Finite Group Databasebeta

← #6850723

Subgroups — #6850723

Order 512.

1551 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36195C211yesmaximal, characteristic
2#36180C211yesmaximal, characteristic, complemented
3#33293C211yesmaximal, characteristic, complemented
4#32052C214yesmaximal, complemented
5#32048C214yesmaximal, complemented
6#29815C212yesmaximal, complemented
7#29147C211yesmaximal, characteristic, complemented
8#28520C211yesmaximal, characteristic, complemented
9#27978C212yesmaximal, complemented
10#27896C212yesmaximal, complemented
11#24698C212yesmaximal, complemented
12#24696C212yesmaximal, complemented
13#24134C212yesmaximal, complemented
14#24130C212yesmaximal
15#21987C212yesmaximal, complemented
16#21702C211yesmaximal, characteristic, complemented
17#21271C211yesmaximal, characteristic, complemented
18C2×#554C2212yes—
19C2×#497C2211yescharacteristic
20C2×#475C2212yes—
21C2×#474C2212yes—
22C2×#441C2212yes—
23C2×#440C2212yes—
24C2×#432C2212yes—
25#2868C2211yescharacteristic
26#2862C2211yescharacteristic
27#2857C2212yes—
28#2850C2212yes—
29#2572—22no—
30#2567C2212yes—
31#2552—22no—
32#2536C2211yescharacteristic
33#2535C2211yescharacteristic
34#2532—22no—
35#2502C2214yes—
36#2492C2212yes—
37#2491C2212yes—
38#2459—22no—
39#2440—22no—
40#2440—22no—
41#2372—22no—
42#2357C2211yescharacteristic
43#2356C2211yescharacteristic
44D4×C4⋊C4—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#124C2211yescharacteristic
47C2×#384C2212yes—
48#2283C2212yes—
49C42×D4C2211yescharacteristic
50#2095C2212yes—
51#2094C2212yes—
52#2072C2214yes—
53#2071C2214yes—
54#2056C2214yes—
55#2055C2214yes—
56#2036C2214yes—
57#2035C2214yes—
58#2024C2214yes—
59#2001C2214yes—
60#2000C2214yes—
61#1993C2212yes—
62#1993C2212yes—
63#1990C2214yes—
64#1969C2212yes—
65#1968C2212yes—
66#1960C2212yes—
67#1960C2211yescharacteristic
68#1959C2211yescharacteristic
69#1959C2212yes—
70#1955C2212yes—
71#1948C2214yes—
72#1947C2214yes—
73#1940C2212yes—
74#1928C2214yes—
75#1927C2214yes—
76#1923C2214yes—
77#1874C2214yes—
78#1872C2214yes—
79#1853C2214yes—
80#1852C2214yes—
81#1847C2212yes—
82#1847C2211yescharacteristic
83#1836C2212yes—
84#1836C2212yes—
85#1836C2212yes—
86#1805C2212yes—
87#1805C2212yes—
88#1767C2211yescharacteristic
89#1749C2212yes—
90C4×#97C2211yescharacteristic
91C22×Q16C2312yes—
92C22×SD16C2312yes—
93#554—44no—
94#554—44no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2312yes—
99C2×#125—22no—
100C2×#124—22no—
101C2×#124C2311yescharacteristic
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2314yes—
105C2×#121C2312yes—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2312yes—
108C2×C4×D4—22no—
109C2×C4×D4D412yes—
110C2×C4×D4C2311yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C22×C4⋊C4—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4C2314yes—
115C22×C42C2312yes—
116#497—24no—
117#475—44no—
118#474—44no—
119#441—44no—
120#440—44no—
121#432—28no—
122C2×#107C2314yes—
123C2×#107C2312yes—
124C2×#106C2314yes—
125C2×#106C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#102C2314yes—
135C2×#102C2312yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2311yescharacteristic
139C2×#97C2312yes—
140C2×#97C2312yes—
141C2×#97C2311yescharacteristic
142C2×C4×C8C2312yes—
143#400—22no—
144#398—21no—
145#398—24no—
146#398—24no—
147#397—21no—
148#395—22no—
149#395—22no—
150#394—21no—
151#394—22no—
152#393—22no—
153#393—22no—
154#392—21no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391C2312yes—
159#391C2314yes—
160#391C2312yes—
161#391C2311yescharacteristic
162#390C2312yes—
163#390C2311yescharacteristic
164#389—24no—
165#389—22no—
166#389—22no—
167#389—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#387—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#385—24no—
180#385—24no—
181#384—22no—
182#384—22no—
183#384—22no—
184#384C2311yescharacteristic
185#384C2314yes—
186#384C2312yes—
187#384C2312yes—
188#383—22no—
189#383—21no—
190#382C2311yescharacteristic
191#382C2311yescharacteristic
192#382C2312yes—
193#382C2311yescharacteristic
