Finite Group Databasebeta

← #6993542

Subgroups — #6993542

Order 512.

1471 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36301C211yesmaximal, characteristic
2#35320C211yesmaximal, characteristic, complemented
3#32824C212yesmaximal, complemented
4#32290C214yesmaximal, complemented
5#31941C214yesmaximal
6#30306C212yesmaximal, complemented
7#29982C212yesmaximal, complemented
8#29691C211yesmaximal, characteristic, complemented
9#29554C212yesmaximal, complemented
10#28529C212yesmaximal, complemented
11#27730C212yesmaximal, complemented
12#25811C211yesmaximal, characteristic, complemented
13#25454C212yesmaximal, complemented
14#24816C212yesmaximal, complemented
15#24298C211yesmaximal, characteristic
16#23568C211yesmaximal, characteristic
17#22632C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#498C2212yes—
21C2×#477C2212yes—
22C2×#474C2212yes—
23C2×C4×SD16C2212yes—
24#2864C2212yes—
25#2862C2212yes—
26#2862C2212yes—
27#2857C2211yescharacteristic
28#2642—22no—
29#2571C2211yescharacteristic
30#2568—21no—
31#2568—21no—
32#2568C2212yes—
33#2568C2212yes—
34#2563C2211yescharacteristic
35#2527—22no—
36#2503—22no—
37#2503C2214yes—
38#2494—21no—
39#2494C2212yes—
40#2485—21no—
41#2477—22no—
42#2420C2212yes—
43#2369C2211yescharacteristic
44#2352—21no—
45D4×C4⋊C4C2211yescharacteristic
46#2344—21no—
47#2310—21no—
48#2310—21no—
49C4×#121C2211yescharacteristic
50#2293C2212yes—
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2097C2214yes—
54#2092C2214yes—
55#2085C2214yes—
56#2085C2214yes—
57#2074C2214yes—
58#2068C2214yes—
59#2066C2214yes—
60#2058C2211yescharacteristic
61#2058C2212yes—
62#2058C2212yes—
63#2057C2212yes—
64#2057C2212yes—
65#2043C2211yescharacteristic
66#2043C2212yes—
67#2036C2214yes—
68#2035C2212yes—
69#2023C2211yescharacteristic
70#2023C2212yes—
71#2013C2211yescharacteristic
72#2000C2212yes—
73#1990C2212yes—
74#1988C2212yes—
75#1968C2214yes—
76#1949C2214yes—
77#1939C2212yes—
78#1933C2214yes—
79#1927C2214yes—
80#1923C2214yes—
81#1877C2214yes—
82#1874C2212yes—
83#1853C2214yes—
84#1852C2212yes—
85#1846C2214yes—
86#1835C2214yes—
87#1831C2212yes—
88#1804C2214yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C23×Q8—21no—
96C22×SD16C2312yes—
97#533—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#125—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8C2312yes—
110C2×#121C2311yescharacteristic
111C2×C4×Q8—21no—
112C2×C4×Q8C2312yes—
113C2×C4×D4—21no—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C2×#117—21no—
117C2×#117—22no—
118C2×#117D411yescharacteristic
119C2×#117D411yescharacteristic
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2312yes—
123C22×C42C2311yescharacteristic
124#498—44no—
125#477—44no—
126#474—44no—
127C4×SD16—44no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2312yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C4×C8C2314yes—
145C2×C4×C8C2312yes—
146#401—22no—
147#400—24no—
148#400—24no—
149#400—24no—
150#400—24no—
151#400—22no—
152#397—22no—
153#397—22no—
154#394—22no—
155#393—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—21no—
167#391—22no—
168#391C2314yes—
169#391C2314yes—
170#390C2312yes—
171#390C2311yescharacteristic
172#389—21no—
173#389—21no—
174#389—21no—
175#389C2311yescharacteristic
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—21no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#386—21no—
190#386—22no—
191#385—22no—
192#384C2311yescharacteristic
193#382—21no—
194#382—22no—
195#382—22no—
196#382C2312yes—
197#382C2314yes—
198#382C2314yes—
199#382C2312yes—
200#381—24no—
201#381—24no—
202#381—22no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380C2312yes—
208#380C2311yescharacteristic
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×Q8C2412yes—
