Finite Group Databasebeta

← #538044

Subgroups — #538044

Order 512.

3842 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×#2360C212yesmaximal, complemented
3C2×Q8×C22⋊C4C212yesmaximal, complemented
4#14575C212yesmaximal, complemented
5#14524C214yesmaximal, complemented
6#14402C212yesmaximal, complemented
7#14213C212yesmaximal, complemented
8#14143C212yesmaximal, complemented
9#14017C212yesmaximal, complemented
10#13994C212yesmaximal, complemented
11#13988C214yesmaximal
12#13969C212yesmaximal, complemented
13#13872C211yesmaximal, characteristic
14#13847C211yesmaximal, characteristic
15#13786C212yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2409C4116yescomplemented
23C2×#395C2212yes—
24C2×#393C2212yes—
25#2360—216no—
26Q8×C22⋊C4—216no—
27C2×#391C2214yes—
28C2×#391C2212yes—
29C2×#391C2214yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2214yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#387C2214yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#385C2214yes—
39C2×#385C2212yes—
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2214yes—
49C2×C4×C22⋊C4C2214yes—
50C2×#376C2211yescharacteristic
51#1445C2212yes—
52#1444C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1440C2214yes—
58#1439C2211yescharacteristic
59#1439C2211yescharacteristic
60#1439C2211yescharacteristic
61#1439C2211yescharacteristic
62#1439C2212yes—
63#1439C2214yes—
64#1438C2212yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75#1434C2214yes—
76#1434C2214yes—
77C22⋊#95C2212yes—
78C23×Q8—28no—
79C23×Q8—28no—
80C23×Q8—42no—
81C23×Q8—42no—
82C23×Q8C2×C418yescomplemented
83C23×Q8C2312yes—
84C23×Q8C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8C2×C4116yes—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×Q8—44no—
98C2×C4×Q8—44no—
99C2×C4×D4—42no—
100C2×C4×D4—42no—
101C2×C4×D4C2×C418yes—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—42no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2314yes—
131#395C2×C4116yescomplemented
132#393C2×C4116yes—
133#391—28no—
134#391—216no—
135#391—216no—
136#391C2×C4116yescomplemented
137#391D4116yescomplemented
138#389—28no—
139#389—216no—
140#388—48no—
141#387—216no—
142#386—44no—
143#386C2×C4116yes—
144#385—216no—
145#385C2×C4116yescomplemented
146#380—28no—
147#380—216no—
148#380—42no—
149#380C2×C418yescomplemented
150C4×C4⋊C4—28no—
151C4×C4⋊C4—42no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—216no—
155C4×C22⋊C4—216no—
156#376—42no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2311yescharacteristic
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×Q8—264no—
185C22×Q8—24no—
186C22×Q8—28no—
187C22×Q8—24no—
188C22×Q8—28no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—416no—
194C22×Q8—416no—
195C22×Q8—432no—
196C22×Q8—416no—
197C22×Q8C22×C418yes—
198C22×Q8C22×C418yes—
199C22×Q8C2×D418yes—
200C22×Q8#4118yes—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—41no—
206C22×D4—41no—
207C22×D4C22×C414yes—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—42no—
225C23×C4C22×C414yes—
226C23×C4C22×C414yes—
227C23×C4C22×C418yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2411yescharacteristic
241C22⋊Q8—264no—
242C4×Q8—432no—
243C4×Q8—432no—
244C4×D4—416no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C22×C418yes—
273C2×C4⋊C4C22×C4116yes—
274C2×C4⋊C4C22×C414yes—
275C2×C4⋊C4C2×D418yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
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—41no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4C22×C4116yes—
303C2×C22⋊C4C22×C418yes—
304C2×C22⋊C4C22×C414yes—
305C2×C22⋊C4C22×C414yes—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42C22×C414yes—
323C2×C42C2×D418yes—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—44no—
335#95—44no—
336#95—41no—
337#95—42no—
338#95—41no—
339#95—42no—
340#95—42no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95C22×C418yes—
345#95C22×C4116yes—
346#95C22×C4116yes—
347#95C2×D4116yes—
348#95C2×D4116yes—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—21no—
