Finite Group Databasebeta

← #554194

Subgroups — #554194

Order 512.

2816 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic, complemented
3C2×#2290C211yesmaximal, characteristic
4#14697C212yesmaximal, complemented
5#14575C214yesmaximal, complemented
6#14522C214yesmaximal, complemented
7#14429C214yesmaximal
8#14426C214yesmaximal, complemented
9#14264C212yesmaximal, complemented
10#14155C211yesmaximal, characteristic, complemented
11#13994C212yesmaximal, complemented
12#13993C212yesmaximal, complemented
13#13954C211yesmaximal, characteristic, complemented
14#13905C211yesmaximal, characteristic
15#13797C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2563C4116yescomplemented
21C2×#393C2212yes—
22C2×#393C2212yes—
23#2331—28no—
24C2×#391C2214yes—
25C2×#391C2214yes—
26C2×#390C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30#2290—28no—
31C2×#388C2212yes—
32C2×#388C2214yes—
33C2×#387C2214yes—
34C2×#387C2212yes—
35C2×#386C2212yes—
36C2×#385C2212yes—
37C2×#385C2214yes—
38C2×#382C2211yescharacteristic
39C2×#382C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×#376C2211yescharacteristic
51#1445C2214yes—
52#1445C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1443C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1437C2212yes—
64#1437C2212yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75#1434C2214yes—
76#1434C2214yes—
77C22⋊#95C2214yes—
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8C2×C418yescomplemented
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4C2×C418yes—
89C2×#117—24no—
90C2×#117—44no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42—24no—
112C22×C42—44no—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2314yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122#393C2×C4116yes—
123#393C2×C4116yescomplemented
124#391—216no—
125#391—48no—
126#390C2×C4116yes—
127#389—28no—
128#389—44no—
129#389—44no—
130#388—28no—
131#388—48no—
132#387—216no—
133#387—28no—
134#386—28no—
135#385—28no—
136#385—48no—
137#382—24no—
138#382C2×C4116yescomplemented
139#380—44no—
140#380—44no—
141#380C2×C4116yes—
142#380C2×C4116yescomplemented
143C4×C4⋊C4—28no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4C2×C418yescomplemented
147C4×C22⋊C4—24no—
148C4×C22⋊C4—24no—
149#376D418yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
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—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4C22×C414yes—
203C23×C4C22×C414yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C4⋊Q8—232no—
218C4×D4—232no—
219#117—416no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
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—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C418yes—
251C2×C4⋊C4C22×C418yes—
252C2×C4⋊C4C22×C418yes—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4C22×C414yes—
277C2×C22⋊C4C22×C414yes—
278C2×C22⋊C4C22×C4116yes—
279C2×C22⋊C4C22×C4116yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—24no—
289C2×C42—416no—
290C2×C42—42no—
291C2×C42—42no—
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—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C22×C418yes—
323#95C22×C4116yes—
324#95C22×C4116yes—
325#95C2×D418yes—
326#95C2×D418yes—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24C23×C412yes—
343C24C23×C412yes—
344C242−1+414yes—
345C24C2511yescharacteristic
346C2×Q8—216no—
347C2×Q8—416no—
348C2×Q8—88no—
349C2×Q8—84no—
350C2×Q8—84no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—48no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—216no—
401C22×C4—216no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4C23×C412yes—
437C22×C4C23×C412yes—
438C22×C4C23×C418yes—
439C22×C4C23×C418yes—
440C22×C4C23×C412yes—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C4H(5,𝔽2)14yes—
453C22×C42−1+414yes—
454C4⋊C4—28no—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C4⋊C4—88no—
465C22⋊C4—28no—
466C22⋊C4—28no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C42—28no—
475C42—28no—
476C42—48no—
477C42—48no—
478C42—44no—
479C42—44no—
480C42—84no—
481C42—88no—
482C42C4×D4116yes—
483C23—22no—
484C23—24no—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—24no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—41no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23#51914yes—
525C23C24×C411yescharacteristic
526C23C23×D411yescharacteristic
527C23C22×#4111yescharacteristic
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+412yes—
531C23C2×2−1+412yes—
532C23#58512yes—
533C23#58512yes—
534Q8—816no—
535D4—416no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—432no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
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—42no—
589C2×C4—44no—
590C2×C4—416no—
591C2×C4—416no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4C2×C4×D414yes—
614C2×C4C2×C4×D414yes—
615C2×C4C2×C4×D414yes—
616C2×C4#52018yes—
617C2×C4#53514yes—
618C2×C4#53614yes—
619C2×C4#54218yes—
620C22—24no—
621C22—28no—
622C22—42no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—44no—
632C22—44no—
633C22—42no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22C22×C4×D411yescharacteristic
639C22C2×#51911yescharacteristic
640C22C2×#52012yes—
641C22#342811yescharacteristic
642C22C4×H(5,𝔽2)12yes—
643C22C2×#53511yescharacteristic
644C22C2×#53611yescharacteristic
645C22C2×#54212yes—
646C22#348812yes—
647C22#348912yes—
648C22#349512yes—
649C22#350811yescharacteristic
650C22#351011yescharacteristic
651C22#353612yes—
652C22#354012yes—
653C22#354112yes—
654C22#354212yes—
655C22#354712yes—
656C22#355212yes—
657C22#356312yes—
658C22#356512yes—
659C4—24no—
660C4—24no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—48no—
667C4—48no—
668C4—82no—
669C4—84no—
670C4—84no—
671C4#232618yes—
672C2—44no—
673C2—82no—
674C2C2×#232611yescharacteristic
675C2#3420311yescharacteristic
676C2#3426911yescharacteristic
677C2#3446412yes—
678C2#3446712yes—
679C2#3727112yes—
680C2#3727912yes—
681C2#3733312yes—
682C2#3740812yes—
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. #14697C2semidirect
5. #14575C2semidirect
6. #14522C2semidirect
8. #14426C2semidirect
9. #14264C2semidirect
10. #14155C2semidirect
11. #13994C2semidirect
12. #13993C2semidirect
13. #13954C2semidirect
15. #13797C2semidirect
20. #2563C4semidirect
85. C2×C4⋊Q8C2×C4semidirect
123. #393C2×C4semidirect
138. #382C2×C4semidirect
142. #380C2×C4semidirect
146. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#23312
C2×#22902
#146972
#145752
#145222
#144292
#144262
#142642
#141552
#139942
#139932
#139542
#139052
#137972