Finite Group Databasebeta

← #6660633

Subgroups — #6660633

Order 512.

1512 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#34627C211yesmaximal, characteristic
3#32794C212yesmaximal, complemented
4#32780C212yesmaximal, complemented
5#30374C214yesmaximal, complemented
6#29941C212yesmaximal, complemented
7#29900C214yesmaximal
8#29688C212yesmaximal, complemented
9#29344C212yesmaximal, complemented
10#29311C212yesmaximal, complemented
11#27682C212yesmaximal, complemented
12#24710C212yesmaximal, complemented
13#24703C212yesmaximal, complemented
14#24301C212yesmaximal, complemented
15#20721C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#475C2212yes—
20C2×#475C2212yes—
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2857C2212yes—
24#2857C2212yes—
25#2638—21no—
26#2638—21no—
27#2634—21no—
28#2633—21no—
29#2628C2212yes—
30#2628C2211yescharacteristic
31#2628C2211yescharacteristic
32#2628C2212yes—
33#2625C2212yes—
34#2621C2211yescharacteristic
35#2621C2211yescharacteristic
36#2616C2212yes—
37#2568—21no—
38#2567—21no—
39#2563—21no—
40#2542—21no—
41#2487—21no—
42#2487—21no—
43#2409—21no—
44#2409—21no—
45#2399—21no—
46#2398—21no—
47#2395—21no—
48#2395—21no—
49#2317C2212yes—
50#2311C2212yes—
51C4×C22⋊Q8C2212yes—
52C4×C22⋊Q8C2212yes—
53#2285C2211yescharacteristic
54#2283C2211yescharacteristic
55#2101C2214yes—
56#2101C2214yes—
57#2097C2214yes—
58#2085C2214yes—
59#2077C2214yes—
60#2077C2214yes—
61#2077C2214yes—
62#2076C2214yes—
63#2058C2214yes—
64#2058C2214yes—
65#2043C2214yes—
66#2036C2214yes—
67#2035C2214yes—
68#2035C2214yes—
69#2035C2214yes—
70#2024C2214yes—
71#1988C2212yes—
72#1988C2212yes—
73#1948C2212yes—
74#1947C2212yes—
75#1943C2214yes—
76#1941C2214yes—
77#1935C2214yes—
78#1933C2214yes—
79#1873C2214yes—
80#1872C2214yes—
81#1854C2214yes—
82#1853C2214yes—
83#1818C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1797C2212yes—
87#1797C2212yes—
88#1780C2212yes—
89#1750C2212yes—
90#1750C2212yes—
91C22×#41C2311yescharacteristic
92C23×Q8—21no—
93#533—42no—
94#533—42no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—42no—
98C2×C4⋊Q8—42no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#125—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C4×#41—42no—
111C4×#41—42no—
112C2×C4×Q8—21no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×#117—21no—
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#475—44no—
133#475—44no—
134#432—42no—
135#431—42no—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×#97C2312yes—
151C2×C4×C8C2312yes—
152#400—22no—
153#400—22no—
154#400—22no—
155#400—21no—
156#400—21no—
157#399—21no—
158#399—21no—
159#399—22no—
160#399—22no—
161#398—21no—
162#398—21no—
163#398—21no—
164#398—21no—
165#398—22no—
166#397—21no—
167#397—21no—
168#397—21no—
169#397—21no—
170#397—22no—
171#397—22no—
172#395—22no—
173#395—22no—
174#395—22no—
175#395—21no—
176#395—21no—
177#394—22no—
178#394—22no—
179#394—22no—
180#393—22no—
181#393—22no—
182#393—21no—
183#393—21no—
184#391—22no—
185#391—22no—
186#391—22no—
187#391C2312yes—
188#391C2312yes—
189#390—22no—
190#389—22no—
191#389—22no—
192#389C2312yes—
193#389C2314yes—
194#389C2314yes—
195#389C2312yes—
196#387—22no—
197#387—22no—
198#386—22no—
199#386—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#385—22no—
204#384—22no—
205#384C2312yes—
206#383—21no—
207#382—22no—
208#382C2311yescharacteristic
209#382C2312yes—
210#382C2311yescharacteristic
211#381—21no—
212#380—22no—
213#380—22no—
214#380—21no—
215#380—21no—
216#380C2314yes—
217#380C2314yes—
218#380C2312yes—
219#380C2312yes—
220#380C2312yes—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4C2312yes—
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229#376—21no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#336C2314yes—
239#336C2314yes—
240C2×#41—24no—
241C2×#41—41no—
242C2×#41—41no—
243C2×#41—42no—
244C22×Q8—22no—
245C22×Q8—21no—
246C22×Q8—22no—
247C22×Q8—44no—
248C22×Q8—84no—
249C22×Q8C2×D411yescharacteristic
