Finite Group Databasebeta

← #6526713

Subgroups — #6526713

Order 512.

1926 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×#3238C211yesmaximal, characteristic
2C2×#3108C211yesmaximal, characteristic
3C2×#2567C211yesmaximal, characteristic, complemented
4#34243C211yesmaximal, characteristic
5#28124C212yesmaximal, complemented
6#28118C212yesmaximal
7#27647C212yesmaximal, complemented
8#27644C212yesmaximal
9#25467C214yesmaximal, complemented
10#25336C214yesmaximal, complemented
11#24817C212yesmaximal, complemented
12#24815C212yesmaximal, complemented
13#24682C212yesmaximal, complemented
14#24680C212yesmaximal, complemented
15#21953C211yesmaximal, characteristic, complemented
16#21833C211yesmaximal, characteristic, complemented
17#20740C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3238—28no—
22C2×#501C2212yes—
23C2×#489C2212yes—
24#3108—28no—
25C2×#483C2211yescharacteristic
26C2×#483C2212yes—
27C2×#480C2211yescharacteristic
28C2×#480C2212yes—
29C2×#479C2212yes—
30C2×#477C2212yes—
31C2×#476C2212yes—
32C2×#464C2212yes—
33C2×#451C2211yescharacteristic
34C2×#448C2211yescharacteristic
35C2×C4×Q16C2212yes—
36C2×C4×SD16C2212yes—
37C2×#432C2211yescharacteristic
38C2×#431C2211yescharacteristic
39#2567—21no—
40#2567—22no—
41#2567—21no—
42#2567C2212yes—
43#2567C2212yes—
44#2567C2214yes—
45C2×#395—22no—
46#2370C2212yes—
47#2367C2212yes—
48#2359—22no—
49#2355C2211yescharacteristic
50#2354C2211yescharacteristic
51#2352—21no—
52Q8×C4⋊C4C2211yescharacteristic
53D4×C4⋊C4—21no—
54D4×C4⋊C4C2211yescharacteristic
55#2334—22no—
56#2314—21no—
57C4×C22⋊Q8C2212yes—
58C4×#121C2212yes—
59C2×#391C2212yes—
60C2×#389—21no—
61C2×#389—21no—
62#2285C2211yescharacteristic
63C2×#384C2212yes—
64#2274—21no—
65C2×C4×C4⋊C4C2211yescharacteristic
66#2084C2214yes—
67#2083C2214yes—
68#2072C2214yes—
69#2071C2214yes—
70#2068C2214yes—
71#2066C2214yes—
72#2056C2214yes—
73#2055C2214yes—
74#1990C2212yes—
75#1990C2212yes—
76#1988C2214yes—
77#1948C2214yes—
78#1947C2214yes—
79#1940C2214yes—
80#1939C2212yes—
81#1939C2212yes—
82#1928C2214yes—
83#1927C2214yes—
84#1878C2214yes—
85#1877C2214yes—
86#1823C2212yes—
87#1823C2212yes—
88#1823C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1803C2214yes—
92#1802C2214yes—
93#1797C2212yes—
94#1797C2212yes—
95#1787C2212yes—
96#1763C2214yes—
97C23×Q8—21no—
98C23×Q8C2311yescharacteristic
99C22×Q16C2312yes—
100C22×SD16C2312yes—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107#529—24no—
108#529D418yes—
109C2×#126—21no—
110C2×#125C2311yescharacteristic
111C2×#124C2311yescharacteristic
112C2×#123—21no—
113C2×C22⋊Q8C2311yescharacteristic
114C2×#121C2311yescharacteristic
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—24no—
118C2×C4×Q8—24no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2312yes—
127C2×#117—21no—
128C2×#117—21no—
129C2×#117—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4D411yescharacteristic
136C22×C4⋊C4D411yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2312yes—
139C22×C42—21no—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142#501—28no—
143#489—28no—
144#483—24no—
145#483—28no—
146#480—24no—
147#480—28no—
148#479—28no—
149#477—28no—
150#476—28no—
151#464—28no—
152#451—24no—
153#448—24no—
154C4×Q16—28no—
155C4×SD16—28no—
156#432—24no—
157#431—24no—
158C2×#107C2314yes—
159C2×#107C2312yes—
160C2×#106C2312yes—
161C2×#106C2314yes—
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2314yes—
165C2×C4⋊C8C2314yes—
166C2×#103C2314yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2314yes—
