Finite Group Databasebeta

← #6971032

Subgroups — #6971032

Order 512.

2081 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36260C211yesmaximal, characteristic
2#36151C211yesmaximal, characteristic, complemented
3#32770C212yesmaximal, complemented
4#31960C214yesmaximal, complemented
5#31004C214yesmaximal, complemented
6#30213C211yesmaximal, characteristic, complemented
7#29950C212yesmaximal, complemented
8#29661C212yesmaximal, complemented
9#29145C211yesmaximal, characteristic, complemented
10#28386C211yesmaximal, characteristic, complemented
11#24827C212yesmaximal, complemented
12#24742C212yesmaximal, complemented
13#24252C212yesmaximal, complemented
14#24132C212yesmaximal, complemented
15#23933C212yesmaximal, complemented
16#23803C211yesmaximal, characteristic, complemented
17#22014C211yesmaximal, characteristic, complemented
18C2×#547C2211yescharacteristic
19C2×#546—22no—
20C2×D42C2211yescharacteristic
21C2×#533C2212yescomplemented
22C2×#477C2212yescomplemented
23C2×#460C2212yes—
24C2×#460C2214yescomplemented
25C2×#450C2214yes—
26C2×#440C2212yes—
27C2×#440C2212yescomplemented
28#2862C2211yescharacteristic, complemented
29#2858C2211yescharacteristic
30#2857C2212yescomplemented
31#2856C2212yes—
32#2542C2211yescharacteristic
33#2541C2211yescharacteristic
34#2540—22no—
35#2539—22no—
36#2539—22no—
37#2536C2212yes—
38#2501C2212yes—
39#2489—22no—
40#2453—22no—
41#2433C2212yes—
42#2421C2214yescomplemented
43#2415—22no—
44#2362—22no—
45#2351C2212yescomplemented
46C4×C4⋊Q8C2211yescharacteristic, complemented
47C4×#127C2211yescharacteristic
48C2×#390C2212yes—
49C42×D4C2211yescharacteristic
50#2095C2212yescomplemented
51#2093C2211yescharacteristic
52#2093C2211yescharacteristic, complemented
53#2093C2212yescomplemented
54#2083C2214yescomplemented
55#2073C2214yescomplemented
56#2067C2214yes—
57#2067C2214yescomplemented
58#2041C2214yescomplemented
59#2033C2214yes—
60#2033C2214yescomplemented
61#2023C2212yes—
62#2023C2212yescomplemented
63#2023C2212yescomplemented
64#2023C2212yescomplemented
65#2014C2212yes—
66#2013C2211yescharacteristic
67#2013C2212yescomplemented
68#2013C2211yescharacteristic, complemented
69#2000C2214yes—
70#2000C2214yescomplemented
71#1993C2212yescomplemented
72#1973C2214yescomplemented
73#1961C2212yes—
74#1961C2212yes—
75#1955C2212yes—
76#1940C2212yescomplemented
77#1924C2214yescomplemented
78#1880C2214yes—
79#1880C2214yescomplemented
80#1872C2214yescomplemented
81#1852C2214yescomplemented
82#1836C2212yescomplemented
83#1817C2212yescomplemented
84#1804C2212yes—
85#1804C2212yescomplemented
86#1803C2214yescomplemented
87#1797C2212yescomplemented
88#1786C2214yes—
89#1767C2211yescharacteristic, complemented
90#1745C2211yescharacteristic, complemented
91C22×#41—21no—
92C23×D4—21no—
93C23×D4C2312yes—
94C22×SD16C2312yes—
95C22×SD16C2314yes—
96C22×SD16C2312yes—
97#547—44no—
98#546—416no—
99D42—44no—
100#533—48no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#127C2312yes—
107C2×#125—22no—
108C2×#124—22no—
109C2×#123—22no—
110C2×#123—22no—
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8C2314yes—
117C2×#121—22no—
118C2×#121—22no—
119C2×#121—21no—
120C2×#121—22no—
121C2×#121C2312yes—
122C2×#121C2312yes—
123C2×#120—22no—
124C2×#120—22no—
125C2×#120—22no—
126C2×C4×Q8—22no—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2314yes—
129C2×C4×D4—22no—
130C2×C4×D4D412yes—
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2311yescharacteristic
133C2×C4×D4C2311yescharacteristic
134C2×#117—22no—
135C22×C4⋊C4C2312yes—
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C42C2312yes—
