Finite Group Databasebeta

← #6803201

Subgroups — #6803201

Order 512.

1407 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#34945C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic
4#32542C214yesmaximal, complemented
5#31922C214yesmaximal
6#29830C212yesmaximal, complemented
7#28511C211yesmaximal, characteristic, complemented
8#28384C212yesmaximal
9#28282C212yesmaximal
10#28272C212yesmaximal
11#28255C211yesmaximal, characteristic, complemented
12#26093C212yesmaximal, complemented
13#25302C212yesmaximal, complemented
14#24629C212yesmaximal
15#22722C211yesmaximal, characteristic, complemented
16#22119C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic
18C2×#477C2212yes—
19C2×#477C2212yes—
20C2×#476C2212yes—
21C2×#462C2212yes—
22C2×#462C2212yes—
23C2×#461C2211yescharacteristic
24C2×#446C2212yes—
25C2×C4×SD16C2212yes—
26#2868C2211yescharacteristic
27#2862C2212yes—
28#2582C2212yes—
29#2573—22no—
30#2571—21no—
31#2571C2211yescharacteristic
32#2570—22no—
33#2568—21no—
34#2568C2211yescharacteristic
35#2547C2211yescharacteristic
36#2538C2211yescharacteristic
37#2514C2212yes—
38#2511C2214yes—
39#2494—21no—
40#2447C2212yes—
41#2446—22no—
42#2421—21no—
43#2401—22no—
44#2365—21no—
45#2355—21no—
46#2355C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×#124C2212yes—
51C4×C22⋊Q8—21no—
52#2295C2212yes—
53#2283—21no—
54C42×Q8C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2097C2214yes—
57#2095C2214yes—
58#2068C2214yes—
59#2068C2214yes—
60#2067C2214yes—
61#2066C2214yes—
62#2056C2214yes—
63#2056C2214yes—
64#2056C2214yes—
65#2015C2214yes—
66#1987C2212yes—
67#1983C2214yes—
68#1981C2211yescharacteristic
69#1973C2214yes—
70#1971C2214yes—
71#1956C2212yes—
72#1949C2214yes—
73#1948C2212yes—
74#1948C2211yescharacteristic
75#1948C2212yes—
76#1943C2214yes—
77#1940C2214yes—
78#1939C2212yes—
79#1935C2214yes—
80#1927C2212yes—
81#1927C2214yes—
82#1918C2214yes—
83#1877C2212yes—
84#1845C2212yes—
85#1842C2211yescharacteristic
86#1842C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1802C2212yes—
90#1802C2211yescharacteristic
91#1802C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2314yes—
96C22×SD16C2312yes—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#124C2312yes—
103C2×C22⋊Q8—21no—
104C2×C4×Q8—22no—
105C2×C4×Q8D411yescharacteristic
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4C2311yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C22×C42—21no—
114#477—28no—
115#477—44no—
116#476—44no—
117#462—44no—
118#462—44no—
119#461—24no—
120#446—44no—
121C4×SD16—44no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×#106C2312yes—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#97C2314yes—
143C2×C4×C8C2312yes—
144#401—22no—
145#400—24no—
146#400—22no—
147#400—22no—
148#400—24no—
149#400—22no—
150#398—24no—
151#397—22no—
152#397—22no—
153#397—22no—
154#396—22no—
155#391—22no—
156#391—21no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#391C2314yes—
165#391C2314yes—
166#391C2312yes—
167#390C2312yes—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389C2314yes—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—24no—
182#387—22no—
183#387—22no—
184#387—24no—
185#387—24no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#383—22no—
194#382—22no—
195#382—21no—
196#382—21no—
197#382—22no—
198#382—22no—
199#382C2312yes—
200#381—22no—
201#380—21no—
202#380—21no—
203#380—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226#376—22no—
227C43C2311yescharacteristic
228C43C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C2×Q16—44no—