170C2×#103C2312yes—
171C2×#103C2314yes—
172C2×#102C2314yes—
173C2×#102C2312yes—
174C2×C22⋊C8C2312yes—
175C2×C22⋊C8C2311yescharacteristic
176C2×C22⋊C8C2311yescharacteristic
177C2×#97C2312yes—
178C2×C4×C8C2312yes—
179#395—24no—
180#395—24no—
181#395—24no—
182#395—24no—
183#391—22no—
184#391—22no—
185#391C2314yes—
186#391C2314yes—
187#391C2312yes—
188#390C2312yes—
189#389—22no—
190#389—22no—
191#389—24no—
192#389—24no—
193#389—22no—
194#389—22no—
195#389—22no—
196#389—22no—
197#389C2312yes—
198#389C2312yes—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#386—22no—
208#385—22no—
209#385—22no—
210#385—22no—
211#384—21no—
212#384—22no—
213#384—22no—
214#384C2314yes—
215#384C2314yes—
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#382—21no—
219#382—21no—
220#380—21no—
221#380—21no—
222#380—21no—
223#380C2312yes—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4D411yescharacteristic
230C4×C4⋊C4D412yes—
231C4×C4⋊C4D411yescharacteristic
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2312yes—
234C4×C4⋊C4C2312yes—
235C4×C4⋊C4C2311yescharacteristic
236C4×C4⋊C4C2311yescharacteristic
237C4×C4⋊C4C2312yes—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240C4×C22⋊C4—22no—
241C2×#95—22no—
242C2×#95—22no—
243C2×#95D412yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#336C2314yes—
251#336C2314yes—
252C22×Q8—24no—
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—42no—
259C22×Q8—44no—
260C22×Q8—82no—
261C22×Q8—82no—
262C22×Q8C2411yescharacteristic
263C22×Q8C2412yes—
264C22×D4—21no—
265C22×D4C2411yescharacteristic
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4C2×D412yes—
272C23×C4C2×D412yes—
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2411yescharacteristic
276C2×Q16—24no—
277C2×Q16—24no—
278C2×Q16—42no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—42no—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2414yes—
285C4⋊Q8—24no—
286C4⋊Q8—24no—
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—44no—
290C4⋊Q8—44no—
291#126—28no—
292#125—22no—
293#125C2×D414yes—
294#124—22no—
295#124C2×D414yes—
296#123—28no—
297C22⋊Q8—22no—
298C22⋊Q8C2×D414yes—
299#121—22no—
300#121C2×D414yes—
301C4×Q8—22no—
302C4×Q8—24no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—24no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8C2×D414yes—
310C4×D4—28no—
311C4×D4—22no—
312C4×D4—44no—
313C4×D4—84no—
314C4×D4C2×D414yes—
315#117—28no—
316#117—44no—
317#117—82no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4—44no—
347C2×C4⋊C4—44no—
348C2×C4⋊C4—84no—
349C2×C4⋊C4—84no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D412yes—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—44no—
379C2×C42—44no—
380C2×C42C2×D412yes—
381C2×C42C2×D412yes—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D414yes—
385C2×C42#4114yes—
386C2×C42C2412yes—
387C2×C42C2412yes—
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390#107—24no—
391#107—28no—
392#106—24no—
393#106—28no—
394C4⋊C8—24no—
395C4⋊C8—24no—
396C4⋊C8—44no—
397C4⋊C8—44no—
398#103—24no—
399#103—28no—
400#103—28no—
401#103—24no—
402#103—24no—
403#103—44no—
404#102—24no—
405#102—28no—
406C22⋊C8—22no—
407C22⋊C8—22no—
408C22⋊C8—24no—
409#97—24no—
410C4×C8—24no—
411#95—24no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95C2×D412yes—
424#95C2×D412yes—
425#95C2×D412yes—
426#95C2×D412yes—
427C24—21no—
428C24C22×D411yescharacteristic
429C2×Q8—24no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—48no—
