Finite Group Databasebeta

← #537846

Subgroups — #537846

Order 512.

3404 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×#2542C211yesmaximal, characteristic
2C2×#2360C212yesmaximal, complemented
3C2×#2323C212yesmaximal, complemented
4#14551C212yesmaximal, complemented
5#14529C214yesmaximal, complemented
6#14424C212yesmaximal, complemented
7#14215C212yesmaximal, complemented
8#14151C212yesmaximal, complemented
9#14036C214yesmaximal
10#14024C212yesmaximal, complemented
11#13997C212yesmaximal, complemented
12#13997C212yesmaximal, complemented
13#13872C211yesmaximal, characteristic
14#13839C211yesmaximal, characteristic
15#13821C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2542C4116yescomplemented
24C2×#399C2212yes—
25C2×#399C2212yes—
26#2360—216no—
27#2323—216no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#390C2212yes—
32C2×#389C2214yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2214yes—
37C2×#388C2214yes—
38C2×#387C2214yes—
39C2×#387C2214yes—
40C2×#386C2212yes—
41C2×#385C2214yes—
42C2×#384C2212yes—
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2214yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2214yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52#1445C2214yes—
53#1445C2214yes—
54#1441C2211yescharacteristic
55#1441C2211yescharacteristic
56#1441C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2212yes—
60#1439C2211yescharacteristic
61#1439C2211yescharacteristic
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2212yes—
66#1437C2214yes—
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2214yes—
74C4⋊#95C2214yes—
75#1434C2214yes—
76C22⋊#95C2212yes—
77C4×#95C2212yes—
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C2×C4⋊Q8—42no—
84C2×C4⋊Q8—42no—
85C2×C4⋊Q8C2×C418yescomplemented
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8C2×C4116yes—
91C2×C22⋊Q8C2×C4116yescomplemented
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—44no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C2×C4×D4C2×C418yes—
98C2×#117—28no—
99C2×#117—42no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4—42no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2314yes—
133#399C2×C4116yes—
134#399C2×C4116yescomplemented
135#391—28no—
136#391—28no—
137#391—44no—
138#390C2×C4116yes—
139#389—216no—
140#389—216no—
141#389—28no—
142#389D4116yescomplemented
143#388—216no—
144#388—216no—
145#387—216no—
146#387—216no—
147#386—44no—
148#385—216no—
149#384C2×C4116yescomplemented
150#380—28no—
151#380—42no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—48no—
155C4×C4⋊C4C2×C418yescomplemented
156C4×C22⋊C4—28no—
157C4×C22⋊C4—216no—
158C4×C22⋊C4—42no—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2311yescharacteristic
180C2×#95C2312yes—
181C2×#95C2314yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8C22×C418yes—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4C22×C414yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4C22×C414yes—
212C23×C4C22×C414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C4⋊Q8—416no—
227C22⋊Q8—264no—
228C22⋊Q8—264no—
229C4×Q8—432no—
230C4×D4—416no—
231#117—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C22×C4116yes—
263C2×C4⋊C4C22×C4116yes—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C418yes—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×D4116yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4#4118yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
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—28no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4C22×C4116yes—
299C2×C22⋊C4C22×C4116yes—
300C2×C22⋊C4C22×C414yes—
301C2×C22⋊C4C22×C414yes—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—28no—
309C2×C42—28no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—28no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42C22×C414yes—
321C2×C42C22×C418yes—
322C2×C42C2×D418yes—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—44no—
338#95—42no—
339#95—41no—
340#95—41no—
341#95—42no—
342#95—44no—
343#95—44no—
