Finite Group Databasebeta

← #7510593

Subgroups — #7510593

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34953C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic
3#32645C212yesmaximal, complemented
4#32644C212yesmaximal, complemented
5#32641C214yesmaximal, complemented
6#32634C214yesmaximal, complemented
7#32631C212yesmaximal, complemented
8#32630C212yesmaximal, complemented
9#23619C212yesmaximal
10#23507C212yesmaximal
11#23455C212yesmaximal
12#21303C211yesmaximal, characteristic
13#21094C212yesmaximal
14#21061C212yesmaximal
15#21033C212yesmaximal
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#428C2212yes—
20C2×#423C2211yescharacteristic
21C2×#419C2212yes—
22C2×#419C2212yes—
23C2×#419C2212yes—
24C2×#419C2212yes—
25C2×C4×M16C2211yescharacteristic
26#2584C2212yes—
27#2584C2212yes—
28#2583—21no—
29#2583—21no—
30#2568—21no—
31#2568—21no—
32#2568—21no—
33#2568—21no—
34#2568C2212yes—
35#2568C2212yes—
36#2568C2212yes—
37#2568C2212yes—
38#2567C2211yescharacteristic
39#2567C2211yescharacteristic
40#2546C2211yescharacteristic
41#2546C2211yescharacteristic
42#2502—21no—
43#2434—21no—
44#2399—21no—
45#2399—21no—
46C2×#400—21no—
47C2×#400—21no—
48C4×C4⋊Q8—21no—
49C4×C4⋊Q8—21no—
50#2296C2212yes—
51#2296C2212yes—
52C2×#391—21no—
53C2×#391C2212yes—
54C2×#391C2212yes—
55C2×#382—21no—
56C4×#117C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58#2068C2214yes—
59#2068C2214yes—
60#2066C2214yes—
61#2066C2214yes—
62#2043C2214yes—
63#2043C2214yes—
64#2043C2214yes—
65#2043C2214yes—
66#2026C2214yes—
67#2026C2214yes—
68#2024C2214yes—
69#2024C2214yes—
70#1985C2212yes—
71#1985C2212yes—
72#1984C2212yes—
73#1984C2212yes—
74#1935C2214yes—
75#1935C2214yes—
76#1933C2214yes—
77#1933C2214yes—
78#1928C2214yes—
79#1928C2214yes—
80#1927C2214yes—
81#1927C2214yes—
82#1921C2214yes—
83#1920C2214yes—
84#1919C2214yes—
85#1918C2214yes—
86#1736C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1728C2212yes—
90#1728C2212yes—
91C23×Q8D411yescharacteristic
92C23×Q8C2312yes—
93C22×M16C2312yes—
94C22×M16C2312yes—
95#533—42no—
96#533—42no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—42no—
102C2×C4⋊Q8—42no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125C2312yes—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8—42no—
113C2×C4×Q8—42no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×#117C2312yes—
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C42—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#428—28no—
135C2×#107C2314yes—
136C2×#106C2314yes—
137#423—42no—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140#419—28no—
141#419—28no—
142#419—44no—
143#419—44no—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C4×M16—42no—
153C2×#97C2312yes—
154C2×C4×C8C2312yes—
155#400—22no—
156#400—22no—
157#400—22no—
158#400—22no—
159#400—22no—
160#400—22no—
161#400—22no—
162#400—22no—
163#398—21no—
164#398—21no—
165#398—21no—
166#398—21no—
167#398—22no—
168#398—22no—
169#395—22no—
170#395—22no—
171#395—21no—
172#395—21no—
173#395—21no—
174#395—21no—
175#391—21no—
176#391—21no—
177#391—22no—
178#391—22no—
179#391—22no—
180#391—22no—
181#391—21no—
182#391—21no—
183#391—22no—
184#391—22no—
185#391—22no—
186#391—22no—
187#391C2314yes—
188#391C2314yes—
189#391C2314yes—
190#391C2314yes—
191#389—22no—
192#389—22no—
193#389—22no—
194#389—22no—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#385—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#384—21no—
208#384—21no—
209#382—22no—
210#382—22no—
211#382—22no—
212#382—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—21no—
226#376—21no—
227C2×#95—21no—
228C2×#95—22no—
229C2×#95—22no—
230C2×#95—22no—
231C2×#95—22no—
232C43—21no—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2314yes—
240#336C2314yes—
241#336C2314yes—
242#336C2314yes—
243C22×Q8—24no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—21no—
