Finite Group Databasebeta

← #554189

Subgroups — #554189

Order 512.

3962 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2C2×#2329C211yesmaximal, characteristic, complemented
3C22×#385C211yesmaximal, characteristic
4#14693C212yesmaximal
5#14575C214yesmaximal, complemented
6#14541C214yesmaximal, complemented
7#14433C214yesmaximal, complemented
8#14405C214yesmaximal, complemented
9#14263C212yesmaximal, complemented
10#14139C211yesmaximal, characteristic, complemented
11#13992C212yesmaximal, complemented
12#13991C212yesmaximal
13#13903C211yesmaximal, characteristic
14#13880C211yesmaximal, characteristic, complemented
15#13796C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2561C4116yescomplemented
23C2×#397C2212yes—
24C2×#393C2212yes—
25#2329—28no—
26C2×#391C2212yescomplemented
27C2×#391C2212yescomplemented
28C2×#389C2212yescomplemented
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#387C2212yescomplemented
32C2×#387C2212yescomplemented
33C2×#385—28no—
34C2×#385C2212yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#384C2212yescomplemented
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#380C2211yescharacteristic
45C2×#379C2212yes—
46C2×#379C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52#1444C2214yescomplemented
53#1444C2214yescomplemented
54#1444C2214yescomplemented
55#1443C2214yescomplemented
56#1442C2214yescomplemented
57#1442C2214yescomplemented
58#1442C2214yescomplemented
59#1441C2214yescomplemented
60#1440C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2212yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yescomplemented
67C4⋊#95C2212yescomplemented
68C4⋊#95C2212yescomplemented
69#1434C2214yescomplemented
70#1434C2214yescomplemented
71#1434C2214yescomplemented
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C26C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C24×C4—24no—
81C24×C4—44no—
82C24×C4—44no—
83C24×C4—44no—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2314yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8C2×C418yescomplemented
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—44no—
113C22×C22⋊C4C2×C418yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129#397C2×C4116yescomplemented
130#393C2×C4116yescomplemented
131#391—28no—
132#391—44no—
133#389—28no—
134#388—48no—
135#388C2×C4132yescomplemented
136#387—28no—
137#387—28no—
138#385—28no—
139#385—216no—
140#385—216no—
141#385—48no—
142#385—44no—
143#385—44no—
144#385—48no—
145#384—44no—
146#381—44no—
147#381C2×C4116yescomplemented
148#380—24no—
149#379—44no—
150#379C2×C4116yes—
151C4×C22⋊C4—28no—
152C4×C22⋊C4—24no—
153C4×C22⋊C4C2×C418yescomplemented
154C2×#95—24no—
155C2×#95D418yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C25—22no—
181C25—22no—
182C25—24no—
183C25—24no—
184C25—24no—
185C25—24no—
186C25—42no—
187C25—42no—
188C25C22×C414yes—
189C25C2412yes—
190C25C2411yescharacteristic
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8C22×C414yescomplemented
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—416no—
224C23×C4—416no—
225C23×C4—416no—
226C23×C4—464no—
227C23×C4C22×C414yes—
228C23×C4C22×C414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4#4114yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C22⋊Q8—232no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—416no—
266C2×C4⋊C4—416no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4C22×C418yescomplemented
277C2×C4⋊C4C22×C414yescomplemented
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—232no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—416no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4C22×C418yes—
308C2×C22⋊C4C22×C418yes—
309C2×C22⋊C4C22×C418yes—
310C2×C22⋊C4C22×C418yes—
311C2×C22⋊C4C22×C4116yes—
312C2×C22⋊C4C22×C4116yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—44no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—44no—
322C2×C42C22×C418yescomplemented
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95C22×C418yescomplemented
343#95C22×C4116yescomplemented
344#95C22×C4116yescomplemented
345#95C2×D418yes—
346#95C2×D418yes—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—28no—
352C24—28no—
353C24—216no—
354C24—28no—
355C24—216no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—216no—
365C24—48no—
366C24—416no—
367C24—416no—
368C24—416no—