439C2×Q8—48no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—82no—
445C2×Q8—82no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8—816no—
449C2×Q8C22×D412yes—
450C2×Q82−1+414yes—
451C2×D4—24no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4C22×D412yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—44no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4#12114yes—
494C22×C4#12114yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C2×#4112yes—
506C22×C4C2511yescharacteristic
507Q16—48no—
508SD16—48no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—24no—
516C2×C8—22no—
517C2×C8—22no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4—84no—
546C4⋊C4—84no—
547C4⋊C4—81no—
548C4⋊C4—81no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4#13714yes—
552C4⋊C4#13714yes—
553C4⋊C4C22×D412yes—
554C4⋊C4C22×D412yes—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—24no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C42—24no—
565C42—24no—
566C42—24no—
567C42—42no—
568C42—42no—
569C42—44no—
570C42—44no—
571C42—44no—
572C42—82no—
573C42—82no—
574C42—82no—
575C42#12114yes—
576C42#12114yes—
577C42C22×D412yes—
578C42C22×D412yes—
579C42C2×#4112yes—
580C42C2×#4112yes—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—22no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23C2×#12112yes—
593C23#57612yes—
594C23#57712yes—
595C23C23×D411yescharacteristic
596Q8—24no—
597Q8—24no—
598Q8—24no—
599Q8—48no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—88no—
604Q8—88no—
605D4—24no—
606D4—48no—
607D4—42no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—28no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—48no—
635C2×C4—48no—
636C2×C4—48no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—48no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—84no—
659C2×C4—88no—
660C2×C4—164no—
661C2×C4C2×#12112yes—
662C2×C4C2×#12112yes—
663C2×C4C2×#12112yes—
664C2×C4C2×#12112yes—
665C2×C4C2×#12112yes—
666C2×C4C2×#13714yes—
667C2×C4C2×#13714yes—
668C2×C4C2×#13712yes—
669C2×C4C2×#13712yes—
670C2×C4C23×D411yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C2×2−1+412yes—
673C2×C4#58512yes—
674C8—44no—
675C8—42no—
676C8—42no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22#304514yes—
691C22C22×#12111yescharacteristic
692C22#348511yescharacteristic
693C22#348611yescharacteristic
694C22C22×#13711yescharacteristic
695C22C22×#13711yescharacteristic
696C22C2×#57611yescharacteristic
697C22C2×#57711yescharacteristic
698C4—22no—
699C4—22no—
700C4—24no—
701C4—24no—
702C4—22no—
703C4—22no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—44no—
708C4—42no—
709C4—42no—
710C4—82no—
711C4—84no—
712C4—82no—
713C4—82no—
714C4#303914yes—
715C4#348912yes—
716C4#349512yes—
717C2—24no—
718C2—82no—
719C2#3563311yescharacteristic
720C2C2×#303911yescharacteristic
721C2C2×#304511yescharacteristic
722C2#6172611yescharacteristic
723C2#6173511yescharacteristic
724C2#6183811yescharacteristic
725C2#6184411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2567C2semidirect
5. #28124C2semidirect
7. #27647C2semidirect
9. #25467C2semidirect
10. #25336C2semidirect
11. #24817C2semidirect
12. #24815C2semidirect
13. #24682C2semidirect
14. #24680C2semidirect
15. #21953C2semidirect
16. #21833C2semidirect
17. #20740C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32382
C2×#31082
C2×#25672
#342432
#281242
#281182
#276472
#276442
#254672
#253362
#248172
#248152
#246822
#246802
#219532
#218332
#207402