Finite Group Databasebeta

← #4928990

Subgroups — #4928990

Order 512.

3418 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic, complemented
2C2×#2482C211yesmaximal, characteristic
3C22×#396C211yesmaximal, characteristic
4C2×#2274C211yesmaximal, characteristic, complemented
5#14864C214yesmaximal, complemented
6#14853C214yesmaximal, complemented
7#14805C214yesmaximal, complemented
8#14788C214yesmaximal, complemented
9#14339C211yesmaximal, characteristic, complemented
10#14130C212yesmaximal, complemented
11#13992C212yesmaximal, complemented
12#13991C212yesmaximal
13#13903C212yesmaximal
14#13870C211yesmaximal, characteristic, complemented
15#13868C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24#2484—28no—
25#2482—28no—
26C2×#400C2212yescomplemented
27C2×#399C2212yes—
28C2×#398C2212yescomplemented
29C2×#397C2212yes—
30C2×#396—28no—
31C2×#396C2214yes—
32C2×#396C2214yes—
33C2×#389C2214yescomplemented
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#387C2214yescomplemented
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#381C2211yescharacteristic
40C2×#381C2211yescharacteristic
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#380C2212yes—
44C2×#379C2212yes—
45C2×#379C2212yes—
46#2274—28no—
47C2×C4×C4⋊C4C2212yescomplemented
48C2×C4×C4⋊C4C2212yescomplemented
49C2×C4×C4⋊C4C2212yescomplemented
50C2×C4×C4⋊C4C2212yescomplemented
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2212yes—
53C22×#95C2211yescharacteristic
54#1444C2214yescomplemented
55#1443C2214yescomplemented
56#1442C2214yescomplemented
57#1442C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2212yescomplemented
61#1441C2212yescomplemented
62#1440C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1436C2214yescomplemented
73#1436C2214yescomplemented
74C4⋊#95C2214yescomplemented
75#1434C2214yescomplemented
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C26C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—24no—
89C2×#117—28no—
90C2×#117D418yes—
91C22×C4⋊C4—28no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4D418yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—28no—
116C22×C22⋊C4—44no—
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×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135#400—28no—
136#399—28no—
137#398—28no—
138#397—28no—
139#396—216no—
140#396—216no—
141#389—48no—
142#388—216no—
143#388—48no—
144#387—216no—
145#385—28no—
146#385—28no—
147#381—24no—
148#381—24no—
149#381—28no—
150#381—28no—
151#380—28no—
152#379—28no—
153#379—28no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C22⋊C4—28no—
159C4×C22⋊C4—28no—
160C2×#95—24no—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C25—22no—
184C25—24no—
185C25—22no—
186C25—22no—
187C25—24no—
188C25—24no—
189C25—42no—
190C25—42no—
191C25—42no—
192C25C2412yes—
193C25C2411yescharacteristic
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4#4114yes—
221C23×C4#4118yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235#117—232no—
236#117—232no—
237C2×C4⋊C4—232no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—416no—
267C2×C4⋊C4—416no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
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—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—416no—
305C2×C22⋊C4—416no—
306C2×C22⋊C4—416no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×D418yes—
312C2×C22⋊C4#4118yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42C2×D418yes—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—28no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—48no—
365C24—48no—
366C24—416no—
367C24—416no—
368C24—44no—
369C24—44no—
370C24—44no—
371C24—44no—
372C24—44no—
373C24—44no—
374C24—48no—
375C24—48no—
376C24—84no—
377C24—84no—
378C24—88no—
379C24—88no—
380C24C22×D412yes—
381C24C22×D412yes—
382C24C2×#4112yes—
383C24C2×#4114yes—
384C24C2×#4114yes—
385C242−1+414yes—
386C24C2511yescharacteristic
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
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—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—416no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—88no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4#121116yes—
470C22×C4#123116yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4114yes—
485C22×C4H(5,𝔽2)14yes—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—416no—
493C4⋊C4—416no—
494C4⋊C4—416no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—416no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C42—216no—
511C42—216no—
512C42—84no—
513C42—84no—
514C42—84no—
515C42—84no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—416no—
534C23—416no—
535C23—416no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—42no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—832no—
555C23—832no—
556C23—88no—
557C23—88no—
558C23—88no—
559C23—88no—
560C23C2×#12114yes—
561C23C2×#12314yes—
562C23#52514yes—
563C23#52918yes—
564C23#53218yes—
565C23C23×D411yescharacteristic
566C23C23×D411yescharacteristic
567C23C22×#4111yescharacteristic
568C23C22×#4112yes—
569C23C22×#4112yes—
570C23C2×H(5,𝔽2)12yes—
571C23C2×2−1+412yes—
572C23#58512yes—
573C23#58512yes—
574C2×C4—232no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
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—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
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—44no—
644C2×C4—44no—
645C2×C4—88no—
646C2×C4—88no—
647C2×C4—88no—
648C2×C4C2×#12114yes—
649C2×C4C2×#12114yes—
650C2×C4C2×#12318yes—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—216no—
658C22—216no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—48no—
663C22—48no—
664C22—832no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—84no—
670C22—84no—
671C22#285418yes—
672C22C22×#12111yescharacteristic
673C22C22×#12311yescharacteristic
674C22C2×#52511yescharacteristic
675C22C2×#52912yes—
676C22C2×#53212yes—
677C22#348411yescharacteristic
678C22#348511yescharacteristic
679C22#348611yescharacteristic
680C22#348711yescharacteristic
681C22#348812yes—
682C22#348912yes—
683C22#349412yes—
684C22#349512yes—
685C22#349514yes—
686C22#350514yes—
687C22#352014yes—
688C22#352114yes—
689C4—24no—
690C4—24no—
691C4—28no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—44no—
696C4—44no—
697C4—44no—
698C4—82no—
699C4—82no—
700C4—84no—
701C4—84no—
702C2—28no—
703C2—84no—
704C2#3535111yescharacteristic
705C2#3535711yescharacteristic
706C2#3545411yescharacteristic
707C2#3556311yescharacteristic
708C2#3563311yescharacteristic
709C2#3567211yescharacteristic
710C2#3582414yes—
711C2#3582514yes—
712C2C2×#285411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2484C2semidirect
4. C2×#2274C2semidirect
5. #14864C2semidirect
6. #14853C2semidirect
7. #14805C2semidirect
8. #14788C2semidirect
9. #14339C2semidirect
10. #14130C2semidirect
11. #13992C2semidirect
14. #13870C2semidirect
26. C2×#400C22semidirect
28. C2×#398C22semidirect
33. C2×#389C22semidirect
36. C2×#387C22semidirect
47. C2×C4×C4⋊C4C22semidirect
48. C2×C4×C4⋊C4C22semidirect
49. C2×C4×C4⋊C4C22semidirect
50. C2×C4×C4⋊C4C22semidirect
54. #1444C22semidirect
55. #1443C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
61. #1441C22semidirect
72. #1436C22semidirect
73. #1436C22semidirect
74. C4⋊#95C22semidirect
75. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C2×#24822
C22×#3962
C2×#22742
#148642
#148532
#148052
#147882
#143392
#141302
#139922
#139912
#139032
#138702
#138682