242C22×D4—21no—
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×SD16—24no—
251C2×SD16—24no—
252C2×SD16—42no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—44no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264#125—44no—
265C22⋊Q8—42no—
266C22⋊Q8—42no—
267C22⋊Q8—44no—
268C22⋊Q8—82no—
269#121—22no—
270#121—41no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—44no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—42no—
277C4×D4—44no—
278#117—24no—
279#117—42no—
280#117—44no—
281#117—84no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
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—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—22no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338#107—44no—
339#107—44no—
340#106—42no—
341#106—44no—
342#106—44no—
343C4⋊C8—24no—
344C4⋊C8—44no—
345#103—24no—
346#103—44no—
347#103—42no—
348#103—42no—
349#103—44no—
350#102—42no—
351#102—44no—
352C22⋊C8—24no—
353C22⋊C8—42no—
354C4×C8—28no—
355C4×C8—24no—
356#95—22no—
357#95—21no—
358#95—22no—
359#95—21no—
360#95—22no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—21no—
382C24C22×D411yescharacteristic
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—81no—
400C2×Q8—81no—
401C2×Q8—816no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—81no—
408C2×D42−1+412yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—82no—
435C22×C4#13514yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2511yescharacteristic
446SD16—48no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C82−1+414yes—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—81no—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—44no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42—24no—
503C42—22no—
504C42—24no—
505C42—22no—
506C42—22no—
507C42—42no—
508C42—42no—
509C42—44no—
510C42—42no—
511C42—82no—
512C42H(5,𝔽2)14yes—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×#13512yes—
527C23C23×D411yescharacteristic
528Q8—42no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—84no—
533Q8—88no—
534Q8—88no—
535Q8—162no—
536D4—24no—
537D4—42no—
538D4—84no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—88no—
589C2×C4—88no—
590C2×C4—84no—
591C2×C4#53512yes—
592C2×C4#53512yes—
593C2×C4C2×#13512yes—
594C2×C4C2×#13512yes—
595C2×C4#57512yes—
596C2×C4#57712yes—
597C2×C4C2×H(5,𝔽2)12yes—
598C2×C4C2×2−1+411yescharacteristic
599C2×C4C2×2−1+412yes—
600C2×C4#58511yescharacteristic
601C8—24no—
602C8—42no—
603C8—44no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22C2×#53511yescharacteristic
619C22#348911yescharacteristic
620C22#349911yescharacteristic
621C22C22×#13511yescharacteristic
622C22C2×#57511yescharacteristic
623C22C2×#57711yescharacteristic
624C22#358611yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—84no—
636C4—82no—
637C4—82no—
638C4—81no—
639C4—82no—
640C4—82no—
641C4#321114yes—
642C4#353812yes—
643C4#354712yes—
644C2—42no—
645C2—82no—
646C2#4299411yescharacteristic
647C2C2×#321111yescharacteristic
648C2#6081311yescharacteristic
649C2#6209711yescharacteristic
650C2#6214411yescharacteristic
651C2#6247011yescharacteristic
652C2#6250711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. #35320C2semidirect
3. #32824C2semidirect
4. #32290C2semidirect
6. #30306C2semidirect
7. #29982C2semidirect
8. #29691C2semidirect
9. #29554C2semidirect
10. #28529C2semidirect
11. #27730C2semidirect
12. #25811C2semidirect
13. #25454C2semidirect
14. #24816C2semidirect
17. #22632C2semidirect

Maximal subgroups

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

subgroupindex
#363012
#353202
#328242
#322902
#319412
#303062
#299822
#296912
#295542
#285292
#277302
#258112
#254542
#248162
#242982
#235682
#226322