Finite Group Databasebeta

← #657577

Subgroups — #657577

Order 512.

4304 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#2503C211yesmaximal, characteristic, complemented
3C2×#2486C211yesmaximal, characteristic
4C22×#384C211yesmaximal, characteristic, complemented
5#15928C214yesmaximal, complemented
6#15850C214yesmaximal, complemented
7#15228C212yesmaximal, complemented
8#15182C212yesmaximal, complemented
9#15069C214yesmaximal
10#14702C211yesmaximal, characteristic, complemented
11#13976C212yesmaximal, complemented
12#13904C211yesmaximal, characteristic, complemented
13#13879C212yesmaximal, complemented
14#13870C214yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C2×#552—28no—
17C2×#552—28no—
18C2×#552—28no—
19C22×#126C2212yes—
20C22×#125C2211yescharacteristic
21C22×#125C2212yes—
22C22×#123C2212yes—
23C22×#123C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C22×#117C2212yes—
29C22×#117C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31C23×C4⋊C4C2214yes—
32C23×C4⋊C4C2211yescharacteristic
33C23×C42C2211yescharacteristic
34#2503—28no—
35#2486—28no—
36C2×#400C2214yes—
37C2×#398C2212yes—
38C2×#397C2212yes—
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#389C2212yes—
42C2×#386C2212yes—
43C2×#384—28no—
44C2×#384C2212yes—
45C2×#384C2214yes—
46C2×#384C2212yes—
47C2×#382C2212yes—
48C2×#381C2212yes—
49C2×#381C2214yes—
50C2×#381C2214yes—
51C2×#380C2212yes—
52C2×#380C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55#1445C2214yes—
56#1444C2218yes—
57#1444C2218yes—
58#1441C2218yes—
59#1441C2218yes—
60#1439C2218yes—
61#1439C2218yes—
62#1437C2212yes—
63#1436C2218yes—
64#1436C2218yes—
65#1436C2214yes—
66#1436C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C4×#95C2212yes—
73C23×Q8C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80#552—2128no—
81C2×#126—28no—
82C2×#126—28no—
83C2×#126—28no—
84C2×#125—28no—
85C2×#125—28no—
86C2×#125—28no—
87C2×#125—24no—
88C2×#125—24no—
89C2×#125—24no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—28no—
93C2×#123—28no—
94C2×#123—48no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8D418yes—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4D418yes—
107C2×#117—24no—
108C2×#117—28no—
109C2×#117—24no—
110C2×#117—24no—
111C2×#117—28no—
112C2×#117—28no—
113C22×C4⋊C4—24no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—216no—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—216no—
120C22×C4⋊C4—48no—
121C22×C4⋊C4D418yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2318yes—
130C22×C4⋊C4C2318yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2314yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2314yes—
143C22×C22⋊C4C2311yescharacteristic
144C22×C22⋊C4C2311yescharacteristic
145C22×C22⋊C4C2312yes—
146C22×C22⋊C4C2314yes—
147C22×C42—24no—
148C22×C42—28no—
149C22×C42C2312yes—
150C22×C42C2312yes—
151C22×C42C2312yes—
152C22×C42C2312yes—
153C22×C42C2311yescharacteristic
154C22×C42C2312yes—
155#400—216no—
156#398—28no—
157#397—28no—
158#391—28no—
159#391—28no—
160#389—28no—
161#386—28no—
162#384—28no—
163#384—28no—
164#384—216no—
165#382—28no—
166#381—28no—
167#381—216no—
168#381—48no—
169#380—28no—
170#380—24no—
171C4×C22⋊C4—24no—
172C4×C22⋊C4—28no—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2318yes—
178C2×#95C2318yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2318yes—
182C2×#95C2314yes—
183C2×#95C2318yes—
184C2×#95C2318yes—
185C2×#95C2318yes—
186C25C2411yescharacteristic
187C25C2411yescharacteristic
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8C2×D414yes—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4C2×D414yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—28no—
217C23×C4C2×D418yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4#4118yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236#126—264no—
237#125—264no—
238#125—232no—
239#123—264no—
240#123—432no—
241C22⋊Q8—232no—
242C22⋊Q8—232no—
243C4×Q8—232no—
244C4×D4—232no—
245#117—232no—
246#117—264no—
247C2×C4⋊C4—232no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
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—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—216no—
275C2×C4⋊C4—216no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—216no—