250C22×Q8C2412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4C2×D411yescharacteristic
253C22×D4C2×D411yescharacteristic
254C23×C4—22no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C22×C8C2414yes—
265C22×C8C2414yes—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—42no—
269C4⋊Q8—42no—
270C4⋊Q8—41no—
271C4⋊Q8—41no—
272C4⋊Q8—41no—
273C4⋊Q8—41no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276#125—44no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—42no—
280C22⋊Q8—42no—
281C22⋊Q8—42no—
282C22⋊Q8—42no—
283C22⋊Q8—44no—
284C22⋊Q8—84no—
285C22⋊Q8—84no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×Q8—44no—
291C4×D4—42no—
292C4×D4—42no—
293C4×D4—42no—
294C4×D4—42no—
295C4×D4—42no—
296C4×D4—42no—
297#117—41no—
298#117—41no—
299#117—41no—
300#117—41no—
301#117—41no—
302#117—41no—
303#117—44no—
304#117—44no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2414yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C2×D411yescharacteristic
345C2×C22⋊C4C2×D411yescharacteristic
346C2×C22⋊C4C2×D411yescharacteristic
347C2×C22⋊C4C2×D411yescharacteristic
348C2×C22⋊C4C2×D411yescharacteristic
349C2×C22⋊C4C2×D411yescharacteristic
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#107—44no—
370#106—44no—
371#106—44no—
372#106—44no—
373C4⋊C8—24no—
374C4⋊C8—24no—
375#103—44no—
376#103—44no—
377#103—44no—
378#103—44no—
379C22⋊C8—28no—
380C22⋊C8—44no—
381C22⋊C8—42no—
382C22⋊C8—42no—
383#97—42no—
384C4×C8—42no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
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#95—22no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405C24C22×D411yescharacteristic
406C24C22×D411yescharacteristic
407C24C22×D411yescharacteristic
408#41—48no—
409#41—48no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8—42no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—82no—
425C2×Q8—84no—
426C2×Q8—816no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—41no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
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—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—84no—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C42−1+414yes—
479C22×C4C2511yescharacteristic
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—44no—
546C42—44no—
547C42—44no—
548C42H(5,𝔽2)12yes—
549C42H(5,𝔽2)12yes—
550C23—22no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23#53612yes—
565C23C23×D411yescharacteristic
566Q8—44no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—84no—
572Q8—88no—
573Q8—88no—
574D4—44no—
575D4—44no—
576D4—44no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4—88no—
629C2×C4#53412yes—
630C2×C4#53412yes—
631C2×C4#53612yes—
632C2×C4#57714yes—
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+412yes—
636C2×C4#58512yes—
637C8—44no—
638C8—44no—
639C22—22no—
640C22—24no—
641C22—42no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—42no—
647C22—42no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—42no—
653C22—42no—
654C22C2×#53411yescharacteristic
655C22C2×#53611yescharacteristic
656C22#344911yescharacteristic
657C22C2×#57712yes—
658C22#358612yes—
659C4—22no—
660C4—22no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—82no—
670C4—82no—
671C4—84no—
672C4—84no—
673C4—82no—
674C4—82no—
675C4#320014yes—
676C4#354112yes—
677C4#354112yes—
678C2—42no—
679C2—81no—
680C2—81no—
681C2#3743811yescharacteristic
682C2C2×#320011yescharacteristic
683C2#6075411yescharacteristic
684C2#6088912yes—
685C2#6099212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32794C2semidirect
4. #32780C2semidirect
5. #30374C2semidirect
6. #29941C2semidirect
8. #29688C2semidirect
9. #29344C2semidirect
10. #29311C2semidirect
11. #27682C2semidirect
12. #24710C2semidirect
13. #24703C2semidirect
14. #24301C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#346272
#327942
#327802
#303742
#299412
#299002
#296882
#293442
#293112
#276822
#247102
#247032
#243012
#207212