Finite Group Databasebeta

← #848149

Subgroups — #848149

Order 512.

2614 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×C4×#123C211yesmaximal, characteristic
3#16668C212yesmaximal, complemented
4#16321C212yesmaximal, complemented
5#15890C214yesmaximal, complemented
6#15359C214yesmaximal, complemented
7#15183C212yesmaximal, complemented
8#15183C212yesmaximal, complemented
9#15133C212yesmaximal, complemented
10#14697C212yesmaximal, complemented
11#14505C212yesmaximal
12#14440C212yesmaximal
13#14130C212yesmaximal, complemented
14#14045C212yesmaximal, complemented
15#13906C211yesmaximal, characteristic
16C22×#123C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2487—28no—
21C2×#400C2212yes—
22C2×#399C2212yes—
23C2×#398C2214yes—
24C4×#123—28no—
25C2×#390C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#388C2212yes—
31C2×#387C2212yes—
32C2×#386C2211yescharacteristic
33C2×#386C2212yes—
34C2×#385C2211yescharacteristic
35C2×#385C2212yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#384C2211yescharacteristic
39C2×#381C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48#1445C2214yes—
49#1445C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×#123—24no—
84C2×#123—24no—
85C2×#123Q818yes—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4D418yes—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C42—24no—
117C22×C42—42no—
118C22×C42—42no—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124#400—28no—
125#399—28no—
126#398—48no—
127#390—28no—
128#389—28no—
129#389—28no—
130#389—44no—
131#389—44no—
132#388—28no—
133#387—28no—
134#386—28no—
135#386—24no—
136#385—28no—
137#385—24no—
138#385—48no—
139#385—48no—
140#384—42no—
141#381—24no—
142#380—28no—
143#380—24no—
144#380—28no—
145#380—28no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—42no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—24no—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4C2×D414yes—
181C22×D4C2×Q814yes—
182C22×D4#4114yes—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—42no—
199C23×C4—41no—
200C23×C4—41no—
201C23×C4—41no—
202C23×C4—41no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×Q814yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223#123—232no—
224C4×D4—416no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
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—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—416no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×Q818yes—
266C2×C4⋊C4#4118yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4C2×D414yes—
294C2×C22⋊C4C2×D414yes—
295C2×C22⋊C4C2×Q814yes—
296C2×C22⋊C4C2×Q818yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—416no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42C2×D414yes—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—42no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—21no—
352C24—21no—
353C24C22×D412yes—
354C24C22×D412yes—
355C24C22×Q812yes—
356C24C22×Q812yes—
357C24C2×#4112yes—
358C24C2×#4112yes—
359C24C2511yescharacteristic
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4C22⋊Q8116yes—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
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—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—88no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—88no—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×Q814yes—
468C22×C4C22×Q814yes—
469C22×C4C22×Q812yes—
470C22×C4C2×#4118yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4114yes—
473C22×C4H(5,𝔽2)14yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C42−1+414yes—
477C4⋊C4—216no—
478C4⋊C4—216no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—84no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C4⋊C4—84no—
491C22⋊C4—28no—
492C22⋊C4—216no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—88no—
501C42—28no—
502C42—48no—
503C42—48no—
504C42—48no—
505C42—84no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23C2×C22⋊Q814yes—
551C23C2×C22⋊Q814yes—
552C23C23×D411yescharacteristic
553C23C23×Q811yescharacteristic
554C23C22×#4111yescharacteristic
555C23C22×#4112yes—
556C23C2×H(5,𝔽2)11yescharacteristic
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×H(5,𝔽2)12yes—
559C23C2×2−1+412yes—
560C23#58511yescharacteristic
561C23#58511yescharacteristic
562C23#58512yes—
563D4—416no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—416no—
616C2×C4—416no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—816no—
646C2×C4C2×C22⋊Q814yes—
647C2×C4#53414yes—
648C2×C4#53918yes—
649C2×C4#54718yes—
650C2×C4#55914yes—
651C2×C4#56314yes—
652C22—28no—
653C22—28no—
654C22—24no—
655C22—24no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22C22×C22⋊Q811yescharacteristic
675C22C2×#53411yescharacteristic
676C22#344911yescharacteristic
677C22C2×#53912yes—
678C22C2×#54712yes—
679C22Q8×#4112yes—
680C22C2×#55911yescharacteristic
681C22#348112yes—
682C22#348812yes—
683C22#350111yescharacteristic
684C22#350111yescharacteristic
685C22#350312yes—
686C22#350511yescharacteristic
687C22#350511yescharacteristic
688C22#351512yes—
689C22#351712yes—
690C22#352112yes—
691C22C2×#56311yescharacteristic
692C22#353711yescharacteristic
693C22#354011yescharacteristic
694C22#354212yes—
695C22#354712yes—
696C22#356512yes—
697C4—24no—
698C4—42no—
699C4—44no—
700C4—44no—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—48no—
705C4—84no—
706C4—82no—
707C4—84no—
708C4—84no—
709C4#272418yes—
710C2—44no—
711C2—44no—
712C2#3501011yescharacteristic
713C2#3627512yes—
714C2#3726911yescharacteristic
715C2#3804912yes—
716C2#3965812yes—
717C2#4014412yes—
718C2#4550211yescharacteristic
719C2#4568411yescharacteristic
720C2C2×#272411yescharacteristic
721C2#5962911yescharacteristic
722C2#5963211yescharacteristic
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. #16668C2semidirect
4. #16321C2semidirect
5. #15890C2semidirect
6. #15359C2semidirect
7. #15183C2semidirect
8. #15183C2semidirect
9. #15133C2semidirect
10. #14697C2semidirect
13. #14130C2semidirect
14. #14045C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×C4×#1232
#166682
#163212
#158902
#153592
#151832
#151332
#146972
#145052
#144402
#141302
#140452
#139062