Finite Group Databasebeta

← #4380257

Subgroups — #4380257

Order 512.

3414 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic, complemented
2C2×#2406C211yesmaximal, characteristic
3C2×#2616C211yesmaximal, characteristic
4C2×#2316C211yesmaximal, characteristic, complemented
5#16257C214yesmaximal, complemented
6#15623C214yesmaximal, complemented
7#15615C214yesmaximal, complemented
8#15614C214yesmaximal, complemented
9#15436C214yesmaximal, complemented
10#14890C214yesmaximal, complemented
11#14660C211yesmaximal, characteristic, complemented
12#14297C211yesmaximal, characteristic, complemented
13#13937C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C22×#123C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20#2616—28no—
21#2583—28no—
22#2406—28no—
23C2×#398C2212yes—
24C2×#398C2214yes—
25C2×#396C2214yes—
26C2×#396C2214yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#394C2214yes—
31C2×#394C2212yes—
32C2×#394C2212yes—
33#2316—28no—
34C2×#391C2212yes—
35C2×#390C2212yes—
36C2×#389C2214yes—
37C2×#389C2214yes—
38C2×#389C2214yes—
39C2×#388C2214yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2214yes—
44C2×#381C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#1446C2212yes—
51#1445C2212yes—
52#1444C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2214yes—
66#1437C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2214yes—
70C22⋊#95C2214yes—
71C22×#41—24no—
72C22×#41—24no—
73C22×#41—24no—
74C22×#41—44no—
75C22×#41—44no—
76C23×Q8C2311yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#123—28no—
85C2×#123—44no—
86C2×#123—44no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—44no—
92C2×C22⋊Q8—44no—
93C2×#117—28no—
94C2×#117—44no—
95C2×#117—44no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4D418yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125#398—28no—
126#398—48no—
127#396—48no—
128#396—48no—
129#395—28no—
130#395—28no—
131#395—28no—
132#394—28no—
133#394—216no—
134#394—28no—
135#391—44no—
136#390—44no—
137#389—216no—
138#389—216no—
139#389—216no—
140#388—48no—
141#386—28no—
142#385—28no—
143#385—28no—
144#385—216no—
145#381—24no—
146#380—24no—
147#380—28no—
148#380—28no—
149#380D418yes—
150C4×C22⋊C4—24no—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C25C2412yes—
171C25C2411yescharacteristic
172C2×#41—432no—
173C2×#41—432no—
174C2×#41—416no—
175C2×#41—416no—
176C2×#41—416no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—22no—
183C22×Q8—24no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—42no—
197C22×D4—42no—
198C22×D4—42no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4#4114yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232#123—432no—
233C22⋊Q8—432no—
234C22⋊Q8—416no—
235#117—432no—
236C2×C4⋊C4—232no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D418yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4C2×D418yes—
308C2×C42—28no—
309C2×C42—28no—
310C2×C42—24no—
311C2×C42—44no—
312C2×C42—42no—
313C2×C42—42no—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95C2×D418yes—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24C22×D412yes—
349C24C22×D412yes—
350C24C2×#4112yes—
351C24H(5,𝔽2)14yes—
352C24C2511yescharacteristic
353#41—4128no—
354C2×Q8—28no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—44no—
358C2×Q8—48no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—84no—
362C2×Q8—82no—
363C2×Q8—82no—
364C2×D4—216no—
365C2×D4—28no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—44no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—44no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—216no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
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—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—84no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4#121116yes—
452C22×C4C22×D418yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4H(5,𝔽2)14yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C42−1+418yes—
464C22×C42−1+414yes—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—48no—
468C4⋊C4—416no—
469C4⋊C4—416no—
470C4⋊C4—416no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—84no—
475C4⋊C4—88no—
476C4⋊C4—88no—
477C4⋊C4—88no—
478C22⋊C4—216no—
479C22⋊C4—48no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—416no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C42—416no—
490C42—84no—
491C23—24no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23C2×#12114yes—
530C23C23×D411yescharacteristic
531C23C23×D411yescharacteristic
532C23C22×#4111yescharacteristic
533C23C2×H(5,𝔽2)12yes—
534C23C2×H(5,𝔽2)12yes—
535C23C2×H(5,𝔽2)12yes—
536C23C2×2−1+412yes—
537C23C2×2−1+414yes—
538Q8—416no—
539Q8—88no—
540D4—416no—
541D4—432no—
542C2×C4—232no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—416no—
569C2×C4—432no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
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—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
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—88no—
610C2×C4C2×#12114yes—
611C2×C4C2×#12114yes—
612C2×C4#53718yes—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—48no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—48no—
625C22—48no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—84no—
631C22—84no—
632C22C22×#12111yescharacteristic
633C22C2×#53712yes—
634C22#348412yes—
635C22#348412yes—
636C22#348411yescharacteristic
637C22#348612yes—
638C22#348611yescharacteristic
639C22#349011yescharacteristic
640C22#349011yescharacteristic
641C22#349212yes—
642C22#349912yes—
643C22#350912yes—
644C22#353614yes—
645C22#354414yes—
646C22#354814yes—
647C22#355214yes—
648C4—24no—
649C4—24no—
650C4—28no—
651C4—44no—
652C4—44no—
653C4—48no—
654C4—48no—
655C4—82no—
656C4—84no—
657C4—82no—
658C4—84no—
659C2—44no—
660C2—84no—
661C2#3522311yescharacteristic
662C2#3534811yescharacteristic
663C2#3566711yescharacteristic
664C2#4247714yes—
665C2#4642312yes—
666C2#4642612yes—
667C2#4959112yes—
668C2#4959412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2583C2semidirect
4. C2×#2316C2semidirect
5. #16257C2semidirect
6. #15623C2semidirect
7. #15615C2semidirect
8. #15614C2semidirect
9. #15436C2semidirect
10. #14890C2semidirect
11. #14660C2semidirect
12. #14297C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#24062
C2×#26162
C2×#23162
#162572
#156232
#156152
#156142
#154362
#148902
#146602
#142972
#139372