Finite Group Databasebeta

← #4307559

Subgroups — #4307559

Order 512.

2506 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2360C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#15213C212yesmaximal, complemented
4#15194C211yesmaximal, characteristic, complemented
5#15186C212yesmaximal, complemented
6#15179C212yesmaximal
7#15153C211yesmaximal, characteristic, complemented
8#15135C211yesmaximal, characteristic
9#15024C212yesmaximal, complemented
10#15016C212yesmaximal
11#14781C214yesmaximal, complemented
12#14746C214yesmaximal
13C4×#399C212yesmaximal
14C4×#397C211yesmaximal, characteristic
15#13821C212yesmaximal, complemented
16#13817C211yesmaximal, characteristic, complemented
17C22⋊C4×C4⋊C4C212yesmaximal
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#400C2214yes—
22C2×#399C2212yes—
23C2×#397C2211yescharacteristic
24#2360—28no—
25#2355—28no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#387C2214yes—
34C2×#387C2212yes—
35C2×#385C2212yes—
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2214yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2214yes—
49#1445C2212yes—
50#1445C2211yescharacteristic
51#1445C2212yes—
52#1445C2214yes—
53#1445C2212yes—
54#1445C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1443C2212yes—
58#1443C2214yes—
59#1443C2214yes—
60#1442C2211yescharacteristic
61#1442C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2212yes—
66#1440C2214yes—
67#1439C2212yes—
68#1439C2211yescharacteristic
69#1439C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2212yes—
80#1434C2214yes—
81C22⋊#95C2214yes—
82C4×#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C4×Q8—28no—
92C2×C4×Q8D418yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2314yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2314yes—
123C22×C42C2312yes—
124#400—48no—
125#399—28no—
126#397—24no—
127#391—28no—
128#391—24no—
129#391—28no—
130#389—28no—
131#389—28no—
132#389—28no—
133#389—24no—
134#387—216no—
135#387—28no—
136#385—28no—
137#384—24no—
138#382—28no—
139#380—28no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—24no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—48no—
146C4×C22⋊C4—28no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—48no—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C25C2411yescharacteristic
184C22×Q8—24no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8C2×D414yes—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4#4114yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C22⋊Q8—232no—
216C4×Q8—232no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D418yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4#4114yes—
254C2×C4⋊C4#4118yes—
255C2×C4⋊C4#4118yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4#4114yes—
274C2×C42—24no—
275C2×C42—28no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42C2×D418yes—
293C2×C42C2×D414yes—
294#95—22no—
295#95—24no—
296#95—22no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—22no—
320#95—24no—
321#95—42no—
322#95—44no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—21no—
333C24—22no—
334C24—21no—
335C24C2×#4112yes—
336C24C2×#4112yes—
337C242−1+412yes—
338C24C2511yescharacteristic
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8—48no—
343C2×Q8—48no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—216no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—84no—
427C22×C4—88no—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C2×#4118yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4118yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4114yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C42−1+414yes—
445C22×C42−1+412yes—
446C4⋊C4—28no—
447C4⋊C4—216no—
448C4⋊C4—216no—
449C4⋊C4—28no—
450C4⋊C4—216no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4#123116yes—
461C22⋊C4—216no—
462C22⋊C4—216no—
463C22⋊C4—28no—
464C22⋊C4—416no—
465C22⋊C4—44no—
466C22⋊C4—48no—
467C42—28no—
468C42—216no—
469C42—416no—
470C42—48no—
471C42—48no—
472C42—44no—
473C42—84no—
474C42—88no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—41no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23#55614yes—
498C23C23×D411yescharacteristic
499C23C22×#4112yes—
500C23C22×#4111yescharacteristic
501C23C22×#4111yescharacteristic
502C23C22×#4112yes—
503C23C22×#4111yescharacteristic
504C23C2×H(5,𝔽2)12yes—
505C23C2×2−1+411yescharacteristic
506C23C2×2−1+411yescharacteristic
507C23C2×2−1+412yes—
508C23#58511yescharacteristic
509Q8—416no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—416no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—816no—
597C2×C4C2×#12318yes—
598C2×C4C2×#12314yes—
599C2×C4#52514yes—
600C2×C4#52918yes—
601C2×C4#52918yes—
602C2×C4#53214yes—
603C2×C4#54814yes—
604C22—28no—
605C22—24no—
606C22—44no—
607C22—44no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22C22×#12311yescharacteristic
616C22C2×#52511yescharacteristic
617C22C2×#52912yes—
618C22C2×#52912yes—
619C22C2×#53211yescharacteristic
620C22C2×#54811yescharacteristic
621C22C2×#55611yescharacteristic
622C22#348011yescharacteristic
623C22#348112yes—
624C22#348212yes—
625C22#348611yescharacteristic
626C22#348911yescharacteristic
627C22#349212yes—
628C22#349811yescharacteristic
629C22#350711yescharacteristic
630C22#351312yes—
631C22#351512yes—
632C22#352012yes—
633C22#352012yes—
634C22#352212yes—
635C22#352311yescharacteristic
636C22#352312yes—
637C22#352312yes—
638C4—24no—
639C4—28no—
640C4—48no—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—44no—
648C4—81no—
649C4—82no—
650C4—84no—
651C4—82no—
652C4—82no—
653C4#239518yes—
654C2—44no—
655C2C2×#239511yescharacteristic
656C2#3570111yescharacteristic
657C2#3572012yes—
658C2#3591912yes—
659C2#3594911yescharacteristic
660C2#3598711yescharacteristic
661C2#3645212yes—
662C2#3648211yescharacteristic
663C2#3648912yes—
664C2#3649611yescharacteristic
665C2#3657911yescharacteristic
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
2. C2×#2355C2semidirect
3. #15213C2semidirect
4. #15194C2semidirect
5. #15186C2semidirect
7. #15153C2semidirect
9. #15024C2semidirect
11. #14781C2semidirect
15. #13821C2semidirect
16. #13817C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23602
C2×#23552
#152132
#151942
#151862
#151792
#151532
#151352
#150242
#150162
#147812
#147462
C4×#3992
C4×#3972
#138212
#138172
C22⋊C4×C4⋊C42