Finite Group Databasebeta

← #737663

Subgroups — #737663

Order 512.

3904 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3C2×#2542C211yesmaximal, characteristic
4C2×#2642C211yesmaximal, characteristic, complemented
5#16674C214yesmaximal, complemented
6#15850C214yesmaximal, complemented
7#15464C214yesmaximal
8#15178C212yesmaximal, complemented
9#15178C212yesmaximal, complemented
10#14893C214yesmaximal, complemented
11#14160C211yesmaximal, characteristic, complemented
12#14139C212yesmaximal, complemented
13#14138C212yesmaximal, complemented
14#13847C211yesmaximal, characteristic
15#13786C211yesmaximal, characteristic, complemented
16C2×#552—28no—
17C2×#552—28no—
18C2×#552—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2212yes—
21C22×#126C2212yes—
22C22×#125C2212yes—
23C22×#125C2211yescharacteristic
24C22×#123C2212yes—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2212yes—
28C22×C22⋊Q8C2212yes—
29C22×C4×Q8C2211yescharacteristic
30C22×C4×D4C2211yescharacteristic
31C22×#117C2212yes—
32C22×#117C2211yescharacteristic
33C23×C4⋊C4C2211yescharacteristic
34#2642—28no—
35#2546—28no—
36#2542—28no—
37C2×#400C2214yes—
38C2×#400C2214yes—
39C2×#399C2212yes—
40C2×#399C2212yes—
41C2×#398C2214yes—
42C2×#391C2212yes—
43C2×#391C2212yes—
44C2×#391C2212yes—
45C2×#391C2212yes—
46C2×#390C2212yes—
47C2×#389C2214yes—
48C2×#384C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2212yes—
53C2×#376C2211yescharacteristic
54#1445C2212yes—
55#1444C2218yes—
56#1444C2218yes—
57#1441C2214yes—
58#1441C2218yes—
59#1440C2218yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2218yes—
63#1439C2218yes—
64#1438C2212yes—
65#1436C2218yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2218yes—
73C23×Q8C2312yes—
74C23×Q8C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79#552—2128no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—44no—
85C2×C4⋊Q8—44no—
86C2×#126—28no—
87C2×#126—28no—
88C2×#126—28no—
89C2×#125—28no—
90C2×#125—28no—
91C2×#125—28no—
92C2×#125—24no—
93C2×#125—24no—
94C2×#125D418yes—
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—28no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—28no—
108C2×C22⋊Q8—44no—
109C2×C22⋊Q8—44no—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×Q8D418yes—
113C2×C4×D4—24no—
114C2×C4×D4—24no—
115C2×C4×D4D418yes—
116C2×#117—24no—
117C2×#117—24no—
118C2×#117—24no—
119C2×#117—28no—
120C2×#117—28no—
121C2×#117—28no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—24no—
124C22×C4⋊C4—24no—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2318yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2314yes—
146C22×C22⋊C4C2314yes—
147C22×C22⋊C4C2314yes—
148C22×C22⋊C4C2311yescharacteristic
149C22×C22⋊C4C2311yescharacteristic
150C22×C22⋊C4C2314yes—
151C22×C42C2312yes—
152C22×C42C2312yes—
153C22×C42C2312yes—
154C22×C42C2312yes—
155C22×C42C2312yes—
156C22×C42C2311yescharacteristic
157#400—216no—
158#400—216no—
159#399—28no—
160#399—28no—
161#398—216no—
162#391—28no—
163#391—28no—
164#391—28no—
165#391—28no—
166#390—28no—
167#389—216no—
168#384—28no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—24no—
171C4×C22⋊C4—44no—
172C4×C22⋊C4—44no—
173#376—24no—
174C2×#95C2312yes—
175C2×#95C2318yes—
176C2×#95C2312yes—
177C2×#95C2318yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2318yes—
181C2×#95C2318yes—
182C2×#95C2318yes—
183C2×#95C2318yes—
184C2×#95C2318yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25C2411yescharacteristic
188C25C2411yescharacteristic
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8C2×D414yes—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—42no—
205C22×D4C2×D414yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—44no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—432no—
235C4⋊Q8—416no—
236#126—264no—
237#125—264no—
238#125—232no—
239#123—264no—
240C22⋊Q8—232no—
