Finite Group Databasebeta

← #545382

Subgroups — #545382

Order 512.

3038 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3C2×#2320C211yesmaximal, characteristic, complemented
4#14631C212yesmaximal, complemented
5#14403C214yesmaximal, complemented
6#14389C214yesmaximal, complemented
7#14328C212yesmaximal, complemented
8#14300C212yesmaximal
9#14277C211yesmaximal, characteristic, complemented
10#14190C212yesmaximal
11#14139C212yesmaximal, complemented
12#14130C212yesmaximal, complemented
13#13997C212yesmaximal, complemented
14#13994C212yesmaximal, complemented
15#13926C211yesmaximal, characteristic, complemented
16#13847C211yesmaximal, characteristic
17#13804C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C4⋊C4C2212yes—
23#2536C4116yescomplemented
24C2×#398C2212yes—
25C2×#394C2212yes—
26D4×C4⋊C4—28no—
27#2320—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#384C2211yescharacteristic
42C2×#383C2212yes—
43C2×#382C2212yes—
44C2×#382C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2211yescharacteristic
53#1445C2211yescharacteristic
54#1445C2211yescharacteristic
55#1445C2214yes—
56#1444C2214yes—
57#1444C2212yes—
58#1444C2212yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2212yes—
62#1442C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2212yes—
67#1440C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1439C2214yes—
71#1438C2211yescharacteristic
72#1438C2211yescharacteristic
73#1437C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2214yes—
80#1434C2212yes—
81#1434C2214yes—
82C22⋊#95C2214yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2314yes—
90C2×C22⋊Q8—44no—
91C2×C22⋊Q8—44no—
92C2×C22⋊Q8C2×C4116yescomplemented
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8C2×C418yescomplemented
96C2×C4×D4—24no—
97C2×C4×D4—28no—
98C2×C4×D4—24no—
99C2×C4×D4—48no—
100C2×C4×D4C2×C418yes—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134#398C2×C4116yescomplemented
135#394C2×C4116yes—
136#391—24no—
137#391—24no—
138#391—44no—
139#389—24no—
140#389—24no—
141#389—28no—
142#389—44no—
143#388—48no—
144#388C2×C4132yescomplemented
145#387—28no—
146#387—28no—
147#385—216no—
148#385—48no—
149#384—24no—
150#383—28no—
151#382—28no—
152#382C2×C418yescomplemented
153#380—28no—
154#380C2×C4116yescomplemented
155C4×C4⋊C4—24no—
156C4×C4⋊C4—44no—
157C4×C4⋊C4D418yes—
158C4×C22⋊C4—44no—
159C4×C22⋊C4—44no—
160#376—24no—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C25C2412yes—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8C22×C414yes—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4C22×C414yes—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4C22×C418yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C22⋊Q8—432no—
236C4×Q8—232no—
237C4×D4—232no—
238C4×D4—432no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—432no—
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⋊C4C22×C414yes—
275C2×C4⋊C4C22×C414yes—
276C2×C4⋊C4C22×C418yes—
277C2×C4⋊C4C22×C418yes—
278C2×C4⋊C4C22×C4116yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4C22×C418yes—
303C2×C22⋊C4C22×C4116yes—
304C2×C22⋊C4C22×C4116yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—28no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42C22×C418yes—
321C2×C42C22×C414yes—
322C2×C42C2×D418yes—
323C2×C42C2×D414yes—
324C2×C42#4114yes—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—44no—
339#95—44no—
340#95—42no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95C22×C4116yes—
353#95C22×C418yes—
354#95C22×C418yes—
355#95C2×D418yes—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24C23×C414yes—
369C24H(5,𝔽2)14yes—
370C242−1+414yes—
371C24C2511yescharacteristic
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—28no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—216no—
382C2×D4—216no—
383C2×D4—28no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
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—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—216no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4C23×C418yes—
473C22×C4C23×C414yes—
474C22×C4C23×C414yes—
475C22×C4C23×C414yes—
476C22×C4C23×C412yes—
477C22×C4C23×C412yes—
478C22×C4C23×C412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+414yes—
491C22×C42−1+412yes—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—416no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—88no—
507C22⋊C4—232no—
508C22⋊C4—216no—
509C22⋊C4—416no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C42—216no—
516C42—48no—
517C42—44no—
518C42—44no—
519C42—48no—
520C42—88no—
521C42C4×D4116yes—
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—24no—
539C23—24no—
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—42no—
555C23—42no—
556C23#52018yes—
557C23C24×C411yescharacteristic
558C23C23×D411yescharacteristic
559C23C22×#4111yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)11yescharacteristic
563C23C2×H(5,𝔽2)12yes—
564C23C2×2−1+411yescharacteristic
565C23C2×2−1+412yes—
566C23C2×2−1+412yes—
567C23#58512yes—
568Q8—416no—
569D4—232no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—432no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—832no—
657C2×C4C2×C4×D414yes—
658C2×C4C2×C4×D414yes—
659C2×C4C2×C4×D414yes—
660C2×C4#51814yes—
661C2×C4#52014yes—
662C2×C4#53514yes—
663C2×C4#53714yes—
664C2×C4#54014yes—
665C2×C4#54214yes—
666C22—28no—
667C22—28no—
668C22—28no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22C22×C4×D411yescharacteristic
682C22C2×#51811yescharacteristic
683C22C2×#52011yescharacteristic
684C22C2×#52012yes—
685C22C4×2−1+412yes—
686C22C2×#53511yescharacteristic
687C22C2×#53711yescharacteristic
688C22C2×#54011yescharacteristic
689C22C2×#54211yescharacteristic
690C22#348411yescharacteristic
691C22#348611yescharacteristic
692C22#349212yes—
693C22#349512yes—
694C22#349911yescharacteristic
695C22#350011yescharacteristic
696C22#353612yes—
697C22#353812yes—
698C22#354112yes—
699C22#354212yes—
700C22#354412yes—
701C22#355212yes—
702C22#355812yes—
703C22#356312yes—
704C4—24no—
705C4—24no—
706C4—44no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4—48no—
711C4—44no—
712C4—44no—
713C4—82no—
714C4—81no—
715C4—81no—
716C4—81no—
717C4—84no—
718C4—81no—
719C4#232418yes—
720C2—48no—
721C2C2×#232411yescharacteristic
722C2#3420011yescharacteristic
723C2#3427611yescharacteristic
724C2#3446512yes—
725C2#3447212yes—
726C2#4242611yescharacteristic
727C2#4242811yescharacteristic
728C2#4251012yes—
729C2#4251212yes—
730C2#4279011yescharacteristic
731C2#4279211yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#2320C2semidirect
4. #14631C2semidirect
5. #14403C2semidirect
6. #14389C2semidirect
7. #14328C2semidirect
9. #14277C2semidirect
11. #14139C2semidirect
12. #14130C2semidirect
13. #13997C2semidirect
14. #13994C2semidirect
15. #13926C2semidirect
17. #13804C2semidirect
23. #2536C4semidirect
92. C2×C22⋊Q8C2×C4semidirect
95. C2×C4×Q8C2×C4semidirect
134. #398C2×C4semidirect
144. #388C2×C4semidirect
152. #382C2×C4semidirect
154. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
C2×D4×C4⋊C42
C2×#23202
#146312
#144032
#143892
#143282
#143002
#142772
#141902
#141392
#141302
#139972
#139942
#139262
#138472
#138042