Finite Group Databasebeta

← #6435621

Subgroups — #6435621

Order 512.

1584 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#34060C211yesmaximal, characteristic
3#30004C214yesmaximal, complemented
4#30003C214yesmaximal, complemented
5#29998C212yesmaximal, complemented
6#29997C212yesmaximal, complemented
7#29996C212yesmaximal, complemented
8#29995C212yesmaximal, complemented
9#23671C212yesmaximal, complemented
10#23668C212yesmaximal, complemented
11#23616C212yesmaximal
12#23610C212yesmaximal
13#21170C211yesmaximal, characteristic, complemented
14C2×#1948C211yesmaximal, characteristic, complemented
15#20793C211yesmaximal, characteristic, complemented
16#20648C211yesmaximal, characteristic
17#20645C211yesmaximal, characteristic
18C2×C4×#41C2211yescharacteristic
19C2×#428C2212yes—
20C22×#106C2212yes—
21C2×#424C2212yes—
22C2×#424C2212yes—
23C2×#423C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C2×#417C2212yes—
26C2×#416C2212yes—
27#2859C2212yes—
28#2859C2212yes—
29#2647C2214yes—
30#2571—21no—
31#2571—21no—
32#2552—21no—
33#2552—21no—
34#2442—22no—
35#2442—22no—
36#2320C2212yes—
37#2319C2212yes—
38#2318—22no—
39#2317C2211yescharacteristic
40#2317C2211yescharacteristic
41#2317C2212yes—
42#2316C2212yes—
43#2314—21no—
44#2314—21no—
45#2311C2211yescharacteristic
46#2311C2211yescharacteristic
47#2295C2212yes—
48#2293C2212yes—
49#2290—22no—
50#2285—21no—
51#2285C2211yescharacteristic
52C2×#384C2211yescharacteristic
53#2283—21no—
54C4×#117C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#2088C2214yes—
57#2088C2214yes—
58#2088C2214yes—
59#2088C2214yes—
60#2088C2214yes—
61#2088C2214yes—
62#2087C2214yes—
63#2086C2214yes—
64#2068C2214yes—
65#2067C2214yes—
66#2066C2214yes—
67#2066C2214yes—
68#2066C2214yes—
69#2065C2214yes—
70#2065C2214yes—
71#2065C2214yes—
72#1949C2214yes—
73#1948C2212yes—
74#1948C2212yes—
75#1948C2212yes—
76#1948C2214yes—
77#1947C2212yes—
78#1843C2214yes—
79#1842C2212yes—
80#1842C2212yes—
81#1805C2212yes—
82#1804C2212yes—
83#1803C2211yescharacteristic
84#1803C2212yes—
85#1803C2211yescharacteristic
86#1803C2212yes—
87#1802C2211yescharacteristic
88#1802C2212yes—
89#1802C2211yescharacteristic
90#1802C2212yes—
91#1736C2212yes—
92#1736C2212yes—
93#1735C2212yes—
94C2×#340C2212yes—
95C22×#41C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98C2×#125—22no—
99C2×#125—22no—
100C4×#41—24no—
101C4×#41Q818yes—
102C2×C4×Q8C2311yescharacteristic
103C2×C4×Q8C2311yescharacteristic
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2311yescharacteristic
108C2×#117—21no—
109C2×#117—22no—
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122#428—44no—
123C2×#107C2314yes—
124C2×#106—44no—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128#424—44no—
129#424—44no—
130#423—24no—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141#417—44no—
142#416—44no—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149#400—24no—
150#400—24no—
151#398—24no—
152#398—24no—
153#391C2312yes—
154#391C2312yes—
155#390C2312yes—
156#390C2312yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389C2314yes—
164#389C2314yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#389C2314yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#386—22no—
180#386—22no—
181#385—22no—
182#385—22no—
183#385—24no—
184#384—21no—
185#384—21no—
186#384—22no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2312yes—
193#383—22no—
194#382—21no—
195#382—21no—
196#382—22no—
197#382—22no—
198#382—22no—
199#382—22no—
200#382C2312yes—
201#381—22no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380—21no—
208#380—24no—
209#380—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224#376—22no—
225#376C2312yes—
226C2×#95—21no—
227C43C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41C2×Q814yes—
244C22×Q8C2411yescharacteristic
245C22×D4—21no—
246C22×D4C2411yescharacteristic
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×M16—24no—
257C2×M16—24no—
258C2×M16—24no—
