Finite Group Databasebeta

← #759879

Subgroups — #759879

Order 512.

2906 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2581C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×C4×C22⋊Q8C211yesmaximal, characteristic
4C2×#2285C211yesmaximal, characteristic
5#16400C212yesmaximal
6#16264C212yesmaximal
7#15184C212yesmaximal, complemented
8#15133C212yesmaximal, complemented
9#14997C214yesmaximal
10#14872C212yesmaximal, complemented
11#14869C212yesmaximal, complemented
12#14203C214yesmaximal, complemented
13#14150C211yesmaximal, characteristic, complemented
14#14138C212yesmaximal
15#13951C212yesmaximal
16#13942C211yesmaximal, characteristic, complemented
17#13925C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2212yes—
23C23×C42C2211yescharacteristic
24#2581—28no—
25C2×#400C2212yes—
26C2×#398C2214yes—
27C2×#395C2212yes—
28C4×C22⋊Q8—28no—
29C4×C22⋊Q8—28no—
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#387C2212yes—
39C2×#387C2214yes—
40C2×#387C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43#2285—28no—
44C2×#384C2212yes—
45C2×#382C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C22⋊C4C2212yes—
58C2×C4×C22⋊C4C2212yes—
59#1446C2212yes—
60#1445C2212yes—
61#1444C2212yes—
62#1442C2212yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1440C2214yes—
73#1440C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80C4⋊#95C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85#1434C2214yes—
86#1434C2214yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8Q818yescomplemented
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×#117—28no—
104C2×#117—28no—
105C2×#117—44no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C42—24no—
130C22×C42—24no—
131C22×C42D418yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2314yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141#400—28no—
142#398—216no—
143#395—28no—
144#391—28no—
145#391—24no—
146#391—24no—
147#391—44no—
148#389—28no—
149#389—24no—
150#389—24no—
151#389—44no—
152#387—28no—
153#387—28no—
154#387—216no—
155#385—28no—
156#385—28no—
157#384—28no—
158#382—24no—
159#382D418yes—
160#380—24no—
161#380—24no—
162#380—28no—
163#380—28no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—24no—
168C4×C4⋊C4—28no—
169C4×C4⋊C4—28no—
170C4×C22⋊C4—28no—
171C4×C22⋊C4—28no—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C25C2411yescharacteristic
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—42no—
212C22×Q8C2×Q814yes—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×Q814yes—
229C23×C4#4114yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C22⋊Q8—232no—
247C22⋊Q8—232no—
248C4×Q8—416no—
249C4×Q8—416no—
250#117—432no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D418yes—
299C2×C4⋊C4C2×Q814yes—
300C2×C4⋊C4C2×Q818yes—
301C2×C4⋊C4#4118yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×Q818yes—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—232no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—28no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42#4114yes—
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—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95C2×D418yes—
370#95C2×D418yes—
371C24—21no—
372C24—22no—
373C24—21no—
374C24—21no—
375C24—22no—
376C24—21no—
377C24—22no—
378C24—22no—
379C24C22×D412yes—
380C24C22×Q812yes—
381C24C2×#4112yes—
382C24C2511yescharacteristic
383C2×Q8—28no—
384C2×Q8—28no—
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
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—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4C22⋊Q8116yes—
485C22×C4C22×D418yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×Q814yes—
496C22×C4C22×Q814yes—
497C22×C4C22×Q812yes—
498C22×C4C22×Q812yes—
499C22×C4C2×#4114yes—
500C22×C4C2×#4114yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4114yes—
505C22×C4C2×#4112yes—
506C22×C4C2×#4112yes—
507C22×C4H(5,𝔽2)12yes—
508C22×C42−1+414yes—
509C22×C42−1+412yes—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—216no—
513C4⋊C4—216no—
514C4⋊C4—216no—
515C4⋊C4—416no—
516C4⋊C4—416no—
517C4⋊C4—416no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—88no—
528C22⋊C4—216no—
529C22⋊C4—216no—
530C22⋊C4—416no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C42—28no—
534C42—28no—
535C42—28no—
536C42—416no—
537C42—48no—
538C42—48no—
539C42—48no—
540C42—48no—
541C42#121116yes—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23C2×C22⋊Q814yes—
565C23C23×D411yescharacteristic
566C23C23×D411yescharacteristic
567C23C23×Q811yescharacteristic
568C23C22×#4111yescharacteristic
569C23C22×#4111yescharacteristic
570C23C22×#4112yes—
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×2−1+412yes—
573C23C2×2−1+411yescharacteristic
574C23#58512yes—
575C23#58512yes—
576Q8—416no—
577Q8—416no—
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—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—232no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—416no—
616C2×C4—416no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
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—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4C2×#12114yes—
668C2×C4C2×#12114yes—
669C2×C4C2×#12114yes—
670C2×C4C2×C22⋊Q814yes—
671C2×C4C2×C22⋊Q814yes—
672C2×C4#52514yes—
673C2×C4#53614yes—
674C2×C4#53714yes—
675C2×C4#54614yes—
676C2×C4D4×Q814yes—
677C2×C4#55214yes—
678C2×C4#55414yes—
679C22—24no—
680C22—24no—
681C22—24no—
682C22—44no—
683C22—44no—
684C22—42no—
685C22—42no—
686C22—44no—
687C22—44no—
688C22—42no—
689C22—42no—
690C22C22×#12111yescharacteristic
691C22C22×C22⋊Q811yescharacteristic
692C22C2×#52511yescharacteristic
693C22C2×#53611yescharacteristic
694C22C2×#53711yescharacteristic
695C22#344911yescharacteristic
696C22#344911yescharacteristic
697C22C2×#54611yescharacteristic
698C22C2×D4×Q811yescharacteristic
699C22D4×#4112yes—
700C22C2×#55211yescharacteristic
701C22C2×#55411yescharacteristic
702C22#347512yes—
703C22Q8×#4112yes—
704C22#348012yes—
705C22#348912yes—
706C22#349512yes—
707C22#350212yes—
708C22#350712yes—
709C22#350912yes—
710C22#351112yes—
711C22#356212yes—
712C22#356512yes—
713C4—24no—
714C4—24no—
715C4—24no—
716C4—24no—
717C4—24no—
718C4—48no—
719C4—44no—
720C4—44no—
721C4—44no—
722C4—44no—
723C4—84no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4#241718yes—
729C2—44no—
730C2C2×#241711yescharacteristic
731C2#3473711yescharacteristic
732C2#3495211yescharacteristic
733C2#3517911yescharacteristic
734C2#3529711yescharacteristic
735C2#3530611yescharacteristic
736C2#3755311yescharacteristic
737C2#5914612yes—
738C2#5915212yes—
739C2#5932412yes—
740C2#5933012yes—
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×#2581C2semidirect
7. #15184C2semidirect
8. #15133C2semidirect
10. #14872C2semidirect
11. #14869C2semidirect
12. #14203C2semidirect
13. #14150C2semidirect
16. #13942C2semidirect
17. #13925C2semidirect
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×#25812
C2×C4×C22⋊Q82
C2×#22852
#164002
#162642
#151842
#151332
#149972
#148722
#148692
#142032
#141502
#141382
#139512
#139422
#139252