Finite Group Databasebeta

← #6561810

Subgroups — #6561810

Order 512.

1867 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36410C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic, complemented
3#29920C211yesmaximal, characteristic, complemented
4#29907C212yesmaximal, complemented
5#27973C211yesmaximal, characteristic, complemented
6#27862C212yesmaximal, complemented
7#27647C211yesmaximal, characteristic, complemented
8#27587C212yesmaximal, complemented
9#26318C212yesmaximal, complemented
10#26214C214yesmaximal, complemented
11#24996C212yesmaximal, complemented
12#24879C214yesmaximal, complemented
13#23756C211yesmaximal, characteristic, complemented
14#23630C212yesmaximal, complemented
15#22715C212yesmaximal, complemented
16#22473C211yesmaximal, characteristic, complemented
17#22432C212yesmaximal, complemented
18C2×#550C2211yescharacteristic
19C2×#520—21no—
20C2×C4×#41C2211yescharacteristic
21C2×#483C2212yes—
22C2×#436C2212yes—
23C2×#417C2214yes—
24#2856C2211yescharacteristic
25#2854C2212yes—
26#2582C2211yescharacteristic
27#2548C2211yescharacteristic
28#2476—22no—
29#2464—22no—
30#2445—21no—
31#2431—21no—
32#2414C2212yes—
33#2407C2212yes—
34#2367—21no—
35#2363—21no—
36#2354C2211yescharacteristic
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4C2212yes—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—22no—
41D4×C4⋊C4C2214yes—
42D4×C4⋊C4C2212yes—
43#2317—21no—
44#2317—22no—
45C4×C22⋊Q8C2211yescharacteristic
46#2297C2212yes—
47#2285C2212yes—
48#2285C2211yescharacteristic
49C2×#384C2211yescharacteristic
50C2×#384C2212yes—
51#2100C2214yes—
52#2099C2211yescharacteristic
53#2096C2211yescharacteristic
54#2094C2214yes—
55#2093C2212yes—
56#2084C2211yescharacteristic
57#2084C2212yes—
58#2077C2212yes—
59#2072C2214yes—
60#2072C2211yescharacteristic
61#2072C2212yes—
62#2071C2212yes—
63#2071C2212yes—
64#2071C2214yes—
65#2071C2212yes—
66#2035C2212yes—
67#2033C2214yes—
68#1984C2212yes—
69#1982C2212yes—
70#1957C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1928C2214yes—
74#1926C2214yes—
75#1926C2214yes—
76#1921C2214yes—
77#1895C2214yes—
78#1874C2214yes—
79#1874C2214yes—
80#1870C2214yes—
81#1854C2214yes—
82#1853C2214yes—
83#1834C2212yes—
84#1823C2212yes—
85#1817C2212yes—
86#1809C2214yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89#1788C2214yes—
90C4×#106C2212yes—
91C4×#103C2212yes—
92C4×C22⋊C8C2211yescharacteristic
93#1736C2214yes—
94#1728C2212yes—
95C22×#41D411yescharacteristic
96C22×#41C2312yes—
97C22×M16C2314yes—
98#550—28no—
99C2×#127—21no—
100C2×#127C2311yescharacteristic
101C2×#124—21no—
102C2×#123—22no—
103C2×C22⋊Q8C2311yescharacteristic
104C2×#121—21no—
105C2×#121—21no—
106C2×#121—22no—
107C2×#121C2312yes—
108C2×#121C2312yes—
109#520—48no—
110C4×#41—28no—
111C2×C4×Q8—22no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—24no—
116C2×C4×D4—22no—
117C2×C4×D4—21no—
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D412yes—
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×#117—22no—
123C2×#117—21no—
124C2×#117D411yescharacteristic
125C2×#117D412yes—
126C2×#117C2312yes—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#483—44no—
136#436—28no—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2312yes—
140C2×#106C2312yes—
141C2×C4⋊C8C2312yes—
142#417—48no—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C4×C8C2312yes—
154#398—22no—
155#394—22no—
156#394—22no—
157#394—22no—
158#392—22no—
159#392—22no—
160#391—21no—
161#391C2312yes—
162#391C2311yescharacteristic
163#390—21no—
164#390—21no—
165#390—22no—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#389—22no—
172#389—24no—
173#389—24no—
174#389—22no—
175#389—21no—
176#389C2312yes—
177#389C2311yescharacteristic
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—21no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#385—21no—
188#385—22no—
189#385—24no—
190#385—24no—
191#385—24no—
192#384—22no—
193#384—21no—
194#384—22no—
195#384—21no—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2314yes—
201#384C2311yescharacteristic
202#382—22no—
203#382—21no—
204#380—21no—
205#380—22no—
206#380C2312yes—
207#380C2314yes—
208#380C2311yescharacteristic
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C2×#41—24no—
235C2×#41—24no—
236C2×#41—24no—
237C2×#41—24no—
238C2×#41—28no—
239C2×#41—42no—
240C2×#41—44no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2411yescharacteristic
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2×D412yes—
249C22×D4C2×D411yescharacteristic
250C22×D4C2412yes—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2412yes—
260C2×M16—28no—
261C2×M16—44no—
262C2×M16—44no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266#127—24no—
