Finite Group Databasebeta

← #4930238

Subgroups — #4930238

Order 512.

2426 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic
3#15102C212yesmaximal, complemented
4#15101C212yesmaximal, complemented
5#15088C212yesmaximal, complemented
6#15080C212yesmaximal, complemented
7#14829C214yesmaximal
8#14740C214yesmaximal
9#14258C211yesmaximal, characteristic
10#14142C212yesmaximal, complemented
11#14139C211yesmaximal, characteristic
12#14130C211yesmaximal, characteristic
13#14021C212yesmaximal, complemented
14#13981C212yesmaximal, complemented
15#13939C211yesmaximal, characteristic
16#13885C212yesmaximal, complemented
17#13844C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2499—28no—
23C2×#397C2212yes—
24C2×#397C2212yes—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#389C2212yes—
30C2×#387C2212yes—
31C2×#387C2212yes—
32C2×#387C2214yes—
33C2×#385C2212yes—
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#380C2211yescharacteristic
37#2274—28no—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2214yes—
46#1445C2214yes—
47#1444C2211yescharacteristic
48#1444C2211yescharacteristic
49#1444C2212yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2211yescharacteristic
54#1442C2211yescharacteristic
55#1442C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2212yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2212yes—
67#1439C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77#1434C2214yes—
78#1434C2214yes—
79#1434C2212yes—
80#1434C2212yes—
81#1434C2214yes—
82C22⋊#95C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—28no—
95C2×#117D418yes—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2314yes—
126#397—28no—
127#397—28no—
128#391—24no—
129#391—24no—
130#391—28no—
131#391—28no—
132#389—28no—
133#387—28no—
134#387—28no—
135#387—48no—
136#385—44no—
137#384—24no—
138#382—24no—
139#380—24no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—28no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—216no—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2411yescharacteristic
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4C2×D414yes—
196C23×C4#4114yes—
197C23×C4#4114yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C22⋊Q8—416no—
215#117—232no—
216#117—232no—
217C2×C4⋊C4—232no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4#4118yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4C2×D418yes—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—28no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42C2×D418yes—
296C2×C42#4118yes—
297#95—28no—
298#95—28no—
299#95—22no—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—22no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—24no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—21no—
335C24—22no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24C22×D412yes—
340C24C2×#4112yes—
341C24C2×#4112yes—
342C24C2511yescharacteristic
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×Q8—48no—
347C2×Q8—48no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—216no—
353C22×C4—216no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—216no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4#123116yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4118yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C42−1+414yes—
453C22×C42−1+412yes—
454C4⋊C4—216no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—216no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C42—216no—
476C42—216no—
477C42—44no—
478C42—44no—
479C42—48no—
480C42—88no—
481C42—88no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23C2×#12314yes—
505C23C23×D411yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4111yescharacteristic
508C23C22×#4112yes—
509C23C22×#4112yes—
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×2−1+411yescharacteristic
512C23C2×2−1+412yes—
513C23#58511yescharacteristic
514C23#58511yescharacteristic
515C23#58512yes—
516Q8—88no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—232no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—816no—
609C2×C4C2×#12314yes—
610C2×C4C2×#12314yes—
611C2×C4#52514yes—
612C2×C4#52514yes—
613C2×C4#52918yes—
614C2×C4#52918yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—42no—
625C22—44no—
626C22C22×#12311yescharacteristic
627C22C2×#52511yescharacteristic
628C22C2×#52511yescharacteristic
629C22C2×#52912yes—
630C22C2×#52912yes—
631C22#348811yescharacteristic
632C22#348911yescharacteristic
633C22#349312yes—
634C22#349512yes—
635C22#349512yes—
636C22#349911yescharacteristic
637C22#350011yescharacteristic
638C22#350611yescharacteristic
639C22#350712yes—
640C22#350712yes—
641C22#350711yescharacteristic
642C22#351111yescharacteristic
643C22#351112yes—
644C22#351211yescharacteristic
645C22#351312yes—
646C22#351612yes—
647C22#352112yes—
648C22#352312yes—
649C4—24no—
650C4—44no—
651C4—44no—
652C4—42no—
653C4—48no—
654C4—42no—
655C4—44no—
656C4—44no—
657C4—44no—
658C4—82no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4—81no—
663C4—84no—
664C4#285318yes—
665C2—44no—
666C2#3551711yescharacteristic
667C2#3559612yes—
668C2#3561911yescharacteristic
669C2#3564712yes—
670C2#3575911yescharacteristic
671C2#3578912yes—
672C2#3583812yes—
673C2#3598211yescharacteristic
674C2#3599711yescharacteristic
675C2#3608911yescharacteristic
676C2C2×#285311yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #15102C2semidirect
4. #15101C2semidirect
5. #15088C2semidirect
6. #15080C2semidirect
10. #14142C2semidirect
13. #14021C2semidirect
14. #13981C2semidirect
16. #13885C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
C2×#22742
#151022
#151012
#150882
#150802
#148292
#147402
#142582
#141422
#141392
#141302
#140212
#139812
#139392
#138852
#138442