Finite Group Databasebeta

← #6660906

Subgroups — #6660906

Order 512.

1791 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35316C211yesmaximal, characteristic, complemented
2#35254C211yesmaximal, characteristic
3#32723C212yesmaximal, complemented
4#32722C212yesmaximal, complemented
5#30333C214yesmaximal, complemented
6#29914C212yesmaximal, complemented
7#29764C212yesmaximal, complemented
8#29723C212yesmaximal, complemented
9#29624C212yesmaximal, complemented
10#29338C211yesmaximal, characteristic, complemented
11#29332C211yesmaximal, characteristic, complemented
12#28311C211yesmaximal, characteristic, complemented
13#28302C211yesmaximal, characteristic, complemented
14#24518C214yesmaximal, complemented
15#24014C212yesmaximal, complemented
16#23926C212yesmaximal
17#20720C211yesmaximal, characteristic, complemented
18C2×#536C2212yes—
19C2×#535C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic, complemented
21C2×C4×#41C2211yescharacteristic
22C2×#484C2212yes—
23C2×#483C2212yes—
24C2×#448C2214yes—
25C2×#450C2212yes—
26C2×#450C2212yes—
27C2×#433C2212yescomplemented
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic, complemented
31#2857C2211yescharacteristic, complemented
32#2854C2212yes—
33#2854C2212yes—
34#2624C2212yes—
35#2624C2212yes—
36#2621C2212yes—
37#2621C2212yescomplemented
38#2547—22no—
39#2546—22no—
40#2486—22no—
41#2486—22no—
42#2448—22no—
43#2445—22no—
44#2418—22no—
45#2418—22no—
46#2316C2211yescharacteristic
47#2311C2211yescharacteristic
48#2310C2212yescomplemented
49#2309C2211yescharacteristic
50#2296C2211yescharacteristic
51C2×#390C2211yescharacteristic
52#2285C2211yescharacteristic, complemented
53#2100C2212yescomplemented
54#2099C2212yescomplemented
55#2094C2212yes—
56#2093C2212yes—
57#2085C2214yescomplemented
58#2073C2214yes—
59#2068C2214yescomplemented
60#2066C2214yescomplemented
61#2055C2214yes—
62#2055C2214yes—
63#2043C2214yescomplemented
64#2043C2212yescomplemented
65#2043C2212yescomplemented
66#2036C2214yescomplemented
67#2035C2214yescomplemented
68#2013C2214yes—
69#2013C2212yes—
70#2013C2212yes—
71#2000C2214yes—
72#2000C2214yes—
73#1973C2214yes—
74#1968C2214yes—
75#1935C2214yescomplemented
76#1933C2214yescomplemented
77#1875C2214yes—
78#1873C2212yes—
79#1873C2212yes—
80#1873C2212yes—
81#1873C2212yes—
82#1854C2214yescomplemented
83#1817C2212yescomplemented
84#1798C2212yes—
85#1797C2212yescomplemented
86#1787C2214yescomplemented
87#1780C2212yes—
88#1750C2212yescomplemented
89#1749C2211yescharacteristic
90C4×C22⋊C8C2211yescharacteristic
91C22×#41C2311yescharacteristic
92C23×Q8C2312yes—
93C23×D4C2311yescharacteristic
94C22×SD16C2314yes—
95C22×SD16C2314yes—
96#536—44no—
97#536—44no—
98#535—44no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127—21no—
106C2×#124—21no—
107C2×#124—22no—
108C2×#123—22no—
109C2×#123—22no—
110C2×#123—22no—
111C2×#123—22no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×#121C2312yes—
117C2×#121C2312yes—
118C2×#120—21no—
119C2×#120—21no—
120C4×#41—44no—
121C2×C4×Q8D412yes—
122C2×C4×D4D412yes—
123C2×C4×D4D412yes—
124C2×C4×D4D412yes—
125C2×#117C2311yescharacteristic
126C2×#117C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—22no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—22no—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#484—44no—
138#483—44no—
139#450—44no—
140#450—44no—
141#448—48no—
142#433—44no—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160#399—22no—
161#399—22no—
162#398—24no—
163#398—24no—
164#397—22no—
165#397—22no—
166#397—24no—
167#397—24no—
168#394—22no—
169#394—22no—
170#394—22no—
171#394—22no—
172#391—21no—
173#391—24no—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#390—21no—
179#390—22no—
180#390—22no—
181#390C2312yes—
182#390C2312yes—
183#390C2312yes—
184#390C2312yes—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389—22no—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#386—21no—
199#386—21no—
200#386—22no—
201#386—22no—
202#385—22no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#382C2312yes—
206#381—24no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380—22no—
212#380—22no—
213#380C2312yes—
214#380C2312yes—
215#379—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#336C2312yes—
223#336C2312yes—
224#336C2314yes—
225C25—21no—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—42no—
229C22×Q8—22no—
230C22×Q8—28no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8—44no—
234C22×Q8C2×D411yescharacteristic
235C22×Q8C2412yes—
236C22×Q8C2412yes—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—24no—
241C22×D4—42no—
242C22×D4—42no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2×D411yescharacteristic
245C22×D4C2×D411yescharacteristic
