Finite Group Databasebeta

← #6507917

Subgroups — #6507917

Order 512.

1832 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic
2C2×#2423C211yesmaximal, characteristic, complemented
3#29105C211yesmaximal, characteristic, complemented
4#29005C214yesmaximal, complemented
5#28867C214yesmaximal, complemented
6#28305C211yesmaximal, characteristic, complemented
7#27973C211yesmaximal, characteristic, complemented
8#27950C211yesmaximal, characteristic, complemented
9#26987C214yesmaximal, complemented
10#25045C214yesmaximal, complemented
11#24557C212yesmaximal, complemented
12#24387C212yesmaximal, complemented
13#22082C212yesmaximal, complemented
14#21454C212yesmaximal, complemented
15#20524C211yesmaximal, characteristic, complemented
16C2×#538—21no—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#481C2212yes—
20C2×#468C2214yes—
21C2×#448C2212yes—
22C2×#440C2214yescomplemented
23C2×#407C2212yes—
24#2857C2211yescharacteristic
25#2856C2211yescharacteristic
26#2847C2212yes—
27#2639—21no—
28#2617—21no—
29#2559—22no—
30#2557—22no—
31#2496C2212yes—
32#2491C2212yes—
33#2428C2212yes—
34#2423—22no—
35#2423—22no—
36#2423C2214yes—
37#2423C2214yes—
38#2422C2212yes—
39C2×#399—21no—
40C2×#398—21no—
41#2325—21no—
42C4×#121C2212yes—
43C2×#391C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#388—21no—
46C2×#387—21no—
47C2×#384C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2096C2212yescomplemented
51#2083C2214yescomplemented
52#2077C2214yes—
53#2072C2212yescomplemented
54#2071C2214yescomplemented
55#2036C2214yes—
56#2033C2214yes—
57#2024C2212yescomplemented
58#2014C2212yescomplemented
59#2000C2214yes—
60#1990C2212yescomplemented
61#1968C2214yescomplemented
62#1957C2212yescomplemented
63#1944C2214yescomplemented
64#1936C2214yescomplemented
65#1933C2214yescomplemented
66#1919C2214yescomplemented
67#1895C2214yes—
68#1894C2214yes—
69#1880C2214yescomplemented
70#1874C2212yes—
71#1874C2212yes—
72#1863C2214yescomplemented
73#1862C2214yescomplemented
74#1852C2212yes—
75#1852C2212yes—
76#1845C2212yescomplemented
77#1834C2212yescomplemented
78#1824C2212yes—
79#1809C2214yes—
80#1787C2212yes—
81#1763C2214yescomplemented
82#1753C2214yes—
83#1753C2214yes—
84#1749C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×SD16C2314yes—
90C22×M16C2314yes—
91#538—48no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#126—22no—
95C2×#124—22no—
96C2×#123—22no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—44no—
100C2×C22⋊Q8C2314yes—
101C2×#121—21no—
102C2×#121—21no—
103C2×#121—22no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#121C2312yes—
107C2×#120—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—44no—
110C2×C4×Q8C2314yes—
111C2×C4×D4—22no—
112C2×C4×D4C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4D411yescharacteristic
124C22×C22⋊C4D411yescharacteristic
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#481—28no—
129#468—48no—
130#448—28no—
131#440—48no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141#407—28no—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148#400—22no—
149#399—24no—
150#399—24no—
151#398—24no—
152#398—24no—
153#397—22no—
154#397—22no—
155#394—22no—
156#394—22no—
157#394—22no—
158#393—22no—
159#392—22no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391C2312yes—
164#391C2312yes—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#389—21no—
170#389—22no—
171#389C2312yes—
172#389C2314yes—
173#389C2312yes—
174#388—24no—
175#388—24no—
176#388—22no—
177#388—22no—
178#387—22no—
179#387—24no—
180#387—24no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2311yescharacteristic
193#380—21no—
194#380—22no—
195#380C2312yes—
196#380C2314yes—
197#380C2312yes—
198#380C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4C2312yes—
204C4×C22⋊C4C2312yes—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C25C2×D411yescharacteristic
221C22×Q8—24no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—24no—
225C22×Q8—42no—
226C22×Q8C2412yes—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—24no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C2×SD16—28no—
250C2×SD16—28no—
251C2×SD16—44no—
252C2×M16—28no—
253C2×M16—44no—
254C2×M16—44no—
255C22×C8C2414yes—
256C22×C8C2414yes—
257C4⋊Q8—42no—
258#127—42no—
