Finite Group Databasebeta

← #515981

Subgroups — #515981

Order 512.

3110 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×#2290C211yesmaximal, characteristic
3C2×#2274C211yesmaximal, characteristic, complemented
4#14180C214yesmaximal, complemented
5#14178C214yesmaximal, complemented
6#14129C211yesmaximal, characteristic, complemented
7#14042C212yesmaximal, complemented
8#14008C214yesmaximal, complemented
9#13993C212yesmaximal, complemented
10#13984C214yesmaximal
11#13904C211yesmaximal, characteristic, complemented
12#13871C211yesmaximal, characteristic
13#13783C212yesmaximal, complemented
14#13777C211yesmaximal, characteristic, complemented
15C4×#380C212yesmaximal, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2487C4116yescomplemented
22C2×#400C2212yes—
23C2×#399C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26#2290—28no—
27C2×#388C2212yes—
28C2×#387C2212yes—
29C2×#387C2214yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#381C2214yes—
35C2×#380C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39#2274—28no—
40C2×C4×C4⋊C4C2214yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2214yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2214yes—
49C2×#376C2211yescharacteristic
50#1445C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1441C2214yes—
54#1440C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2212yes—
58#1437C2212yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74#1434C2214yes—
75#1434C2214yes—
76C22⋊#95C2214yes—
77C4×#95C2214yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4C2×C418yes—
86C2×#117—24no—
87C2×#117—28no—
88C2×#117—28no—
89C2×#117D418yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4C2×C418yescomplemented
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C42—24no—
114C22×C42—28no—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2314yes—
122C22×C42C2314yes—
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125#400C2×C4116yescomplemented
126#399C2×C4116yes—
127#389—28no—
128#389C2×C4116yescomplemented
129#388—28no—
130#387—28no—
131#387—48no—
132#386—28no—
133#386C2×C4116yes—
134#385—28no—
135#385C2×C4116yescomplemented
136#381—48no—
137#380—24no—
138#380—28no—
139#380C2×C4116yes—
140#380C2×C418yescomplemented
141C4×C4⋊C4—216no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—28no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—216no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4—48no—
150#376—24no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—42no—
181C22×D4C22×C414yes—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4C22×C414yes—
200C23×C4C22×C414yes—
201C23×C4C22×C418yes—
202C23×C4C2×D414yes—
203C23×C4#4114yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C4×D4—232no—
218#117—232no—
219#117—232no—
220C2×C4⋊C4—232no—
221C2×C4⋊C4—24no—
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—22no—
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—28no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C22×C418yes—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C414yes—
247C2×C4⋊C4C22×C418yes—
248C2×C4⋊C4C22×C4116yes—
249C2×C4⋊C4C2×D418yes—
250C2×C4⋊C4#4118yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4C22×C414yes—
274C2×C22⋊C4C22×C414yes—
275C2×C22⋊C4C22×C4116yes—
276C2×C22⋊C4C22×C418yes—
277C2×C22⋊C4C2×D418yes—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—416no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42C22×C414yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42#4118yes—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95C22×C4116yes—
326#95C22×C418yes—
327#95C22×C4116yes—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24C23×C412yes—
342C24C23×C412yes—
343C24C22×D412yes—
344C24C2×#4112yes—
345C242−1+414yes—
346C24C2511yescharacteristic
347C2×D4—28no—
348C2×D4—28no—
349C2×D4—44no—
350C2×D4—48no—
351C2×D4—48no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—216no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—216no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—416no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—88no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4C4×D4116yes—
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×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C2×#4118yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4H(5,𝔽2)14yes—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—28no—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—88no—
465C22⋊C4—216no—
466C22⋊C4—28no—
467C22⋊C4—28no—
468C22⋊C4—216no—
469C22⋊C4—48no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—84no—
475C42—216no—
476C42—28no—
477C42—28no—
478C42—48no—
479C42—44no—
480C42—44no—
481C42—88no—
482C42—88no—
483C42—88no—
484C42#123116yes—
485C23—22no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23C2×C4×D414yes—
526C23#51914yes—
527C23C24×C411yescharacteristic
528C23C23×D411yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4112yes—
531C23C22×#4112yes—
532C23C2×H(5,𝔽2)12yes—
533C23C2×2−1+412yes—
534C23#58512yes—
535C23#58512yes—
536D4—416no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—232no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—28no—
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—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—416no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—816no—
615C2×C4C2×C4×D414yes—
616C2×C4C2×C4×D414yes—
617C2×C4C4×#4118yes—
618C2×C4C2×#12318yes—
619C2×C4C2×#12314yes—
620C2×C4#52514yes—
621C2×C4#52918yes—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—28no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22—42no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22C22×C4×D411yescharacteristic
641C22C2×C4×#4112yes—
642C22C2×#51911yescharacteristic
643C22#342811yescharacteristic
644C22C4×H(5,𝔽2)12yes—
645C22C22×#12311yescharacteristic
646C22C2×#52511yescharacteristic
647C22C2×#52912yes—
648C22#348812yes—
649C22#348912yes—
650C22#349212yes—
651C22#349312yes—
652C22#349412yes—
653C22#349512yes—
654C22#349512yes—
655C22#350811yescharacteristic
656C22#351011yescharacteristic
657C22#351612yes—
658C22#351712yes—
659C22#352012yes—
660C22#352112yes—
661C4—28no—
662C4—24no—
663C4—42no—
664C4—44no—
665C4—44no—
666C4—48no—
667C4—42no—
668C4—48no—
669C4—42no—
670C4—48no—
671C4—84no—
672C4—82no—
673C4C4×#12318yes—
674C2—44no—
675C2—44no—
676C2C2×C4×#12311yescharacteristic
677C2#3420311yescharacteristic
678C2#3426911yescharacteristic
679C2#3429912yes—
680C2#3430112yes—
681C2#3551612yes—
682C2#3552112yes—
683C2#3559112yes—
684C2#3564812yes—
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
3. C2×#2274C2semidirect
4. #14180C2semidirect
5. #14178C2semidirect
6. #14129C2semidirect
7. #14042C2semidirect
8. #14008C2semidirect
9. #13993C2semidirect
11. #13904C2semidirect
13. #13783C2semidirect
14. #13777C2semidirect
15. C4×#380C2semidirect
21. #2487C4semidirect
92. C22×C4⋊C4C2×C4semidirect
125. #400C2×C4semidirect
128. #389C2×C4semidirect
135. #385C2×C4semidirect
140. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#22902
C2×#22742
#141802
#141782
#141292
#140422
#140082
#139932
#139842
#139042
#138712
#137832
#137772
C4×#3802