259C22×C8—24no—
260C22×C8—24no—
261C22×C8—24no—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2414yes—
265#125—84no—
266#125—84no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8C2×Q814yes—
270C4×Q8C2×Q814yes—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4C2×Q814yes—
276C4×D4C2×Q814yes—
277#117—22no—
278#117—22no—
279#117—22no—
280#117—22no—
281#117—22no—
282#117—28no—
283#117—24no—
284#117—24no—
285#117—84no—
286#117C2×Q814yes—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—28no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—82no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×Q814yes—
352C2×C42#4114yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—44no—
358#106—44no—
359#106—44no—
360#106—44no—
361C4⋊C8—22no—
362C4⋊C8—22no—
363C4⋊C8—24no—
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367C4⋊C8—44no—
368C4⋊C8—44no—
369C4⋊C8—44no—
370#103—42no—
371#103—42no—
372#103—44no—
373#102—42no—
374#102—42no—
375#102—44no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—21no—
397C24C22×D411yescharacteristic
398#41—28no—
399#41—28no—
400#41—44no—
401#41—44no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8C22×Q812yes—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4C22×Q812yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4C22⋊Q814yes—
448C22×C4C22⋊Q814yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×Q812yes—
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C22×C4C2511yescharacteristic
465M16—48no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—48no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4C22×Q812yes—
514C4⋊C4C22×Q812yes—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—44no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C42—24no—
526C42—24no—
527C42—24no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—44no—
538C42—82no—
539C42C22⋊Q814yes—
540C42C22⋊Q814yes—
541C42C22×Q812yes—
542C42C22×Q812yes—
543C42C2×#4112yes—
544C42C2×#4112yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23C2×C22⋊Q812yes—
557C23#53712yes—
558C23#53712yes—
559C23C23×D411yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564D4—22no—
565D4—22no—
566D4—48no—
567D4—41no—
568D4—41no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—28no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—164no—
619C2×C4C2×C22⋊Q812yes—
620C2×C4C2×C22⋊Q812yes—
621C2×C4C2×C22⋊Q812yes—
622C2×C4C2×C22⋊Q812yes—
623C2×C4C2×C22⋊Q812yes—
624C2×C4#53712yes—
625C2×C4#53712yes—
626C2×C4#54014yes—
627C2×C4#54114yes—
628C2×C4#54414yes—
629C2×C4#54414yes—
630C2×C4#57712yes—
631C2×C4#57712yes—
632C2×C4C23×Q811yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×2−1+411yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C8—42no—
639C8—42no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22#262914yes—
655C22C22×C22⋊Q811yescharacteristic
656C22C2×#53711yescharacteristic
657C22C2×#53711yescharacteristic
658C22C2×#57711yescharacteristic
659C22C2×#57711yescharacteristic
660C22#358511yescharacteristic
661C22#358611yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—84no—
677C4#262914yes—
678C4#308414yes—
679C4C2×#54011yescharacteristic
680C4C2×#54111yescharacteristic
681C4C2×#54411yescharacteristic
682C4C2×#54411yescharacteristic
683C2—24no—
684C2—82no—
685C2C2×#262911yescharacteristic
686C2C2×#308411yescharacteristic
687C2#6049311yescharacteristic
688C2#6092311yescharacteristic
689C2#6092411yescharacteristic
690C2#6092911yescharacteristic
691C2#6093111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #30004C2semidirect
4. #30003C2semidirect
5. #29998C2semidirect
6. #29997C2semidirect
7. #29996C2semidirect
8. #29995C2semidirect
9. #23671C2semidirect
10. #23668C2semidirect
13. #21170C2semidirect
14. C2×#1948C2semidirect
15. #20793C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#340602
#300042
#300032
#299982
#299972
#299962
#299952
#236712
#236682
#236162
#236102
#211702
C2×#19482
#207932
#206482
#206452