Finite Group Databasebeta

← #6508271

Subgroups — #6508271

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2C2×#2446C211yesmaximal, characteristic, complemented
3#29328C211yesmaximal, characteristic
4#29296C211yesmaximal, characteristic, complemented
5#29034C214yesmaximal, complemented
6#28900C214yesmaximal, complemented
7#26836C214yesmaximal, complemented
8#26260C214yesmaximal, complemented
9#25856C211yesmaximal, characteristic, complemented
10#25718C211yesmaximal, characteristic
11#24509C212yesmaximal, complemented
12#24509C212yesmaximal, complemented
13#22047C212yesmaximal, complemented
14#21454C212yesmaximal, complemented
15#20534C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#492C2212yes—
20C2×#487C2212yes—
21C2×#448C2212yes—
22C2×#448C2212yes—
23C2×#440C2214yes—
24C2×#432C2212yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854C2212yes—
28#2639—21no—
29#2618—21no—
30#2560—22no—
31#2558—22no—
32#2512C2212yes—
33#2506C2212yes—
34#2452C2212yes—
35#2446—22no—
36#2446—22no—
37#2446C2214yes—
38#2446C2214yes—
39#2445C2212yes—
40C2×#400—21no—
41C2×#398—21no—
42#2325—21no—
43C4×#121C2212yes—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#387—21no—
49C2×#387—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2099C2212yes—
52#2083C2212yes—
53#2068C2214yes—
54#2068C2214yes—
55#2057C2212yes—
56#2055C2214yes—
57#2055C2214yes—
58#2043C2214yes—
59#2041C2212yes—
60#2013C2214yes—
61#1973C2214yes—
62#1973C2214yes—
63#1973C2214yes—
64#1956C2212yes—
65#1956C2212yes—
66#1942C2214yes—
67#1935C2214yes—
68#1935C2214yes—
69#1935C2214yes—
70#1934C2214yes—
71#1928C2214yes—
72#1880C2214yes—
73#1877C2214yes—
74#1874C2212yes—
75#1874C2212yes—
76#1863C2214yes—
77#1852C2212yes—
78#1852C2212yes—
79#1787C2212yes—
80#1787C2212yes—
81#1771C2214yes—
82#1771C2214yes—
83#1763C2214yes—
84#1749C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C22×SD16C2314yes—
90C22×SD16C2314yes—
91#539—48no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127C2311yescharacteristic
99C2×#126—22no—
100C2×#125—22no—
101C2×#123—22no—
102C2×#123—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×#121—21no—
106C2×#121—21no—
107C2×#121C2312yes—
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⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#492—44no—
126#487—44no—
127#448—28no—
128#448—28no—
129#440—48no—
130#432—28no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×#97C2314yes—
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148#400—22no—
149#400—24no—
150#400—24no—
151#398—22no—
152#398—24no—
153#398—24no—
154#397—22no—
155#396—22no—
156#396—22no—
157#394—22no—
158#394—22no—
159#393—22no—
160#391—21no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391C2314yes—
165#391C2314yes—
166#391C2314yes—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#390C2312yes—
173#389—21no—
174#389—21no—
175#389—21no—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—24no—
187#387—24no—
188#387—22no—
189#387—24no—
190#387—24no—
191#386—22no—
192#386—22no—
193#386—22no—
194#385—22no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#380—21no—
198#380C2312yes—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C22×Q8—24no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—24no—
226C22×Q8—24no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8C2412yes—
231C22×Q8C2412yes—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×SD16—28no—
249C2×SD16—28no—
250C2×SD16—28no—
251C2×SD16—44no—
252C2×SD16—44no—
253C2×SD16—44no—
254C22×C8C2414yes—
255C22×C8C2414yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—42no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263#127—42no—
264#126—48no—
265#125—48no—
266#123—48no—
267#123—48no—
268C22⋊Q8—48no—
269C22⋊Q8—44no—
270#121—24no—
271#121—24no—
272#121—44no—
273#121—44no—
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—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
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—22no—
324C2×C22⋊C4—22no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—44no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2412yes—
339#107—44no—
340#107—44no—
341C4⋊C8—28no—
342#103—28no—
343#103—28no—
344#103—28no—
345#103—44no—
346#103—44no—
347#103—44no—
348#102—44no—
349#102—44no—
350C22⋊C8—24no—
351C22⋊C8#4118yes—
352#97—28no—
353#97—44no—
354C4×C8—42no—
355C4×C8—42no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—22no—
375C24C22×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—416no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×Q8—88no—
394C2×Q8H(5,𝔽2)14yes—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—81no—
405C2×D4—81no—
406C2×D4—81no—
407C2×D4—81no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—41no—
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—41no—
434C22×C4—44no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
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—82no—
447C22×C4—82no—
448C22×C4AGL(1,ℤ/8)14yes—
449C22×C4#13714yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4114yes—
460C22×C4C2511yescharacteristic
461SD16—416no—
462SD16—88no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8C2×#4118yes—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—24no—
515C42—42no—
516C42—41no—
517C42—41no—
518C42—48no—
519C42—44no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23#53212yes—
532C23C2×AGL(1,ℤ/8)12yes—
533C23C2×#13712yes—
534C23C23×D411yescharacteristic
535Q8—28no—
536Q8—48no—
537Q8—44no—
538Q8—44no—
539Q8—44no—
540Q8—84no—
541Q8—162no—
542D4—44no—
543D4—44no—
544D4—88no—
545D4—162no—
546D4—162no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—164no—
602C2×C4#53212yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4C2×#13712yes—
606C2×C4C2×#13712yes—
607C2×C4C22×#4112yes—
608C2×C4C2×H(5,𝔽2)12yes—
609C2×C4#58512yes—
610C8—28no—
611C8—44no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22#318214yes—
626C22C2×#53211yescharacteristic
627C22#348411yescharacteristic
628C22#348711yescharacteristic
629C22C22×AGL(1,ℤ/8)11yescharacteristic
630C22C22×#13711yescharacteristic
631C22#358611yescharacteristic
632C22#358611yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4#351314yes—
650C2—24no—
651C2—162no—
652C2#3539611yescharacteristic
653C2C2×#318211yescharacteristic
654C2#6070511yescharacteristic
655C2#6157211yescharacteristic
656C2#6157411yescharacteristic
657C2#6188811yescharacteristic
658C2#6189311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2446C2semidirect
4. #29296C2semidirect
5. #29034C2semidirect
6. #28900C2semidirect
7. #26836C2semidirect
8. #26260C2semidirect
9. #25856C2semidirect
11. #24509C2semidirect
12. #24509C2semidirect
13. #22047C2semidirect
14. #21454C2semidirect
15. #20534C2semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#24462
#293282
#292962
#290342
#289002
#268362
#262602
#258562
#257182
#245092
#220472
#214542
#205342