Finite Group Databasebeta

← #781057

Subgroups — #781057

Order 512.

3822 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2440C212yesmaximal, complemented
2C2×#2463C212yesmaximal
3C2×D4×C4⋊C4C212yesmaximal, complemented
4#16318C212yesmaximal, complemented
5#15966C212yesmaximal, complemented
6#15778C212yesmaximal, complemented
7#15354C214yesmaximal, complemented
8#15129C212yesmaximal, complemented
9#15125C211yesmaximal, characteristic, complemented
10#14801C214yesmaximal, complemented
11#14130C211yesmaximal, characteristic, complemented
12#14049C212yesmaximal, complemented
13#13994C212yesmaximal, complemented
14#13872C212yesmaximal, complemented
15C4×#392C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2214yes—
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23#2463—216no—
24#2440—216no—
25C2×#398C2214yes—
26C2×#395C2214yes—
27C2×#394C2214yes—
28C2×#393C2212yes—
29C2×#392C2211yescharacteristic
30D4×C4⋊C4—216no—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#388C2214yes—
35C2×#387C2214yes—
36C2×#387C2214yes—
37C2×#386C2214yes—
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#384C2212yes—
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51#1446C2212yes—
52#1445C2214yes—
53#1445C2214yes—
54#1444C2211yescharacteristic
55#1444C2212yes—
56#1442C2212yes—
57#1442C2212yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2212yes—
65#1439C2211yescharacteristic
66#1439C2212yes—
67#1439C2214yes—
68#1438C2211yescharacteristic
69#1438C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C4×#95C2211yescharacteristic
78C23×Q8C2312yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2314yes—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8—42no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—48no—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4—216no—
93C2×C4×D4—28no—
94C2×C4×D4—216no—
95C2×C4×D4—28no—
96C2×C4×D4—48no—
97C2×C4×D4—42no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—42no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#398—216no—
133#395—216no—
134#394—216no—
135#393—28no—
136#392D418yes—
137#389—216no—
138#389—28no—
139#389—28no—
140#388—216no—
141#387—216no—
142#387—48no—
143#386—216no—
144#385—216no—
145#385—216no—
146#385—216no—
147#385—28no—
148#384D4116yescomplemented
149#380—24no—
150#380—28no—
151#380—28no—
152#380—28no—
153#380—28no—
154C4×C4⋊C4—28no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—42no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C25C2412yes—
181C25C2411yescharacteristic
182C22×Q8—24no—
183C22×Q8—28no—
184C22×Q8—44no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4C2×D418yes—
198C22×D4C2×D414yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4#4114yes—
222C23×C4#4118yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C4⋊Q8—416no—
237C22⋊Q8—464no—
238C4×D4—264no—
239C4×D4—464no—
240C4×D4—416no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
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—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—416no—
267C2×C4⋊C4—432no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D4116yes—
278C2×C4⋊C4C2×D4116yes—
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—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—41no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—41no—
312C2×C22⋊C4—41no—
313C2×C22⋊C4C2×D418yes—
314C2×C22⋊C4C2×D414yes—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—28no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—28no—
323C2×C42—22no—
324C2×C42C2×D418yes—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—41no—
342#95—41no—
343#95—41no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95C2×D414yes—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—21no—
367C24—22no—
368C24C22×D414yes—
369C24C22×D412yes—
370C24C2×#4112yes—
371C24C2×#4114yes—
372C242−1+414yes—
373C242−1+412yes—
374C24C2511yescharacteristic
375C2×Q8—416no—
376C2×Q8—48no—
377C2×Q8—84no—
378C2×Q8—88no—
379C2×Q8—84no—
380C2×D4—216no—
381C2×D4—216no—
382C2×D4—216no—
383C2×D4—216no—
384C2×D4—216no—
385C2×D4—216no—
386C2×D4—216no—