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—21no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×SD16—44no—
256C2×SD16—44no—
257C2×SD16—44no—
258C2×SD16—44no—
259C2×SD16—44no—
260C2×SD16—44no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263C4⋊Q8—42no—
264C4⋊Q8—42no—
265C4⋊Q8—44no—
266C4⋊Q8—48no—
267C4⋊Q8—48no—
268#127—44no—
269#124—48no—
270#124—44no—
271#123—48no—
272#123—48no—
273#123—84no—
274#123—84no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—42no—
280C22⋊Q8—42no—
281C22⋊Q8—42no—
282C22⋊Q8—42no—
283#121—28no—
284#121—44no—
285#120—44no—
286#120—44no—
287C4×Q8—44no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—44no—
291#117—24no—
292#117—42no—
293#117—42no—
294#117—42no—
295#117—42no—
296#117—42no—
297#117—42no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—84no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—24no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—42no—
350C2×C42—42no—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—44no—
359#106—44no—
360C4⋊C8—44no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#103—44no—
367#102—44no—
368#102—44no—
369C22⋊C8—28no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372C22⋊C8—44no—
373#97—42no—
374C4×C8—42no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386C24—22no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—82no—
391C24—82no—
392C24C22×D411yescharacteristic
393C24C22×D411yescharacteristic
394C24C22×D411yescharacteristic
395#41—416no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—41no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—88no—
411C2×Q8—88no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—42no—
430C2×D4—88no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—82no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—42no—
462C22×C4—82no—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C4H(5,𝔽2)14yes—
472C22×C4C2511yescharacteristic
473SD16—88no—
474SD16—88no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—42no—
537C42—44no—
538C23—22no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—84no—
559C23—88no—
560C23—162no—
561C23—162no—
562C23—164no—
563C23#53412yes—
564C23#53512yes—
565C23C23×D411yescharacteristic
566Q8—44no—
567Q8—44no—
568Q8—44no—
569Q8—44no—
570Q8—44no—
571Q8—88no—
572D4—44no—
573D4—44no—
574D4—44no—
575D4—44no—
576D4—44no—
577D4—84no—
578D4—164no—
579D4—164no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—48no—
601C2×C4—44no—
602C2×C4—42no—
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—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—88no—
624C2×C4—162no—
625C2×C4#53412yes—
626C2×C4#53512yes—
627C2×C4#57714yes—
628C2×C4C2×H(5,𝔽2)12yes—
629C2×C4C2×H(5,𝔽2)12yes—
630C2×C4#58512yes—
631C8—44no—
632C8—44no—
633C22—22no—
634C22—22no—
635C22—24no—
636C22—22no—
637C22—22no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—41no—
642C22—41no—
643C22—42no—
644C22—41no—
645C22—42no—
646C22—41no—
647C22—41no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—168no—
656C22C2×#53411yescharacteristic
657C22C2×#53511yescharacteristic
658C22#344911yescharacteristic
659C22C2×#57712yes—
660C22#358612yes—
661C4—22no—
662C4—22no—
663C4—24no—
664C4—41no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4#353414yes—
676C2—42no—
677C2—42no—
678C2—81no—
679C2—162no—
680C2#3744011yescharacteristic
681C2#6075812yes—
682C2#6082011yescharacteristic
683C2#6082811yescharacteristic
684C2#6099112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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
1. #35316C2semidirect
3. #32723C2semidirect
4. #32722C2semidirect
5. #30333C2semidirect
6. #29914C2semidirect
7. #29764C2semidirect
8. #29723C2semidirect
9. #29624C2semidirect
10. #29338C2semidirect
11. #29332C2semidirect
12. #28311C2semidirect
13. #28302C2semidirect
14. #24518C2semidirect
15. #24014C2semidirect
17. #20720C2semidirect
20. C22×C4⋊Q8C22semidirect
27. C2×#433C22semidirect
30. #2857C22semidirect
31. #2857C22semidirect
37. #2621C22semidirect
48. #2310C22semidirect
52. #2285C22semidirect
53. #2100C22semidirect
54. #2099C22semidirect
57. #2085C22semidirect
59. #2068C22semidirect
60. #2066C22semidirect
63. #2043C22semidirect
64. #2043C22semidirect
65. #2043C22semidirect
66. #2036C22semidirect
67. #2035C22semidirect
75. #1935C22semidirect
76. #1933C22semidirect
82. #1854C22semidirect
83. #1817C22semidirect
85. #1797C22semidirect
86. #1787C22semidirect
88. #1750C22semidirect

Maximal subgroups

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

subgroupindex
#353162
#352542
#327232
#327222
#303332
#299142
#297642
#297232
#296242
#293382
#293322
#283112
#283022
#245182
#240142
#239262
#207202