241C22⋊Q8—232no—
242C22⋊Q8—432no—
243C22⋊Q8—432no—
244C4×Q8—232no—
245C4×D4—232no—
246#117—232no—
247#117—264no—
248C2×C4⋊C4—232no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
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—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—216no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4—48no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×D418yes—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2×D418yes—
298C2×C4⋊C4#4118yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4C2×D414yes—
325C2×C22⋊C4C2×D414yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42C2×D418yes—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42C2×D414yes—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—216no—
348#95—216no—
349#95—216no—
350#95—216no—
351#95—216no—
352#95—48no—
353#95—48no—
354#95—44no—
355#95—44no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—22no—
368C24—22no—
369C24C22×D412yes—
370C24C22×D412yes—
371C242−1+414yes—
372C24C2511yescharacteristic
373C2×Q8—28no—
374C2×Q8—28no—
375C2×Q8—28no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—44no—
388C2×D4—48no—
389C2×D4—48no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—216no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
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—28no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C2×#4118yes—
471C22×C4C2×#4114yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C42−1+418yes—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—432no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C42—216no—
501C42—216no—
502C42—48no—
503C42—48no—
504C42—48no—
505C42#120116yes—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
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—44no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23#53614yes—
538C23C23×D411yescharacteristic
539C23C23×D411yescharacteristic
540C23C23×D411yescharacteristic
541C23C22×#4112yes—
542C23C2×H(5,𝔽2)12yes—
543C23C2×2−1+412yes—
544C23C2×2−1+414yes—
545C23#58512yes—
546Q8—416no—
547Q8—816no—
548D4—416no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—216no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—432no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4C2×#12014yes—
615C2×C4C2×#12014yes—
616C2×C4C2×#12014yes—
617C2×C4#53214yes—
618C2×C4#53514yes—
619C2×C4#54618yes—
620C2×C4#55018yes—
621C22—28no—
622C22—24no—
623C22—44no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22—48no—
628C22—44no—
629C22—42no—
630C22—42no—
631C22—44no—
632C22—44no—
633C22—42no—
634C22—82no—
635C22—82no—
636C22—84no—
637C22C22×#12011yescharacteristic
638C22C2×#53211yescharacteristic
639C22C2×#53511yescharacteristic
640C22C2×#53611yescharacteristic
641C22#344911yescharacteristic
642C22C2×#54612yes—
643C22C2×#55012yes—
644C22#348612yes—
645C22#348912yes—
646C22#348912yes—
647C22#349912yes—
648C22#349912yes—
649C22#351514yes—
650C22#352314yes—
651C22#353814yes—
652C22#354114yes—
653C4—24no—
654C4—24no—
655C4—48no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—48no—
661C4—44no—
662C4—44no—
663C4—84no—
664C4—84no—
665C4#243118yes—
666C2—44no—
667C2—82no—
668C2C2×#243111yescharacteristic
669C2#3530511yescharacteristic
670C2#3566611yescharacteristic
671C2#3594912yes—
672C2#3808112yes—
673C2#3808812yes—
674C2#4280812yes—
675C2#4521714yes—
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×#2546C2semidirect
4. C2×#2642C2semidirect
5. #16674C2semidirect
6. #15850C2semidirect
8. #15178C2semidirect
9. #15178C2semidirect
10. #14893C2semidirect
11. #14160C2semidirect
12. #14139C2semidirect
13. #14138C2semidirect
15. #13786C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#25462
C2×#25422
C2×#26422
#166742
#158502
#154642
#151782
#148932
#141602
#141392
#141382
#138472
#137862