245C2×Q16—44no—
246C2×Q16—44no—
247C2×SD16—24no—
248C2×SD16—24no—
249C2×SD16—42no—
250C22×C8C2414yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—44no—
260C4⋊Q8—82no—
261#124—24no—
262#124—42no—
263C22⋊Q8—44no—
264C4×Q8—24no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—42no—
272C4×Q8—84no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—41no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2411yescharacteristic
332C2×C42C2412yes—
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—28no—
337#107—44no—
338#106—42no—
339#106—44no—
340C4⋊C8—24no—
341C4⋊C8—24no—
342C4⋊C8—22no—
343C4⋊C8—42no—
344C4⋊C8—44no—
345C4⋊C8—44no—
346C4⋊C8#4114yes—
347#103—24no—
348#103—28no—
349#103—42no—
350#103—42no—
351#103—44no—
352#103—44no—
353#103—44no—
354#102—42no—
355#102—44no—
356#97—44no—
357C4×C8—24no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D42−1+412yes—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—81no—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2511yescharacteristic
437Q16—88no—
438SD16—48no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8C2×#4114yes—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C22⋊C4—42no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—82no—
491C42—24no—
492C42—22no—
493C42—22no—
494C42—24no—
495C42—22no—
496C42—21no—
497C42—22no—
498C42—22no—
499C42—21no—
500C42—22no—
501C42—42no—
502C42—42no—
503C42—41no—
504C42—44no—
505C42—44no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—84no—
510C42#13714yes—
511C42C2×#4112yes—
512C42H(5,𝔽2)12yes—
513C42H(5,𝔽2)12yes—
514C422−1+412yes—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C23×D411yescharacteristic
523Q8—44no—
524Q8—42no—
525Q8—42no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—88no—
533D4—22no—
534D4—22no—
535D4—41no—
536D4—41no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4#53712yes—
586C2×C4#53712yes—
587C2×C4#55012yes—
588C2×C4#55012yes—
589C2×C4C2×#13712yes—
590C2×C4C2×#13712yes—
591C2×C4C2×#13712yes—
592C2×C4#57512yes—
593C2×C4#57812yes—
594C2×C4C22×#4111yescharacteristic
595C2×C4C2×H(5,𝔽2)11yescharacteristic
596C2×C4C2×H(5,𝔽2)11yescharacteristic
597C2×C4C2×2−1+411yescharacteristic
598C2×C4C2×2−1+411yescharacteristic
599C2×C4#58511yescharacteristic
600C8—24no—
601C8—42no—
602C8—44no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22C2×#53711yescharacteristic
611C22C2×#55011yescharacteristic
612C22#348911yescharacteristic
613C22C22×#13711yescharacteristic
614C22C2×#57511yescharacteristic
615C22C2×#57811yescharacteristic
616C22#358511yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—42no—
621C4—42no—
622C4—41no—
623C4—42no—
624C4—42no—
625C4—82no—
626C4—81no—
627C4—81no—
628C4—82no—
629C4—84no—
630C4—82no—
631C4—82no—
632C4—81no—
633C4#325214yes—
634C4#339814yes—
635C4#351811yescharacteristic
636C4#352011yescharacteristic
637C4#354611yescharacteristic
638C4#355211yescharacteristic
639C2—82no—
640C2#3913611yescharacteristic
641C2C2×#325211yescharacteristic
642C2#6093011yescharacteristic
643C2C2×#339811yescharacteristic
644C2#6140511yescharacteristic
645C2#6210811yescharacteristic
646C2#6213311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
4. #32542C2semidirect
6. #29830C2semidirect
7. #28511C2semidirect
11. #28255C2semidirect
12. #26093C2semidirect
13. #25302C2semidirect
15. #22722C2semidirect
16. #22119C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#349452
#331972
#325422
#319222
#298302
#285112
#283842
#282822
#282722
#282552
#260932
#253022
#246292
#227222
#221192
C4×#4612