139#477—44no—
140#460—44no—
141#460—48no—
142#450—48no—
143#440—44no—
144#440—44no—
145C2×#107C2314yes—
146C2×#107C2312yes—
147C2×#107C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160C2×C22⋊C8C2314yes—
161C2×C22⋊C8C2314yes—
162C2×#97C2312yes—
163C2×#97C2312yes—
164#400—24no—
165#399—22no—
166#399—22no—
167#399—21no—
168#399—21no—
169#398—22no—
170#397—24no—
171#394—22no—
172#392—22no—
173#392—22no—
174#392—21no—
175#392—21no—
176#391—22no—
177#391—22no—
178#391—22no—
179#391C2312yes—
180#390—22no—
181#390—22no—
182#390—22no—
183#390—22no—
184#390—22no—
185#390—22no—
186#390C2311yescharacteristic
187#390C2311yescharacteristic
188#390C2312yes—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#389—24no—
193#389—22no—
194#389—22no—
195#389C2314yes—
196#388—24no—
197#388—24no—
198#387—24no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2312yes—
202#382—22no—
203#382C2312yes—
204#382C2312yes—
205#380—22no—
206#380C2314yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4D412yes—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4C2312yes—
214C43C2311yescharacteristic
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2311yescharacteristic
220#336C2312yes—
221#336C2311yescharacteristic
222#336C2314yes—
223#336C2312yes—
224C25—22no—
225C2×#41—48no—
226C2×#41—44no—
227C22×Q8—22no—
228C22×Q8—21no—
229C22×Q8C2412yes—
230C22×D4—24no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—44no—
240C22×D4—42no—
241C22×D4—44no—
242C22×D4—44no—
243C22×D4—48no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2411yescharacteristic
246C22×D4C2412yes—
247C22×D4C2411yescharacteristic
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—28no—
258C2×SD16—42no—
259C2×SD16—42no—
260C2×SD16—42no—
261C2×SD16—44no—
262C2×SD16—44no—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—44no—
269#127—22no—
270#127—24no—
271#127—41no—
272#127—44no—
273#125—48no—
274#124—48no—
275#123—48no—
276#123—48no—
277#123—48no—
278C22⋊Q8—28no—
279C22⋊Q8—44no—
280C22⋊Q8—48no—
281C22⋊Q8—48no—
282C22⋊Q8—44no—
283C22⋊Q8—48no—
284#121—44no—
285#121—44no—
286#121—44no—
287#121—48no—
288#121—48no—
289#121—48no—
290#120—48no—
291#120—48no—
292#120—48no—
293C4×Q8—28no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—44no—
297C4×Q8—48no—
298C4×D4—28no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—48no—
303#117—48no—
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—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
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—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—48no—
341C2×C22⋊C4—48no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—44no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—42no—
357#107—42no—
358#107—44no—
359C4⋊C8—24no—
360C4⋊C8—44no—
361#103—42no—
362#103—42no—
363#103—44no—
364#102—42no—
365#102—42no—
366#102—44no—
367#102—44no—
368#102—44no—
369#102—44no—
370#102—44no—
371C22⋊C8—44no—
372C22⋊C8—44no—
373#97—24no—
374#97—42no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95C2×D411yescharacteristic
382#95C2×D411yescharacteristic
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24—22no—
387C24—21no—
388C24—44no—
389C24—44no—
390C24—44no—
391C24—44no—
392C24—82no—
393C24—82no—
394C24—82no—
395C24C22×D412yes—
396#41—816no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8H(5,𝔽2)14yes—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—21no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—82no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—88no—
444C2×D4—88no—
445C2×D4—816no—
