Finite Group Databasebeta

← #532292

Subgroups — #532292

Order 512.

3946 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2#14515C212yesmaximal, complemented
3#14514C212yesmaximal, complemented
4#14143C212yesmaximal, complemented
5#14075C214yesmaximal, complemented
6#14067C214yesmaximal, complemented
7#13903C211yesmaximal, characteristic
8#13882C212yesmaximal, complemented
9#13874C212yesmaximal, complemented
10#13873C212yesmaximal
11#13868C211yesmaximal, characteristic
12#13823C212yesmaximal, complemented
13C22⋊C4×C4⋊C4C212yesmaximal, complemented
14#13799C212yesmaximal, complemented
15C4×#379C212yesmaximal
16C25×C4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C2×#388C2214yes—
22C2×#387C2214yescomplemented
23C2×#385C4116yescomplemented
24C2×#385C2211yescharacteristic
25C2×#385C2211yescharacteristic
26C2×#385C2211yescharacteristic
27C2×#385C2212yes—
28C2×#385C2212yes—
29C2×#385C2211yescharacteristic
30C2×#381C2212yes—
31C2×#381C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×#379C2212yes—
35C2×#379C2212yes—
36C2×C4×C4⋊C4C2214yescomplemented
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2214yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2214yes—
43C22×#95C2212yes—
44#1445C2214yescomplemented
45#1443C2214yescomplemented
46#1442C2214yescomplemented
47#1441C2214yescomplemented
48#1440C2214yes—
49#1439C2212yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2212yes—
53#1439C2212yes—
54#1437C2212yes—
55#1437C2212yes—
56#1437C2214yes—
57#1437C2212yes—
58#1437C2212yes—
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65C4⋊#95C2214yescomplemented
66#1434C2214yescomplemented
67#1434C2214yescomplemented
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C4×#95C2214yescomplemented
76C26C2311yescharacteristic
77C24×C4—48no—
78C24×C4—42no—
79C24×C4—42no—
80C24×C4C2×C418yes—
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#117—28no—
89C2×#117—42no—
90C22×C4⋊C4—48no—
91C22×C4⋊C4C2×C4116yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—42no—
101C22×C22⋊C4—42no—
102C22×C22⋊C4C2×C418yes—
103C22×C22⋊C4C2×C418yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2314yes—
121C22×C42C2314yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124#388—216no—
125#387—48no—
126#385C2×C418yes—
127#385C2×C4116yes—
128#385C2×C4116yes—
129#385C2×C418yes—
130#385C2×C418yes—
131#385C2×C418yes—
132#381—28no—
133#381—42no—
134#380—28no—
135#380—42no—
136#379—28no—
137#379—28no—
138C4×C4⋊C4—48no—
139C4×C22⋊C4—28no—
140C4×C22⋊C4—48no—
141C4×C22⋊C4—48no—
142C4×C22⋊C4—44no—
143C4×C22⋊C4—42no—
144C4×C22⋊C4—44no—
145C2×#95C2×C4116yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C25—24no—
171C25—22no—
172C25—22no—
173C25—24no—
174C25—24no—
175C25—24no—
176C25—44no—
177C25C22×C414yes—
178C25C2411yescharacteristic
179C25C2411yescharacteristic
180C25C2411yescharacteristic
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—432no—
196C23×C4—44no—
197C23×C4—416no—
198C23×C4—41no—
199C23×C4—41no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—41no—
205C23×C4—41no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—432no—
209C23×C4—432no—
210C23×C4C22×C414yes—
211C23×C4C22×C418yes—
212C23×C4C22×C418yes—
213C23×C4C22×C414yes—
214C23×C4C22×C414yes—
215C23×C4C22×C418yes—
216C23×C4C22×C414yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229#117—416no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—432no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C22×C418yes—
243C2×C4⋊C4C22×C418yes—
244C2×C4⋊C4C22×C418yes—
245C2×C4⋊C4C22×C418yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—232no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—416no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4C22×C414yes—
281C2×C22⋊C4C22×C418yes—
282C2×C22⋊C4C22×C414yes—
283C2×C22⋊C4C22×C414yes—
284C2×C22⋊C4C22×C414yes—
285C2×C22⋊C4C22×C414yes—
286C2×C22⋊C4C22×C414yes—
287C2×C22⋊C4#4118yes—
288C2×C22⋊C4#4118yes—
289C2×C22⋊C4#4118yes—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—28no—
295C2×C42—42no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—42no—
302C2×C42—44no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—44no—
312#95—42no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95—42no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C22×C418yes—
