Finite Group Databasebeta

← #1394412

Subgroups — #1394412

Order 512.

2762 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2511C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#16988C212yesmaximal, complemented
4#16981C212yesmaximal, complemented
5#16339C214yesmaximal, complemented
6#16322C214yesmaximal, complemented
7#15199C211yesmaximal, characteristic, complemented
8#15179C212yesmaximal, complemented
9#15177C211yesmaximal, characteristic
10#15152C211yesmaximal, characteristic, complemented
11#15131C212yesmaximal, complemented
12#14812C212yesmaximal
13#14300C212yesmaximal
14#14139C212yesmaximal, complemented
15#13933C211yesmaximal, characteristic, complemented
16#13880C212yesmaximal, complemented
17C4×#394C211yesmaximal, characteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C4⋊C4C2212yes—
22#2511—28no—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#396C2214yes—
27C2×#394C2211yescharacteristic
28D4×C4⋊C4—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#391C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#387C2212yes—
37C2×#387C2212yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#384C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50#1445C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2212yes—
56#1444C2214yes—
57#1442C2211yescharacteristic
58#1442C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2211yescharacteristic
64#1441C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1440C2212yes—
69#1440C2214yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2214yes—
73#1439C2214yes—
74#1438C2211yescharacteristic
75#1438C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82C22⋊#95C2214yes—
83C22⋊#95C2214yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2314yes—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—44no—
91C2×C22⋊Q8—44no—
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C2×C4×D4—28no—
95C2×C4×D4—48no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#400—28no—
131#400—28no—
132#400—44no—
133#396—48no—
134#394Q818yescomplemented
135#391—24no—
136#391—24no—
137#391—24no—
138#391—44no—
139#389—24no—
140#389—28no—
141#389—44no—
142#387—28no—
143#387—28no—
144#385—216no—
145#385—48no—
146#385—48no—
147#384—24no—
148#384—24no—
149#384—44no—
150#380—28no—
151#380—44no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4D418yes—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—44no—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2311yescharacteristic
186C25C2412yes—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—22no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4C2×Q814yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4#4118yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C22⋊Q8—432no—
228C4×D4—232no—
229C4×D4—432no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—432no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
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⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×Q814yes—
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—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4C2×Q818yes—
302C2×C22⋊C4C2×Q818yes—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—44no—
313C2×C42C2×D414yes—
314C2×C42C2×D418yes—
315#95—22no—
316#95—22no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—24no—
329#95—22no—
330#95—24no—
331#95—42no—
332#95—42no—
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#95—42no—
343#95C2×D418yes—
344#95C2×Q814yes—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24C22×Q814yes—
358C24C2×#4114yes—
359C24H(5,𝔽2)14yes—
360C24C2511yescharacteristic
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—216no—
369C2×D4—216no—
370C2×D4—28no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×Q814yes—
466C22×C4C22×Q812yes—
467C22×C4C22×Q812yes—
468C22×C4C22×Q812yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C42−1+414yes—
475C22×C42−1+412yes—
476C22×C42−1+412yes—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—416no—
484C4⋊C4—48no—
485C4⋊C4—416no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4—88no—
493C4⋊C4—88no—
494C4⋊C4—88no—
495C22⋊C4—216no—
496C22⋊C4—216no—
497C22⋊C4—216no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—416no—
501C22⋊C4—416no—
502C22⋊C4—88no—
503C42—28no—
504C42—48no—
505C42—44no—
506C42—44no—
507C42—48no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23#55418yes—
543C23C23×D411yescharacteristic
544C23C23×Q811yescharacteristic
545C23C22×#4112yes—
546C23C2×H(5,𝔽2)12yes—
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)12yes—
549C23C2×H(5,𝔽2)11yescharacteristic
550C23C2×2−1+411yescharacteristic
551C23C2×2−1+411yescharacteristic
552C23C2×2−1+412yes—
553C23#58511yescharacteristic
554Q8—88no—
555D4—232no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—432no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—832no—
644C2×C4#52514yes—
645C2×C4#53514yes—
646C2×C4#53714yes—
647C2×C4#53818yes—
648C2×C4#53918yes—
649C2×C4D4×Q814yes—
650C22—28no—
651C22—28no—
652C22—28no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22C2×#52511yescharacteristic
666C22C2×#53511yescharacteristic
667C22C2×#53711yescharacteristic
668C22C2×#53812yes—
669C22C2×#53912yes—
670C22C2×D4×Q811yescharacteristic
671C22C2×#55412yes—
672C22#348411yescharacteristic
673C22#348811yescharacteristic
674C22#349212yes—
675C22#349212yes—
676C22#349911yescharacteristic
677C22#350111yescharacteristic
678C22#350311yescharacteristic
679C22#350312yes—
680C22#350712yes—
681C22#353612yes—
682C22#354312yes—
683C22#354412yes—
684C22#354811yescharacteristic
685C22#355212yes—
686C22#355511yescharacteristic
687C22#355812yes—
688C4—24no—
689C4—44no—
690C4—44no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—42no—
696C4—44no—
697C4—44no—
698C4—84no—
699C4—82no—
700C4—81no—
701C4—82no—
702C4—84no—
703C4#263618yes—
704C2—48no—
705C2C2×#263611yescharacteristic
706C2#3568611yescharacteristic
707C2#3654611yescharacteristic
708C2#3959811yescharacteristic
709C2#3960412yes—
710C2#3961611yescharacteristic
711C2#3963212yes—
712C2#4242811yescharacteristic
713C2#4245212yes—
714C2#4248411yescharacteristic
715C2#4250412yes—
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×#2511C2semidirect
3. #16988C2semidirect
4. #16981C2semidirect
5. #16339C2semidirect
6. #16322C2semidirect
7. #15199C2semidirect
8. #15179C2semidirect
10. #15152C2semidirect
11. #15131C2semidirect
14. #14139C2semidirect
15. #13933C2semidirect
16. #13880C2semidirect
134. #394Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25112
C2×D4×C4⋊C42
#169882
#169812
#163392
#163222
#151992
#151792
#151772
#151522
#151312
#148122
#143002
#141392
#139332
#138802
C4×#3942