Finite Group Databasebeta

← #6640548

Subgroups — #6640548

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2C2×#2485C211yesmaximal, characteristic
3#32984C212yesmaximal, complemented
4#32266C214yesmaximal, complemented
5#32034C214yesmaximal, complemented
6#32009C214yesmaximal, complemented
7#32007C214yesmaximal, complemented
8#29894C212yesmaximal, complemented
9#27611C211yesmaximal, characteristic, complemented
10#27610C211yesmaximal, characteristic, complemented
11#24552C212yesmaximal, complemented
12#24551C212yesmaximal, complemented
13#24064C211yesmaximal, characteristic, complemented
14#24063C211yesmaximal, characteristic, complemented
15#21738C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#484C2212yes—
20C2×#483C2212yes—
21C22×C22⋊C8C2212yes—
22#2848C2212yes—
23#2848C2212yes—
24#2626—21no—
25#2626—21no—
26#2580—22no—
27#2566—22no—
28#2563C2212yes—
29#2542C2212yes—
30#2538C2212yes—
31#2538C2212yes—
32#2485—22no—
33#2485—22no—
34#2485C2214yes—
35#2485C2214yes—
36C2×#397—21no—
37C2×#394—21no—
38#2317—21no—
39#2309—21no—
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43C2×#387—21no—
44#2285C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48#2099C2212yescomplemented
49#2099C2212yescomplemented
50#2084C2212yescomplemented
51#2083C2212yescomplemented
52#2077C2214yescomplemented
53#2077C2214yescomplemented
54#2076C2214yescomplemented
55#2075C2214yescomplemented
56#2058C2212yescomplemented
57#2057C2212yescomplemented
58#2043C2212yescomplemented
59#2043C2212yescomplemented
60#2036C2214yescomplemented
61#2036C2214yescomplemented
62#2035C2214yescomplemented
63#2035C2214yescomplemented
64#1920C2214yescomplemented
65#1918C2214yescomplemented
66#1892C2214yescomplemented
67#1892C2214yescomplemented
68#1873C2214yescomplemented
69#1873C2214yescomplemented
70#1872C2212yes—
71#1872C2212yes—
72#1869C2214yes—
73#1869C2214yes—
74#1865C2214yescomplemented
75#1863C2214yescomplemented
76#1854C2214yescomplemented
77#1854C2214yescomplemented
78#1853C2212yes—
79#1852C2212yes—
80#1824C2212yes—
81#1823C2212yes—
82#1817C2211yescharacteristic
83#1817C2211yescharacteristic
84#1808C2214yescomplemented
85#1807C2214yescomplemented
86#1779C2212yes—
87C23×D4D411yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C23×C8C2314yes—
91#533—44no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#125—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C4×Q8—21no—
102C2×C4×Q8C2312yes—
103C2×C4×D4—44no—
104C2×C4×D4D412yes—
105C2×C4×D4D412yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117—21no—
109C2×#117—21no—
110C2×#117—22no—
111C2×#117—44no—
112C2×#117C2314yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128#484—44no—
129#483—44no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×C22⋊C8—44no—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144#399—22no—
145#399—22no—
146#397—24no—
147#397—24no—
148#396—22no—
149#396—22no—
150#396—22no—
151#396—22no—
152#394—24no—
153#394—24no—
154#393—22no—
155#393—22no—
156#390—22no—
157#390—22no—
158#390—22no—
159#390—22no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#388—24no—
171#388—24no—
172#387—24no—
173#387—24no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#384—22no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#382—22no—
184#382—21no—
185#380—21no—
186#380—22no—
187#380—22no—
188#380—22no—
189#380C2312yes—
190#380C2312yes—
191#380C2312yes—
192#380C2312yes—
193#380C2314yes—
194#380C2314yes—
195#380C2314yes—
196#380C2312yes—
197#379—22no—
198#379—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4C2312yes—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218C25C2×D411yescharacteristic
219C25C2×D411yescharacteristic
220C22×Q8—21no—
221C22×Q8C2412yes—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—42no—
225C22×D4C2×D412yes—
226C22×D4C2×D412yes—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C22×C8—44no—
247C22×C8—44no—
248C22×C8—44no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—42no—
