Finite Group Databasebeta

← #4306672

Subgroups — #4306672

Order 512.

3230 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2360C212yesmaximal, complemented
2C2×#2356C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#15220C212yesmaximal, complemented
5#15185C211yesmaximal, characteristic, complemented
6#15179C212yesmaximal, complemented
7#15177C211yesmaximal, characteristic
8#15151C211yesmaximal, characteristic, complemented
9#15122C212yesmaximal, complemented
10#14925C214yesmaximal, complemented
11#14818C212yesmaximal, complemented
12#14801C212yesmaximal
13#14753C214yesmaximal, complemented
14C4×#394C211yesmaximal, characteristic
15#13847C212yesmaximal
16#13821C211yesmaximal, characteristic, complemented
17C22⋊C4×C4⋊C4C212yesmaximal, complemented
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C4⋊C4C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#394C2211yescharacteristic
26#2360—216no—
27#2356—28no—
28D4×C4⋊C4—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#387C2214yes—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#384C2211yescharacteristic
42C2×#383C2212yes—
43C2×#382C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2214yes—
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2214yes—
55C2×#376C2212yes—
56#1445C2212yes—
57#1445C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1444C2212yes—
61#1443C2214yes—
62#1442C2211yescharacteristic
63#1442C2214yes—
64#1442C2214yes—
65#1441C2211yescharacteristic
66#1441C2214yes—
67#1441C2214yes—
68#1440C2212yes—
69#1440C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1438C2211yescharacteristic
76#1438C2212yes—
77#1436C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80C4⋊#95C2211yescharacteristic
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C22⋊#95C2214yes—
85C4×#95C2211yescharacteristic
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—28no—
98C2×C4×D4—24no—
99C2×C4×D4—28no—
100C2×C4×D4—44no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2314yes—
135#398—44no—
136#398—44no—
137#394—24no—
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#383—28no—
152#382—24no—
153#380—28no—
154#380—28no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—44no—
159C4×C4⋊C4D418yes—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—28no—
162C4×C22⋊C4—216no—
163C4×C22⋊C4—48no—
164#376—28no—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C25C2412yes—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8C2×D414yes—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4#4114yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4#4118yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C22⋊Q8—264no—
237C4×Q8—232no—
238C4×D4—232no—
239C4×D4—432no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—432no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
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—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—42no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42C2×D418yes—
326C2×C42C2×D414yes—
327C2×C42C2×D418yes—
328C2×C42C2×D414yes—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—28no—
337#95—22no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—22no—
350#95—28no—
351#95—22no—
352#95—24no—
353#95—24no—
354#95—44no—
355#95—44no—
356#95—42no—
357#95—44no—
358#95C2×D418yes—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24C2×#4114yes—
372C24C2×#4114yes—
373C242−1+414yes—
374C24C2511yescharacteristic
375C2×Q8—28no—
376C2×Q8—216no—
377C2×Q8—44no—
378C2×Q8—48no—
379C2×Q8—44no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—216no—
383C2×D4—216no—
384C2×D4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—216no—
436C22×C4—216no—
437C22×C4—216no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—416no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C2×#4118yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4114yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C42−1+414yes—
493C4⋊C4—28no—
494C4⋊C4—232no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—28no—
498C4⋊C4—44no—
499C4⋊C4—416no—
500C4⋊C4—416no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—84no—
506C4⋊C4—88no—
507C4⋊C4#121116yes—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—216no—
511C22⋊C4—232no—
512C22⋊C4—48no—
513C22⋊C4—416no—
514C22⋊C4—416no—
515C22⋊C4—48no—
516C42—28no—
517C42—216no—
518C42—28no—
519C42—216no—
520C42—48no—
521C42—44no—
522C42—48no—
523C42—88no—
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—24no—
539C23—24no—
540C23—24no—
541C23—24no—
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—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23#55218yes—
559C23C23×D411yescharacteristic
560C23C23×D411yescharacteristic
561C23C22×#4111yescharacteristic
562C23C22×#4112yes—
563C23C22×#4112yes—
564C23C22×#4111yescharacteristic
565C23C2×H(5,𝔽2)11yescharacteristic
566C23C2×H(5,𝔽2)11yescharacteristic
567C23C2×2−1+412yes—
568C23C2×2−1+412yes—
569C23#58511yescharacteristic
570Q8—416no—
571D4—232no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—28no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—432no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—832no—
661C2×C4C2×#12118yes—
662C2×C4C2×#12114yes—
663C2×C4#52514yes—
664C2×C4#52518yes—
665C2×C4#52918yes—
666C2×C4#53214yes—
667C2×C4#54614yes—
668C22—28no—
669C22—28no—
670C22—28no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22C22×#12111yescharacteristic
684C22C2×#52511yescharacteristic
685C22C2×#52512yes—
686C22C2×#52912yes—
687C22C2×#53211yescharacteristic
688C22C2×#54611yescharacteristic
689C22D4×#4111yescharacteristic
690C22C2×#55212yes—
691C22#348111yescharacteristic
692C22#348212yes—
693C22#348611yescharacteristic
694C22#348612yes—
695C22#348811yescharacteristic
696C22#349212yes—
697C22#349212yes—
698C22#349811yescharacteristic
699C22#349911yescharacteristic
700C22#350712yes—
701C22#351411yescharacteristic
702C22#351612yes—
703C22#352012yes—
704C22#352012yes—
705C22#352312yes—
706C4—24no—
707C4—28no—
708C4—44no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—48no—
713C4—44no—
714C4—42no—
715C4—44no—
716C4—44no—
717C4—81no—
718C4—82no—
719C4—82no—
720C4—84no—
721C4#239218yes—
722C2—48no—
723C2C2×#239211yescharacteristic
724C2#3568611yescharacteristic
725C2#3570011yescharacteristic
726C2#3570112yes—
727C2#3572612yes—
728C2#3590911yescharacteristic
729C2#3616311yescharacteristic
730C2#3623212yes—
731C2#3625612yes—
732C2#3625811yescharacteristic
733C2#3640611yescharacteristic
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×#2356C2semidirect
4. #15220C2semidirect
5. #15185C2semidirect
6. #15179C2semidirect
8. #15151C2semidirect
9. #15122C2semidirect
10. #14925C2semidirect
11. #14818C2semidirect
13. #14753C2semidirect
16. #13821C2semidirect
17. C22⋊C4×C4⋊C4C2semidirect
95. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23602
C2×#23562
C2×D4×C4⋊C42
#152202
#151852
#151792
#151772
#151512
#151222
#149252
#148182
#148012
#147532
C4×#3942
#138472
#138212
C22⋊C4×C4⋊C42