Finite Group Databasebeta

← #6519068

Subgroups — #6519068

Order 512.

1818 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35273C211yesmaximal, characteristic
2#34368C211yesmaximal, characteristic, complemented
3#27606C214yesmaximal, complemented
4#25017C212yesmaximal, complemented
5#25015C212yesmaximal, complemented
6#25013C212yesmaximal, complemented
7#25011C212yesmaximal, complemented
8#24971C212yesmaximal, complemented
9#24969C212yesmaximal, complemented
10#24967C212yesmaximal, complemented
11#24965C212yesmaximal, complemented
12#24017C214yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21981C211yesmaximal, characteristic, complemented
15#21787C211yesmaximal, characteristic, complemented
16#20737C211yesmaximal, characteristic, complemented
17#20737C211yesmaximal, characteristic, complemented
18C2×#537C2212yes—
19C2×#532C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#485C2214yes—
22C2×#482C2214yes—
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2857—21no—
26#2856—21no—
27#2620—22no—
28#2538C2212yes—
29#2538C2212yes—
30#2537C2212yes—
31#2537C2212yes—
32#2416—22no—
33#2416—22no—
34#2379—22no—
35#2369—21no—
36#2366—21no—
37#2354C2212yes—
38#2354C2212yes—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2212yes—
41#2324—22no—
42#2324—22no—
43#2311C2214yes—
44#2311C2212yes—
45#2285C2211yescharacteristic
46#2285C2211yescharacteristic
47C2×#384C2211yescharacteristic
48#2077C2214yes—
49#2077C2214yes—
50#2076C2214yes—
51#2075C2214yes—
52#2036C2214yes—
53#2035C2214yes—
54#2034C2214yes—
55#2033C2214yes—
56#1954C2212yes—
57#1954C2212yes—
58#1954C2212yes—
59#1954C2212yes—
60#1948C2212yes—
61#1948C2212yes—
62#1947C2212yes—
63#1947C2212yes—
64#1874C2214yes—
65#1872C2214yes—
66#1854C2214yes—
67#1854C2214yes—
68#1853C2214yes—
69#1852C2214yes—
70#1845C2212yes—
71#1844C2212yes—
72#1834C2212yes—
73#1834C2212yes—
74#1825C2212yes—
75#1825C2212yes—
76#1824C2214yes—
77#1823C2214yes—
78#1803C2211yescharacteristic
79#1803C2211yescharacteristic
80#1803C2211yescharacteristic
81#1803C2211yescharacteristic
82#1802C2211yescharacteristic
83#1802C2211yescharacteristic
84#1802C2211yescharacteristic
85#1802C2211yescharacteristic
86#1788C2212yes—
87#1788C2212yes—
88#1780C2214yes—
89C4×#103C2212yes—
90C4×#103C2212yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×#41C2311yescharacteristic
94C22×#41C2312yes—
95#537—44no—
96#537—44no—
97#532—24no—
98#532—24no—
99C2×C4⋊Q8D411yescharacteristic
100C2×#127D411yescharacteristic
101C2×#124D412yes—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8C2314yes—
104C2×#121—22no—
105C2×#121—24no—
106C2×#121—22no—
107C2×#121—22no—
108C2×#121D412yes—
109C2×#121D412yes—
110C2×#121C2314yes—
111C4×#41—24no—
112C4×#41—24no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C2×#117—21no—
124C2×#117—21no—
125C2×#117—21no—
126C2×#117C2311yescharacteristic
127C2×#117C2314yes—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2312yes—
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134#485—48no—
135#482—48no—
136#431—24no—
137#431—24no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2311yescharacteristic
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C4×C8C2312yes—
161C2×C4×C8C2312yes—
162#396—24no—
163#396—24no—
164#394—24no—
165#394—24no—
166#391—22no—
167#391—24no—
168#390—24no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#388—24no—
179#388—24no—
180#386—24no—
181#386—24no—
182#386—24no—
183#385—24no—
184#385—24no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#382—21no—
194#382—21no—
195#380—21no—
196#380—21no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200#380C2312yes—
201#380C2314yes—
202#380C2314yes—
203#380C2314yes—
204#380C2314yes—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#340C2312yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—24no—
226C2×#41—24no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—42no—
230C2×#41—42no—
231C2×#41—42no—
232C2×#41—42no—
233C2×#41—41no—
234C2×#41—41no—
235C22×Q8C2×D412yes—
236C22×Q8C2411yescharacteristic
237C22×D4—21no—
238C22×D4—24no—
239C22×D4C2×D412yes—
240C22×D4C2×D412yes—
241C22×D4C2×D414yes—
242C22×D4C2×D412yes—
243C22×D4C2411yescharacteristic
244C23×C4—21no—
245C23×C4—22no—
246C23×C4—24no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2414yes—
252C23×C4C2411yescharacteristic
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258#127—24no—
259#124—28no—
260C22⋊Q8—28no—
261C22⋊Q8—44no—
262C22⋊Q8—48no—
263#121—28no—
