Finite Group Databasebeta

← #641725

Subgroups — #641725

Order 512.

2810 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×#2627C211yesmaximal, characteristic
3C22×#387C211yesmaximal, characteristic
4C2×#2285C211yesmaximal, characteristic
5#16567C212yesmaximal, complemented
6#16565C212yesmaximal, complemented
7#16563C214yesmaximal, complemented
8#14935C212yesmaximal
9#14920C212yesmaximal
10#14872C212yesmaximal, complemented
11#14869C212yesmaximal, complemented
12#14861C214yesmaximal, complemented
13#14339C211yesmaximal, characteristic
14#14281C211yesmaximal, characteristic
15#14275C211yesmaximal, characteristic
16#13962C212yesmaximal
17#13870C212yesmaximal
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2855—28no—
24#2627—28no—
25C2×#400C2212yes—
26C2×#398C2214yes—
27C2×#395C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#387—28no—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42#2285—28no—
43C2×#384C2212yes—
44C2×#382C2212yes—
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2212yes—
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C22×#95C2211yescharacteristic
55C22×#95C2211yescharacteristic
56C22×#95C2211yescharacteristic
57C22×#95C2211yescharacteristic
58#1446C2212yes—
59#1445C2214yes—
60#1445C2212yes—
61#1444C2212yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2212yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2214yes—
76#1441C2214yes—
77#1441C2214yes—
78#1441C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2214yes—
82#1434C2214yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×#117—28no—
99C2×#117—28no—
100C2×#117—44no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4Q818yescomplemented
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C42—24no—
128C22×C42—24no—
129C22×C42D418yes—
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#400—28no—
136#398—216no—
137#395—28no—
138#391—24no—
139#391—24no—
140#389—24no—
141#389—24no—
142#387—28no—
143#387—28no—
144#387—28no—
145#387—28no—
146#387—28no—
147#387—28no—
148#385—28no—
149#385—28no—
150#385—44no—
151#384—28no—
152#382—28no—
153#381—28no—
154#381—28no—
155#381—44no—
156#380—24no—
157#380—24no—
158#380—28no—
159#380D418yes—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—28no—
162C2×#95—24no—
163C2×#95—24no—
164C2×#95—24no—
165C2×#95—24no—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C25C2411yescharacteristic
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—42no—
203C22×Q8—42no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×Q814yes—
239C23×C4C2×Q814yes—
240C23×C4C2×Q814yes—
241C23×C4#4114yes—
242C23×C4#4114yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2414yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2414yes—
251C23×C4C2411yescharacteristic
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C22⋊Q8—416no—
260#117—432no—
261C2×C4⋊C4—232no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—416no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2×D418yes—
305C2×C4⋊C4C2×Q818yes—
306C2×C4⋊C4C2×Q818yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4C2×D418yes—
319C2×C42—232no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—28no—
329C2×C42—28no—
330C2×C42—44no—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—24no—
354#95—28no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95C2×D418yes—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24C22×D412yes—
372C24C22×D412yes—
373C24C22×Q812yes—
374C24C2×#4112yes—
375C24C2×#4112yes—
376C24H(5,𝔽2)12yes—
377C242−1+412yes—
378C24C2511yescharacteristic
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—84no—
383C2×Q8—84no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—48no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—84no—
490C22×C4—84no—
491C22×C4—84no—
492C22×C4—84no—
493C22×C4#121116yes—
494C22×C4C22⋊Q8116yes—
495C22×C4C22×D412yes—
496C22×C4C22×D418yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×Q814yes—
505C22×C4C22×Q814yes—
506C22×C4C22×Q812yes—
507C22×C4C22×Q812yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C42−1+414yes—
513C4⋊C4—216no—
514C4⋊C4—216no—
515C4⋊C4—48no—
516C4⋊C4—416no—
517C4⋊C4—416no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—48no—
527C4⋊C4—88no—
528C4⋊C4—88no—
529C22⋊C4—216no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C42—28no—
534C42—28no—
535C42—28no—
536C42—28no—
537C42—416no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23C2×#12114yes—
555C23C2×C22⋊Q814yes—
556C23#53714yes—
557C23#54614yes—
558C23D4×Q814yes—
559C23#55214yes—
560C23#55414yes—
561C23C23×D411yescharacteristic
562C23C23×D411yescharacteristic
563C23C23×Q811yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4111yescharacteristic
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×2−1+412yes—
568C23C2×2−1+411yescharacteristic
569C23#58512yes—
570C23#58512yes—
571C23#58512yes—
572Q8—88no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—232no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—416no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—88no—
664C2×C4—88no—
665C2×C4C2×#12114yes—
666C2×C4C2×#12114yes—
667C2×C4C2×C22⋊Q814yes—
668C2×C4C2×C22⋊Q814yes—
669C2×C4#53614yes—
670C22—24no—
671C22—28no—
672C22—28no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—28no—
679C22—28no—
680C22#241718yes—
681C22C22×#12111yescharacteristic
682C22C22×C22⋊Q811yescharacteristic
683C22C2×#53611yescharacteristic
684C22C2×#53711yescharacteristic
685C22#344911yescharacteristic
686C22#344911yescharacteristic
687C22#344911yescharacteristic
688C22#345512yes—
689C22#345812yes—
690C22C2×#54611yescharacteristic
691C22C2×D4×Q811yescharacteristic
692C22C2×#55211yescharacteristic
693C22C2×#55411yescharacteristic
694C22#348512yes—
695C22#348912yes—
696C22#349112yes—
697C22#349512yes—
698C22#350212yes—
699C22#350712yes—
700C22#355912yes—
701C22#356012yes—
702C22#356312yes—
703C22#356412yes—
704C4—24no—
705C4—24no—
706C4—24no—
707C4—24no—
708C4—24no—
709C4—48no—
710C4—44no—
711C4—44no—
712C4—42no—
713C4—82no—
714C4—82no—
715C4—84no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C2—28no—
721C2#3463411yescharacteristic
722C2C2×#241711yescharacteristic
723C2#3508911yescharacteristic
724C2#3517911yescharacteristic
725C2#3530611yescharacteristic
726C2#3753811yescharacteristic
727C2#3754911yescharacteristic
728C2#3768912yes—
729C2#3769812yes—
730C2#3787712yes—
731C2#3788612yes—
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. #16567C2semidirect
6. #16565C2semidirect
7. #16563C2semidirect
10. #14872C2semidirect
11. #14869C2semidirect
12. #14861C2semidirect
106. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#26272
C22×#3872
C2×#22852
#165672
#165652
#165632
#149352
#149202
#148722
#148692
#148612
#143392
#142812
#142752
#139622
#138702