Finite Group Databasebeta

← #6640633

Subgroups — #6640633

Order 512.

1808 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3#32979C212yesmaximal, complemented
4#32280C214yesmaximal, complemented
5#32048C214yesmaximal, complemented
6#31934C214yesmaximal, complemented
7#31838C214yesmaximal, complemented
8#29593C212yesmaximal, complemented
9#29105C211yesmaximal, characteristic, complemented
10#28305C211yesmaximal, characteristic, complemented
11#28115C212yesmaximal, complemented
12#27961C211yesmaximal, characteristic, complemented
13#27906C211yesmaximal, characteristic, complemented
14#27645C212yesmaximal, complemented
15#21902C211yesmaximal, characteristic, complemented
16C2×#538—21no—
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#481C2212yes—
20C2×#481C2212yes—
21C2×#481C2212yes—
22C2×#448C2212yes—
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2847C2212yes—
26#2639—21no—
27#2617—21no—
28#2559—22no—
29#2557—22no—
30#2502—22no—
31#2502—22no—
32#2502C2214yes—
33#2502C2214yes—
34#2496C2212yes—
35#2491C2212yes—
36#2428C2212yes—
37#2422C2212yes—
38C2×#398—21no—
39C2×#398—21no—
40#2325—21no—
41C4×#121C2212yes—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#385—21no—
45C2×#385—21no—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2100C2214yescomplemented
50#2096C2212yescomplemented
51#2094C2214yescomplemented
52#2084C2214yescomplemented
53#2072C2212yescomplemented
54#2071C2214yescomplemented
55#2066C2214yescomplemented
56#2055C2214yescomplemented
57#2035C2214yes—
58#2035C2214yes—
59#2024C2212yescomplemented
60#2014C2212yescomplemented
61#2000C2214yes—
62#2000C2214yes—
63#1990C2212yescomplemented
64#1968C2214yescomplemented
65#1957C2212yescomplemented
66#1948C2214yescomplemented
67#1933C2214yescomplemented
68#1927C2214yescomplemented
69#1923C2214yes—
70#1923C2214yes—
71#1877C2214yescomplemented
72#1874C2214yes—
73#1874C2214yes—
74#1853C2214yes—
75#1853C2214yes—
76#1844C2212yescomplemented
77#1834C2212yescomplemented
78#1824C2212yes—
79#1824C2212yes—
80#1824C2212yes—
81#1802C2214yescomplemented
82#1797C2212yes—
83#1787C2212yes—
84#1779C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C22×SD16C2314yes—
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×D4—22no—
109C2×C4×D4C2312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—44no—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4—21no—
123C22×C22⋊C4D411yescharacteristic
124C22×C22⋊C4D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#481—44no—
128#481—44no—
129#481—44no—
130#448—44no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2312yes—
147#400—22no—
148#398—24no—
149#398—24no—
150#398—24no—
151#398—24no—
152#397—22no—
153#397—22no—
154#394—22no—
155#394—22no—
156#394—22no—
157#393—22no—
158#392—22no—
159#391—21no—
160#391—22no—
161#391—21no—
162#391—21no—
163#391C2314yes—
164#391C2312yes—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#387—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#385—24no—
182#385—24no—
183#385—24no—
184#385—24no—
185#385—22no—
186#384—22no—
187#384—21no—
188#384—21no—
189#384C2314yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#380—21no—
195#380C2312yes—
196#380C2312yes—
197#380C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4C2312yes—
203C4×C22⋊C4C2312yes—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C25C2×D411yescharacteristic
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—24no—
223C22×Q8—42no—
224C22×Q8—42no—
225C22×Q8C2412yes—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2412yes—
232C23×C4—24no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—24no—
238C23×C4—22no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—44no—
242C23×C4—82no—
243C23×C4—82no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D412yes—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C2×SD16—44no—
253C2×SD16—44no—
254C2×SD16—44no—
255C22×C8C2414yes—
256C22×C8C2414yes—
257C4⋊Q8—42no—
258#127—42no—
259#126—48no—
260#124—48no—
261#123—48no—
262C22⋊Q8—48no—
263C22⋊Q8—48no—
264C22⋊Q8—44no—
265#121—24no—
266#121—24no—
267#121—44no—
268#121—44no—
269#121—48no—
270#121—44no—
271#121—44no—
272#120—48no—
273C4×D4—44no—
274C4×D4—48no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—48no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—48no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
