Finite Group Databasebeta

← #4930386

Subgroups — #4930386

Order 512.

2486 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2562C211yesmaximal, characteristic
3#15047C212yesmaximal, complemented
4#15027C214yesmaximal, complemented
5#14999C214yesmaximal
6#14972C212yesmaximal, complemented
7#14933C212yesmaximal, complemented
8#14933C212yesmaximal, complemented
9#14283C211yesmaximal, characteristic, complemented
10#14160C211yesmaximal, characteristic, complemented
11#13994C212yesmaximal, complemented
12#13993C212yesmaximal, complemented
13#13906C211yesmaximal, characteristic
14#13876C212yesmaximal, complemented
15#13876C212yesmaximal, complemented
16#13847C211yesmaximal, characteristic
17#13750C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×#125C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2562—28no—
22#2552—28no—
23C2×#400C2212yes—
24C2×#398C2214yes—
25C2×#394C2212yes—
26C2×#390C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2212yes—
30C2×#388C2212yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#384C2212yes—
35C2×#384C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×#376C2211yescharacteristic
50#1445C2212yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2212yes—
57#1441C2212yes—
58#1441C2212yes—
59#1441C2214yes—
60#1440C2214yes—
61#1439C2211yescharacteristic
62#1439C2211yescharacteristic
63#1439C2211yescharacteristic
64#1439C2211yescharacteristic
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2212yes—
69#1439C2214yes—
70#1438C2212yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2214yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2214yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#125—24no—
89C2×#125—28no—
90C2×#125—24no—
91C2×#125—24no—
92C2×#125—48no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#400—28no—
132#398—216no—
133#394—28no—
134#390—44no—
135#389—28no—
136#389—28no—
137#388—28no—
138#388—28no—
139#388—48no—
140#388—48no—
141#388—48no—
142#384—28no—
143#384—24no—
144#380—24no—
145#380—24no—
146#380—24no—
147#380—24no—
148#380—28no—
149#380—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4D418yes—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—28no—
157#376—24no—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C25C2411yescharacteristic
186C25C2411yescharacteristic
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—41no—
193C22×D4—41no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219#125—432no—
220#125—416no—
221C4×D4—416no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4#4118yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—44no—
314C2×C42C2×D418yes—
315C2×C42C2×D414yes—
316C2×C42#4118yes—
317C2×C42#4114yes—
318C2×C42#4114yes—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95C2×D418yes—
346C24—21no—
347C24—21no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—21no—
359C24—22no—
360C24—22no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24H(5,𝔽2)12yes—
367C242−1+412yes—
368C24C2511yescharacteristic
369C2×D4—28no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4118yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)14yes—
476C22×C42−1+414yes—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—216no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—416no—
483C4⋊C4—416no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—84no—
492C4⋊C4—88no—
493C22⋊C4—216no—
494C22⋊C4—216no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—416no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—88no—
505C42—216no—
506C42—48no—
507C42—48no—
508C42—84no—
509C42—88no—
510C42—88no—
511C42#124116yes—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23C23×D411yescharacteristic
559C23C22×#4111yescharacteristic
560C23C22×#4111yescharacteristic
561C23C22×#4112yes—
562C23C22×#4112yes—
563C23C2×H(5,𝔽2)12yes—
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×2−1+412yes—
566C23C2×2−1+411yescharacteristic
567C23#58511yescharacteristic
568C23#58511yescharacteristic
569D4—416no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
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—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
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—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—88no—
655C2×C4—88no—
656C2×C4C2×#12414yes—
657C2×C4C2×#12414yes—
658C2×C4C2×#12414yes—
659C2×C4#52918yes—
660C2×C4#52918yes—
661C2×C4#53214yes—
662C2×C4#53214yes—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—44no—
669C22—44no—
670C22—42no—
671C22—42no—
672C22—44no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—82no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22C22×#12411yescharacteristic
686C22C2×#52912yes—
687C22C2×#52912yes—
688C22C2×#53211yescharacteristic
689C22C2×#53211yescharacteristic
690C22#348411yescharacteristic
691C22#348611yescharacteristic
692C22#348811yescharacteristic
693C22#348911yescharacteristic
694C22#349212yes—
695C22#349212yes—
696C22#349211yescharacteristic
697C22#349311yescharacteristic
698C22#349412yes—
699C22#349411yescharacteristic
700C22#349511yescharacteristic
701C22#349512yes—
702C22#351112yes—
703C22#351112yes—
704C22#351512yes—
705C22#351712yes—
706C22#351912yes—
707C22#352312yes—
708C4—24no—
709C4—24no—
710C4—44no—
711C4—44no—
712C4—44no—
713C4—44no—
714C4—44no—
715C4—44no—
716C4—48no—
717C4—82no—
718C4—81no—
719C4—81no—
720C4—84no—
721C4—82no—
722C4#286118yes—
723C2—82no—
724C2—82no—
725C2#3554211yescharacteristic
726C2#3555112yes—
727C2#3559211yescharacteristic
728C2#3560912yes—
729C2#3569912yes—
730C2#3592311yescharacteristic
731C2#3592611yescharacteristic
732C2#3594111yescharacteristic
733C2#3594911yescharacteristic
734C2#3602612yes—
735C2C2×#286111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2552C2semidirect
3. #15047C2semidirect
4. #15027C2semidirect
6. #14972C2semidirect
7. #14933C2semidirect
8. #14933C2semidirect
9. #14283C2semidirect
10. #14160C2semidirect
11. #13994C2semidirect
12. #13993C2semidirect
14. #13876C2semidirect
15. #13876C2semidirect
17. #13750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#25622
#150472
#150272
#149992
#149722
#149332
#142832
#141602
#139942
#139932
#139062
#138762
#138472
#137502