Finite Group Databasebeta

← #847686

Subgroups — #847686

Order 512.

3016 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C22×#384C211yesmaximal, characteristic, complemented
3#16570C212yesmaximal, complemented
4#16514C212yesmaximal, complemented
5#15798C212yesmaximal, complemented
6#15643C212yesmaximal, complemented
7#15396C214yesmaximal, complemented
8#15091C212yesmaximal, complemented
9#14729C212yesmaximal, complemented
10#14719C212yesmaximal
11#14696C212yesmaximal, complemented
12#14413C214yesmaximal, complemented
13#14039C212yesmaximal, complemented
14#14019C212yesmaximal, complemented
15#13923C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2591—28no—
23C2×#400C2211yescharacteristic
24C2×#400C2211yescharacteristic
25C2×#396C2214yes—
26C2×#394C2211yescharacteristic
27C2×#394C2211yescharacteristic
28C2×#394C2214yes—
29C2×#388C2212yes—
30C2×#388C2212yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#387C2214yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#385C2212yes—
41C2×#384—28no—
42C2×#384C2212yes—
43C2×#384C2212yes—
44C2×#384C2212yes—
45C2×#384C2212yes—
46C2×#380C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2212yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1437C2214yes—
66#1436C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2212yes—
70#1436C2214yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#121—24no—
85C2×#121—42no—
86C2×#121—42no—
87C22×C4⋊C4—28no—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—48no—
91C22×C4⋊C4—48no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4—42no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4D418yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42—24no—
123C22×C42—28no—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128#400—24no—
129#400—24no—
130#396—48no—
131#394—24no—
132#394—24no—
133#394—48no—
134#388—28no—
135#388—28no—
136#388—216no—
137#388—48no—
138#387—48no—
139#386—28no—
140#386—28no—
141#386—28no—
142#386—28no—
143#386—44no—
144#386—44no—
145#385—28no—
146#384—28no—
147#384—28no—
148#384—28no—
149#384—28no—
150#380—42no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—42no—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
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×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
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—41no—
187C22×D4—41no—
188C22×D4—44no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4#4118yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2414yes—
225#121—416no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—432no—
260C2×C4⋊C4—432no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D414yes—
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—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
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×C42—232no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—42no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42#4118yes—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333C24—22no—
334C24—24no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—22no—
346C24—22no—
347C24C22×D412yes—
348C24C2×#4114yes—
349C24H(5,𝔽2)12yes—
350C24H(5,𝔽2)12yes—
351C242−1+414yes—
352C24C2511yescharacteristic
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4—84no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—232no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—28no—
370C22×C4—216no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—232no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4118yes—
455C22×C4C2×#4118yes—
456C22×C4H(5,𝔽2)18yes—
457C22×C42−1+414yes—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—416no—
463C4⋊C4—416no—
464C4⋊C4—416no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—416no—
470C4⋊C4—84no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—416no—
478C22⋊C4—416no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C42—216no—
484C42—216no—
485C42—84no—
486C42—84no—
487C23—22no—
488C23—22no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23#53414yes—
522C23#54218yes—
523C23#54818yes—
524C23#55514yes—
525C23#56314yes—
526C23C23×D411yescharacteristic
527C23C22×#4112yes—
528C23C22×#4112yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×H(5,𝔽2)14yes—
532C23C2×2−1+412yes—
533C23C2×2−1+412yes—
534D4—88no—
535D4—816no—
536C2×C4—264no—
537C2×C4—232no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
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—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
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—88no—
607C2×C4—88no—
608C2×C4—816no—
609C2×C4#52514yes—
610C2×C4#53614yes—
611C2×C4#553116yes—
612C22—216no—
613C22—28no—
614C22—24no—
615C22—28no—
616C22—28no—
617C22—24no—
618C22—28no—
619C22—84no—
620C22—84no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—84no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—84no—
630C22#273118yes—
631C22C2×#52511yescharacteristic
632C22C2×#53411yescharacteristic
633C22C2×#53611yescharacteristic
634C22C2×#54212yes—
635C22C2×#54812yes—
636C22C2×#55314yes—
637C22C2×#55511yescharacteristic
638C22#348412yes—
639C22#349212yes—
640C22#349214yes—
641C22#349212yes—
642C22#351114yes—
643C22C2×#56311yescharacteristic
644C22#354312yes—
645C22#354312yes—
646C22#355214yes—
647C4—216no—
648C4—24no—
649C4—24no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—84no—
655C4—82no—
656C4—84no—
657C4—84no—
658C4—82no—
659C4—82no—
660C2—28no—
661C2—162no—
662C2#3505711yescharacteristic
663C2#3658112yes—
664C2#3723211yescharacteristic
665C2#3789214yes—
666C2#3972012yes—
667C2#4632612yes—
668C2C2×#273111yescharacteristic
669C2#5930312yes—
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×#384C2semidirect
3. #16570C2semidirect
4. #16514C2semidirect
5. #15798C2semidirect
6. #15643C2semidirect
7. #15396C2semidirect
8. #15091C2semidirect
9. #14729C2semidirect
11. #14696C2semidirect
12. #14413C2semidirect
13. #14039C2semidirect
14. #14019C2semidirect
15. #13923C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C22×#3842
#165702
#165142
#157982
#156432
#153962
#150912
#147292
#147192
#146962
#144132
#140392
#140192
#139232