Finite Group Databasebeta

← #4452741

Subgroups — #4452741

Order 512.

2756 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic
2C22×#398C211yesmaximal, characteristic, complemented
3#16999C214yesmaximal, complemented
4#16956C214yesmaximal, complemented
5#16863C214yesmaximal, complemented
6#16861C214yesmaximal, complemented
7#15091C212yesmaximal, complemented
8#15015C211yesmaximal, characteristic, complemented
9#14913C211yesmaximal, characteristic
10#14822C211yesmaximal, characteristic, complemented
11#14776C212yesmaximal, complemented
12#14719C211yesmaximal, characteristic
13#14135C212yesmaximal, complemented
14#14135C212yesmaximal, complemented
15#13795C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2618—28no—
24C2×#400C2212yes—
25C2×#398—28no—
26C2×#398C2214yes—
27C2×#398C2212yes—
28C2×#398C2214yes—
29C2×#397C2214yes—
30C2×#396C2212yes—
31C2×#396C2212yes—
32C2×#394C2212yes—
33C2×#394C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#380C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43#1445C2212yes—
44#1445C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2212yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2212yes—
55#1441C2212yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2212yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#123—28no—
87C2×#123—28no—
88C2×#123—28no—
89C2×#121—24no—
90C2×#121—44no—
91C2×#117—28no—
92C2×#117—44no—
93C2×#117—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4D418yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C42C2314yes—
127#400—44no—
128#398—28no—
129#398—216no—
130#398—216no—
131#397—48no—
132#396—28no—
133#396—28no—
134#394—28no—
135#394—44no—
136#386—28no—
137#386—44no—
138#386—44no—
139#386—44no—
140#386—44no—
141#380—24no—
142C2×#95—24no—
143C2×#95—24no—
144C2×#95—24no—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—42no—
181C22×D4—42no—
182C22×D4—42no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4#4118yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227#123—432no—
228#121—416no—
229#117—432no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—416no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—416no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4C2×D418yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
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—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
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×C42—28no—
302C2×C42—44no—
303C2×C42—44no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—42no—
328#95—44no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24C22×D412yes—
345C24C2×#4114yes—
346C24H(5,𝔽2)14yes—
347C24H(5,𝔽2)12yes—
348C242−1+412yes—
349C24C2511yescharacteristic
350C2×D4—44no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—82no—
359C2×D4—82no—
360C2×D4—88no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—416no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—416no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—88no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—88no—
456C22×C4#124116yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4114yes—
463C22×C4H(5,𝔽2)14yes—
464C22×C42−1+414yes—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—416no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—416no—
471C4⋊C4—416no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C42—416no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23C2×#12414yes—
526C23#53414yes—
527C23#53714yes—
528C23#53818yes—
529C23#53918yes—
530C23C23×D411yescharacteristic
531C23C22×#4112yes—
532C23C2×H(5,𝔽2)12yes—
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×H(5,𝔽2)12yes—
535C23C2×2−1+411yescharacteristic
536C23C2×2−1+412yes—
537C23#58512yes—
538C23#58512yes—
539D4—88no—
540D4—816no—
541C2×C4—232no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—416no—
563C2×C4—416no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
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—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—88no—
618C2×C4—88no—
619C2×C4—88no—
620C2×C4C2×#12418yes—
621C22—24no—
622C22—216no—
623C22—216no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—82no—
630C22—82no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22#263918yes—
641C22C22×#12411yescharacteristic
642C22C2×#53411yescharacteristic
643C22C2×#53711yescharacteristic
644C22C2×#53812yes—
645C22C2×#53912yes—
646C22#348411yescharacteristic
647C22#348511yescharacteristic
648C22#348611yescharacteristic
649C22#348711yescharacteristic
650C22#351114yes—
651C22#351214yes—
652C22#353412yes—
653C22#353512yes—
654C22#353514yes—
655C22#354214yes—
656C22#355912yes—
657C22#356312yes—
658C4—28no—
659C4—44no—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4—84no—
668C4—82no—
669C4—82no—
670C4—82no—
671C2—28no—
672C2—162no—
673C2C2×#263911yescharacteristic
674C2#3559211yescharacteristic
675C2#3564911yescharacteristic
676C2#3671511yescharacteristic
677C2#3672211yescharacteristic
678C2#3696811yescharacteristic
679C2#3718611yescharacteristic
680C2#4202214yes—
681C2#4202414yes—
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
2. C22×#398C2semidirect
3. #16999C2semidirect
4. #16956C2semidirect
5. #16863C2semidirect
6. #16861C2semidirect
7. #15091C2semidirect
8. #15015C2semidirect
10. #14822C2semidirect
11. #14776C2semidirect
13. #14135C2semidirect
14. #14135C2semidirect
15. #13795C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
C22×#3982
#169992
#169562
#168632
#168612
#150912
#150152
#149132
#148222
#147762
#147192
#141352
#137952