324#95C22×C418yes—
325#95C22×C418yes—
326#95C22×C418yes—
327C24—22no—
328C24—24no—
329C24—22no—
330C24—216no—
331C24—216no—
332C24—28no—
333C24—28no—
334C24—28no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—216no—
348C24—48no—
349C24—416no—
350C24—416no—
351C24—416no—
352C24—416no—
353C24—48no—
354C24—48no—
355C24—44no—
356C24—48no—
357C24—44no—
358C24C23×C412yes—
359C24C23×C412yes—
360C24C23×C412yes—
361C24C2×#4114yes—
362C24C2×#4114yes—
363C24C2×#4114yes—
364C24H(5,𝔽2)12yes—
365C24H(5,𝔽2)12yes—
366C242−1+414yes—
367C24C2511yescharacteristic
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—28no—
377C22×C4—216no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—216no—
416C22×C4—216no—
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—24no—
430C22×C4—24no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—4128no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4C23×C412yes—
456C22×C4C23×C412yes—
457C22×C4C23×C414yes—
458C22×C4C23×C414yes—
459C22×C4C23×C412yes—
460C22×C4C23×C412yes—
461C22×C4C23×C414yes—
462C22×C4C23×C414yes—
463C22×C4C2×#4118yes—
464C22×C4C2×#4118yes—
465C22×C4C2×#4118yes—
466C22×C4H(5,𝔽2)18yes—
467C4⋊C4—216no—
468C4⋊C4—216no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—84no—
474C22⋊C4—28no—
475C22⋊C4—216no—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4#117116yescomplemented
489C42—416no—
490C42—48no—
491C42—88no—
492C42—88no—
493C42—84no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—28no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—28no—
503C23—24no—
504C23—24no—
505C23—232no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—28no—
522C23—432no—
523C23—432no—
524C23—416no—
525C23—432no—
526C23—432no—
527C23—464no—
528C23—416no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—41no—
534C23—41no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23C2×#11714yes—
544C23C4×#4118yes—
545C23#51814yes—
546C23#51914yes—
547C23#55114yes—
548C23#55318yes—
549C23#55914yes—
550C23C24×C411yescharacteristic
551C23C22×#4112yes—
552C23C22×#4112yes—
553C23C22×#4112yes—
554C23C2×H(5,𝔽2)11yescharacteristic
555C23C2×H(5,𝔽2)11yescharacteristic
556C23C2×H(5,𝔽2)14yes—
557C23C2×2−1+412yes—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—416no—
587C2×C4—432no—
588C2×C4—432no—
589C2×C4—432no—
590C2×C4—416no—
591C2×C4—416no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
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—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—88no—
629C2×C4C2×#11718yes—
630C2×C4#51818yes—
631C22—24no—
632C22—28no—
633C22—24no—
634C22—28no—
635C22—28no—
636C22—28no—
637C22—216no—
638C22—28no—
639C22—24no—
640C22—24no—
641C22—28no—
642C22—464no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—48no—
654C22—48no—
655C22#237418yes—
656C22C22×#11711yescharacteristic
657C22C2×C4×#4112yes—
658C22C2×#51812yes—
659C22C2×#51811yescharacteristic
660C22C2×#51911yescharacteristic
661C22C2×#55111yescharacteristic
662C22C2×#55312yes—
663C22C2×#55911yescharacteristic
664C22#348114yes—
665C22#348114yes—
666C22#349212yes—
667C22#349212yes—
668C22#352014yes—
669C22#352014yes—
670C22#353212yes—
671C22#353212yes—
672C4—28no—
673C4—28no—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—42no—
679C4—42no—
680C4—48no—
681C4—48no—
682C4—84no—
683C4—84no—
684C2—28no—
685C2—44no—
686C2—44no—
687C2#3412612yes—
688C2C2×#237411yescharacteristic
689C2#3417811yescharacteristic
690C2#3424512yes—
691C2#3435611yescharacteristic
692C2#5937112yes—
693C2#5938514yes—
694C2#5962412yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14515C2semidirect
3. #14514C2semidirect
4. #14143C2semidirect
5. #14075C2semidirect
6. #14067C2semidirect
8. #13882C2semidirect
9. #13874C2semidirect
12. #13823C2semidirect
13. C22⋊C4×C4⋊C4C2semidirect
14. #13799C2semidirect
22. C2×#387C22semidirect
23. C2×#385C4semidirect
36. C2×C4×C4⋊C4C22semidirect
44. #1445C22semidirect
45. #1443C22semidirect
46. #1442C22semidirect
47. #1441C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. C4⋊#95C22semidirect
66. #1434C22semidirect
67. #1434C22semidirect
75. C4×#95C22semidirect
488. C22⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C22×#3852
#145152
#145142
#141432
#140752
#140672
#139032
#138822
#138742
#138732
#138682
#138232
C22⋊C4×C4⋊C42
#137992
C4×#3792