Finite Group Databasebeta

← #637129

Subgroups — #637129

Order 512.

3170 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3C2×C4×#121C211yesmaximal, characteristic
4C2×#2283C211yesmaximal, characteristic, complemented
5#16665C214yesmaximal, complemented
6#16263C214yesmaximal, complemented
7#15233C212yesmaximal, complemented
8#15220C214yesmaximal, complemented
9#15130C212yesmaximal
10#15129C214yesmaximal, complemented
11#14276C212yesmaximal, complemented
12#14260C212yesmaximal, complemented
13#14161C211yesmaximal, characteristic, complemented
14#13910C211yesmaximal, characteristic, complemented
15#13906C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C42C2211yescharacteristic
22#2857—28no—
23#2563—28no—
24C2×#394C2218yes—
25C2×#393C2214yes—
26C4×#121—28no—
27C2×#391C2212yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2214yes—
35C2×#388C2212yes—
36C2×#386C2212yes—
37C2×#384C2211yescharacteristic
38C2×#384C2212yes—
39#2283—28no—
40C2×#383C2212yes—
41C2×#382C2212yes—
42C2×#382C2212yes—
43C2×#381C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2214yes—
48C2×#380C2211yescharacteristic
49C2×#380C2214yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C2×#376C2212yes—
56#1446C2214yes—
57#1444C2214yes—
58#1443C2218yes—
59#1441C2218yes—
60#1441C2218yes—
61#1440C2218yes—
62#1440C2218yes—
63#1439C2218yes—
64#1439C2214yes—
65#1439C2218yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C23×D4C2312yes—
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—48no—
83C2×#121—24no—
84C2×#121—24no—
85C2×#121—24no—
86C2×C4×D4—28no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4—48no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2318yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2318yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C42—24no—
114C22×C42—24no—
115C22×C42D418yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123#394—416no—
124#393—216no—
125#391—28no—
126#390—28no—
127#390—28no—
128#390—24no—
129#389—28no—
130#389—28no—
131#389—24no—
132#389—48no—
133#388—28no—
134#386—28no—
135#384—28no—
136#384—24no—
137#383—28no—
138#382—28no—
139#382—28no—
140#381—28no—
141#380—28no—
142#380—28no—
143#380—24no—
144#380—216no—
145#380—28no—
146#380—48no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4D418yescomplemented
150C4×C22⋊C4—28no—
151C4×C22⋊C4—28no—
152#376—28no—
153C2×#95C2318yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2318yes—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2312yes—
168C25C2412yes—
169C25C2411yescharacteristic
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—28no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—44no—
186C22×D4#4114yes—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—42no—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C4⋊Q8—416no—
223C22⋊Q8—432no—
224#121—232no—
225C4×D4—432no—
226C4×D4—416no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4—48no—
259C2×C4⋊C4—48no—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4#4114yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—216no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—232no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42C2×D418yes—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42#4114yes—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—216no—
314#95—28no—
315#95—28no—
316#95—48no—
317#95—48no—
318#95—48no—
319#95—48no—
320#95C2×D418yes—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24C22×D412yes—
341C24C2×#4114yes—
342C24H(5,𝔽2)14yes—
343C24C2511yescharacteristic
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—88no—
348C2×Q8—88no—
349C2×D4—216no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—416no—
354C2×D4—416no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—28no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4C22×D412yes—
437C22×C4C22×D418yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)14yes—
453C22×C42−1+414yes—
454C22×C42−1+412yes—
455C22×C42−1+412yes—
456C4⋊C4—28no—
457C4⋊C4—216no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—416no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—816no—
469C22⋊C4—216no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—416no—
475C22⋊C4—48no—
476C22⋊C4—816no—
477C22⋊C4—816no—
478C42—28no—
479C42—28no—
480C42—28no—
481C42—216no—
482C42—48no—
483C42—48no—
484C42#121116yes—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23#53514yes—
531C23C23×D411yescharacteristic
532C23C23×D411yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4111yescharacteristic
535C23C22×#4111yescharacteristic
536C23C2×H(5,𝔽2)12yes—
537C23C2×H(5,𝔽2)12yes—
538C23C2×2−1+412yes—
539C23C2×2−1+411yescharacteristic
540C23C2×2−1+411yescharacteristic
541C23#58512yes—
542Q8—816no—
543D4—432no—
544D4—416no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—416no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—416no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—816no—
609C2×C4C2×#12114yes—
610C2×C4C2×#12114yes—
611C2×C4C2×#12114yes—
612C2×C4#52514yes—
613C2×C4#53514yes—
614C2×C4#53614yes—
615C2×C4#53614yes—
616C2×C4#53718yes—
617C2×C4#54218yes—
618C2×C4#54714yes—
619C2×C4#54814yes—
620C2×C4#55214yes—
621C2×C4#55614yes—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—42no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—44no—
632C22—44no—
633C22—48no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—48no—
638C22—82no—
639C22—82no—
640C22—84no—
641C22—82no—
642C22—82no—
643C22—84no—
644C22C22×#12111yescharacteristic
645C22C2×#52511yescharacteristic
646C22C2×#53511yescharacteristic
647C22C2×#53511yescharacteristic
648C22C2×#53611yescharacteristic
649C22C2×#53611yescharacteristic
650C22C2×#53712yes—
651C22#344911yescharacteristic
652C22C2×#54212yes—
653C22C2×#54711yescharacteristic
654C22C2×#54811yescharacteristic
655C22C2×#55211yescharacteristic
656C22C2×#55611yescharacteristic
657C22#348812yes—
658C22#349412yes—
659C22#351512yes—
660C22#351812yes—
661C22#351912yes—
662C22#351912yes—
663C22#354412yes—
664C22#354412yes—
665C22#354712yes—
666C22#354712yes—
667C4—24no—
668C4—48no—
669C4—42no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—82no—
674C4—82no—
675C4—84no—
676C4—84no—
677C4—84no—
678C4#241518yes—
679C4#262018yes—
680C2—44no—
681C2—84no—
682C2C2×#262011yescharacteristic
683C2C2×#241511yescharacteristic
684C2#3520011yescharacteristic
685C2#3523711yescharacteristic
686C2#3525211yescharacteristic
687C2#3763211yescharacteristic
688C2#3763411yescharacteristic
689C2#3892212yes—
690C2#3893212yes—
691C2#3912612yes—
692C2#3915812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2857C2semidirect
2. C2×#2563C2semidirect
4. C2×#2283C2semidirect
5. #16665C2semidirect
6. #16263C2semidirect
7. #15233C2semidirect
8. #15220C2semidirect
10. #15129C2semidirect
11. #14276C2semidirect
12. #14260C2semidirect
13. #14161C2semidirect
14. #13910C2semidirect
15. #13906C2semidirect
149. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#25632
C2×C4×#1212
C2×#22832
#166652
#162632
#152332
#152202
#151302
#151292
#142762
#142602
#141612
#139102
#139062