Finite Group Databasebeta

← #823154

Subgroups — #823154

Order 512.

2798 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2489C212yesmaximal, complemented
3C2×#2370C211yesmaximal, characteristic, complemented
4#17063C212yesmaximal, complemented
5#15739C214yesmaximal, complemented
6#15675C214yesmaximal, complemented
7#15188C211yesmaximal, characteristic, complemented
8#15181C211yesmaximal, characteristic
9#15180C211yesmaximal, characteristic, complemented
10#15112C212yesmaximal
11#14813C212yesmaximal
12#13997C212yesmaximal, complemented
13#13997C212yesmaximal, complemented
14#13997C212yesmaximal, complemented
15#13911C212yesmaximal, complemented
16C4×#394C211yesmaximal, characteristic
17#13821C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C4×D4C2211yescharacteristic
23#2542—28no—
24#2489—216no—
25C2×#400C2214yes—
26C2×#399C2214yes—
27C2×#397C2214yes—
28C2×#394C2211yescharacteristic
29#2370—28no—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#388C2214yes—
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#384C2212yes—
43C2×#382C2214yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55#1445C2214yes—
56#1445C2214yes—
57#1445C2214yes—
58#1442C2212yes—
59#1442C2212yes—
60#1442C2211yescharacteristic
61#1442C2212yes—
62#1442C2212yes—
63#1442C2212yes—
64#1441C2211yescharacteristic
65#1441C2212yes—
66#1441C2214yes—
67#1441C2212yes—
68#1440C2212yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1438C2211yescharacteristic
75#1438C2212yes—
76#1437C2214yes—
77#1437C2214yes—
78#1437C2214yes—
79#1436C2214yes—
80C4⋊#95C2214yes—
81C4⋊#95C2214yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2214yes—
84#1434C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—44no—
91C2×#125—24no—
92C2×#125—24no—
93C2×#125D418yescomplemented
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C4×D4—24no—
101C2×C4×D4—44no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136#400—216no—
137#399—216no—
138#397—216no—
139#394—24no—
140#391—28no—
141#391—28no—
142#391—44no—
143#391—44no—
144#390—28no—
145#389—24no—
146#389—28no—
147#389—24no—
148#389—28no—
149#388—216no—
150#387—28no—
151#387—28no—
152#384—28no—
153#382—216no—
154#380—28no—
155#380—44no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4D418yes—
163C4×C22⋊C4—28no—
164C4×C22⋊C4—44no—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2311yescharacteristic
193C25C2412yes—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—42no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C4⋊Q8—416no—
230#125—232no—
231C22⋊Q8—432no—
232C22⋊Q8—432no—
233C4×D4—416no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4#4114yes—
278C2×C4⋊C4#4118yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—28no—
314C2×C42—28no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42C2×D418yes—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—24no—
341#95—28no—
342#95—22no—
343#95—42no—
344#95—42no—
345#95—42no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95C2×D418yes—
353#95#4114yes—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C242−1+414yes—
369C24C2511yescharacteristic
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×D4—28no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—44no—
381C2×D4—44no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4118yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4114yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C42−1+412yes—
482C22×C42−1+412yes—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—28no—
486C4⋊C4—416no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—416no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—84no—
498C4⋊C4#121116yes—
499C4⋊C4#124116yes—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—416no—
507C22⋊C4—416no—
508C22⋊C4—88no—
509C42—28no—
510C42—28no—
511C42—216no—
512C42—416no—
513C42—44no—
514C42—48no—
515C42—84no—
516C42—84no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
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—
552C23C23×D411yescharacteristic
553C23C23×D411yescharacteristic
554C23C22×#4111yescharacteristic
555C23C22×#4112yes—
556C23C22×#4111yescharacteristic
557C23C2×H(5,𝔽2)12yes—
558C23C2×2−1+411yescharacteristic
559C23C2×2−1+411yescharacteristic
560C23C2×2−1+412yes—
561C23#58511yescharacteristic
562C23#58512yes—
563Q8—816no—
564D4—416no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—816no—
652C2×C4C2×#12118yes—
653C2×C4C2×#12114yes—
654C2×C4C2×#12418yes—
655C2×C4C2×#12414yes—
656C2×C4#53514yes—
657C2×C4#54714yes—
658C2×C4#55018yes—
659C2×C4#55918yes—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22C22×#12111yescharacteristic
676C22C22×#12411yescharacteristic
677C22C2×#53511yescharacteristic
678C22C2×#54711yescharacteristic
679C22C2×#55012yes—
680C22D4×#4111yescharacteristic
681C22#347512yes—
682C22C2×#55912yes—
683C22#348211yescharacteristic
684C22#348611yescharacteristic
685C22#348911yescharacteristic
686C22#348912yes—
687C22#349911yescharacteristic
688C22#350011yescharacteristic
689C22#351512yes—
690C22#351511yescharacteristic
691C22#351612yes—
692C22#351612yes—
693C22#352012yes—
694C22#352312yes—
695C22#353812yes—
696C22#354112yes—
697C22#355112yes—
698C4—24no—
699C4—28no—
700C4—28no—
701C4—42no—
702C4—44no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—44no—
707C4—81no—
708C4—82no—
709C4—84no—
710C4—82no—
711C4—82no—
712C4—84no—
713C4#254818yes—
714C2—84no—
715C2C2×#254811yescharacteristic
716C2#3573711yescharacteristic
717C2#3591011yescharacteristic
718C2#3616411yescharacteristic
719C2#3625911yescharacteristic
720C2#3891711yescharacteristic
721C2#4291011yescharacteristic
722C2#4521712yes—
723C2#4522612yes—
724C2#5905912yes—
725C2#5959312yes—
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
2. C2×#2489C2semidirect
3. C2×#2370C2semidirect
4. #17063C2semidirect
5. #15739C2semidirect
6. #15675C2semidirect
7. #15188C2semidirect
9. #15180C2semidirect
12. #13997C2semidirect
13. #13997C2semidirect
14. #13997C2semidirect
15. #13911C2semidirect
17. #13821C2semidirect
93. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#24892
C2×#23702
#170632
#157392
#156752
#151882
#151812
#151802
#151122
#148132
#139972
#139112
C4×#3942
#138212