Finite Group Databasebeta

← #515977

Subgroups — #515977

Order 512.

3578 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2483C211yesmaximal, characteristic
2C22×#388C211yesmaximal, characteristic
3C2×#2273C211yesmaximal, characteristic
4#14184C214yesmaximal, complemented
5#14183C214yesmaximal, complemented
6#14127C211yesmaximal, characteristic
7#14103C214yesmaximal, complemented
8#14039C212yesmaximal, complemented
9#14008C214yesmaximal, complemented
10#13992C212yesmaximal, complemented
11#13869C211yesmaximal, characteristic
12#13869C211yesmaximal, characteristic
13#13782C212yesmaximal, complemented
14#13775C211yesmaximal, characteristic
15C4×#381C212yesmaximal, complemented
16C23×C4⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2483C4116yescomplemented
24C2×#397C2214yes—
25C2×#388—28no—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#387C2214yescomplemented
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#381C2212yes—
34C2×#381C2212yes—
35C2×#381C2212yes—
36C2×#381C2212yes—
37C2×#379C2211yescharacteristic
38C2×#379C2211yescharacteristic
39#2273—28no—
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1443C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1441C2214yescomplemented
53#1440C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62C4⋊#95C2214yescomplemented
63#1434C2214yescomplemented
64#1434C2214yescomplemented
65#1434C2214yescomplemented
66#1434C2214yescomplemented
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C4×#95C2214yescomplemented
77C4×#95C2214yescomplemented
78C26C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×C4⋊C4—24no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C22⋊C4—24no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4C2×C418yes—
104C22×C22⋊C4C2×C418yes—
105C22×C22⋊C4D418yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42—24no—
123C22×C42—28no—
124C22×C42C2314yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130#397C2×C4132yescomplemented
131#388—216no—
132#388—216no—
133#387—48no—
134#385—28no—
135#385—44no—
136#385C2×C4116yescomplemented
137#385C2×C4116yescomplemented
138#381—28no—
139#381—44no—
140#381C2×C4116yescomplemented
141#381C2×C4116yescomplemented
142#379—24no—
143#379C2×C418yes—
144C4×C22⋊C4—28no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4—28no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—28no—
150C4×C22⋊C4—28no—
151C2×#95—24no—
152C2×#95—24no—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C25—24no—
177C25—24no—
178C25—42no—
179C25—42no—
180C25—42no—
181C25—42no—
182C25C22×C414yes—
183C25C2×D414yes—
184C25#4114yes—
185C25C2411yescharacteristic
186C25C2411yescharacteristic
187C25C2411yescharacteristic
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4C22×C414yes—
226C23×C4C22×C414yes—
227C23×C4C22×C414yes—
228C23×C4C22×C414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4#4118yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—416no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C22×C4116yes—
254C2×C22⋊C4—232no—
255C2×C22⋊C4—232no—
256C2×C22⋊C4—232no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—232no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—416no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4C22×C418yes—
298C2×C22⋊C4C22×C418yes—
299C2×C22⋊C4C22×C418yes—
300C2×C22⋊C4C22×C418yes—
301C2×C22⋊C4C22×C418yes—
302C2×C22⋊C4C22×C418yes—
303C2×C22⋊C4C22×C418yes—
304C2×C22⋊C4C2×D418yes—
305C2×C22⋊C4C2×D418yes—
306C2×C22⋊C4#4118yes—
307C2×C42—28no—
308C2×C42—28no—
309C2×C42—28no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—416no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—44no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
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×C4116yes—
343#95C22×C4116yes—
344#95C22×C4116yes—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—216no—
362C24—28no—
363C24—28no—
364C24—216no—
365C24—44no—
366C24—48no—
367C24—48no—
368C24—44no—
369C24—44no—
370C24—44no—
371C24—48no—
372C24—48no—
373C24—88no—
374C24—88no—
375C24—88no—
376C24—88no—
377C24C4×D4116yes—
378C24C23×C412yes—
379C24C23×C412yes—
