Finite Group Databasebeta

← #4307546

Subgroups — #4307546

Order 512.

2490 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic, complemented
2C2×#2345C211yesmaximal, characteristic
3#15225C212yesmaximal, complemented
4#15220C212yesmaximal
5#15202C211yesmaximal, characteristic, complemented
6#15161C212yesmaximal
7#15153C211yesmaximal, characteristic, complemented
8#15135C211yesmaximal, characteristic
9#15105C212yesmaximal, complemented
10#15053C212yesmaximal, complemented
11#14791C214yesmaximal, complemented
12#14774C214yesmaximal
13C4×#399C212yesmaximal
14C4×#393C211yesmaximal, characteristic
15#13821C212yesmaximal, complemented
16#13818C211yesmaximal, characteristic, complemented
17C22⋊C4×C4⋊C4C212yesmaximal
18C22×C4×Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#400C2214yes—
21C2×#399C2212yes—
22C2×#393C2211yescharacteristic
23#2355—28no—
24#2345—28no—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#389C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#387C2212yes—
32C2×#387C2212yes—
33C2×#387C2214yes—
34C2×#384C2211yescharacteristic
35C2×#382C2212yes—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2214yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2214yes—
49#1446C2211yescharacteristic
50#1446C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1445C2214yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2214yes—
58#1443C2214yes—
59#1442C2211yescharacteristic
60#1442C2212yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1439C2212yes—
66#1439C2211yescharacteristic
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2214yes—
81C22⋊#95C2214yes—
82C4×#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4×Q8—24no—
89C2×C4×Q8—28no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4Q818yescomplemented
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2314yes—
122C22×C42C2312yes—
123#400—48no—
124#399—28no—
125#393—24no—
126#391—28no—
127#389—24no—
128#389—28no—
129#389—28no—
130#389—24no—
131#389—28no—
132#387—28no—
133#387—28no—
134#387—216no—
135#384—24no—
136#382—28no—
137#380—24no—
138#380—28no—
139#380—28no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—28no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—48no—
145C4×C4⋊C4D418yes—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—48no—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2311yescharacteristic
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C25C2411yescharacteristic
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—42no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C2×Q814yes—
197C23×C4C2×Q818yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2414yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C4×Q8—232no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—232no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×Q818yes—
254C2×C4⋊C4C2×Q814yes—
255C2×C4⋊C4C2×Q814yes—
256C2×C4⋊C4#4118yes—
257C2×C4⋊C4#4114yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4#4114yes—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—28no—
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×C42—44no—
293C2×C42C2×D414yes—
294C2×C42C2×D418yes—
295C2×C42#4118yes—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—22no—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—22no—
320#95—42no—
321#95—44no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95C2×D418yes—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—22no—
333C24—21no—
334C24—21no—
335C24—22no—
336C24C22×Q812yes—
337C24C2×#4112yes—
338C242−1+412yes—
339C24C2511yescharacteristic
340C2×Q8—28no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—44no—
344C2×Q8—44no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—24no—
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—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—416no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4—88no—
427C22×C4C22×D414yes—
428C22×C4C22×D412yes—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×Q812yes—
433C22×C4C22×Q814yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q812yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4118yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4114yes—
445C22×C4H(5,𝔽2)14yes—
446C22×C42−1+412yes—
447C4⋊C4—28no—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—28no—
451C4⋊C4—216no—
452C4⋊C4—28no—
453C4⋊C4—416no—
454C4⋊C4—416no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—88no—
462C4⋊C4C22⋊Q8116yes—
463C22⋊C4—216no—
464C22⋊C4—28no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C42—28no—
470C42—216no—
471C42—48no—
472C42—416no—
473C42—48no—
474C42—44no—
475C42—84no—
476C42—88no—
477C23—22no—
478C23—24no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—41no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23#55714yes—
500C23C23×D411yescharacteristic
501C23C23×Q811yescharacteristic
502C23C22×#4111yescharacteristic
503C23C22×#4112yes—
504C23C22×#4112yes—
505C23C22×#4111yescharacteristic
506C23C2×H(5,𝔽2)12yes—
507C23C2×2−1+411yescharacteristic
508C23C2×2−1+411yescharacteristic
509C23#58512yes—
510C23#58511yescharacteristic
511Q8—416no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—416no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—816no—
599C2×C4C2×C22⋊Q818yes—
600C2×C4C2×C22⋊Q814yes—
601C2×C4#52514yes—
602C2×C4#52918yes—
603C2×C4#53214yes—
604C2×C4#53318yes—
605C2×C4#54814yes—
606C22—28no—
607C22—24no—
608C22—42no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22C22×C22⋊Q811yescharacteristic
618C22C2×#52511yescharacteristic
619C22C2×#52912yes—
620C22C2×#53211yescharacteristic
621C22C2×#53312yes—
622C22C2×#54811yescharacteristic
623C22#347512yes—
624C22C2×#55711yescharacteristic
625C22#348011yescharacteristic
626C22#348112yes—
627C22#348511yescharacteristic
628C22#348911yescharacteristic
629C22#349212yes—
630C22#349512yes—
631C22#349811yescharacteristic
632C22#350112yes—
633C22#350211yescharacteristic
634C22#350612yes—
635C22#351312yes—
636C22#351612yes—
637C22#352112yes—
638C22#352212yes—
639C22#352311yescharacteristic
640C4—24no—
641C4—28no—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—44no—
646C4—42no—
647C4—44no—
648C4—48no—
649C4—42no—
650C4—44no—
651C4—81no—
652C4—82no—
653C4—84no—
654C4—82no—
655C4#239618yes—
656C2—44no—
657C2C2×#239611yescharacteristic
658C2#3568911yescharacteristic
659C2#3578312yes—
660C2#3583611yescharacteristic
661C2#3585611yescharacteristic
662C2#3591112yes—
663C2#3649611yescharacteristic
664C2#3651611yescharacteristic
665C2#3651912yes—
666C2#3652612yes—
667C2#3657711yescharacteristic
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
1. C2×#2355C2semidirect
3. #15225C2semidirect
5. #15202C2semidirect
7. #15153C2semidirect
9. #15105C2semidirect
10. #15053C2semidirect
11. #14791C2semidirect
15. #13821C2semidirect
16. #13818C2semidirect
92. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#23452
#152252
#152202
#152022
#151612
#151532
#151352
#151052
#150532
#147912
#147742
C4×#3992
C4×#3932
#138212
#138182
C22⋊C4×C4⋊C42