Finite Group Databasebeta

← #823779

Subgroups — #823779

Order 512.

3036 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2492C211yesmaximal, characteristic
2C2×#2451C212yesmaximal, complemented
3C2×#2429C212yesmaximal
4C2×#2365C211yesmaximal, characteristic, complemented
5#16984C212yesmaximal
6#15825C214yesmaximal
7#15804C214yesmaximal, complemented
8#15193C211yesmaximal, characteristic, complemented
9#15193C212yesmaximal, complemented
10#15190C211yesmaximal, characteristic, complemented
11#15178C211yesmaximal, characteristic
12#15173C212yesmaximal, complemented
13#14231C212yesmaximal
14#14155C212yesmaximal, complemented
15#13997C212yesmaximal
16C4×#399C211yesmaximal, characteristic
17#13818C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2212yes—
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C22×C4×Q8C2212yes—
24#2492—28no—
25#2451—216no—
26#2429—216no—
27C2×#400C2212yes—
28C2×#399C2211yescharacteristic
29C2×#398C2214yes—
30C2×#398C2214yes—
31C2×#396C2214yes—
32C2×#395C2214yes—
33C2×#393C2212yes—
34#2365—28no—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2211yescharacteristic
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#391C2211yescharacteristic
41C2×#391C2212yes—
42C2×#391C2214yes—
43C2×#389C2212yes—
44C2×#389C2214yes—
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#388C2212yes—
48C2×#388C2214yes—
49C2×#387C2214yes—
50C2×#387C2214yes—
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4C2212yes—
59#1445C2211yescharacteristic
60#1445C2212yes—
61#1445C2214yes—
62#1445C2214yes—
63#1445C2212yes—
64#1445C2214yes—
65#1445C2214yes—
66#1444C2211yescharacteristic
67#1444C2212yes—
68#1444C2212yes—
69#1444C2212yes—
70#1444C2212yes—
71#1444C2212yes—
72#1444C2214yes—
73#1439C2211yescharacteristic
74#1439C2212yes—
75#1439C2212yes—
76#1439C2212yes—
77#1439C2214yes—
78#1437C2214yes—
79#1436C2214yes—
80C4⋊#95C2214yes—
81C4⋊#95C2214yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2214yes—
84C4⋊#95C2212yes—
85C4×#95C2211yescharacteristic
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C23×Q8C2312yes—
90C24×C4C2311yescharacteristic
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—44no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—44no—
103C2×C4×Q8—28no—
104C2×C4×Q8—28no—
105C2×C4×Q8—28no—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2314yes—
137C22×C42C2312yes—
138#400—28no—
139#399Q818yescomplemented
140#398—216no—
141#398—216no—
142#396—216no—
143#395—216no—
144#393—28no—
145#391—28no—
146#391—28no—
147#391—216no—
148#391—28no—
149#391—24no—
150#391—28no—
151#391—28no—
152#391—24no—
153#389—216no—
154#389—28no—
155#389—28no—
156#389D4116yes—
157#388—28no—
158#388—216no—
159#387—216no—
160#387—216no—
161#380—24no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4—44no—
166C4×C4⋊C4D418yes—
167C4×C22⋊C4—28no—
168C4×C22⋊C4—24no—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2314yes—
197C2×#95C2311yescharacteristic
198C25C2411yescharacteristic
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—28no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—24no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—22no—
213C22×Q8—44no—
214C22×Q8—42no—
215C22×Q8#4118yes—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C4⋊Q8—232no—
238C4⋊Q8—432no—
239C4⋊Q8—416no—
240C22⋊Q8—432no—
241C22⋊Q8—416no—
242C4×Q8—432no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
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—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D4116yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×Q818yes—
289C2×C4⋊C4C2×Q814yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4C2×Q818yes—
308C2×C22⋊C4C2×Q814yes—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—28no—
317C2×C42—28no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42C2×D414yes—
325C2×C42C2×D418yes—
326C2×C42C2×D418yes—
327#95—24no—