259#126—48no—
260#124—48no—
261#123—48no—
262C22⋊Q8—28no—
263C22⋊Q8—44no—
264C22⋊Q8—48no—
265C22⋊Q8—44no—
266#121—24no—
267#121—24no—
268#121—48no—
269#121—44no—
270#121—44no—
271#121—44no—
272#121—44no—
273#120—48no—
274C4×Q8—28no—
275C4×Q8—44no—
276C4×Q8—48no—
277C4×D4—44no—
278C4×D4—48no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—44no—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—44no—
343#107—44no—
344#106—28no—
345#103—28no—
346#103—28no—
347#103—44no—
348#102—28no—
349#102—44no—
350#102—44no—
351C22⋊C8—24no—
352C22⋊C8—28no—
353C22⋊C8#4118yes—
354#97—44no—
355C4×C8—42no—
356C4×C8—42no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—22no—
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#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—41no—
378C24—41no—
379C24—41no—
380C24—41no—
381C24C22×D412yes—
382C24C22×D411yescharacteristic
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—416no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×Q8H(5,𝔽2)14yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—81no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4—81no—
409C2×D4—82no—
410C2×D4—82no—
411C2×D4—82no—
412C2×D4—82no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—88no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4AGL(1,ℤ/8)14yes—
459C22×C4#13714yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4114yes—
469C22×C4C2511yescharacteristic
470SD16—416no—
471M16—88no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8C2×#4118yes—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—84no—
501C4⋊C4—81no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—41no—
523C42—41no—
524C42—42no—
525C42—48no—
526C42—44no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—82no—
549C23—82no—
550C23—82no—
551C23—82no—
552C23#53212yes—
553C23C2×AGL(1,ℤ/8)12yes—
554C23C2×#13712yes—
555C23C23×D411yescharacteristic
556Q8—28no—
557Q8—48no—
558Q8—44no—
559Q8—162no—
560D4—44no—
561D4—44no—
562D4—88no—
563D4—162no—
564D4—162no—
565D4—164no—
566D4—164no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
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—84no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—84no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—164no—
621C2×C4#53212yes—
622C2×C4C2×AGL(1,ℤ/8)12yes—
623C2×C4C2×AGL(1,ℤ/8)12yes—
624C2×C4C2×#13712yes—
625C2×C4C2×#13712yes—
626C2×C4C22×#4112yes—
627C2×C4C2×H(5,𝔽2)12yes—
628C2×C4#58512yes—
629C8—28no—
630C8—44no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—88no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22#318214yes—
653C22C2×#53211yescharacteristic
654C22#348411yescharacteristic
655C22#348511yescharacteristic
656C22C22×AGL(1,ℤ/8)11yescharacteristic
657C22C22×#13711yescharacteristic
658C22#358511yescharacteristic
659C22#358611yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—42no—
666C4—44no—
667C4—42no—
668C4—42no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4#351314yes—
676C2—24no—
677C2—82no—
678C2—162no—
679C2#3539411yescharacteristic
680C2C2×#318211yescharacteristic
681C2#6070311yescharacteristic
682C2#6158411yescharacteristic
683C2#6158711yescharacteristic
684C2#6170411yescharacteristic
685C2#6170911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35005C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2423C2semidirect
3. #29105C2semidirect
4. #29005C2semidirect
5. #28867C2semidirect
6. #28305C2semidirect
7. #27973C2semidirect
8. #27950C2semidirect
9. #26987C2semidirect
10. #25045C2semidirect
11. #24557C2semidirect
12. #24387C2semidirect
13. #22082C2semidirect
14. #21454C2semidirect
15. #20524C2semidirect
22. C2×#440C22semidirect
50. #2096C22semidirect
51. #2083C22semidirect
53. #2072C22semidirect
54. #2071C22semidirect
57. #2024C22semidirect
58. #2014C22semidirect
60. #1990C22semidirect
61. #1968C22semidirect
62. #1957C22semidirect
63. #1944C22semidirect
64. #1936C22semidirect
65. #1933C22semidirect
66. #1919C22semidirect
69. #1880C22semidirect
72. #1863C22semidirect
73. #1862C22semidirect
76. #1845C22semidirect
77. #1834C22semidirect
81. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#24232
#291052
#290052
#288672
#283052
#279732
#279502
#269872
#250452
#245572
#243872
#220822
#214542
#205242