Finite Group Databasebeta

← #825712

Subgroups — #825712

Order 512.

2504 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×#2546C211yesmaximal, characteristic, complemented
2C2×#2362C211yesmaximal, characteristic
3#16602C212yesmaximal
4#15889C214yesmaximal, complemented
5#15826C214yesmaximal
6#15201C211yesmaximal, characteristic, complemented
7#15191C211yesmaximal, characteristic, complemented
8#15125C211yesmaximal, characteristic
9#15089C212yesmaximal, complemented
10#14866C212yesmaximal
11#14161C212yesmaximal, complemented
12#13997C212yesmaximal
13#13951C212yesmaximal
14#13935C212yesmaximal, complemented
15#13933C212yesmaximal, complemented
16C4×#393C211yesmaximal, characteristic
17#13821C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×#117C2211yescharacteristic
24#2546—28no—
25C2×#398C2214yes—
26C2×#393C2211yescharacteristic
27C2×#393C2212yes—
28#2362—28no—
29C2×#391C2212yes—
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#387C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47#1446C2212yes—
48#1446C2212yes—
49#1446C2211yescharacteristic
50#1446C2212yes—
51#1446C2212yes—
52#1445C2214yes—
53#1442C2212yes—
54#1442C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2211yescharacteristic
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1439C2211yescharacteristic
64#1439C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71C4⋊#95C2214yes—
72C4⋊#95C2214yes—
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75C4⋊#95C2214yes—
76C4⋊#95C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2214yes—
80#1434C2214yes—
81#1434C2214yes—
82#1434C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8—44no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8Q818yescomplemented
97C2×C4×Q8—24no—
98C2×C4×Q8—28no—
99C2×C4×Q8—24no—
100C2×C4×Q8—44no—
101C2×#117—24no—
102C2×#117—28no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2314yes—
132#398—216no—
133#393—24no—
134#393—44no—
135#391—24no—
136#391—28no—
137#391—44no—
138#391—44no—
139#389—24no—
140#389—28no—
141#389—28no—
142#389—28no—
143#388—28no—
144#388—28no—
145#387—28no—
146#380—44no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—44no—
150C4×C4⋊C4D418yes—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—44no—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8C2×Q814yes—
200C22×Q8#4114yes—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4C2×Q814yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C4⋊Q8—432no—
223C4⋊Q8—416no—
224C22⋊Q8—232no—
225C4×Q8—416no—
226C4×Q8—416no—
227#117—416no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
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⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×Q818yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C4⋊C4#4114yes—
264C2×C4⋊C4#4118yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4C2×Q818yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C2×D414yes—
304C2×C42C2×D418yes—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—22no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—42no—
329#95—42no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95C2×D418yes—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24C22×Q812yes—
348C24C2511yescharacteristic
349C2×Q8—28no—
350C2×Q8—28no—
351C2×Q8—216no—
352C2×Q8—48no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—88no—
363C2×Q8—88no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—48no—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q812yes—
451C22×C4C22×Q814yes—
452C22×C4C22×Q812yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4118yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4114yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C42−1+412yes—
462C22×C42−1+414yes—
463C4⋊C4—216no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—416no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4C22⋊Q8116yes—
476C4⋊C4#124116yes—
477C22⋊C4—216no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—44no—
483C22⋊C4—88no—
484C42—28no—
485C42—216no—
486C42—48no—
487C42—48no—
488C42—44no—
489C42—48no—
490C42—48no—
491C42—88no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23C23×D411yescharacteristic
516C23C23×Q811yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4112yes—
519C23C22×#4111yescharacteristic
520C23C2×H(5,𝔽2)12yes—
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×2−1+411yescharacteristic
523C23C2×2−1+412yes—
524C23#58511yescharacteristic
525C23#58512yes—
526Q8—416no—
527Q8—416no—
528Q8—816no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—416no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4C2×C22⋊Q818yes—
617C2×C4C2×C22⋊Q814yes—
618C2×C4C2×#12418yes—
619C2×C4C2×#12414yes—
620C2×C4#53514yes—
621C2×C4#54614yes—
622C2×C4#55418yes—
623C2×C4#55518yes—
624C22—42no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—81no—
630C22—81no—
631C22—82no—
632C22—81no—
633C22—82no—
634C22—81no—
635C22C22×C22⋊Q811yescharacteristic
636C22C22×#12411yescharacteristic
637C22C2×#53511yescharacteristic
638C22C2×#54611yescharacteristic
639C22C2×#55412yes—
640C22C2×#55512yes—
641C22Q8×#4111yescharacteristic
642C22#348212yes—
643C22#348211yescharacteristic
644C22#348711yescharacteristic
645C22#348911yescharacteristic
646C22#349911yescharacteristic
647C22#350212yes—
648C22#350311yescharacteristic
649C22#350512yes—
650C22#351411yescharacteristic
651C22#351612yes—
652C22#351612yes—
653C22#352012yes—
654C22#352412yes—
655C22#354112yes—
656C22#354212yes—
657C22#355012yes—
658C4—24no—
659C4—28no—
660C4—28no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—44no—
665C4—42no—
666C4—44no—
667C4—82no—
668C4—84no—
669C4—82no—
670C4—82no—
671C4—84no—
672C4—81no—
673C4#256318yes—
674C2—82no—
675C2C2×#256311yescharacteristic
676C2#3584511yescharacteristic
677C2#3585411yescharacteristic
678C2#3627411yescharacteristic
679C2#3630311yescharacteristic
680C2#3946711yescharacteristic
681C2#4291111yescharacteristic
682C2#4499312yes—
683C2#4499712yes—
684C2#5992212yes—
685C2#5995112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2546C2semidirect
4. #15889C2semidirect
6. #15201C2semidirect
7. #15191C2semidirect
9. #15089C2semidirect
11. #14161C2semidirect
14. #13935C2semidirect
15. #13933C2semidirect
17. #13821C2semidirect
96. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#23622
#166022
#158892
#158262
#152012
#151912
#151252
#150892
#148662
#141612
#139972
#139512
#139352
#139332
C4×#3932
#138212