Finite Group Databasebeta

← #554255

Subgroups — #554255

Order 512.

3686 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2406C211yesmaximal, characteristic
2C2×#2311C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic, complemented
4#14696C212yesmaximal, complemented
5#14563C214yesmaximal, complemented
6#14554C214yesmaximal, complemented
7#14408C214yesmaximal, complemented
8#14392C214yesmaximal, complemented
9#14256C212yesmaximal, complemented
10#14128C211yesmaximal, characteristic, complemented
11#14019C212yesmaximal, complemented
12#13996C212yesmaximal, complemented
13#13956C211yesmaximal, characteristic
14#13870C211yesmaximal, characteristic, complemented
15#13776C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#123C2212yes—
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2406C4116yescomplemented
25C2×#395C2212yes—
26C2×#394C2212yes—
27#2311—28no—
28C2×#391C2212yes—
29C2×#390C2212yes—
30C2×#389—28no—
31C2×#389C2214yes—
32C2×#389C2214yes—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#385C2212yes—
41C2×#385C2214yes—
42C2×#381C2211yescharacteristic
43C2×#381C2212yes—
44C2×#381C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1446C2212yes—
51#1445C2214yes—
52#1444C2212yes—
53#1442C2212yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2214yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2212yes—
78C22×#41—24no—
79C22×#41—24no—
80C22×#41—44no—
81C22×#41—44no—
82C22×#41C2×C418yes—
83C23×Q8C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×#123—44no—
95C2×#123—44no—
96C2×#123C2×C4116yes—
97C2×#117—24no—
98C2×#117—44no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4—44no—
105C22×C4⋊C4—44no—
106C22×C4⋊C4C2×C418yes—
107C22×C4⋊C4D418yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42—24no—
130C22×C42—44no—
131C22×C42C2314yes—
132C22×C42C2312yes—
133C22×C42C2314yes—
134#395C2×C4116yescomplemented
135#394C2×C4116yescomplemented
136#391—28no—
137#390—28no—
138#389—216no—
139#389—216no—
140#388—48no—
141#388—48no—
142#386—28no—
143#386—28no—
144#386—44no—
145#386—44no—
146#386C2×C4116yes—
147#385—44no—
148#385C2×C4132yescomplemented
149#381—24no—
150#381—44no—
151#381C2×C418yes—
152#380—28no—
153#380—28no—
154C4×C22⋊C4—44no—
155C2×#95—24no—
156C2×#95—24no—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C25C2412yes—
182C25C2411yescharacteristic
183C2×#41—232no—
184C2×#41—432no—
185C2×#41—416no—
186C2×#41—416no—
187C2×#41—432no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8C22×C414yes—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4C22×C418yes—
204C22×D4C22×C414yes—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4C22×C414yes—
236C23×C4C22×C414yes—
237C23×C4C22×C414yes—
238C23×C4C22×C418yes—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4#4114yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2414yes—
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256#123—432no—
257#117—416no—
258C2×C4⋊C4—232no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—416no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—416no—
282C2×C4⋊C4—416no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4C22×C418yes—
286C2×C4⋊C4C22×C418yes—
287C2×C4⋊C4C22×C4116yes—
288C2×C4⋊C4C2×D418yes—
289C2×C4⋊C4C2×D418yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4C22×C418yes—
317C2×C22⋊C4C22×C4116yes—
318C2×C22⋊C4C22×C4116yes—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—28no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42—44no—
325C2×C42—416no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42—44no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95C22×C418yes—
351#95C22×C418yes—
352#95C22×C4116yes—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24C23×C414yes—
368C24C23×C412yes—
369C24C22×D412yes—
370C24C2×#4112yes—
371C24H(5,𝔽2)14yes—
372C24H(5,𝔽2)14yes—
373C24C2511yescharacteristic
374#41—4128no—
375C2×Q8—28no—
376C2×Q8—48no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—48no—
380C2×D4—216no—
381C2×D4—216no—
382C2×D4—216no—
383C2×D4—28no—
384C2×D4—48no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—216no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—216no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—216no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—416no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—416no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—48no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—88no—
486C22×C4—88no—
487C22×C4C4×D4116yes—
488C22×C4C23×C418yes—
489C22×C4C23×C414yes—
490C22×C4C23×C414yes—
491C22×C4C23×C412yes—
492C22×C4C23×C412yes—
493C22×C4C23×C414yes—
494C22×C4C22×D414yes—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)14yes—
501C22×C42−1+414yes—
502C22×C42−1+414yes—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—416no—
508C4⋊C4—416no—
509C4⋊C4—416no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—88no—
513C22⋊C4—216no—
514C22⋊C4—232no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C42—416no—
525C42—48no—
526C42—88no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—28no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23C2×C4×D414yes—
563C23#51818yes—
564C23#53414yes—
565C23#53514yes—
566C23#53818yes—
567C23C24×C411yescharacteristic
568C23C23×D411yescharacteristic
569C23C22×#4111yescharacteristic
570C23C2×H(5,𝔽2)12yes—
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)12yes—
573C23C2×2−1+412yes—
574C23C2×2−1+412yes—
575C23#58512yes—
576Q8—416no—
577D4—416no—
578D4—432no—
579C2×C4—232no—
580C2×C4—232no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—432no—
612C2×C4—416no—
613C2×C4—416no—
614C2×C4—416no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—88no—
656C2×C4—88no—
657C2×C4C2×C4×D414yes—
658C2×C4C2×C4×D414yes—
659C2×C4#52018yes—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—216no—
666C22—216no—
667C22—24no—
668C22—44no—
669C22—44no—
670C22—48no—
671C22—48no—
672C22—44no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—84no—
678C22—84no—
679C22#232118yes—
680C22C22×C4×D411yescharacteristic
681C22C2×#51812yes—
682C22C2×#52012yes—
683C22C4×2−1+412yes—
684C22C2×#53411yescharacteristic
685C22C2×#53511yescharacteristic
686C22C2×#53812yes—
687C22#348411yescharacteristic
688C22#348511yescharacteristic
689C22#348611yescharacteristic
690C22#348611yescharacteristic
691C22#349212yes—
692C22#350712yes—
693C22#353614yes—
694C22#354214yes—
695C22#354414yes—
696C22#355914yes—
697C4—24no—
698C4—24no—
699C4—28no—
700C4—44no—
701C4—44no—
702C4—44no—
703C4—44no—
704C4—44no—
705C4—48no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—84no—
710C2—28no—
711C2—84no—
712C2C2×#232111yescharacteristic
713C2#3420011yescharacteristic
714C2#3426611yescharacteristic
715C2#3446014yes—
716C2#3670411yescharacteristic
717C2#3670711yescharacteristic
718C2#3682314yes—
719C2#3695811yescharacteristic
720C2#3725811yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#389C2semidirect
4. #14696C2semidirect
5. #14563C2semidirect
6. #14554C2semidirect
7. #14408C2semidirect
8. #14392C2semidirect
9. #14256C2semidirect
10. #14128C2semidirect
11. #14019C2semidirect
12. #13996C2semidirect
14. #13870C2semidirect
15. #13776C2semidirect
24. #2406C4semidirect
134. #395C2×C4semidirect
135. #394C2×C4semidirect
148. #385C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24062
C2×#23112
C22×#3892
#146962
#145632
#145542
#144082
#143922
#142562
#141282
#140192
#139962
#139562
#138702
#137762