Finite Group Databasebeta

← #814038

Subgroups — #814038

Order 512.

2800 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic, complemented
2C2×#2429C212yesmaximal
3C2×#2362C211yesmaximal, characteristic
4#16821C212yesmaximal
5#15735C214yesmaximal
6#15518C214yesmaximal, complemented
7#15188C211yesmaximal, characteristic, complemented
8#15178C211yesmaximal, characteristic
9#15166C212yesmaximal, complemented
10#15149C211yesmaximal, characteristic, complemented
11#14972C212yesmaximal
12#14226C212yesmaximal, complemented
13#14142C212yesmaximal, complemented
14#14138C212yesmaximal
15#14024C212yesmaximal, complemented
16C4×#397C211yesmaximal, characteristic
17Q8×#95C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2211yescharacteristic
22C22×C4×Q8C2212yes—
23C22×C4×Q8C2211yescharacteristic
24C22×#117C2211yescharacteristic
25#2546—28no—
26#2429—216no—
27C2×#398C2212yes—
28C2×#398C2214yes—
29C2×#398C2214yes—
30C2×#397C2211yescharacteristic
31C2×#396C2214yes—
32#2362—28no—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2214yes—
43C2×#388C2212yes—
44C2×#388C2212yes—
45C2×#388C2214yes—
46C2×#387C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52#1445C2211yescharacteristic
53#1444C2212yes—
54#1444C2212yes—
55#1443C2212yes—
56#1443C2214yes—
57#1442C2211yescharacteristic
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1440C2212yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2211yescharacteristic
70#1439C2212yes—
71#1439C2212yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82#1434C2214yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—44no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—44no—
99C2×C22⋊Q8D418yes—
100C2×C4×Q8—28no—
101C2×C4×Q8—28no—
102C2×C4×Q8—44no—
103C2×C4×Q8—44no—
104C2×C4×Q8D418yes—
105C2×#117—24no—
106C2×#117—44no—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2314yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137C22×C42C2312yes—
138#398—216no—
139#398—216no—
140#398—28no—
141#397—24no—
142#396—216no—
143#391—24no—
144#391—28no—
145#391—28no—
146#391—28no—
147#391—44no—
148#389—216no—
149#389—28no—
150#389—24no—
151#389—28no—
152#389—28no—
153#388—28no—
154#388—216no—
155#388—28no—
156#387—28no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—24no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—24no—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C25C2411yescharacteristic
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8C2×D414yes—
204C22×Q8C2×D414yes—
205C22×Q8#4114yes—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4C2×D414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C4⋊Q8—416no—
228C4⋊Q8—432no—
229C22⋊Q8—232no—
230C22⋊Q8—432no—
231C4×Q8—416no—
232C4×Q8—432no—
233#117—416no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4#4118yes—
275C2×C4⋊C4#4114yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4C2×D418yes—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—42no—
314C2×C42C2×D414yes—
315C2×C42C2×D418yes—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—22no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—44no—
344#95—42no—
345#95—44no—
346#95—44no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—21no—
357C24C22×D412yes—
358C24C2511yescharacteristic
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—216no—
362C2×Q8—216no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8#121116yes—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—216no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4118yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4114yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C42−1+414yes—
467C22×C42−1+414yes—
468C22×C42−1+412yes—
469C22×C42−1+412yes—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—216no—
473C4⋊C4—28no—
474C4⋊C4—216no—
475C4⋊C4—28no—
476C4⋊C4—48no—
477C4⋊C4—416no—
478C4⋊C4—416no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—416no—
483C4⋊C4—416no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C22⋊C4—216no—
487C22⋊C4—216no—
488C22⋊C4—44no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C42—28no—
494C42—48no—
495C42—48no—
496C42—416no—
497C42—48no—
498C42—48no—
499C42—44no—
500C42—48no—
501C23—22no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23C23×D411yescharacteristic
525C23C23×D411yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4112yes—
528C23C22×#4111yescharacteristic
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+411yescharacteristic
531C23C2×2−1+412yes—
532C23C2×2−1+411yescharacteristic
533C23C2×2−1+412yes—
534C23#58511yescharacteristic
535Q8—232no—
536Q8—816no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—416no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4C2×#12118yes—
626C2×C4C2×#12114yes—
627C2×C4#53214yes—
628C2×C4#53414yes—
629C2×C4#54618yes—
630C2×C4#54814yes—
631C2×C4#55318yes—
632C22—42no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—82no—
638C22—81no—
639C22—82no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22C22×#12111yescharacteristic
644C22C2×#53211yescharacteristic
645C22C2×#53411yescharacteristic
646C22C2×#54612yes—
647C22C2×#54811yescharacteristic
648C22D4×#4111yescharacteristic
649C22C2×#55312yes—
650C22#348211yescharacteristic
651C22#348212yes—
652C22#348612yes—
653C22#348611yescharacteristic
654C22#348911yescharacteristic
655C22#349911yescharacteristic
656C22#349911yescharacteristic
657C22#351112yes—
658C22#351512yes—
659C22#351811yescharacteristic
660C22#352312yes—
661C22#352312yes—
662C22#352312yes—
663C22#355212yes—
664C22#355212yes—
665C22#355412yes—
666C4—28no—
667C4—24no—
668C4—42no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—48no—
677C4—84no—
678C4—82no—
679C4—81no—
680C4—82no—
681C4#254318yes—
682C2—82no—
683C2C2×#254311yescharacteristic
684C2#3570011yescharacteristic
685C2#3594911yescharacteristic
686C2#3644411yescharacteristic
687C2#3650811yescharacteristic
688C2#3972111yescharacteristic
689C2#4291711yescharacteristic
690C2#4532212yes—
691C2#4534212yes—
692C2#5909812yes—
693C2#5969512yes—
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
6. #15518C2semidirect
7. #15188C2semidirect
9. #15166C2semidirect
10. #15149C2semidirect
12. #14226C2semidirect
13. #14142C2semidirect
15. #14024C2semidirect
17. Q8×#95C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#24292
C2×#23622
#168212
#157352
#155182
#151882
#151782
#151662
#151492
#149722
#142262
#141422
#141382
#140242
C4×#3972
Q8×#952