283C2×C4⋊C4—216no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—48no—
289C2×C4⋊C4—464no—
290C2×C4⋊C4—416no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D418yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4C2×D414yes—
320C2×C22⋊C4C2×D418yes—
321C2×C22⋊C4C2×D414yes—
322C2×C42—232no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—28no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42C2×D414yes—
339C2×C42#4118yes—
340#95—24no—
341#95—28no—
342#95—24no—
343#95—216no—
344#95—28no—
345#95—216no—
346#95—28no—
347#95—28no—
348#95—216no—
349#95—28no—
350#95—216no—
351#95—48no—
352#95—48no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—22no—
363C24C22×D412yes—
364C24C22×D412yes—
365C24C22×D412yes—
366C24C22×D412yes—
367C24C2×#4114yes—
368C24H(5,𝔽2)14yes—
369C24C2511yescharacteristic
370C2×Q8—28no—
371C2×Q8—216no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—48no—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—48no—
379C2×D4—48no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—232no—
383C22×C4—216no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—232no—
395C22×C4—216no—
396C22×C4—232no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—216no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—24no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4#120116yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C2×#4118yes—
468C22×C4C2×#4118yes—
469C22×C4C2×#4114yes—
470C22×C42−1+418yes—
471C22×C42−1+414yes—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—216no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—216no—
482C4⋊C4—432no—
483C4⋊C4—432no—
484C4⋊C4—416no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C22⋊C4—216no—
489C22⋊C4—216no—
490C22⋊C4—28no—
491C22⋊C4—216no—
492C22⋊C4—28no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C42—216no—
498C42—216no—
499C42—216no—
500C42—28no—
501C42—216no—
502C42—48no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—28no—
508C23—24no—
509C23—28no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—42no—
526C23—44no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23C2×#12014yes—
531C23#53214yes—
532C23#53514yes—
533C23#54618yes—
534C23#55018yes—
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C23×D411yescharacteristic
538C23C22×#4112yes—
539C23C22×#4112yes—
540C23C2×H(5,𝔽2)12yes—
541C23C2×2−1+412yes—
542C23C2×2−1+414yes—
543Q8—416no—
544D4—416no—
545C2×C4—264no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—232no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—432no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—416no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—816no—
610C2×C4C2×#12014yes—
611C2×C4C2×#12014yes—
612C2×C4#52514yes—
613C2×C4#53614yes—
614C2×C4#54718yes—
615C2×C4#54718yes—
616C2×C4#552116yescomplemented
617C22—28no—
618C22—28no—
619C22—24no—
620C22—28no—
621C22—24no—
622C22—216no—
623C22—28no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—82no—
630C22—82no—
631C22—84no—
632C22#243118yes—
633C22C22×#12011yescharacteristic
634C22C2×#52511yescharacteristic
635C22C2×#53211yescharacteristic
636C22C2×#53511yescharacteristic
637C22C2×#53611yescharacteristic
638C22C2×#54612yes—
639C22C2×#54712yes—
640C22C2×#54712yes—
641C22C2×#55012yes—
642C22C2×#55214yes—
643C22#348612yes—
644C22#349214yes—
645C22#349912yes—
646C22#349912yes—
647C22#350914yes—
648C22#354414yes—
649C4—216no—
650C4—24no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—42no—
656C4—42no—
657C4—44no—
658C4—84no—
659C4—84no—
660C4—84no—
661C4#240618yes—
662C2—28no—
663C2—82no—
664C2C2×#243111yescharacteristic
665C2C2×#240611yescharacteristic
666C2#3566611yescharacteristic
667C2#3616312yes—
668C2#3618112yes—
669C2#3786312yes—
670C2#3946512yes—
671C2#5931214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2503C2semidirect
4. C22×#384C2semidirect
5. #15928C2semidirect
6. #15850C2semidirect
7. #15228C2semidirect
8. #15182C2semidirect
10. #14702C2semidirect
11. #13976C2semidirect
12. #13904C2semidirect
13. #13879C2semidirect
14. #13870C2semidirect
616. C2×C4#552semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#25032
C2×#24862
C22×#3842
#159282
#158502
#152282
#151822
#150692
#147022
#139762
#139042
#138792
#138702
C4×#3862