Finite Group Databasebeta

← #554241

Subgroups — #554241

Order 512.

4002 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic, complemented
3C22×#385C211yesmaximal, characteristic
4#14693C212yesmaximal
5#14551C214yesmaximal, complemented
6#14526C214yesmaximal, complemented
7#14405C214yesmaximal, complemented
8#14393C214yesmaximal, complemented
9#14265C212yesmaximal, complemented
10#14128C211yesmaximal, characteristic, complemented
11#14043C212yesmaximal, complemented
12#13971C212yesmaximal
13#13958C211yesmaximal, characteristic, complemented
14#13868C211yesmaximal, characteristic
15#13776C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2482C4116yescomplemented
24C2×#399C2212yes—
25C2×#397C2212yes—
26#2317—28no—
27C2×#389C2212yescomplemented
28C2×#389C2212yescomplemented
29C2×#389C2214yescomplemented
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#385—28no—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#385C2212yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#385C2212yes—
41C2×#384C2212yescomplemented
42C2×#382C2212yescomplemented
43C2×#381C2211yescharacteristic
44C2×#381C2211yescharacteristic
45C2×#379C2212yes—
46C2×#379C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1445C2214yescomplemented
51#1445C2212yescomplemented
52#1445C2212yescomplemented
53#1444C2212yescomplemented
54#1444C2214yescomplemented
55#1443C2214yescomplemented
56#1442C2212yescomplemented
57#1442C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1440C2214yes—
61#1439C2212yes—
62#1439C2212yes—
63#1439C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69#1436C2214yescomplemented
70#1434C2214yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C26C2311yescharacteristic
79C24×C4—24no—
80C24×C4—44no—
81C24×C4—44no—
82C24×C4—44no—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2314yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#117—24no—
91C2×#117—24no—
92C2×#117—24no—
93C2×#117—44no—
94C2×#117C2×C418yescomplemented
95C22×C4⋊C4—24no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4D418yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—44no—
116C22×C22⋊C4C2×C418yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2314yes—
131#399C2×C4116yescomplemented
132#397C2×C4116yescomplemented
133#389—28no—
134#389—28no—
135#389—48no—
136#388—48no—
137#388C2×C4132yescomplemented
138#385—28no—
139#385—28no—
140#385—216no—
141#385—216no—
142#385—48no—
143#385—44no—
144#385—48no—
145#385C2×C4116yescomplemented
146#384—28no—
147#382—28no—
148#381—24no—
149#381C2×C418yescomplemented
150#379—44no—
151#379C2×C4116yes—
152C4×C22⋊C4—44no—
153C2×#95—24no—
154C2×#95—24no—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
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×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C25—22no—
180C25—22no—
181C25—24no—
182C25—24no—
183C25—24no—
184C25—24no—
185C25—42no—
186C25—42no—
187C25C22×C414yes—
188C25C2412yes—
189C25C2411yescharacteristic
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—416no—
218C23×C4—416no—
219C23×C4—416no—
220C23×C4—464no—
221C23×C4C22×C414yes—
222C23×C4C22×C414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4#4114yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240#117—232no—
241#117—416no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—416no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C22×C414yescomplemented
273C2×C4⋊C4C22×C414yescomplemented
274C2×C4⋊C4C22×C4116yescomplemented
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D418yes—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—232no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—416no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4C22×C418yes—
307C2×C22⋊C4C22×C418yes—
308C2×C22⋊C4C22×C418yes—
309C2×C22⋊C4C22×C418yes—
310C2×C22⋊C4C22×C4116yes—
311C2×C22⋊C4C22×C4116yes—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42C22×C418yescomplemented
321#95—28no—
322#95—24no—
323#95—24no—
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×C418yescomplemented
344#95C22×C4116yescomplemented
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—28no—
350C24—28no—
351C24—216no—
