Finite Group Databasebeta

← #722906

Subgroups — #722906

Order 512.

3418 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×#2583C211yesmaximal, characteristic
2C2×#2434C211yesmaximal, characteristic
3C22×#395C211yesmaximal, characteristic
4C2×Q8×C22⋊C4C211yesmaximal, characteristic
5#17078C212yesmaximal, complemented
6#16993C212yesmaximal, complemented
7#15500C214yesmaximal, complemented
8#15318C214yesmaximal, complemented
9#15158C212yesmaximal, complemented
10#14920C212yesmaximal
11#14880C212yesmaximal, complemented
12#14728C212yesmaximal
13#14281C211yesmaximal, characteristic
14#14044C212yesmaximal
15#14043C212yesmaximal
16#13923C211yesmaximal, characteristic
17#13782C211yesmaximal, characteristic
18C24×Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24#2583—28no—
25#2434—28no—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#395—28no—
29C2×#395C2212yes—
30C2×#395C2211yescharacteristic
31C2×#395C2212yes—
32C2×#395C2211yescharacteristic
33C2×#395C2211yescharacteristic
34C2×#395C2212yes—
35C2×#395C2211yescharacteristic
36C2×#395C2214yes—
37Q8×C22⋊C4—28no—
38C2×#391C2212yes—
39C2×#391C2214yes—
40C2×#391C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2214yes—
44C2×#385C2211yescharacteristic
45C2×#385C2211yescharacteristic
46C2×#385C2211yescharacteristic
47C2×#385C2211yescharacteristic
48C2×#385C2212yes—
49C2×#385C2212yes—
50C2×#385C2212yes—
51C2×#380C2212yes—
52C2×#380C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C22×#95C2212yes—
57C22×#95C2211yescharacteristic
58#1446C2212yes—
59#1445C2214yes—
60#1445C2212yes—
61#1444C2214yes—
62#1444C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C23×Q8—24no—
88C23×Q8—24no—
89C23×Q8—48no—
90C23×Q8D418yes—
91C23×Q8C2311yescharacteristic
92C23×Q8C2312yes—
93C23×Q8C2311yescharacteristic
94C23×Q8C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2312yes—
97C24×C4C2311yescharacteristic
98C24×C4C2312yes—
99C24×C4C2311yescharacteristic
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—48no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—24no—
105C2×C4×Q8—28no—
106C2×C4×Q8—48no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—28no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—48no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2314yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2311yescharacteristic
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2311yescharacteristic
136C22×C42C2314yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139#398—28no—
140#398—48no—
141#395—28no—
142#395—216no—
143#395—28no—
144#395—24no—
145#395—24no—
146#395—24no—
147#395—24no—
148#395—28no—
149#391—28no—
150#391—28no—
151#391—216no—
152#389—28no—
153#389—28no—
154#389—48no—
155#385—28no—
156#385—28no—
157#385—24no—
158#385—24no—
159#385—24no—
160#385—24no—
161#385Q8116yescomplemented
162#380—24no—
163#380—28no—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—28no—
166C4×C22⋊C4D418yes—
167C2×#95—28no—
168C2×#95—24no—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C25C2411yescharacteristic
199C22×Q8—232no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—28no—
210C22×Q8—28no—
211C22×Q8—44no—
212C22×Q8—44no—
213C22×Q8—44no—
214C22×Q8—432no—
215C22×Q8—432no—
216C22×Q8—432no—
217C22×Q8C2×D414yes—
218C22×Q8C2×D418yes—
219C22×Q8C2×D414yes—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—28no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—42no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4C2×D418yes—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×Q818yes—
242C23×C4#4114yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2414yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2414yes—
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2412yes—
258C23×C4C2412yes—
259C4⋊Q8—432no—
260C22⋊Q8—416no—
261C4×Q8—432no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—432no—