247C22×Q8—24no—
248C22×Q8—24no—
249C22×Q8—24no—
250C22×Q8—24no—
251C22×Q8—24no—
252C22×Q8—24no—
253C22×Q8—42no—
254C22×Q8—42no—
255C22×Q8C2×D411yescharacteristic
256C22×Q8C2×D411yescharacteristic
257C22×Q8C2×D411yescharacteristic
258C22×Q8C2×D411yescharacteristic
259C22×Q8C2414yes—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C2×M16—24no—
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16—24no—
274C2×M16—24no—
275C22×C8C2414yes—
276C22×C8C2414yes—
277C4⋊Q8—41no—
278C4⋊Q8—41no—
279C4⋊Q8—41no—
280C4⋊Q8—41no—
281C4⋊Q8—42no—
282C4⋊Q8—42no—
283C4⋊Q8—42no—
284C4⋊Q8—42no—
285C4⋊Q8—44no—
286C4⋊Q8—44no—
287C4⋊Q8—44no—
288C4⋊Q8—44no—
289#125—42no—
290#125—42no—
291C22⋊Q8—44no—
292C22⋊Q8—44no—
293C4×Q8—42no—
294C4×Q8—42no—
295C4×Q8—42no—
296C4×Q8—42no—
297C4×Q8—41no—
298C4×Q8—41no—
299C4×Q8—41no—
300C4×Q8—41no—
301C4×Q8—41no—
302C4×Q8—41no—
303C4×Q8—41no—
304C4×Q8—41no—
305C4×Q8—44no—
306C4×Q8—44no—
307#117—24no—
308#117—24no—
309#117—24no—
310#117#4118yes—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—44no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—28no—
381#106—28no—
382C4⋊C8—42no—
383C4⋊C8—42no—
384#103—28no—
385#103—28no—
386#103—28no—
387#103—28no—
388#103—44no—
389#103—44no—
390#103—44no—
391#103—44no—
392#97—42no—
393C4×C8—42no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—22no—
405#95—21no—
406#95—21no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415C24C22×D411yescharacteristic
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—41no—
428C2×Q8—41no—
429C2×Q8—41no—
430C2×Q8—41no—
431C2×Q8—41no—
432C2×Q8—41no—
433C2×Q8—416no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—48no—
442C2×Q8—48no—
443C2×Q8—48no—
444C2×Q8—48no—
445C2×Q8—82no—
446C2×Q8#13714yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—42no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4114yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+414yes—
491C22×C4C2511yescharacteristic
492M16—48no—
493M16—48no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4C2×#4114yes—
538C4⋊C4C2×#4114yes—
539C22⋊C4—28no—
540C22⋊C4—44no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—28no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—44no—
561C42—42no—
562C42—44no—
563C42—42no—
564C42—44no—
565C42—82no—
566C23—22no—
567C23—21no—
568C23—21no—
569C23#53412yes—
570C23#54814yes—
571C23C23×D411yescharacteristic
572Q8—44no—
573Q8—44no—
574Q8—44no—
575Q8—44no—
576Q8—44no—
577Q8—44no—
578Q8—44no—
579Q8—44no—
580Q8—84no—
581Q8—84no—
582Q8—88no—
583C2×C4—28no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—88no—
634C2×C4—88no—
635C2×C4#53412yes—
636C2×C4#54218yes—
637C2×C4#54814yes—
638C2×C4#55514yes—
639C2×C4#56314yes—
640C2×C4C2×#13712yes—
641C2×C4C2×#13712yes—
642C2×C4C2×#13712yes—
643C2×C4#57512yes—
644C2×C4C22×#4112yes—
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×2−1+412yes—
648C8—44no—
649C8—44no—
650C22—22no—
651C22—22no—
652C22—24no—
653C22—24no—
654C22#273114yes—
655C22C2×#53411yescharacteristic
656C22C2×#54812yes—
657C22C22×#13711yescharacteristic
658C22C2×#57511yescharacteristic
659C22#358612yes—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—44no—
673C4—44no—
674C4—82no—
675C4—82no—
676C4—84no—
677C4—84no—
678C4#273114yes—
679C4C2×#54212yes—
680C4C2×#55511yescharacteristic
681C4C2×#56311yescharacteristic
682C2—24no—
683C2C2×#273111yescharacteristic
684C2#6075111yescharacteristic
685C2#6075411yescharacteristic
686C2#6132212yes—
687C2#6132312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32645C2semidirect
4. #32644C2semidirect
5. #32641C2semidirect
6. #32634C2semidirect
7. #32631C2semidirect
8. #32630C2semidirect

Maximal subgroups

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

subgroupindex
#349532
C2×#25682
#326452
#326442
#326412
#326342
#326312
#326302
#236192
#235072
#234552
#213032
#210942
#210612
#210332