369C24—416no—
370C24—44no—
371C24—44no—
372C24—48no—
373C24—44no—
374C24—44no—
375C24—48no—
376C24C23×C414yes—
377C24C23×C412yes—
378C24C22×D412yes—
379C24C2×#4112yes—
380C24H(5,𝔽2)14yes—
381C24H(5,𝔽2)14yes—
382C242−1+414yes—
383C242−1+414yes—
384C24C2511yescharacteristic
385C2×Q8—28no—
386C2×Q8—48no—
387C2×Q8—82no—
388C2×Q8—82no—
389C2×Q8—84no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
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—28no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—44no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—4128no—
468C22×C4—416no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4C4×D4116yes—
479C22×C4C23×C414yes—
480C22×C4C23×C418yes—
481C22×C4C23×C414yes—
482C22×C4C23×C414yes—
483C22×C4C23×C412yes—
484C22×C4C23×C412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)14yes—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—88no—
502C4⋊C4—88no—
503C22⋊C4—216no—
504C22⋊C4—232no—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C42—48no—
515C42—84no—
516C42—84no—
517C42—84no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—232no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23—28no—
543C23—464no—
544C23—416no—
545C23—416no—
546C23—416no—
547C23—464no—
548C23—416no—
549C23—416no—
550C23—416no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23C2×C4×D414yes—
562C23#52018yes—
563C23#52018yes—
564C23#53514yes—
565C23#53614yes—
566C23#54218yes—
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—88no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—28no—
602C2×C4—416no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—464no—
606C2×C4—416no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
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—48no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—88no—
649C2×C4—88no—
650C2×C4C2×C4×D414yes—
651C2×C4C2×C4×D414yes—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—216no—
657C22—216no—
658C22—24no—
659C22—24no—
660C22—28no—
661C22—216no—
662C22—464no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—48no—
668C22—48no—
669C22—48no—
670C22—44no—
671C22—44no—
672C22#232618yes—
673C22C22×C4×D411yescharacteristic
674C22C2×#52012yes—
675C22C2×#52012yes—
676C22C4×H(5,𝔽2)12yes—
677C22C2×#53511yescharacteristic
678C22C2×#53611yescharacteristic
679C22C2×#54212yes—
680C22#348411yescharacteristic
681C22#348411yescharacteristic
682C22#348511yescharacteristic
683C22#348611yescharacteristic
684C22#349212yes—
685C22#350512yes—
686C22#353214yes—
687C22#353514yes—
688C22#353514yes—
689C22#353814yes—
690C4—24no—
691C4—24no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—44no—
696C4—44no—
697C4—48no—
698C4—44no—
699C4—82no—
700C4—82no—
701C4—84no—
702C4—82no—
703C2—28no—
704C2—48no—
705C2C2×#232611yescharacteristic
706C2#3420011yescharacteristic
707C2#3426411yescharacteristic
708C2#3445714yes—
709C2#3670311yescharacteristic
710C2#3670811yescharacteristic
711C2#3672011yescharacteristic
712C2#3677814yes—
713C2#3711811yescharacteristic
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. C2×#2329C2semidirect
5. #14575C2semidirect
6. #14541C2semidirect
7. #14433C2semidirect
8. #14405C2semidirect
9. #14263C2semidirect
10. #14139C2semidirect
11. #13992C2semidirect
14. #13880C2semidirect
15. #13796C2semidirect
22. #2561C4semidirect
26. C2×#391C22semidirect
27. C2×#391C22semidirect
28. C2×#389C22semidirect
31. C2×#387C22semidirect
32. C2×#387C22semidirect
41. C2×#384C22semidirect
52. #1444C22semidirect
53. #1444C22semidirect
54. #1444C22semidirect
55. #1443C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1442C22semidirect
59. #1441C22semidirect
66. #1436C22semidirect
67. C4⋊#95C22semidirect
68. C4⋊#95C22semidirect
69. #1434C22semidirect
70. #1434C22semidirect
71. #1434C22semidirect
93. C2×C22⋊Q8C2×C4semidirect
129. #397C2×C4semidirect
130. #393C2×C4semidirect
135. #388C2×C4semidirect
147. #381C2×C4semidirect
153. C4×C22⋊C4C2×C4semidirect
195. C22×Q8C22×C4semidirect
276. C2×C4⋊C4C22×C4semidirect
277. C2×C4⋊C4C22×C4semidirect
322. C2×C42C22×C4semidirect
342. #95C22×C4semidirect
343. #95C22×C4semidirect
344. #95C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C2×#23292
C22×#3852
#146932
#145752
#145412
#144332
#144052
#142632
#141392
#139922
#139912
#139032
#138802
#137962