387C2×D4—216no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—48no—
391C2×D4—44no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—216no—
437C22×C4—216no—
438C22×C4—28no—
439C22×C4—216no—
440C22×C4—216no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—48no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—84no—
476C22×C4—88no—
477C22×C4—84no—
478C22×C4—88no—
479C22×C4C22×D412yes—
480C22×C4C22×D414yes—
481C22×C4C22×D418yes—
482C22×C4C22×D418yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4114yes—
491C22×C4H(5,𝔽2)14yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C42−1+412yes—
494C22×C42−1+412yes—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—232no—
499C4⋊C4—416no—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—44no—
503C4⋊C4—416no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—416no—
507C4⋊C4—48no—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—216no—
511C22⋊C4—216no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—416no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C22⋊C4—416no—
520C42—216no—
521C42—416no—
522C42—44no—
523C42—44no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—24no—
546C23—22no—
547C23—24no—
548C23—22no—
549C23—24no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—42no—
575C23—42no—
576C23—41no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23#54714yes—
581C23#54718yes—
582C23C23×D412yes—
583C23C23×D411yescharacteristic
584C23C22×#4111yescharacteristic
585C23C22×#4112yes—
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×H(5,𝔽2)12yes—
588C23C2×2−1+411yescharacteristic
589C23C2×2−1+411yescharacteristic
590C23C2×2−1+412yes—
591C23C2×2−1+411yescharacteristic
592C23#58511yescharacteristic
593Q8—816no—
594D4—264no—
595D4—416no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—22no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—41no—
650C2×C4—416no—
651C2×C4—432no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—41no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—44no—
671C2×C4—816no—
672C2×C4—832no—
673C2×C4#52518yes—
674C2×C4#53914yes—
675C2×C4D4218yes—
676C2×C4D4214yes—
677C2×C4#54818yes—
678C2×C4#55114yes—
679C22—28no—
680C22—28no—
681C22—28no—
682C22—24no—
683C22—28no—
684C22—42no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—42no—
691C22—42no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—42no—
698C22—44no—
699C22—42no—
700C22—44no—
701C22—42no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22C2×#52512yes—
706C22C2×#53911yescharacteristic
707C22C2×D4212yes—
708C22C2×D4211yescharacteristic
709C22C2×#54711yescharacteristic
710C22C2×#54712yes—
711C22C2×#54812yes—
712C22C2×#55111yescharacteristic
713C22#348612yes—
714C22#348612yes—
715C22#349212yes—
716C22#349212yes—
717C22#349512yes—
718C22#349612yes—
719C22#349611yescharacteristic
720C22#349911yescharacteristic
721C22#350711yescharacteristic
722C22#351112yes—
723C22#353811yescharacteristic
724C22#353811yescharacteristic
725C22#354911yescharacteristic
726C22#355212yes—
727C22#356311yescharacteristic
728C4—28no—
729C4—42no—
730C4—42no—
731C4—44no—
732C4—44no—
733C4—42no—
734C4—48no—
735C4—48no—
736C4—42no—
737C4—84no—
738C4—81no—
739C4—84no—
740C4#246318yescomplemented
741C2—44no—
742C2—48no—
743C2C2×#246311yescharacteristic
744C2#3568812yes—
745C2#3610312yes—
746C2#3937611yescharacteristic
747C2#4250211yescharacteristic
748C2#4491411yescharacteristic
749C2#4510912yes—
750C2#4541811yescharacteristic
751C2#5912512yes—
752C2#5919811yescharacteristic
753C2#5981611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2440C2semidirect
3. C2×D4×C4⋊C4C2semidirect
4. #16318C2semidirect
5. #15966C2semidirect
6. #15778C2semidirect
7. #15354C2semidirect
8. #15129C2semidirect
9. #15125C2semidirect
10. #14801C2semidirect
11. #14130C2semidirect
12. #14049C2semidirect
13. #13994C2semidirect
14. #13872C2semidirect
148. #384D4semidirect
740. C4#2463semidirect

Maximal subgroups

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

subgroupindex
C2×#24402
C2×#24632
C2×D4×C4⋊C42
#163182
#159662
#157782
#153542
#151292
#151252
#148012
#141302
#140492
#139942
#138722
C4×#3922