Finite Group Databasebeta

← #6458468

Subgroups — #6458468

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34947C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#27741C214yesmaximal, complemented
4#27739C214yesmaximal, complemented
5#27737C212yesmaximal, complemented
6#27735C212yesmaximal, complemented
7#27729C212yesmaximal, complemented
8#27727C212yesmaximal, complemented
9#23747C212yesmaximal, complemented
10#23630C212yesmaximal
11#21322C211yesmaximal, characteristic, complemented
12#21312C211yesmaximal, characteristic, complemented
13#21093C212yesmaximal
14#20764C212yesmaximal, complemented
15#20648C211yesmaximal, characteristic
16C4⋊C4×M16C211yesmaximal, characteristic, complemented
17C4×#417C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×#125—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#417C2211yescharacteristic
23C2×#417C2212yes—
24C2×#417C2211yescharacteristic
25C2×C4×M16C2212yes—
26C2×C4×M16C2211yescharacteristic
27#2568C2211yescharacteristic
28#2567C2211yescharacteristic
29#2566C2212yes—
30#2565C2212yes—
31#2564—22no—
32#2563C2211yescharacteristic
33#2552—22no—
34#2552—21no—
35#2552—21no—
36#2552C2212yes—
37#2552C2212yes—
38#2552C2214yes—
39#2542C2211yescharacteristic
40#2417—21no—
41#2417—21no—
42#2396—22no—
43C2×#398—22no—
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8—21no—
46#2297C2212yes—
47C2×#389—21no—
48C2×#389C2212yes—
49#2285C2212yes—
50C2×#384C2212yes—
51C2×#384C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yes—
55#2101C2214yes—
56#2100C2214yes—
57#2099C2214yes—
58#2072C2214yes—
59#2072C2214yes—
60#2071C2214yes—
61#2071C2214yes—
62#1992C2214yes—
63#1990C2212yes—
64#1990C2212yes—
65#1989C2214yes—
66#1988C2212yes—
67#1987C2212yes—
68#1936C2214yes—
69#1935C2214yes—
70#1934C2214yes—
71#1933C2214yes—
72#1836C2214yes—
73#1836C2214yes—
74#1834C2214yes—
75#1834C2214yes—
76#1803C2212yes—
77#1803C2212yes—
78#1802C2212yes—
79#1802C2212yes—
80#1771C2214yes—
81#1770C2212yes—
82C8×C4⋊C4C2212yes—
83#1763C2212yes—
84#1762C2212yes—
85C4×#103C2212yes—
86C4×#102C2212yes—
87#1736C2211yescharacteristic
88#1736C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1736C2212yes—
93#1736C2211yescharacteristic
94#1728C2212yes—
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C22×M16C2311yescharacteristic
98C22×M16C2311yescharacteristic
99#533—24no—
100#533—24no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—22no—
104C2×#125—24no—
105C2×#125—22no—
106C2×#125—48no—
107C2×#125C2312yes—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C4×#41—24no—
113C4×#41—24no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117—21no—
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C42—21no—
136C22×C42—21no—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#423—24no—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142#417—24no—
143#417—28no—
144#417—24no—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C4×M16—28no—
152C4×M16—24no—
153C2×#97C2314yes—
154C2×#97C2312yes—
155C2×C4×C8C2314yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#399—22no—
159#398—22no—
160#398—22no—
161#398—24no—
162#398—24no—
163#398—24no—
164#398—24no—
165#396—22no—
166#396—22no—
167#395—22no—
168#393—22no—
169#391—21no—
170#391—21no—
171#391C2312yes—
172#391C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—24no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389C2314yes—
183#389C2314yes—
184#387—22no—
185#387—22no—
186#386—22no—
187#386—22no—
188#385—22no—
189#385—22no—
190#384—22no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#384C2312yes—
198#384C2314yes—
199#384C2314yes—
200#382C2312yes—
201#382C2312yes—
202#380—22no—
203#380—22no—
204#380—21no—
205#380—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C2×#95—21no—
221C2×#95—22no—
222C2×#95D412yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#340C2314yes—
237#336C2314yes—
238#336C2314yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41#4114yes—
245C22×Q8—21no—
246C22×Q8—21no—
247C22×Q8C2411yescharacteristic
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—24no—
264C2×M16—22no—
265C2×M16—24no—
266C2×M16—22no—
267C2×M16—24no—
268C2×M16—22no—
269C2×M16#4114yes—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276C4⋊Q8—22no—
277#125—24no—
278#125—24no—
279#125—416no—
280#125—48no—