307C2×C4⋊C4—416no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4C2×Q8116yescomplemented
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4C2×D414yes—
324C2×C22⋊C4C2×D414yes—
325C2×C22⋊C4C2×Q818yes—
326C2×C22⋊C4C2×Q818yes—
327C2×C42—28no—
328C2×C42—24no—
329C2×C42—28no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42C2×D418yes—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—28no—
339#95—22no—
340#95—22no—
341#95—28no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—28no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95C2×D418yes—
362#95C2×Q8116yescomplemented
363C24—24no—
364C24—22no—
365C24—22no—
366C24C22×D412yes—
367C24C22×D412yes—
368C24C22×Q814yes—
369C24C2×#4112yes—
370C242−1+414yes—
371C24C2511yescharacteristic
372C2×Q8—216no—
373C2×Q8—216no—
374C2×Q8—28no—
375C2×Q8—216no—
376C2×Q8—28no—
377C2×Q8—216no—
378C2×Q8—216no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—416no—
382C2×Q8—416no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—4128no—
387C2×Q8—44no—
388C2×Q8—84no—
389C2×Q8—84no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—216no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—48no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—416no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—44no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4#121116yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×Q814yes—
488C22×C4C22×Q818yes—
489C22×C4C22×Q814yes—
490C22×C4C22×Q818yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4H(5,𝔽2)18yes—
494C22×C42−1+414yes—
495C4⋊C4—416no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—416no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C22⋊C4—28no—
514C22⋊C4—216no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4C4⋊Q8116yescomplemented
519C42—48no—
520C42—416no—
521C42—48no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23C2×#12114yes—
534C23C2×C4⋊Q814yes—
535C23#53614yes—
536C23D4×Q818yes—
537C23#55718yes—
538C23C23×D411yescharacteristic
539C23C23×D411yescharacteristic
540C23C23×Q812yes—
541C23C22×#4111yescharacteristic
542C23C2×H(5,𝔽2)14yes—
543C23C2×2−1+412yes—
544C23C2×2−1+412yes—
545C23#58512yes—
546Q8—232no—
547Q8—432no—
548Q8—416no—
549Q8—88no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—416no—
581C2×C4—432no—
582C2×C4—416no—
583C2×C4—432no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—88no—
633C2×C4C2×#12114yes—
634C2×C4C2×#12114yes—
635C2×C4C2×C4⋊Q818yes—
636C2×C4#53418yes—
637C22—24no—
638C22—28no—
639C22—216no—
640C22—28no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22#254618yes—
645C22C22×#12111yescharacteristic
646C22C22×C4⋊Q811yescharacteristic
647C22C2×#53412yes—
648C22C2×#53611yescharacteristic
649C22C2×D4×Q812yes—
650C22C2×#55712yes—
651C22#348412yes—
652C22#348611yescharacteristic
653C22#348711yescharacteristic
654C22#348912yes—
655C22#349012yes—
656C22#349512yes—
657C22#350114yes—
658C22#350314yes—
659C22#354114yes—
660C22#355214yes—
661C4—28no—
662C4—24no—
663C4—24no—
664C4—28no—
665C4—44no—
666C4—44no—
667C4—44no—
668C4—44no—
669C4—44no—
670C4—48no—
671C4—81no—
672C4—81no—
673C4—84no—
674C4—82no—
675C4—81no—
676C4—81no—
677C2—28no—
678C2C2×#254611yescharacteristic
679C2#3498111yescharacteristic
680C2#3566111yescharacteristic
681C2#3592812yes—
682C2#3959312yes—
683C2#3959512yes—
684C2#4058912yes—
685C2#4312514yes—
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
5. #17078C2semidirect
6. #16993C2semidirect
7. #15500C2semidirect
8. #15318C2semidirect
9. #15158C2semidirect
11. #14880C2semidirect
161. #385Q8semidirect
310. C2×C4⋊C4C2×Q8semidirect
362. #95C2×Q8semidirect
518. C22⋊C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#24342
C22×#3952
C2×Q8×C22⋊C42
#170782
#169932
#155002
#153182
#151582
#149202
#148802
#147282
#142812
#140442
#140432
#139232
#137822