44no—
134#390—28no—
135#390—24no—
136#389—24no—
137#388—216no—
138#385—28no—
139#384—24no—
140#382—28no—
141#381—28no—
142#380—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—44no—
147C2×#95—44no—
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2412yes—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4C2×D414yes—
192C22×D4#4118yes—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C2×D418yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221#124—416no—
222C4×D4—232no—
223C4×D4—432no—
224#117—432no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4#4114yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C2×D418yes—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—28no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—44no—
293C2×C42C2×D414yes—
294C2×C42#4118yes—
295#95—28no—
296#95—22no—
297#95—24no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—22no—
309#95—24no—
310#95—22no—
311#95—24no—
312#95—44no—
313#95—42no—
314#95—44no—
315#95—42no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—22no—
338C24C22×D414yes—
339C24C2×#4118yes—
340C24C2511yescharacteristic
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×D4—28no—
347C2×D4—28no—
348C2×D4—216no—
349C2×D4—216no—
350C2×D4#123116yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—216no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—216no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—48no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C42−1+412yes—
456C22×C42−1+412yes—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C22⋊C4—216no—
471C22⋊C4—48no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—84no—
477C22⋊C4—88no—
478C42—28no—
479C42—216no—
480C42—416no—
481C42—44no—
482C42—48no—
483C42—44no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23C2×#12318yes—
507C23C23×D411yescharacteristic
508C23C22×#4112yes—
509C23C22×#4112yes—
510C23C22×#4111yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×2−1+411yescharacteristic
514C23C2×2−1+411yescharacteristic
515C23#58512yes—
516C23#58512yes—
517C23#58511yescharacteristic
518Q8—88no—
519D4—232no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—432no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—832no—
610C2×C4C2×#12314yes—
611C2×C4#53214yes—
612C2×C4#53714yes—
613C2×C4#54614yes—
614C2×C4#55118yes—
615C2×C4#55218yes—
616C22—28no—
617C22—28no—
618C22—28no—
619C22—48no—
620C22—48no—
621C22—44no—
622C22—48no—
623C22—44no—
624C22—44no—
625C22—48no—
626C22—44no—
627C22C22×#12311yescharacteristic
628C22C2×#53211yescharacteristic
629C22C2×#53711yescharacteristic
630C22C2×#54611yescharacteristic
631C22C2×#55112yes—
632C22C2×#55212yes—
633C22#347512yes—
634C22#348012yes—
635C22#348511yescharacteristic
636C22#348911yescharacteristic
637C22#350011yescharacteristic
638C22#350612yes—
639C22#350712yes—
640C22#351012yes—
641C22#351312yes—
642C22#351511yescharacteristic
643C22#351512yes—
644C22#352011yescharacteristic
645C22#352112yes—
646C22#354012yes—
647C22#354611yescharacteristic
648C22#355211yescharacteristic
649C22#356512yes—
650C4—24no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—84no—
658C4—81no—
659C4—84no—
660C4—82no—
661C4—81no—
662C4—82no—
663C4—81no—
664C4—82no—
665C4#249818yes—
666C2—48no—
667C2C2×#249811yescharacteristic
668C2#3595911yescharacteristic
669C2#3606711yescharacteristic
670C2#3615511yescharacteristic
671C2#3641911yescharacteristic
672C2#3890811yescharacteristic
673C2#4265011yescharacteristic
674C2#4515312yes—
675C2#4536512yes—
676C2#5944812yes—
677C2#5956712yes—
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
3. #16699C2semidirect
4. #15557C2semidirect
5. #15546C2semidirect
6. #15240C2semidirect
7. #15192C2semidirect
8. #15152C2semidirect
9. #14906C2semidirect
10. #14826C2semidirect
11. #14030C2semidirect
13. #14021C2semidirect
14. #13958C2semidirect
16. C4×#398C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24472
C2×#23542
#166992
#155572
#155462
#152402
#151922
#151522
#149062
#148262
#140302
#140272
#140212
#139582
#139412
C4×#3982
D4×#952