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—22no—
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—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4SD1618yescomplemented
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—22no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#106—44no—
339#106—44no—
340#106—44no—
341C4⋊C8—44no—
342#103—44no—
343#103—44no—
344#103—44no—
345#102—44no—
346#102—44no—
347#102—44no—
348C22⋊C8—24no—
349C22⋊C8—24no—
350C22⋊C8—44no—
351C22⋊C8—44no—
352C4×C8—42no—
353C4×C8—42no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—41no—
374C24—41no—
375C24—41no—
376C24—41no—
377C24C2×SD1614yes—
378C24C22×D412yes—
379C24C22×D411yescharacteristic
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—81no—
389C2×Q8—81no—
390C2×Q8—88no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—81no—
403C2×D4—81no—
404C2×D4—81no—
405C2×D4—81no—
406C2×D4—82no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—82no—
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—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—416no—
438C22×C4—48no—
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—42no—
451C22×C4—44no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—816no—
457C22×C4C2×SD1614yes—
458C22×C4C2×SD1614yes—
459C22×C4#13714yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4H(5,𝔽2)14yes—
469C22×C4C2511yescharacteristic
470SD16—88no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C4⋊C4—84no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—84no—
501C4⋊C4—81no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4C2×SD1618yes—
517C22⋊C4C2×SD1618yes—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—41no—
522C42—42no—
523C42—41no—
524C42—44no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—28no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23#53512yes—
550C23C22×SD1614yes—
551C23C22×SD1612yes—
552C23C2×#13712yes—
553C23C23×D411yescharacteristic
554Q8—44no—
555Q8—44no—
556Q8—84no—
557Q8—162no—
558D4—44no—
559D4—44no—
560D4—88no—
561D4—162no—
562D4—162no—
563D4—164no—
564D4—164no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
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—22no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—42no—
592C2×C4—42no—
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—84no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—816no—
617C2×C4—164no—
618C2×C4#53512yes—
619C2×C4C22×SD1614yes—
620C2×C4C22×SD1612yes—
621C2×C4C22×SD1612yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4#58512yes—
626C2×C4#58512yes—
627C8—44no—
628C8—44no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—416no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22#320614yes—
651C22C2×#53511yescharacteristic
652C22#348511yescharacteristic
653C22#348511yescharacteristic
654C22C23×SD1611yescharacteristic
655C22C22×#13711yescharacteristic
656C22#358511yescharacteristic
657C22#358511yescharacteristic
658C4—24no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4#353314yes—
674C2—24no—
675C2—44no—
676C2—162no—
677C2#3695611yescharacteristic
678C2C2×#320611yescharacteristic
679C2#6080611yescharacteristic
680C2#6169711yescharacteristic
681C2#6169711yescharacteristic
682C2#6170911yescharacteristic
683C2#6171911yescharacteristic
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×#2502C2semidirect
3. #32979C2semidirect
4. #32280C2semidirect
5. #32048C2semidirect
6. #31934C2semidirect
7. #31838C2semidirect
8. #29593C2semidirect
9. #29105C2semidirect
10. #28305C2semidirect
11. #28115C2semidirect
12. #27961C2semidirect
13. #27906C2semidirect
14. #27645C2semidirect
15. #21902C2semidirect
49. #2100C22semidirect
50. #2096C22semidirect
51. #2094C22semidirect
52. #2084C22semidirect
53. #2072C22semidirect
54. #2071C22semidirect
55. #2066C22semidirect
56. #2055C22semidirect
59. #2024C22semidirect
60. #2014C22semidirect
63. #1990C22semidirect
64. #1968C22semidirect
65. #1957C22semidirect
66. #1948C22semidirect
67. #1933C22semidirect
68. #1927C22semidirect
71. #1877C22semidirect
76. #1844C22semidirect
77. #1834C22semidirect
81. #1802C22semidirect
324. C2×C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#25022
#329792
#322802
#320482
#319342
#318382
#295932
#291052
#283052
#281152
#279612
#279062
#276452
#219022