328#95—22no—
329#95—24no—
330#95—22no—
331#95—24no—
332#95—28no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—24no—
347#95—28no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—42no—
352#95—44no—
353#95C2×D418yes—
354#95C2×D4116yes—
355#95C2×Q814yes—
356C24—21no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—21no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—21no—
366C24C22×Q812yes—
367C24C2511yescharacteristic
368C2×Q8—216no—
369C2×Q8—216no—
370C2×Q8—216no—
371C2×Q8—48no—
372C2×Q8—416no—
373C2×Q8—44no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—44no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—84no—
384C2×Q8—88no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D418yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D418yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q814yes—
473C22×C4C22×Q812yes—
474C22×C4C22×Q814yes—
475C22×C4C2×#4118yes—
476C22×C4C2×#4114yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C42−1+412yes—
479C22×C42−1+412yes—
480C22×C42−1+414yes—
481C22×C42−1+412yes—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—28no—
485C4⋊C4—216no—
486C4⋊C4—28no—
487C4⋊C4—216no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—416no—
491C4⋊C4—416no—
492C4⋊C4—416no—
493C4⋊C4—44no—
494C4⋊C4—416no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—88no—
499C4⋊C4#127116yes—
500C22⋊C4—28no—
501C22⋊C4—216no—
502C22⋊C4—44no—
503C22⋊C4—416no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C42—28no—
508C42—216no—
509C42—48no—
510C42—416no—
511C42—48no—
512C42—44no—
513C42—88no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23C23×D411yescharacteristic
538C23C23×D412yes—
539C23C23×Q811yescharacteristic
540C23C22×#4112yes—
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×2−1+411yescharacteristic
543C23C2×2−1+411yescharacteristic
544C23C2×2−1+412yes—
545C23C2×2−1+411yescharacteristic
546C23#58512yes—
547C23#58511yescharacteristic
548Q8—432no—
549Q8—816no—
550Q8—88no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—88no—
636C2×C4#52514yes—
637C2×C4C2×#12718yes—
638C2×C4C2×#12714yes—
639C2×C4#53714yes—
640C2×C4#54618yes—
641C2×C4#54818yes—
642C2×C4D4×Q814yes—
643C2×C4D4×Q818yes—
644C22—42no—
645C22—42no—
646C22—44no—
647C22—42no—
648C22—44no—
649C22—82no—
650C22—82no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22C2×#52511yescharacteristic
656C22C22×#12711yescharacteristic
657C22C2×#53711yescharacteristic
658C22C2×#54612yes—
659C22C2×#54812yes—
660C22C2×D4×Q811yescharacteristic
661C22C2×D4×Q812yes—
662C22Q8×#4112yes—
663C22#348512yes—
664C22#348511yescharacteristic
665C22#348912yes—
666C22#348911yescharacteristic
667C22#349912yes—
668C22#349911yescharacteristic
669C22#350211yescharacteristic
670C22#350311yescharacteristic
671C22#351612yes—
672C22#351812yes—
673C22#352312yes—
674C22#355411yescharacteristic
675C22#355512yes—
676C22#355611yescharacteristic
677C22#356112yes—
678C4—24no—
679C4—28no—
680C4—28no—
681C4—44no—
682C4—48no—
683C4—48no—
684C4—44no—
685C4—42no—
686C4—44no—
687C4—42no—
688C4—42no—
689C4—81no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4#254018yes—
694C2—82no—
695C2C2×#254011yescharacteristic
696C2#3600711yescharacteristic
697C2#3606011yescharacteristic
698C2#3654711yescharacteristic
699C2Q8×#12711yescharacteristic
700C2#3959911yescharacteristic
701C2#4264911yescharacteristic
702C2#4509512yes—
703C2#4534312yes—
704C2#5908112yes—
705C2#5909612yes—
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
2. C2×#2451C2semidirect
4. C2×#2365C2semidirect
7. #15804C2semidirect
8. #15193C2semidirect
9. #15193C2semidirect
10. #15190C2semidirect
12. #15173C2semidirect
14. #14155C2semidirect
17. #13818C2semidirect
139. #399Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24922
C2×#24512
C2×#24292
C2×#23652
#169842
#158252
#158042
#151932
#151902
#151782
#151732
#142312
#141552
#139972
C4×#3992
#138182