Finite Group Databasebeta

← #626269

Subgroups — #626269

Order 512.

3088 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×#2413C212yesmaximal, complemented
2C2×#2618C211yesmaximal, characteristic
3C22×#387C211yesmaximal, characteristic, complemented
4#15607C214yesmaximal, complemented
5#15599C214yesmaximal, complemented
6#15531C214yesmaximal, complemented
7#15518C214yesmaximal, complemented
8#15019C211yesmaximal, characteristic, complemented
9#15014C211yesmaximal, characteristic, complemented
10#14926C211yesmaximal, characteristic
11#14913C211yesmaximal, characteristic
12#14706C212yesmaximal, complemented
13#14259C212yesmaximal, complemented
14#14135C212yesmaximal, complemented
15#13795C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#123C2212yes—
18C22×#121C2211yescharacteristic
19C22×#117C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2618—28no—
24#2413—216no—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#396C2212yes—
28C2×#396C2214yes—
29C2×#396C2212yes—
30C2×#394C2212yes—
31C2×#389C2214yes—
32C2×#388C2214yes—
33C2×#387—28no—
34C2×#387C2214yes—
35C2×#387C2214yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#385C2212yes—
41C2×#381C2212yes—
42C2×#380C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1445C2212yes—
48#1445C2212yes—
49#1444C2212yes—
50#1444C2212yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1441C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1438C2212yes—
65#1437C2214yes—
66#1437C2214yes—
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2214yes—
72C4⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C22⋊#95C2212yes—
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—44no—
93C2×#121—24no—
94C2×#121—44no—
95C2×#117—28no—
96C2×#117—28no—
97C2×#117—44no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4D418yes—
102C22×C4⋊C4D418yescomplemented
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C42—24no—
127C22×C42—44no—
128C22×C42C2314yes—
129C22×C42C2314yes—
130#398—28no—
131#398—216no—
132#396—28no—
133#396—28no—
134#396—216no—
135#394—28no—
136#389—216no—
137#388—216no—
138#387—216no—
139#387—216no—
140#386—28no—
141#386—28no—
142#386—28no—
143#386—44no—
144#385—28no—
145#381—28no—
146#380—24no—
147C2×#95—24no—
148C2×#95—24no—
149C2×#95—24no—
150C2×#95—24no—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C25C2412yes—
177C25C2411yescharacteristic
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—42no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4#4114yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233#123—432no—
234#123—432no—
235#121—416no—
236#117—432no—
237C2×C4⋊C4—232no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—416no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C2×D4116yes—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D418yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4#4118yes—
303C2×C42—28no—
304C2×C42—28no—
305C2×C42—28no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—416no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—42no—
326#95—44no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95—42no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—44no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24C22×D412yes—
351C24C22×D412yes—
352C24C2×#4112yes—
353C24C2×#4114yes—
354C24H(5,𝔽2)14yes—
355C24C2511yescharacteristic
356C2×D4—44no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—82no—
365C2×D4—82no—
366C2×D4—88no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—416no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4#121116yes—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4114yes—
468C22×C42−1+414yes—
469C4⋊C4—232no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—416no—
473C4⋊C4—416no—
474C4⋊C4—416no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C22⋊C4—216no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C42—416no—
493C42—416no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23C2×#12114yes—
530C23#53214yes—
531C23#53414yes—
532C23#54618yes—
533C23#55318yes—
534C23C23×D411yescharacteristic
535C23C23×D411yescharacteristic
536C23C22×#4111yescharacteristic
537C23C22×#4112yes—
538C23C2×H(5,𝔽2)12yes—
539C23C2×2−1+412yes—
540C23#58512yes—
541C23#58512yes—
542C23#58512yes—
543D4—88no—
544D4—816no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—232no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—416no—
571C2×C4—416no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—416no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—88no—
621C2×C4—88no—
622C2×C4C2×#12118yes—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—216no—
628C22—24no—
629C22—24no—
630C22—216no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—84no—
637C22—84no—
638C22—82no—
639C22—82no—
640C22—84no—
641C22—84no—
642C22#254318yes—
643C22C22×#12111yescharacteristic
644C22C2×#53211yescharacteristic
645C22C2×#53411yescharacteristic
646C22#344912yes—
647C22#346112yes—
648C22C2×#54612yes—
649C22C2×#55312yes—
650C22#348511yescharacteristic
651C22#348511yescharacteristic
652C22#348611yescharacteristic
653C22#348711yescharacteristic
654C22#349312yes—
655C22#349712yes—
656C22#351514yes—
657C22#352114yes—
658C22#353314yes—
659C22#353514yes—
660C4—28no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—48no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—84no—
673C2—28no—
674C2—162no—
675C2C2×#254311yescharacteristic
676C2#3515011yescharacteristic
677C2#3515311yescharacteristic
678C2#3546811yescharacteristic
679C2#3546911yescharacteristic
680C2#3696511yescharacteristic
681C2#3718611yescharacteristic
682C2#3808714yes—
683C2#3905114yes—
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×#2413C2semidirect
3. C22×#387C2semidirect
4. #15607C2semidirect
5. #15599C2semidirect
6. #15531C2semidirect
7. #15518C2semidirect
8. #15019C2semidirect
9. #15014C2semidirect
12. #14706C2semidirect
13. #14259C2semidirect
14. #14135C2semidirect
15. #13795C2semidirect
102. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24132
C2×#26182
C22×#3872
#156072
#155992
#155312
#155182
#150192
#150142
#149262
#149132
#147062
#142592
#141352
#137952