380C24C23×C412yes—
381C24C22×D412yes—
382C24C22×D412yes—
383C24C22×D412yes—
384C24C2×#4112yes—
385C24C2×#4112yes—
386C24C2×#4112yes—
387C24C2×#4114yes—
388C24C2×#4114yes—
389C24C2511yescharacteristic
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
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—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—416no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—416no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—84no—
482C22×C4—88no—
483C22×C4—84no—
484C22×C4—88no—
485C22×C4#123116yes—
486C22×C4C23×C414yes—
487C22×C4C23×C414yes—
488C22×C4C23×C418yes—
489C22×C4C23×C414yes—
490C22×C4C23×C414yes—
491C22×C4C22×D414yes—
492C22×C4C22×D414yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4114yes—
497C22×C4H(5,𝔽2)14yes—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—88no—
501C22⋊C4—216no—
502C22⋊C4—216no—
503C22⋊C4—216no—
504C22⋊C4—216no—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C42—416no—
518C42—84no—
519C42—84no—
520C42—88no—
521C42—84no—
522C42—84no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—28no—
530C23—24no—
531C23—28no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—24no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—232no—
560C23—416no—
561C23—416no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—88no—
571C23—88no—
572C23—88no—
573C23—832no—
574C23—88no—
575C23—832no—
576C23C2×C4×D414yes—
577C23C2×C4×D414yes—
578C23C2×C4×D414yes—
579C23C4×#4118yes—
580C23C2×#12314yes—
581C23#52514yes—
582C23#52918yes—
583C23C24×C411yescharacteristic
584C23C23×D411yescharacteristic
585C23C22×#4111yescharacteristic
586C23C22×#4112yes—
587C23C22×#4112yes—
588C23C2×H(5,𝔽2)12yes—
589C23#58512yes—
590C23#58512yes—
591C23#58512yes—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—416no—
617C2×C4—416no—
618C2×C4—416no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
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—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—416no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—88no—
660C2×C4—88no—
661C2×C4—88no—
662C2×C4—88no—
663C2×C4—88no—
664C2×C4—88no—
665C2×C4C2×#12318yes—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—216no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—216no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—48no—
687C22—48no—
688C22—42no—
689C22—42no—
690C22—48no—
691C22—48no—
692C22—832no—
693C22C4×#12318yes—
694C22C22×C4×D411yescharacteristic
695C22C2×C4×#4112yes—
696C22#342812yes—
697C22C4×H(5,𝔽2)12yes—
698C22C22×#12311yescharacteristic
699C22C2×#52511yescharacteristic
700C22C2×#52912yes—
701C22#348411yescharacteristic
702C22#348511yescharacteristic
703C22#348511yescharacteristic
704C22#348511yescharacteristic
705C22#349114yes—
706C22#349112yes—
707C22#349414yes—
708C22#350512yes—
709C22#351414yes—
710C22#352114yes—
711C4—28no—
712C4—44no—
713C4—44no—
714C4—44no—
715C4—44no—
716C4—44no—
717C4—44no—
718C4—44no—
719C4—44no—
720C4—48no—
721C4—82no—
722C4—84no—
723C2—28no—
724C2—44no—
725C2—44no—
726C2C2×C4×#12311yescharacteristic
727C2#3419911yescharacteristic
728C2#3426411yescharacteristic
729C2#3430314yes—
730C2#3544811yescharacteristic
731C2#3545111yescharacteristic
732C2#3545211yescharacteristic
733C2#3545714yes—
734C2#3554011yescharacteristic
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
4. #14184C2semidirect
5. #14183C2semidirect
7. #14103C2semidirect
8. #14039C2semidirect
9. #14008C2semidirect
10. #13992C2semidirect
13. #13782C2semidirect
15. C4×#381C2semidirect
23. #2483C4semidirect
28. C2×#387C22semidirect
49. #1443C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. C4⋊#95C22semidirect
63. #1434C22semidirect
64. #1434C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect
76. C4×#95C22semidirect
77. C4×#95C22semidirect
130. #397C2×C4semidirect
136. #385C2×C4semidirect
137. #385C2×C4semidirect
140. #381C2×C4semidirect
141. #381C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24832
C22×#3882
C2×#22732
#141842
#141832
#141272
#141032
#140392
#140082
#139922
#138692
#137822
#137752
C4×#3812