Finite Group Databasebeta

← #6458467

Subgroups — #6458467

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34947C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#27745C214yesmaximal, complemented
4#27743C214yesmaximal, complemented
5#27738C212yesmaximal, complemented
6#27736C212yesmaximal, complemented
7#27730C212yesmaximal, complemented
8#27728C212yesmaximal, complemented
9#23747C212yesmaximal, complemented
10#23631C212yesmaximal
11#21322C211yesmaximal, characteristic, complemented
12#21312C211yesmaximal, characteristic, complemented
13#21093C212yesmaximal
14#20765C212yesmaximal, complemented
15#20648C211yesmaximal, characteristic
16C4⋊C4×M16C211yesmaximal, characteristic, complemented
17C4×#417C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#417C2211yescharacteristic
22C2×#417C2212yes—
23C2×#417C2211yescharacteristic
24C2×C4×M16C2212yes—
25C2×C4×M16C2211yescharacteristic
26#2571—22no—
27#2571—21no—
28#2571—21no—
29#2571C2212yes—
30#2571C2212yes—
31#2571C2214yes—
32#2568C2211yescharacteristic
33#2567C2211yescharacteristic
34#2566C2212yes—
35#2565C2212yes—
36#2564—22no—
37#2563C2211yescharacteristic
38#2542C2211yescharacteristic
39#2417—21no—
40#2417—21no—
41#2396—22no—
42C2×#400—22no—
43C4×C22⋊Q8—21no—
44C4×C22⋊Q8—21no—
45#2297C2212yes—
46C2×#389—21no—
47C2×#389C2212yes—
48#2285C2212yes—
49C2×#384C2211yescharacteristic
50C2×#382—21no—
51C2×#382C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yes—
55#2101C2214yes—
56#2100C2214yes—
57#2099C2214yes—
58#2074C2214yes—
59#2074C2214yes—
60#2073C2214yes—
61#2073C2214yes—
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#1835C2214yes—
75#1835C2214yes—
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×#125C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C4×#41—24no—
109C4×#41—24no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×#117—21no—
119C2×#117C2311yescharacteristic
120C2×#117C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C42—21no—
130C22×C42—21no—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#423—24no—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136#417—24no—
137#417—28no—
138#417—24no—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C4×M16—28no—
146C4×M16—24no—
147C2×#97C2314yes—
148C2×#97C2312yes—
149C2×C4×C8C2314yes—
150C2×C4×C8C2312yes—
151#400—22no—
152#400—24no—
153#400—24no—
154#400—24no—
155#400—24no—
156#399—22no—
157#398—22no—
158#398—22no—
159#396—22no—
160#396—22no—
161#395—22no—
162#393—22no—
163#391—21no—
164#391—21no—
165#391C2312yes—
166#391C2312yes—
167#390C2312yes—
168#390C2312yes—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389—24no—
174#389—22no—
175#389—22no—
176#389C2314yes—
177#389C2314yes—
178#387—22no—
179#387—22no—
180#386—22no—
181#386—22no—
182#385—22no—
183#385—22no—
184#384—21no—
185#384—21no—
186#384C2312yes—
187#384C2312yes—
188#384C2312yes—
189#384C2312yes—
190#382—22no—
191#382—22no—
192#382—24no—
193#382—22no—
194#382—22no—
195#382C2312yes—
196#382C2312yes—
197#382C2314yes—
198#382C2314yes—
199#380—22no—
200#380—22no—
201#380—21no—
202#380—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95D412yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#340C2314yes—
236#336C2314yes—
237#336C2314yes—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41#4114yes—
244C22×Q8—21no—
245C22×Q8—21no—
246C22×Q8C2411yescharacteristic
247C22×D4—21no—
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—24no—
263C2×M16—22no—
264C2×M16—24no—
265C2×M16—22no—
266C2×M16—24no—
267C2×M16—22no—
268C2×M16#4114yes—
269C22×C8C2412yes—
270C22×C8C2414yes—
271C22×C8C2412yes—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—22no—
276#125—24no—