252C4⋊Q8—42no—
253#125—44no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C4×Q8—42no—
262C4×Q8—42no—
263C4×Q8—44no—
264C4×D4—44no—
265C4×D4—44no—
266C4×D4—42no—
267C4×D4—42no—
268C4×D4—42no—
269C4×D4—42no—
270#117—44no—
271#117—44no—
272#117—48no—
273#117—44no—
274#117—44no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4—82no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—44no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—44no—
346#107—44no—
347#106—44no—
348#106—44no—
349#103—44no—
350#103—44no—
351#103—44no—
352#103—44no—
353C22⋊C8—24no—
354C22⋊C8—24no—
355C22⋊C8—44no—
356C22⋊C8—44no—
357C22⋊C8—44no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—41no—
377C24—41no—
378C24—41no—
379C24—41no—
380C24—41no—
381C24—41no—
382C24—41no—
383C24—41no—
384C24#13714yes—
385C24C22×D412yes—
386C24C22×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—42no—
393C2×Q8—81no—
394C2×Q8—81no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—416no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—416no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—88no—
451C22×C4#13514yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4H(5,𝔽2)14yes—
460C22×C4C2511yescharacteristic
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—88no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—48no—
524C42—44no—
525C42—44no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—28no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23—82no—
553C23—82no—
554C23—82no—
555C23—82no—
556C23—82no—
557C23#53412yes—
558C23C2×#13512yes—
559C23C2×#13714yes—
560C23C2×#13712yes—
561C23C23×D411yescharacteristic
562Q8—44no—
563Q8—44no—
564Q8—84no—
565D4—48no—
566D4—48no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—28no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
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—84no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4#53412yes—
625C2×C4C2×#13512yes—
626C2×C4C2×#13512yes—
627C2×C4C2×H(5,𝔽2)12yes—
628C2×C4#58512yes—
629C2×C4#58512yes—
630C8—44no—
631C8—44no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—88no—
653C22—88no—
654C22#319014yes—
655C22C2×#53411yescharacteristic
656C22#348511yescharacteristic
657C22#348511yescharacteristic
658C22C22×#13511yescharacteristic
659C22C22×#13711yescharacteristic
660C22#358511yescharacteristic
661C22#358611yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—41no—
666C4—41no—
667C4—42no—
668C4—42no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4#353314yes—
678C2—24no—
679C2—44no—
680C2—82no—
681C2#3695511yescharacteristic
682C2C2×#319011yescharacteristic
683C2#6075511yescharacteristic
684C2#6170011yescharacteristic
685C2#6170111yescharacteristic
686C2#6170511yescharacteristic
687C2#6170811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32984C2semidirect
4. #32266C2semidirect
5. #32034C2semidirect
6. #32009C2semidirect
7. #32007C2semidirect
8. #29894C2semidirect
9. #27611C2semidirect
10. #27610C2semidirect
11. #24552C2semidirect
12. #24551C2semidirect
13. #24064C2semidirect
14. #24063C2semidirect
48. #2099C22semidirect
49. #2099C22semidirect
50. #2084C22semidirect
51. #2083C22semidirect
52. #2077C22semidirect
53. #2077C22semidirect
54. #2076C22semidirect
55. #2075C22semidirect
56. #2058C22semidirect
57. #2057C22semidirect
58. #2043C22semidirect
59. #2043C22semidirect
60. #2036C22semidirect
61. #2036C22semidirect
62. #2035C22semidirect
63. #2035C22semidirect
64. #1920C22semidirect
65. #1918C22semidirect
66. #1892C22semidirect
67. #1892C22semidirect
68. #1873C22semidirect
69. #1873C22semidirect
74. #1865C22semidirect
75. #1863C22semidirect
76. #1854C22semidirect
77. #1854C22semidirect
84. #1808C22semidirect
85. #1807C22semidirect

Maximal subgroups

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

subgroupindex
#351722
C2×#24852
#329842
#322662
#320342
#320092
#320072
#298942
#276112
#276102
#245522
#245512
#240642
#240632
#217382