Finite Group Databasebeta

← #516046

Subgroups — #516046

Order 512.

3016 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic, complemented
3C2×#2290C211yesmaximal, characteristic
4#14233C214yesmaximal, complemented
5#14183C214yesmaximal, complemented
6#14161C211yesmaximal, characteristic, complemented
7#14098C214yesmaximal
8#14058C214yesmaximal, complemented
9#13974C212yesmaximal, complemented
10#13973C212yesmaximal, complemented
11#13955C211yesmaximal, characteristic, complemented
12#13905C211yesmaximal, characteristic
13#13777C212yesmaximal, complemented
14C4×#380C212yesmaximal, complemented
15#13750C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2547C4116yescomplemented
22C2×#397C2212yes—
23C2×#397C2212yes—
24#2331—28no—
25C2×#391C2214yes—
26C2×#391C2212yes—
27C2×#389C2212yes—
28#2290—28no—
29C2×#388C2212yes—
30C2×#388C2214yes—
31C2×#387C2214yes—
32C2×#387C2212yes—
33C2×#386C2212yes—
34C2×#385C2212yes—
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2214yes—
49C2×#376C2211yescharacteristic
50#1446C2214yes—
51#1445C2212yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1438C2212yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1437C2212yes—
62#1437C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72#1434C2214yes—
73#1434C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2214yes—
76C4×#95C2214yes—
77C4×#95C2214yes—
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8C2×C418yescomplemented
86C2×#124—48no—
87C2×#124C2×C4116yes—
88C2×C4×D4—24no—
89C2×C4×D4—24no—
90C2×C4×D4C2×C418yes—
91C2×#117—24no—
92C2×#117—28no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42—24no—
112C22×C42—28no—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2314yes—
118C22×C42C2314yes—
119C22×C42C2312yes—
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#397C2×C4116yes—
125#397C2×C4116yescomplemented
126#391—216no—
127#391C2×C4116yescomplemented
128#389—28no—
129#388—28no—
130#388—48no—
131#387—216no—
132#387—28no—
133#386—28no—
134#385—28no—
135#382—24no—
136#380—28no—
137#380—28no—
138#380C2×C4116yes—
139#380C2×C4116yescomplemented
140C4×C4⋊C4—28no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4C2×C418yescomplemented
146C4×C22⋊C4—24no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—48no—
149#376D418yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—44no—
180C22×Q8—44no—
181C22×Q8C22×C418yes—
182C22×D4—24no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—42no—
186C22×D4C22×C414yes—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4C22×C414yes—
203C23×C4C22×C414yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C4⋊Q8—232no—
218#124—432no—
219C4×D4—232no—
220#117—416no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C22×C418yes—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C418yes—
247C2×C4⋊C4#4118yes—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C22×C414yes—
272C2×C22⋊C4C22×C414yes—
273C2×C22⋊C4C22×C4116yes—
274C2×C22⋊C4C22×C4116yes—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—28no—
287C2×C42—44no—
288C2×C42—416no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42C22×C418yes—
297C2×C42C22×C414yes—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301C2×C42#4114yes—
302C2×C42#4118yes—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C22×C418yes—
324#95C22×C4116yes—
325#95C22×C4116yes—
326#95C2×D418yes—
327#95C2×D418yes—
328C24—21no—
329C24—22no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24C23×C412yes—
344C24C23×C412yes—
345C24H(5,𝔽2)14yes—
346C24C2511yescharacteristic
347C2×Q8—216no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—88no—
351C2×Q8—88no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—44no—
355C2×D4—48no—
356C2×D4—48no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—216no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4C23×C412yes—
435C22×C4C23×C412yes—
436C22×C4C23×C418yes—
437C22×C4C23×C418yes—
438C22×C4C23×C412yes—
439C22×C4C23×C412yes—
440C22×C4C23×C412yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C2×#4118yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C42−1+414yes—
455C4⋊C4—28no—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—88no—
464C22⋊C4—232no—
465C22⋊C4—28no—
466C22⋊C4—28no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—88no—
473C42—28no—
474C42—48no—
475C42—48no—
476C42—44no—
477C42—44no—
478C42—88no—
479C42—88no—
480C42—84no—
481C42—88no—
482C42C4×D4116yes—
483C42#124116yescomplemented
484C23—22no—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23#51814yes—
526C23C24×C411yescharacteristic
527C23C23×D411yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4112yes—
530C23C22×#4112yes—
531C23C2×H(5,𝔽2)12yes—
532C23C2×2−1+412yes—
533C23#58512yes—
534C23#58512yes—
535Q8—816no—
536D4—416no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—432no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4C2×C4×D414yes—
614C2×C4C2×C4×D414yes—
615C2×C4C2×C4×D414yes—
616C2×C4C4×#4118yes—
617C2×C4C2×#12418yes—
618C2×C4C2×#12414yes—
619C2×C4#52918yes—
620C2×C4#53214yes—
621C22—24no—
622C22—28no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22—44no—
628C22—42no—
629C22—44no—
630C22—44no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22C22×C4×D411yescharacteristic
640C22C2×C4×#4112yes—
641C22C2×#51811yescharacteristic
642C22#342811yescharacteristic
643C22C4×2−1+412yes—
644C22C22×#12411yescharacteristic
645C22C2×#52912yes—
646C22C2×#53211yescharacteristic
647C22#348812yes—
648C22#348912yes—
649C22#349212yes—
650C22#349312yes—
651C22#349412yes—
652C22#349412yes—
653C22#349512yes—
654C22#350911yescharacteristic
655C22#351011yescharacteristic
656C22#351312yes—
657C22#351812yes—
658C22#352112yes—
659C22#352212yes—
660C4—24no—
661C4—28no—
662C4—24no—
663C4—48no—
664C4—44no—
665C4—48no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—48no—
670C4—82no—
671C4—84no—
672C4C4×#12418yes—
673C2—44no—
674C2—82no—
675C2C2×C4×#12411yescharacteristic
676C2#3420411yescharacteristic
677C2#3426811yescharacteristic
678C2#3429812yes—
679C2#3430712yes—
680C2#3554212yes—
681C2#3555012yes—
682C2#3556912yes—
683C2#3560412yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2331C2semidirect
4. #14233C2semidirect
5. #14183C2semidirect
6. #14161C2semidirect
8. #14058C2semidirect
9. #13974C2semidirect
10. #13973C2semidirect
11. #13955C2semidirect
13. #13777C2semidirect
14. C4×#380C2semidirect
15. #13750C2semidirect
21. #2547C4semidirect
85. C2×C4⋊Q8C2×C4semidirect
125. #397C2×C4semidirect
127. #391C2×C4semidirect
139. #380C2×C4semidirect
145. C4×C4⋊C4C2×C4semidirect
483. C42#124semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#23312
C2×#22902
#142332
#141832
#141612
#140982
#140582
#139742
#139732
#139552
#139052
#137772
C4×#3802
#137502