264#121—28no—
265#121—28no—
266#121—44no—
267#121—48no—
268#121—416no—
269#121—84no—
270#121—84no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—84no—
286C4×D4—84no—
287#117—22no—
288#117—22no—
289#117—44no—
290#117—44no—
291#117—44no—
292#117—82no—
293#117—82no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D414yes—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—24no—
346C2×C42—44no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42#4114yes—
350C2×C42#4114yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—44no—
354#107—44no—
355#106—44no—
356#106—44no—
357C4⋊C8—22no—
358C4⋊C8—22no—
359C4⋊C8—22no—
360C4⋊C8—22no—
361#103—24no—
362#103—24no—
363#103—42no—
364#103—42no—
365#102—24no—
366#102—24no—
367#102—42no—
368#102—42no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371C22⋊C8—44no—
372C22⋊C8—44no—
373C22⋊C8—44no—
374C22⋊C8—44no—
375C4×C8—24no—
376C4×C8—24no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385C24—21no—
386C24C22×D414yes—
387C24C22×D411yescharacteristic
388#41—44no—
389#41—44no—
390#41—44no—
391#41—44no—
392#41—84no—
393#41—84no—
394#41—84no—
395#41—84no—
396C2×Q8—24no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8H(5,𝔽2)12yes—
405C2×D4—24no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—48no—
426C2×D4—84no—
427C2×D4—84no—
428C2×D4H(5,𝔽2)12yes—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—84no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4#12314yes—
467C22×C4#12314yes—
468C22×C4C22×D414yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2511yescharacteristic
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—84no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C22⋊C4—28no—
523C22⋊C4—28no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—84no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C42—42no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42—44no—
541C42—84no—
542C42#12314yes—
543C42#12314yes—
544C42C2×#4112yes—
545C42C2×#4112yes—
546C42C2×#4112yes—
547C42C2×#4112yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—24no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23C2×#12312yes—
569C23C23×D411yescharacteristic
570Q8—22no—
571Q8—22no—
572Q8—41no—
573Q8—41no—
574Q8—84no—
575D4—22no—
576D4—22no—
577D4—48no—
578D4—48no—
579D4—48no—
580D4—41no—
581D4—41no—
582D4—84no—
583D4—88no—
584D4—168no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—48no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—164no—
632C2×C4—168no—
633C2×C4C2×#12312yes—
634C2×C4C2×#12312yes—
635C2×C4C2×#12312yes—
636C2×C4C2×#12312yes—
637C2×C4C2×#12312yes—
638C2×C4#57612yes—
639C2×C4#57712yes—
640C2×C4#57712yes—
641C2×C4#57812yes—
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4#58511yescharacteristic
647C2×C4#58511yescharacteristic
648C8—42no—
649C8—42no—
650C8—42no—
651C8—42no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—42no—
664C22—44no—
665C22—44no—
666C22—42no—
667C22—42no—
668C22—44no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22C22×#12311yescharacteristic
674C22#348411yescharacteristic
675C22#348511yescharacteristic
676C22C2×#57611yescharacteristic
677C22C2×#57711yescharacteristic
678C22C2×#57711yescharacteristic
679C22C2×#57811yescharacteristic
680C4—22no—
681C4—22no—
682C4—22no—
683C4—44no—
684C4—42no—
685C4—82no—
686C4—82no—
687C4—81no—
688C4—81no—
689C4—82no—
690C4—82no—
691C4—84no—
692C4#309814yes—
693C4#309914yes—
694C4#349411yescharacteristic
695C4#349411yescharacteristic
696C4#349411yescharacteristic
697C4#349411yescharacteristic
698C2—42no—
699C2—84no—
700C2—82no—
701C2#3554011yescharacteristic
702C2C2×#309811yescharacteristic
703C2C2×#309911yescharacteristic
704C2#6166711yescharacteristic
705C2#6166811yescharacteristic
706C2#6178811yescharacteristic
707C2#6178911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35273C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34368C2semidirect
3. #27606C2semidirect
4. #25017C2semidirect
5. #25015C2semidirect
6. #25013C2semidirect
7. #25011C2semidirect
8. #24971C2semidirect
9. #24969C2semidirect
10. #24967C2semidirect
11. #24965C2semidirect
12. #24017C2semidirect
13. #21982C2semidirect
14. #21981C2semidirect
15. #21787C2semidirect
16. #20737C2semidirect
17. #20737C2semidirect

Maximal subgroups

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

subgroupindex
#352732
#343682
#276062
#250172
#250152
#250132
#250112
#249712
#249692
#249672
#249652
#240172
#219822
#219812
#217872
#207372