352C24—28no—
353C24—216no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—216no—
363C24—48no—
364C24—416no—
365C24—416no—
366C24—416no—
367C24—416no—
368C24—44no—
369C24—44no—
370C24—48no—
371C24—44no—
372C24—44no—
373C24—48no—
374C24C23×C414yes—
375C24C23×C412yes—
376C24C22×D412yes—
377C24C2×#4112yes—
378C24H(5,𝔽2)14yes—
379C24H(5,𝔽2)14yes—
380C242−1+414yes—
381C242−1+414yes—
382C24C2511yescharacteristic
383C22×C4—216no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—48no—
450C22×C4—416no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—4128no—
465C22×C4—416no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4C4×D4116yes—
473C22×C4C23×C418yes—
474C22×C4C23×C414yes—
475C22×C4C23×C414yes—
476C22×C4C23×C414yes—
477C22×C4C23×C412yes—
478C22×C4C23×C412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)14yes—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—416no—
491C4⋊C4—416no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—84no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—88no—
499C22⋊C4—216no—
500C22⋊C4—232no—
501C22⋊C4—216no—
502C22⋊C4—216no—
503C22⋊C4—216no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C42—216no—
511C42—48no—
512C42—88no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—232no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—464no—
539C23—416no—
540C23—416no—
541C23—416no—
542C23—416no—
543C23—464no—
544C23—416no—
545C23—416no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23C2×C4×D414yes—
557C23#52018yes—
558C23#52018yes—
559C23#53414yes—
560C23#53614yes—
561C23#53918yes—
562C23C24×C411yescharacteristic
563C23C23×D411yescharacteristic
564C23C22×#4111yescharacteristic
565C23C2×H(5,𝔽2)12yes—
566C23C2×H(5,𝔽2)12yes—
567C23C2×H(5,𝔽2)12yes—
568C23C2×2−1+412yes—
569C23C2×2−1+412yes—
570C23#58512yes—
571C2×C4—232no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—464no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—88no—
644C2×C4—88no—
645C2×C4C2×C4×D414yes—
646C2×C4C2×C4×D414yes—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—216no—
653C22—216no—
654C22—24no—
655C22—28no—
656C22—216no—
657C22—464no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—48no—
664C22—48no—
665C22—48no—
666C22—44no—
667C22#232218yes—
668C22C22×C4×D411yescharacteristic
669C22C2×#52012yes—
670C22C2×#52012yes—
671C22C4×H(5,𝔽2)12yes—
672C22C2×#53411yescharacteristic
673C22C2×#53611yescharacteristic
674C22C2×#53912yes—
675C22#348411yescharacteristic
676C22#348411yescharacteristic
677C22#348611yescharacteristic
678C22#348711yescharacteristic
679C22#349212yes—
680C22#350612yes—
681C22#353214yes—
682C22#353514yes—
683C22#353514yes—
684C22#353814yes—
685C4—24no—
686C4—24no—
687C4—44no—
688C4—44no—
689C4—44no—
690C4—44no—
691C4—48no—
692C4—44no—
693C4—44no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—84no—
698C2—28no—
699C2—48no—
700C2C2×#232211yescharacteristic
701C2#3420011yescharacteristic
702C2#3426511yescharacteristic
703C2#3445714yes—
704C2#3670211yescharacteristic
705C2#3670811yescharacteristic
706C2#3671111yescharacteristic
707C2#3679214yes—
708C2#3718611yescharacteristic
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×#2317C2semidirect
5. #14551C2semidirect
6. #14526C2semidirect
7. #14405C2semidirect
8. #14393C2semidirect
9. #14265C2semidirect
10. #14128C2semidirect
11. #14043C2semidirect
13. #13958C2semidirect
15. #13776C2semidirect
23. #2482C4semidirect
27. C2×#389C22semidirect
28. C2×#389C22semidirect
29. C2×#389C22semidirect
41. C2×#384C22semidirect
42. C2×#382C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1445C22semidirect
53. #1444C22semidirect
54. #1444C22semidirect
55. #1443C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
70. #1434C22semidirect
94. C2×#117C2×C4semidirect
131. #399C2×C4semidirect
132. #397C2×C4semidirect
137. #388C2×C4semidirect
145. #385C2×C4semidirect
149. #381C2×C4semidirect
272. C2×C4⋊C4C22×C4semidirect
273. C2×C4⋊C4C22×C4semidirect
274. C2×C4⋊C4C22×C4semidirect
320. 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×#24822
C2×#23172
C22×#3852
#146932
#145512
#145262
#144052
#143932
#142652
#141282
#140432
#139712
#139582
#138682
#137762