446C2×D42−1+412yes—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4AGL(1,ℤ/8)18yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C2511yescharacteristic
478SD16—48no—
479SD16—48no—
480SD16—88no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—22no—
490C4⋊C4—28no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—42no—
536C42—42no—
537C42—42no—
538C42—48no—
539C42—44no—
540C42H(5,𝔽2)14yes—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—41no—
556C23—42no—
557C23—41no—
558C23—42no—
559C23—41no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—88no—
565C23—88no—
566C23—88no—
567C23—88no—
568C23—164no—
569C23—164no—
570C23—164no—
571C23C2×AGL(1,ℤ/8)14yes—
572C23C23×D411yescharacteristic
573Q8—24no—
574Q8—24no—
575Q8—42no—
576Q8—42no—
577Q8—88no—
578Q8—84no—
579D4—28no—
580D4—24no—
581D4—42no—
582D4—44no—
583D4—42no—
584D4—42no—
585D4—44no—
586D4—84no—
587D4—88no—
588D4—88no—
589D4—88no—
590D4—88no—
591D4—164no—
592D4—162no—
593D4—164no—
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—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—88no—
631C2×C4#53512yes—
632C2×C4#53512yes—
633C2×C4C2×AGL(1,ℤ/8)14yes—
634C2×C4C2×AGL(1,ℤ/8)14yes—
635C2×C4C2×H(5,𝔽2)12yes—
636C2×C4C2×H(5,𝔽2)12yes—
637C2×C4C2×2−1+411yescharacteristic
638C2×C4#58511yescharacteristic
639C8—42no—
640C8—42no—
641C8—44no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—44no—
650C22—42no—
651C22—81no—
652C22—82no—
653C22—81no—
654C22—81no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—81no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—81no—
666C22—1616no—
667C22C2×#53511yescharacteristic
668C22#348411yescharacteristic
669C22#348911yescharacteristic
670C22C22×AGL(1,ℤ/8)12yes—
671C22#358612yes—
672C4—22no—
673C4—24no—
674C4—24no—
675C4—22no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—82no—
681C4—81no—
682C4—81no—
683C4—82no—
684C4—84no—
685C4#353612yes—
686C4#354212yes—
687C2—44no—
688C2—82no—
689C2—161no—
690C2—162no—
691C2#4247811yescharacteristic
692C2#6079811yescharacteristic
693C2#6082611yescharacteristic
694C2#6157212yes—
695C2#6210912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36260C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36151C2semidirect
3. #32770C2semidirect
4. #31960C2semidirect
5. #31004C2semidirect
6. #30213C2semidirect
7. #29950C2semidirect
8. #29661C2semidirect
9. #29145C2semidirect
10. #28386C2semidirect
11. #24827C2semidirect
12. #24742C2semidirect
13. #24252C2semidirect
14. #24132C2semidirect
15. #23933C2semidirect
16. #23803C2semidirect
17. #22014C2semidirect
21. C2×#533C22semidirect
22. C2×#477C22semidirect
24. C2×#460C22semidirect
27. C2×#440C22semidirect
28. #2862C22semidirect
30. #2857C22semidirect
42. #2421C22semidirect
45. #2351C22semidirect
46. C4×C4⋊Q8C22semidirect
50. #2095C22semidirect
52. #2093C22semidirect
53. #2093C22semidirect
54. #2083C22semidirect
55. #2073C22semidirect
57. #2067C22semidirect
58. #2041C22semidirect
60. #2033C22semidirect
62. #2023C22semidirect
63. #2023C22semidirect
64. #2023C22semidirect
67. #2013C22semidirect
68. #2013C22semidirect
70. #2000C22semidirect
71. #1993C22semidirect
72. #1973C22semidirect
76. #1940C22semidirect
77. #1924C22semidirect
79. #1880C22semidirect
80. #1872C22semidirect
81. #1852C22semidirect
82. #1836C22semidirect
83. #1817C22semidirect
85. #1804C22semidirect
86. #1803C22semidirect
87. #1797C22semidirect
89. #1767C22semidirect
90. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#362602
#361512
#327702
#319602
#310042
#302132
#299502
#296612
#291452
#283862
#248272
#247422
#242522
#241322
#239332
#238032
#220142