Finite Group Databasebeta

← #4307460

Subgroups — #4307460

Order 512.

3230 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2360C212yesmaximal, complemented
2C2×#2355C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#15206C212yesmaximal, complemented
5#15196C211yesmaximal, characteristic, complemented
6#15185C211yesmaximal, characteristic, complemented
7#15179C212yesmaximal, complemented
8#15177C211yesmaximal, characteristic
9#15131C212yesmaximal, complemented
10#15012C212yesmaximal, complemented
11#14923C214yesmaximal, complemented
12#14812C212yesmaximal
13#14752C214yesmaximal, complemented
14C4×#394C211yesmaximal, characteristic
15C4×#394C212yesmaximal
16#13818C211yesmaximal, characteristic, complemented
17C22⋊C4×C4⋊C4C212yesmaximal, complemented
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C4⋊C4C2212yes—
23C2×#396C2214yes—
24C2×#394C2211yescharacteristic
25#2360—216no—
26#2355—28no—
27D4×C4⋊C4—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#387C2214yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#384C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#382C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2214yes—
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2214yes—
54#1445C2214yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2212yes—
58#1442C2212yes—
59#1442C2211yescharacteristic
60#1442C2214yes—
61#1442C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2211yescharacteristic
66#1441C2212yes—
67#1441C2214yes—
68#1440C2214yes—
69#1440C2212yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2214yes—
73#1439C2214yes—
74#1438C2212yes—
75#1438C2211yescharacteristic
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2314yes—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8D418yescomplemented
96C2×C4×D4—28no—
97C2×C4×D4—24no—
98C2×C4×D4—28no—
99C2×C4×D4—48no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2314yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136#396—48no—
137#394Q818yescomplemented
138#391—28no—
139#391—24no—
140#391—24no—
141#389—216no—
142#389—28no—
143#389—28no—
144#389—28no—
145#387—28no—
146#387—216no—
147#387—28no—
148#385—216no—
149#385—216no—
150#384—24no—
151#384—24no—
152#382—28no—
153#380—28no—
154#380—28no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4D418yes—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—28no—
162C4×C22⋊C4—216no—
163C4×C22⋊C4—48no—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2311yescharacteristic
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2311yescharacteristic
192C25C2412yes—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8C2×D414yes—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4C2×Q814yes—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4#4118yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2414yes—
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C22⋊Q8—264no—
234C4×Q8—232no—
235C4×D4—232no—
236C4×D4—432no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—432no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×Q814yes—
281C2×C4⋊C4#4118yes—
282C2×C4⋊C4#4114yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×Q818yes—
306C2×C22⋊C4C2×Q818yes—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42C2×D414yes—
324C2×C42C2×D418yes—
325C2×C42C2×D414yes—
326C2×C42C2×D418yes—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—28no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—22no—
347#95—42no—
348#95—42no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95C2×D418yes—
354#95C2×Q814yes—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24C22×Q814yes—
368C24C2×#4114yes—
369C24H(5,𝔽2)14yes—
370C24C2511yescharacteristic
371C2×Q8—28no—
372C2×Q8—216no—
373C2×Q8—44no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—216no—
379C2×D4—216no—
380C2×D4—28no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—21no—
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—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—216no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—416no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×Q814yes—
476C22×C4C22×Q812yes—
477C22×C4C22×Q812yes—
478C22×C4C22×Q812yes—
479C22×C4C2×#4118yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4114yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4114yes—
485C22×C42−1+412yes—
486C22×C42−1+412yes—
487C4⋊C4—28no—
488C4⋊C4—232no—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—84no—
500C4⋊C4—88no—
501C4⋊C4—84no—
502C4⋊C4—88no—
503C4⋊C4#121116yes—
504C22⋊C4—216no—
505C22⋊C4—232no—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—48no—
509C22⋊C4—416no—
510C22⋊C4—416no—
511C22⋊C4—48no—
512C42—28no—
513C42—216no—
514C42—28no—
515C42—48no—
516C42—44no—
517C42—48no—
518C42—84no—
519C42—88no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23#55418yes—
555C23C23×D411yescharacteristic
556C23C23×D411yescharacteristic
557C23C23×Q811yescharacteristic
558C23C22×#4111yescharacteristic
559C23C22×#4112yes—
560C23C22×#4112yes—
561C23C2×H(5,𝔽2)12yes—
562C23C2×2−1+411yescharacteristic
563C23C2×2−1+411yescharacteristic
564C23#58512yes—
565C23#58511yescharacteristic
566Q8—416no—
567D4—232no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—432no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—832no—
654C2×C4C2×#12118yes—
655C2×C4C2×#12114yes—
656C2×C4#52514yes—
657C2×C4#52518yes—
658C2×C4#52918yes—
659C2×C4#53214yes—
660C2×C4D4×Q814yes—
661C2×C4D4×Q814yes—
662C22—28no—
663C22—28no—
664C22—28no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22C22×#12111yescharacteristic
678C22C2×#52511yescharacteristic
679C22C2×#52512yes—
680C22C2×#52912yes—
681C22C2×#53211yescharacteristic
682C22C2×D4×Q811yescharacteristic
683C22C2×D4×Q811yescharacteristic
684C22C2×#55412yes—
685C22Q8×#4111yescharacteristic
686C22Q8×#4112yes—
687C22#348411yescharacteristic
688C22#348812yes—
689C22#348911yescharacteristic
690C22#349212yes—
691C22#349312yes—
692C22#349711yescharacteristic
693C22#350011yescharacteristic
694C22#350712yes—
695C22#351312yes—
696C22#351312yes—
697C22#351512yes—
698C22#352212yes—
699C22#352311yescharacteristic
700C4—28no—
701C4—24no—
702C4—24no—
703C4—44no—
704C4—42no—
705C4—42no—
706C4—48no—
707C4—44no—
708C4—42no—
709C4—44no—
710C4—44no—
711C4—82no—
712C4—84no—
713C4—81no—
714C4—82no—
715C4#239218yes—
716C2—48no—
717C2C2×#239211yescharacteristic
718C2#3572012yes—
719C2#3572811yescharacteristic
720C2#3575911yescharacteristic
721C2#3577212yes—
722C2#3591211yescharacteristic
723C2Q8×#12111yescharacteristic
724C2#3654412yes—
725C2#3654611yescharacteristic
726C2#3655312yes—
727C2#3656211yescharacteristic
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×#2360C2semidirect
2. C2×#2355C2semidirect
4. #15206C2semidirect
5. #15196C2semidirect
6. #15185C2semidirect
7. #15179C2semidirect
9. #15131C2semidirect
10. #15012C2semidirect
11. #14923C2semidirect
13. #14752C2semidirect
16. #13818C2semidirect
17. C22⋊C4×C4⋊C4C2semidirect
95. C2×C4×Q8D4semidirect
137. #394Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23602
C2×#23552
C2×D4×C4⋊C42
#152062
#151962
#151852
#151792
#151772
#151312
#150122
#149232
#148122
#147522
C4×#3942
#138182
C22⋊C4×C4⋊C42