267#127—44no—
268#124—44no—
269#123—84no—
270C22⋊Q8—22no—
271C22⋊Q8—41no—
272#121—28no—
273#121—44no—
274#121—48no—
275#121—44no—
276#121—82no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—48no—
281C4×D4—24no—
282C4×D4—28no—
283C4×D4—28no—
284C4×D4—416no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4#4118yes—
288#117—24no—
289#117—28no—
290#117—48no—
291#117—44no—
292#117—44no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4—84no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
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—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—28no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—22no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—44no—
362#106—42no—
363#106—42no—
364#106—44no—
365C4⋊C8—24no—
366#103—24no—
367#103—28no—
368#103—42no—
369#102—44no—
370#102—44no—
371#102—44no—
372#102—44no—
373C22⋊C8—28no—
374C22⋊C8—42no—
375C22⋊C8—42no—
376C4×C8—24no—
377#95—21no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—22no—
385#95—21no—
386#95—24no—
387#95—22no—
388#95—21no—
389#95—22no—
390C24—22no—
391C24C22×D412yes—
392C24C22×D411yescharacteristic
393#41—232no—
394#41—416no—
395#41—88no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8#13714yes—
402C2×Q8H(5,𝔽2)12yes—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—82no—
427C2×D4#12314yes—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—84no—
461C22×C4—82no—
462C22×C4—84no—
463C22×C4AGL(1,ℤ/8)14yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4118yes—
473C22×C4C2511yescharacteristic
474M16—88no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—42no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—84no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—84no—
514C4⋊C4—81no—
515C4⋊C4—82no—
516C4⋊C4#12314yes—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—81no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4#12318yes—
534C42—24no—
535C42—22no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—24no—
540C42—48no—
541C42—42no—
542C42—44no—
543C42—42no—
544C42—82no—
545C42C2×#4114yes—
546C23—21no—
547C23—22no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—22no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—21no—
556C23—22no—
557C23—21no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23C2×#12314yes—
566C23C2×AGL(1,ℤ/8)12yes—
567C23C23×D411yescharacteristic
568Q8—28no—
569Q8—24no—
570Q8—42no—
571D4—216no—
572D4—28no—
573D4—44no—
574D4—44no—
575D4—88no—
576D4—84no—
577D4—162no—
578D4—164no—
579D4—164no—
580C2×C4—216no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—88no—
629C2×C4—164no—
630C2×C4—168no—
631C2×C4C2×#12314yes—
632C2×C4C2×#12312yes—
633C2×C4C2×#12312yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C2×AGL(1,ℤ/8)12yes—
636C2×C4C2×#13714yes—
637C2×C4C2×#13712yes—
638C2×C4#57712yes—
639C2×C4C22×#4112yes—
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4#58512yes—
642C2×C4#58511yescharacteristic
643C8—44no—
644C8—42no—
645C8—42no—
646C22—22no—
647C22—22no—
648C22—24no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—42no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—41no—
657C22—41no—
658C22—42no—
659C22—41no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22C22×#12311yescharacteristic
668C22#348811yescharacteristic
669C22#350011yescharacteristic
670C22C22×AGL(1,ℤ/8)11yescharacteristic
671C22C22×#13711yescharacteristic
672C22C2×#57711yescharacteristic
673C22#358511yescharacteristic
674C4—22no—
675C4—24no—
676C4—22no—
677C4—22no—
678C4—24no—
679C4—42no—
680C4—41no—
681C4—42no—
682C4—84no—
683C4—82no—
684C4—81no—
685C4—82no—
686C4—82no—
687C4—81no—
688C4—81no—
689C4#309214yes—
690C4#351312yes—
691C4#352112yes—
692C2—42no—
693C2—44no—
694C2—162no—
695C2#3597611yescharacteristic
696C2C2×#309211yescharacteristic
697C2#6052511yescharacteristic
698C2#6196311yescharacteristic
699C2#6200211yescharacteristic
700C2#6254111yescharacteristic
701C2#6259411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36410C2
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
2. #34183C2semidirect
3. #29920C2semidirect
4. #29907C2semidirect
5. #27973C2semidirect
6. #27862C2semidirect
7. #27647C2semidirect
8. #27587C2semidirect
9. #26318C2semidirect
10. #26214C2semidirect
11. #24996C2semidirect
12. #24879C2semidirect
13. #23756C2semidirect
14. #23630C2semidirect
15. #22715C2semidirect
16. #22473C2semidirect
17. #22432C2semidirect

Maximal subgroups

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

subgroupindex
#364102
#341832
#299202
#299072
#279732
#278622
#276472
#275872
#263182
#262142
#249962
#248792
#237562
#236302
#227152
#224732
#224322