Finite Group Databasebeta

← #6986478

Subgroups — #6986478

Order 512.

1491 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36481C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#32838C212yesmaximal, complemented
4#31936C214yesmaximal
5#31840C214yesmaximal
6#30338C212yesmaximal, complemented
7#30042C212yesmaximal, complemented
8#29664C211yesmaximal, characteristic
9#29141C212yesmaximal, complemented
10#28401C212yesmaximal, complemented
11#27729C212yesmaximal, complemented
12#25902C211yesmaximal, characteristic
13#25324C212yesmaximal, complemented
14#24817C212yesmaximal, complemented
15#24302C211yesmaximal, characteristic
16#23571C211yesmaximal, characteristic
17#22639C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#487C2212yes—
21C2×#477C2212yes—
22C2×#462C2212yes—
23C2×C4×Q16C2212yes—
24#2868C2212yes—
25#2863C2212yes—
26#2862C2212yes—
27#2857C2211yescharacteristic
28#2660—22no—
29#2606—22no—
30#2567C2212yes—
31#2567C2212yes—
32#2552C2211yescharacteristic
33#2546—21no—
34#2546—21no—
35#2546C2211yescharacteristic
36#2522—22no—
37#2507—21no—
38#2502C2214yes—
39#2441—21no—
40#2434—22no—
41#2429C2212yes—
42#2423C2212yes—
43#2370—21no—
44#2365C2211yescharacteristic
45Q8×C4⋊C4C2211yescharacteristic
46#2347—21no—
47#2317—21no—
48#2317—21no—
49C4×C4⋊Q8C2212yes—
50C4×C22⋊Q8C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2214yes—
53#2095C2214yes—
54#2083C2214yes—
55#2072C2214yes—
56#2072C2214yes—
57#2068C2214yes—
58#2057C2211yescharacteristic
59#2057C2212yes—
60#2056C2214yes—
61#2043C2211yescharacteristic
62#2036C2212yes—
63#2024C2211yescharacteristic
64#2024C2212yes—
65#2024C2212yes—
66#2024C2212yes—
67#2024C2212yes—
68#2016C2211yescharacteristic
69#2016C2214yes—
70#2016C2212yes—
71#2001C2212yes—
72#2001C2214yes—
73#1990C2212yes—
74#1987C2212yes—
75#1969C2214yes—
76#1956C2214yes—
77#1939C2212yes—
78#1933C2214yes—
79#1923C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1872C2214yes—
83#1872C2212yes—
84#1852C2212yes—
85#1839C2212yes—
86#1834C2214yes—
87#1803C2212yes—
88#1803C2214yes—
89#1803C2214yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×#125—21no—
111C2×#125—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2311yescharacteristic
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×#117—21no—
122C2×#117D411yescharacteristic
123C2×#117D411yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127#487—44no—
128#477—44no—
129#462—44no—
130C4×Q16—44no—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2314yes—
148#399—22no—
149#398—22no—
150#398—22no—
151#398—22no—
152#398—22no—
153#398—24no—
154#398—24no—
155#396—22no—
156#395—24no—
157#395—24no—
158#391—22no—
159#391—22no—
160#391C2311yescharacteristic
161#391C2312yes—
162#391C2314yes—
163#391C2314yes—
164#391C2312yes—
165#391C2312yes—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389C2311yescharacteristic
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#387—21no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#385—21no—
192#385—24no—
193#385—24no—
194#384—21no—
195#384—22no—
196#384—22no—
197#384C2312yes—
198#384C2314yes—
199#384C2314yes—
200#384C2311yescharacteristic
201#383—22no—
202#382—22no—
203#382—22no—
204#380—21no—
205#380—21no—
206#380C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233#336C2311yescharacteristic
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—44no—
238C22×Q8—82no—
239C22×Q8—82no—
240C22×Q8C2412yes—
241C22×Q8C2412yes—
242C22×Q8C2411yescharacteristic
243C23×C4—22no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×Q16—24no—
248C2×Q16—42no—
249C2×Q16—42no—
250C2×Q16—44no—
251C2×Q16—44no—
252C2×Q16—44no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—44no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—42no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8—84no—
267C4⋊Q8—84no—
268#125—44no—
269#125—44no—
270C22⋊Q8—22no—
271C22⋊Q8—41no—
272C22⋊Q8—42no—
273C22⋊Q8—42no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×Q8—44no—
282#117—24no—
283#117—44no—
284#117Q1618yescomplemented
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347#107—42no—
348#107—44no—
349C4⋊C8—24no—
350C4⋊C8—44no—
351#103—42no—
352#103—42no—
353#103—44no—
354#103—42no—
355#103—42no—
356#103—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360C22⋊C8—42no—
361C22⋊C8—42no—
362C4×C8—24no—
363C4×C8—44no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24C22×D411yescharacteristic
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—816no—
409C2×Q82−1+412yes—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4—44no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4C2×Q1614yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442Q16—48no—
443Q16—88no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—82no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4C2×Q1614yes—
492C4⋊C4C2×Q1614yes—
493C4⋊C4#13714yes—
494C22⋊C4—24no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—81no—
498C22⋊C4—82no—
499C22⋊C4C2×Q1618yes—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—42no—
509C42—44no—
510C42—42no—
511C42—42no—
512C42—82no—
513C42—82no—
514C42C2×Q1618yes—
515C42H(5,𝔽2)14yes—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23C22×Q1612yes—
523C23C23×D411yescharacteristic
524Q8—24no—
525Q8—42no—
526Q8—42no—
527Q8—44no—
528Q8—42no—
529Q8—42no—
530Q8—44no—
531Q8—42no—
532Q8—84no—
533Q8—88no—
534Q8—88no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—88no—
585C2×C4#53512yes—
586C2×C4#53512yes—
587C2×C4C22×Q1612yes—
588C2×C4C22×Q1614yes—
589C2×C4C22×Q1614yes—
590C2×C4C22×Q1612yes—
591C2×C4C2×#13714yes—
592C2×C4C2×#13712yes—
593C2×C4#57712yes—
594C2×C4C2×H(5,𝔽2)12yes—
595C2×C4C2×2−1+411yescharacteristic
596C2×C4#58512yes—
597C2×C4#58511yescharacteristic
598C8—42no—
599C8—42no—
600C8—44no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22C2×#53511yescharacteristic
609C22#348911yescharacteristic
610C22#350011yescharacteristic
611C22C23×Q1611yescharacteristic
612C22C22×#13711yescharacteristic
613C22C2×#57711yescharacteristic
614C22#358511yescharacteristic
615C4—24no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—24no—
620C4—24no—
621C4—42no—
622C4—41no—
623C4—42no—
624C4—81no—
625C4—82no—
626C4—81no—
627C4—84no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—81no—
632C4#320914yes—
633C4#354112yes—
634C4#355112yes—
635C2—42no—
636C2#4291011yescharacteristic
637C2C2×#320911yescharacteristic
638C2#6081211yescharacteristic
639C2#6210011yescharacteristic
640C2#6213111yescharacteristic
641C2#6253011yescharacteristic
642C2#6259411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32838C2semidirect
6. #30338C2semidirect
7. #30042C2semidirect
9. #29141C2semidirect
10. #28401C2semidirect
11. #27729C2semidirect
13. #25324C2semidirect
14. #24817C2semidirect
284. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#364812
#352812
#328382
#319362
#318402
#303382
#300422
#296642
#291412
#284012
#277292
#259022
#253242
#248172
#243022
#235712
#226392