277#125—24no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—44no—
293C4×D4—44no—
294#117—22no—
295#117—22no—
296#117—24no—
297#117—24no—
298#117—44no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4—84no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—44no—
359C2×C42—44no—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42#4114yes—
365C2×C42#4114yes—
366C2×C42C2412yes—
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C4⋊C8—24no—
371C4⋊C8—24no—
372#103—28no—
373#103—24no—
374#103—24no—
375#102—28no—
376#102—24no—
377#102—24no—
378#97—28no—
379#97—24no—
380C4×C8—28no—
381C4×C8—24no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402C24—21no—
403C24C22×D411yescharacteristic
404#41—28no—
405#41—28no—
406#41—44no—
407#41—44no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—81no—
418C2×Q8—81no—
419C2×Q8C2×#4112yes—
420C2×D4—24no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4C2×#4112yes—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4#12414yes—
466C22×C4#12414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C42−1+412yes—
481C22×C42−1+412yes—
482C22×C4C2511yescharacteristic
483M16—28no—
484M16—48no—
485M16—44no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8C2×#4114yes—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—42no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C42—24no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—44no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—44no—
553C42—44no—
554C42#12414yes—
555C42#12414yes—
556C42C2×#4112yes—
557C42C2×#4112yes—
558C42C2×#4112yes—
559C42C2×#4112yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23C2×#12412yes—
572C23#54712yes—
573C23#54812yes—
574C23C23×D411yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—41no—
578Q8—41no—
579Q8—84no—
580Q8—84no—
581D4—22no—
582D4—22no—
583D4—48no—
584D4—41no—
585D4—41no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4—84no—
638C2×C4C2×#12412yes—
639C2×C4C2×#12412yes—
640C2×C4C2×#12412yes—
641C2×C4C2×#12412yes—
642C2×C4C2×#12412yes—
643C2×C4#52914yes—
644C2×C4#54114yes—
645C2×C4#54712yes—
646C2×C4#54812yes—
647C2×C4#55214yes—
648C2×C4#55214yes—
649C2×C4#57512yes—
650C2×C4#57512yes—
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C2×C4C2×2−1+411yescharacteristic
657C8—24no—
658C8—44no—
659C8—42no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22#248914yes—
674C22C22×#12411yescharacteristic
675C22C2×#54711yescharacteristic
676C22C2×#54811yescharacteristic
677C22C2×#57511yescharacteristic
678C22C2×#57511yescharacteristic
679C22#358511yescharacteristic
680C22#358611yescharacteristic
681C4—22no—
682C4—22no—
683C4—22no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—82no—
690C4—82no—
691C4—84no—
692C4—81no—
693C4—82no—
694C4—81no—
695C4—82no—
696C4#248914yes—
697C4#314214yes—
698C4C2×#52911yescharacteristic
699C4C2×#54111yescharacteristic
700C4C2×#55211yescharacteristic
701C4C2×#55211yescharacteristic
702C2—24no—
703C2—82no—
704C2C2×#248911yescharacteristic
705C2C2×#314211yescharacteristic
706C2#6062111yescharacteristic
707C2#6125611yescharacteristic
708C2#6125711yescharacteristic
709C2#6132311yescharacteristic
710C2#6132411yescharacteristic
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×#2571C2semidirect
3. #27745C2semidirect
4. #27743C2semidirect
5. #27738C2semidirect
6. #27736C2semidirect
7. #27730C2semidirect
8. #27728C2semidirect
9. #23747C2semidirect
11. #21322C2semidirect
12. #21312C2semidirect
14. #20765C2semidirect
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×#25712
#277452
#277432
#277382
#277362
#277302
#277282
#237472
#236312
#213222
#213122
#210932
#207652
#206482
C4⋊C4×M162
C4×#4172