281#125—48no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—44no—
297C4×D4—44no—
298#117—22no—
299#117—22no—
300#117—24no—
301#117—24no—
302#117—44no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—48no—
336C2×C4⋊C4—48no—
337C2×C4⋊C4—82no—
338C2×C4⋊C4—84no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C42—24no—
361C2×C42—24no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—44no—
369C2×C42—44no—
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42#4114yes—
375C2×C42#4114yes—
376C2×C42C2412yes—
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C4⋊C8—24no—
381C4⋊C8—24no—
382#103—28no—
383#103—24no—
384#103—24no—
385#102—28no—
386#102—24no—
387#102—24no—
388#97—28no—
389#97—24no—
390C4×C8—28no—
391C4×C8—24no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—21no—
409C24C22×D411yescharacteristic
410#41—28no—
411#41—28no—
412#41—44no—
413#41—44no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—81no—
424C2×Q8—81no—
425C2×Q8C2×#4112yes—
426C2×D4—24no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4C2×#4112yes—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4#12414yes—
472C22×C4#12414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C42−1+412yes—
487C22×C42−1+412yes—
488C22×C4C2511yescharacteristic
489M16—28no—
490M16—48no—
491M16—44no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8C2×#4114yes—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—84no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—84no—
542C4⋊C4AGL(1,ℤ/8)14yes—
543C4⋊C4#13714yes—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—44no—
559C42—42no—
560C42—42no—
561C42—42no—
562C42—44no—
563C42—44no—
564C42#12414yes—
565C42#12414yes—
566C42C2×#4112yes—
567C42C2×#4112yes—
568C42C2×#4112yes—
569C42C2×#4112yes—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—22no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23C2×#12412yes—
582C23#54712yes—
583C23#54812yes—
584C23C23×D411yescharacteristic
585Q8—22no—
586Q8—22no—
587Q8—41no—
588Q8—41no—
589Q8—84no—
590Q8—84no—
591D4—22no—
592D4—22no—
593D4—48no—
594D4—41no—
595D4—41no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—44no—
615C2×C4—48no—
616C2×C4—48no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—48no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—88no—
645C2×C4—84no—
646C2×C4C2×#12412yes—
647C2×C4C2×#12412yes—
648C2×C4C2×#12412yes—
649C2×C4C2×#12412yes—
650C2×C4C2×#12412yes—
651C2×C4#52914yes—
652C2×C4#54114yes—
653C2×C4#54712yes—
654C2×C4#54812yes—
655C2×C4#55214yes—
656C2×C4#55214yes—
657C2×C4C2×AGL(1,ℤ/8)14yes—
658C2×C4C2×AGL(1,ℤ/8)12yes—
659C2×C4C2×#13714yes—
660C2×C4C2×#13712yes—
661C2×C4C22×#4111yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C22×#4111yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C2×2−1+411yescharacteristic
666C2×C4C2×2−1+411yescharacteristic
667C8—24no—
668C8—44no—
669C8—42no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22#248914yes—
684C22C22×#12411yescharacteristic
685C22C2×#54711yescharacteristic
686C22C2×#54811yescharacteristic
687C22C22×AGL(1,ℤ/8)11yescharacteristic
688C22C22×#13711yescharacteristic
689C22#358511yescharacteristic
690C22#358611yescharacteristic
691C4—24no—
692C4—24no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—82no—
700C4—82no—
701C4—84no—
702C4—81no—
703C4—82no—
704C4—81no—
705C4—82no—
706C4#248914yes—
707C4#314114yes—
708C4C2×#52911yescharacteristic
709C4C2×#54111yescharacteristic
710C4C2×#55211yescharacteristic
711C4C2×#55211yescharacteristic
712C2—24no—
713C2—82no—
714C2C2×#248911yescharacteristic
715C2C2×#314111yescharacteristic
716C2#6062111yescharacteristic
717C2#6125211yescharacteristic
718C2#6125411yescharacteristic
719C2#6132111yescharacteristic
720C2#6132211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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
2. C2×#2552C2semidirect
3. #27741C2semidirect
4. #27739C2semidirect
5. #27737C2semidirect
6. #27735C2semidirect
7. #27729C2semidirect
8. #27727C2semidirect
9. #23747C2semidirect
11. #21322C2semidirect
12. #21312C2semidirect
14. #20764C2semidirect
16. C4⋊C4×M16C2semidirect

Maximal subgroups

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

subgroupindex
#349472
C2×#25522
#277412
#277392
#277372
#277352
#277292
#277272
#237472
#236302
#213222
#213122
#210932
#207642
#206482
C4⋊C4×M162
C4×#4172