344#95C22×C418yes—
345#95C2×D4116yes—
346C24—22no—
347C24—22no—
348C24—21no—
349C24—21no—
350C24—21no—
351C24—22no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24C23×C412yes—
365C24C23×C412yes—
366C242−1+412yes—
367C242−1+412yes—
368C24C2511yescharacteristic
369C2×Q8—216no—
370C2×Q8—216no—
371C2×Q8—216no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—216no—
434C22×C4—216no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—48no—
452C22×C4—416no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—88no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4C23×C412yes—
463C22×C4C23×C418yes—
464C22×C4C23×C412yes—
465C22×C4C23×C412yes—
466C22×C4C23×C418yes—
467C22×C4C23×C412yes—
468C22×C4C23×C412yes—
469C22×C4C22×D414yes—
470C22×C4C22×D418yes—
471C22×C4C22×D414yes—
472C22×C4C22×D418yes—
473C22×C4C22×D414yes—
474C22×C4C2×#4118yes—
475C22×C4C2×#4114yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+414yes—
480C22×C42−1+412yes—
481C4⋊C4—232no—
482C4⋊C4—232no—
483C4⋊C4—216no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—416no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—416no—
493C4⋊C4—48no—
494C4⋊C4C4×D4132yes—
495C22⋊C4—216no—
496C22⋊C4—232no—
497C22⋊C4—232no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—88no—
504C22⋊C4—84no—
505C42—28no—
506C42—216no—
507C42—216no—
508C42—416no—
509C42—48no—
510C42—88no—
511C42—88no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23—41no—
556C23—41no—
557C23#51914yes—
558C23C24×C411yescharacteristic
559C23C23×D412yes—
560C23C22×#4112yes—
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)11yescharacteristic
563C23C2×H(5,𝔽2)11yescharacteristic
564C23C2×2−1+411yescharacteristic
565C23C2×2−1+411yescharacteristic
566C23C2×2−1+411yescharacteristic
567C23C2×2−1+412yes—
568C23#58512yes—
569Q8—432no—
570D4—416no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—416no—
634C2×C4—416no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—816no—
650C2×C4C2×C4×D4116yes—
651C2×C4C2×C4×D418yes—
652C2×C4#51814yes—
653C2×C4#52014yes—
654C2×C4D4214yes—
655C2×C4#55018yes—
656C2×C4#55914yes—
657C22—24no—
658C22—28no—
659C22—44no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—42no—
668C22—44no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—44no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—82no—
677C22—82no—
678C22—81no—
679C22C22×C4×D412yes—
680C22C2×#51811yescharacteristic
681C22C2×#51911yescharacteristic
682C22C2×#52011yescharacteristic
683C22C4×2−1+412yes—
684C22C2×D4211yescharacteristic
685C22C2×#55012yes—
686C22C2×#55911yescharacteristic
687C22#348612yes—
688C22#348912yes—
689C22#349612yes—
690C22#349912yes—
691C22#351512yes—
692C22#351612yes—
693C22#352012yes—
694C22#352312yes—
695C22#353611yescharacteristic
696C22#353611yescharacteristic
697C22#353811yescharacteristic
698C22#353811yescharacteristic
699C22#354112yes—
700C22#355011yescharacteristic
701C22#355111yescharacteristic
702C4—216no—
703C4—24no—
704C4—44no—
705C4—44no—
706C4—48no—
707C4—44no—
708C4—42no—
709C4—42no—
710C4—48no—
711C4—42no—
712C4—84no—
713C4—82no—
714C4#236618yes—
715C2—44no—
716C2—82no—
717C2C2×#236611yescharacteristic
718C2#3429112yes—
719C2#3430812yes—
720C2#3438011yescharacteristic
721C2#3441311yescharacteristic
722C2#4510311yescharacteristic
723C2#4511212yes—
724C2#4521712yes—
725C2#4522611yescharacteristic
726C2#4532111yescharacteristic
727C2#4534511yescharacteristic
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
2. C2×#2360C2semidirect
3. C2×#2323C2semidirect
4. #14551C2semidirect
5. #14529C2semidirect
6. #14424C2semidirect
7. #14215C2semidirect
8. #14151C2semidirect
10. #14024C2semidirect
11. #13997C2semidirect
12. #13997C2semidirect
15. #13821C2semidirect
23. #2542C4semidirect
85. C2×C4⋊Q8C2×C4semidirect
91. C2×C22⋊Q8C2×C4semidirect
134. #399C2×C4semidirect
142. #389D4semidirect
149. #384C2×C4semidirect
155. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#23602
C2×#23232
#145512
#145292
#144242
#142152
#141512
#140362
#140242
#139972
#138722
#138392
#138212