Finite Group Databasebeta

← #2736677

Subgroups — #2736677

Order 512.

2516 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic, complemented
2C2×#2427C211yesmaximal, characteristic
3#17199C214yesmaximal, complemented
4#17151C214yesmaximal, complemented
5#17058C212yesmaximal, complemented
6#16600C212yesmaximal, complemented
7#16392C212yesmaximal, complemented
8#15240C211yesmaximal, characteristic, complemented
9#15226C212yesmaximal, complemented
10#15225C212yesmaximal, complemented
11#15219C212yesmaximal
12#15201C211yesmaximal, characteristic, complemented
13#15179C212yesmaximal, complemented
14#15135C211yesmaximal, characteristic, complemented
15#14660C212yesmaximal, complemented
16#13916C211yesmaximal, characteristic, complemented
17C4×#393C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#2538—28no—
25#2427—28no—
26C2×#399C2212yes—
27C2×#397C2212yes—
28C2×#396C2214yes—
29C2×#394C2212yes—
30C2×#393C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#389C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#386C2212yes—
40C2×#384C2211yescharacteristic
41C2×#382C2212yes—
42C2×#381C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48#1446C2211yescharacteristic
49#1446C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1445C2214yes—
53#1445C2214yes—
54#1445C2212yes—
55#1444C2212yes—
56#1442C2214yes—
57#1442C2212yes—
58#1442C2212yes—
59#1442C2211yescharacteristic
60#1442C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1439C2211yescharacteristic
72#1439C2214yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2214yes—
76#1439C2214yes—
77#1438C2214yes—
78#1438C2214yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C22⋊#95C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×#124—24no—
92C2×#124—44no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8—44no—
98C2×#121—24no—
99C2×#121—44no—
100C2×C4×Q8—24no—
101C2×C4×Q8—44no—
102C2×C4×D4—44no—
103C2×C4×D4D418yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#399—44no—
135#397—28no—
136#396—216no—
137#394—28no—
138#393—24no—
139#391—24no—
140#391—44no—
141#391—44no—
142#390—28no—
143#390—44no—
144#389—24no—
145#389—28no—
146#389—44no—
147#386—28no—
148#384—24no—
149#382—44no—
150#381—44no—
151#380—28no—
152#380—28no—
153#380—44no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—44no—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8#4114yes—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—44no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4C2×D414yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4C2×D418yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232#124—416no—
233C22⋊Q8—416no—
234C22⋊Q8—432no—
235#121—416no—
236C4×Q8—416no—
237C4×D4—416no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D414yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4C2×D418yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42C2×D414yes—
315#95—22no—
316#95—22no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—42no—
326#95—44no—
327#95—42no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—22no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—21no—
363C24—22no—
364C24C22×D414yes—
365C242−1+412yes—
366C24C2511yescharacteristic
367C2×Q8—28no—
368C2×Q8—48no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—48no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—84no—
385C2×D4—88no—
386C2×D4—88no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—216no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C2×#4114yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C42−1+412yes—
483C22×C42−1+412yes—
484C4⋊C4—28no—
485C4⋊C4—416no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—88no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C42—28no—
511C42—48no—
512C42—44no—
513C42—48no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—22no—
525C23—24no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—41no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—44no—
558C23—42no—
559C23—44no—
560C23C23×D411yescharacteristic
561C23C22×#4111yescharacteristic
562C23C2×H(5,𝔽2)12yes—
563C23C2×H(5,𝔽2)12yes—
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×2−1+411yescharacteristic
566C23C2×2−1+411yescharacteristic
567C23C2×2−1+411yescharacteristic
568C23#58512yes—
569C23#58511yescharacteristic
570C23#58512yes—
571Q8—416no—
572Q8—88no—
573D4—416no—
574D4—816no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—816no—
657C2×C4—88no—
658C2×C4#53414yes—
659C2×C4#53514yes—
660C2×C4#53614yes—
661C2×C4#54614yes—
662C2×C4#56218yes—
663C2×C4#56218yes—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—44no—
675C22—42no—
676C22—82no—
677C22—84no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—82no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22—81no—
686C22C2×#53411yescharacteristic
687C22C2×#53511yescharacteristic
688C22C2×#53611yescharacteristic
689C22C2×#54611yescharacteristic
690C22#348511yescharacteristic
691C22#348911yescharacteristic
692C22#350011yescharacteristic
693C22#350512yes—
694C22#351712yes—
695C22#351811yescharacteristic
696C22#352311yescharacteristic
697C22C2×#56212yes—
698C22C2×#56212yes—
699C22#354112yes—
700C22#354212yes—
701C22#354612yes—
702C22#355112yes—
703C22#355211yescharacteristic
704C22#355411yescharacteristic
705C22#355712yes—
706C22#356112yes—
707C22#356212yes—
708C22#356512yes—
709C4—24no—
710C4—44no—
711C4—44no—
712C4—42no—
713C4—82no—
714C4—81no—
715C4—82no—
716C4—81no—
717C4—84no—
718C4—84no—
719C4—82no—
720C4—82no—
721C4—81no—
722C4—82no—
723C4#281618yes—
724C2—84no—
725C2—82no—
726C2C2×#281611yescharacteristic
727C2#3891111yescharacteristic
728C2#3895611yescharacteristic
729C2#3946611yescharacteristic
730C2#4264711yescharacteristic
731C2#4268011yescharacteristic
732C2#4290811yescharacteristic
733C2#4570312yes—
734C2#4608312yes—
735C2#5310512yes—
736C2#5312012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2538C2semidirect
3. #17199C2semidirect
4. #17151C2semidirect
5. #17058C2semidirect
6. #16600C2semidirect
7. #16392C2semidirect
8. #15240C2semidirect
9. #15226C2semidirect
10. #15225C2semidirect
12. #15201C2semidirect
13. #15179C2semidirect
14. #15135C2semidirect
15. #14660C2semidirect
16. #13916C2semidirect
17. C4×#393C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#24272
#171992
#171512
#170582
#166002
#163922
#152402
#152262
#152252
#152192
#152012
#151792
#151352
#146602
#139162
C4×#3932