Finite Group Databasebeta

← #729357

Subgroups — #729357

Order 512.

2824 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2C2×#2618C211yesmaximal, characteristic
3C22×#400C211yesmaximal, characteristic, complemented
4C2×#2345C211yesmaximal, characteristic, complemented
5#16595C214yesmaximal, complemented
6#15885C214yesmaximal, complemented
7#15815C214yesmaximal, complemented
8#14927C212yesmaximal, complemented
9#14921C212yesmaximal, complemented
10#14853C214yesmaximal, complemented
11#14152C211yesmaximal, characteristic
12#14149C211yesmaximal, characteristic, complemented
13#14020C212yesmaximal, complemented
14#13996C212yesmaximal, complemented
15#13776C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2618—28no—
21#2504—28no—
22C2×#400—28no—
23C2×#400C2212yes—
24C2×#400C2214yes—
25C2×#400C2212yes—
26C2×#398C2212yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29C2×#396C2214yes—
30C2×#394C2212yes—
31#2345—28no—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#387C2214yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#385C2212yes—
41C2×#382C2212yes—
42C2×#381C2211yescharacteristic
43C2×#381C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C22×#95C2211yescharacteristic
51C22×#95C2212yes—
52C22×#95C2212yes—
53#1445C2212yes—
54#1445C2214yes—
55#1443C2214yes—
56#1443C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2212yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2214yes—
75#1434C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2211yescharacteristic
78C22⋊#95C2211yescharacteristic
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C2×#123—28no—
87C2×#123—28no—
88C2×#123—44no—
89C2×#121—24no—
90C2×#121—24no—
91C2×#121—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4D418yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125#400—216no—
126#400—28no—
127#400—28no—
128#398—28no—
129#397—28no—
130#397—28no—
131#396—216no—
132#394—28no—
133#390—28no—
134#389—28no—
135#387—216no—
136#386—28no—
137#386—28no—
138#386—28no—
139#386—28no—
140#386—44no—
141#385—44no—
142#382—28no—
143#381—24no—
144#381—24no—
145#380—28no—
146#380D418yes—
147C4×C4⋊C4—28no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—28no—
151C2×#95—28no—
152C2×#95—28no—
153C2×#95—24no—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C25C2412yes—
180C25C2411yescharacteristic
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—44no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4C2×D418yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4#4114yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230#123—432no—
231#121—416no—
232C2×C4⋊C4—232no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—416no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D418yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
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—44no—
288C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
300C2×C22⋊C4#4114yes—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—44no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95C2×D418yes—
333C24—24no—
334C24—24no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24C22×D412yes—
347C24C22×D412yes—
348C24C2×#4112yes—
349C24C2×#4112yes—
350C24C2×#4112yes—
351C24H(5,𝔽2)12yes—
352C242−1+412yes—
353C24C2511yescharacteristic
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—44no—
363C2×D4—84no—
364C2×D4—84no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—216no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4#121116yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D418yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C22⋊C4—216no—
472C22⋊C4—28no—
473C22⋊C4—416no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—48no—
478C22⋊C4—88no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C42—48no—
486C42—48no—
487C42—88no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23C2×#12114yes—
522C23#52914yes—
523C23#54214yes—
524C23#54614yes—
525C23#54814yes—
526C23#54814yes—
527C23#55014yes—
528C23C23×D411yescharacteristic
529C23C23×D411yescharacteristic
530C23C22×#4111yescharacteristic
531C23C22×#4111yescharacteristic
532C23C22×#4111yescharacteristic
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×2−1+411yescharacteristic
535C23#58514yes—
536C23#58514yes—
537D4—88no—
538D4—816no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—416no—
560C2×C4—42no—
561C2×C4—416no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
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—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
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—816no—
611C2×C4—88no—
612C2×C4—816no—
613C2×C4C2×#12114yes—
614C2×C4C2×#12114yes—
615C22—24no—
616C22—24no—
617C22—28no—
618C22—28no—
619C22—28no—
620C22—28no—
621C22—28no—
622C22—28no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—82no—
632C22—82no—
633C22—84no—
634C22#242618yes—
635C22C22×#12111yescharacteristic
636C22C2×#52911yescharacteristic
637C22#344912yes—
638C22C2×#54211yescharacteristic
639C22C2×#54611yescharacteristic
640C22C2×#54811yescharacteristic
641C22C2×#54811yescharacteristic
642C22C2×#55011yescharacteristic
643C22#348512yes—
644C22#348512yes—
645C22#349112yes—
646C22#350012yes—
647C22#351012yes—
648C22#351214yes—
649C22#352114yes—
650C22#355314yes—
651C22#356014yes—
652C4—24no—
653C4—24no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—48no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—84no—
665C2—28no—
666C2—162no—
667C2C2×#242611yescharacteristic
668C2#3508911yescharacteristic
669C2#3561711yescharacteristic
670C2#3603412yes—
671C2#3912912yes—
672C2#3971712yes—
673C2#4139212yes—
674C2#4349812yes—
675C2#4444612yes—
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
3. C22×#400C2semidirect
4. C2×#2345C2semidirect
5. #16595C2semidirect
6. #15885C2semidirect
7. #15815C2semidirect
8. #14927C2semidirect
9. #14921C2semidirect
10. #14853C2semidirect
12. #14149C2semidirect
13. #14020C2semidirect
14. #13996C2semidirect
15. #13776C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#26182
C22×#4002
C2×#23452
#165952
#158852
#158152
#149272
#149212
#148532
#141522
#141492
#140202
#139962
#137762