354C24—21no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24C23×C412yes—
366C24C23×C412yes—
367C24H(5,𝔽2)12yes—
368C242−1+412yes—
369C242−1+412yes—
370C242−1+412yes—
371C24C2511yescharacteristic
372C2×Q8—216no—
373C2×Q8—232no—
374C2×Q8—216no—
375C2×Q8—216no—
376C2×Q8—216no—
377C2×Q8—216no—
378C2×Q8—216no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—416no—
382C2×Q8—48no—
383C2×Q8—4128no—
384C2×Q8C4×D4132yes—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—216no—
445C22×C4—232no—
446C22×C4—216no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—416no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—416no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—88no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4C23×C412yes—
479C22×C4C23×C412yes—
480C22×C4C23×C418yes—
481C22×C4C23×C412yes—
482C22×C4C23×C412yes—
483C22×C4C23×C418yes—
484C22×C4C23×C412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D418yes—
488C22×C4C22×D418yes—
489C22×C4C22×D414yes—
490C22×C4C2×#4118yes—
491C22×C4C2×#4114yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4H(5,𝔽2)12yes—
494C22×C42−1+414yes—
495C4⋊C4—216no—
496C4⋊C4—232no—
497C4⋊C4—216no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—416no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—88no—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—216no—
509C22⋊C4—232no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—84no—
516C42—28no—
517C42—216no—
518C42—416no—
519C42—416no—
520C42—48no—
521C42—88no—
522C23—24no—
523C23—24no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23#51914yes—
567C23#52014yes—
568C23C24×C411yescharacteristic
569C23C23×D412yes—
570C23C22×#4112yes—
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×2−1+411yescharacteristic
575C23C2×2−1+411yescharacteristic
576C23C2×2−1+411yescharacteristic
577C23C2×2−1+412yes—
578C23#58512yes—
579Q8—264no—
580Q8—432no—
581D4—416no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—432no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—416no—
647C2×C4—416no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—816no—
663C2×C4C2×C4×D4116yes—
664C2×C4C2×C4×D418yes—
665C2×C4#51814yes—
666C2×C4D4214yes—
667C2×C4#54618yes—
668C2×C4#55114yes—
669C22—24no—
670C22—24no—
671C22—28no—
672C22—28no—
673C22—42no—
674C22—42no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—44no—
690C22—44no—
691C22C22×C4×D412yes—
692C22C2×#51811yescharacteristic
693C22C2×#51911yescharacteristic
694C22C2×#52011yescharacteristic
695C22C4×2−1+412yes—
696C22C2×D4211yescharacteristic
697C22C2×#54612yes—
698C22C2×#55111yescharacteristic
699C22#348612yes—
700C22#348612yes—
701C22#348812yes—
702C22#348912yes—
703C22#351612yes—
704C22#351612yes—
705C22#352212yes—
706C22#352312yes—
707C22#353611yescharacteristic
708C22#353611yescharacteristic
709C22#353811yescharacteristic
710C22#353811yescharacteristic
711C22#353811yescharacteristic
712C22#354011yescharacteristic
713C22#354712yes—
714C4—216no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—44no—
719C4—44no—
720C4—44no—
721C4—42no—
722C4—48no—
723C4—48no—
724C4—84no—
725C4—82no—
726C4#235818yes—
727C2—44no—
728C2—44no—
729C2C2×#235811yescharacteristic
730C2#3429112yes—
731C2#3429412yes—
732C2#3438011yescharacteristic
733C2#3441211yescharacteristic
734C2#4541612yes—
735C2#4541712yes—
736C2#4541811yescharacteristic
737C2#4541911yescharacteristic
738C2#4542011yescharacteristic
739C2#4542211yescharacteristic
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×Q8×C22⋊C4C2semidirect
4. #14575C2semidirect
5. #14524C2semidirect
6. #14402C2semidirect
7. #14213C2semidirect
8. #14143C2semidirect
9. #14017C2semidirect
10. #13994C2semidirect
12. #13969C2semidirect
15. #13786C2semidirect
22. #2409C4semidirect
82. C23×Q8C2×C4semidirect
131. #395C2×C4semidirect
136. #391C2×C4semidirect
137. #391D4semidirect
145. #385C2×C4semidirect
149. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#23602
C2×Q8×C22⋊C42
#145752
#145242
#144022
#142132
#141432
#140172
#139942
#139882
#139692
#138722
#138472
#137862