194#381—22no—
195#381—21no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2312yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4D412yes—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4C2312yes—
211#376—22no—
212#376—21no—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2312yes—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8C2411yescharacteristic
225C22×Q8C2412yes—
226C22×D4—22no—
227C22×D4C2×D411yescharacteristic
228C22×D4C2411yescharacteristic
229C23×C4—24no—
230C23×C4—21no—
231C23×C4C2×D412yes—
232C23×C4C2×D412yes—
233C23×C4C2412yes—
234C2×Q16—24no—
235C2×Q16—42no—
236C2×Q16—42no—
237C2×SD16—24no—
238C2×SD16—42no—
239C2×SD16—42no—
240C22×C8C2412yes—
241C22×C8C2414yes—
242C22×C8C2412yes—
243C4⋊Q8—22no—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8—22no—
247C4⋊Q8—44no—
248C4⋊Q8—84no—
249#125—48no—
250#124—22no—
251#124—22no—
252#124—48no—
253C22⋊Q8—416no—
254C22⋊Q8—44no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257#121—44no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×D4—28no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—48no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—84no—
282C2×C4⋊C4—84no—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2411yescharacteristic
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4C2×D412yes—
302C2×C22⋊C4C2×D412yes—
303C2×C22⋊C4C2×D412yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—21no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2411yescharacteristic
315C2×C42C2412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318#107—42no—
319#107—44no—
320#106—42no—
321#106—44no—
322C4⋊C8—28no—
323C4⋊C8—24no—
324#103—44no—
325#103—44no—
326#103—42no—
327#103—42no—
328#103—42no—
329#103—44no—
330#102—42no—
331#102—44no—
332C22⋊C8—28no—
333C22⋊C8—44no—
334#97—24no—
335#97—24no—
336#97—22no—
337#97#4114yes—
338C4×C8—42no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—21no—
344#95—22no—
345#95—21no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—24no—
353#95—21no—
354#95—21no—
355#95—22no—
356#95C2×D411yescharacteristic
357#95C2×D411yescharacteristic
358#95C2×D412yes—
359#95C2×D412yes—
360C24—21no—
361C24C22×D412yes—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—82no—
375C2×Q8—84no—
376C2×Q8—82no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—44no—
384C2×D4—82no—
385C2×D4—82no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—48no—
405C22×C4—42no—
406C22×C4—82no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C2511yescharacteristic
417Q16—48no—
418SD16—48no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8C2×#4114yes—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—84no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—48no—
455C22⋊C4—44no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4#13718yes—
465C42—24no—
466C42—24no—
467C42—24no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—42no—
472C42—44no—
473C42—42no—
474C42—44no—
475C42—82no—
476C42C2×#4112yes—
477C42H(5,𝔽2)14yes—
478C422−1+412yes—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—42no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23C2×#13714yes—
491C23C23×D411yescharacteristic
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497Q8—42no—
498Q8—164no—
499Q8—164no—
500D4—28no—
501D4—42no—
502D4—42no—
503D4—164no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—48no—
519C2×C4—48no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—162no—
544C2×C4—168no—
545C2×C4#53512yes—
546C2×C4#53512yes—
547C2×C4#54714yes—
548C2×C4C2×#13714yes—
549C2×C4C2×#13714yes—
550C2×C4C22×#4111yescharacteristic
551C2×C4C2×H(5,𝔽2)12yes—
552C2×C4C2×2−1+411yescharacteristic
553C2×C4#58512yes—
554C8—24no—
555C8—44no—
556C8—42no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—81no—
565C22—82no—
566C22—82no—
567C22—81no—
568C22—81no—
569C22C2×#53511yescharacteristic
570C22C2×#54711yescharacteristic
571C22#350011yescharacteristic
572C22C22×#13712yes—
573C22#358512yes—
574C4—24no—
575C4—24no—
576C4—24no—
577C4—22no—
578C4—22no—
579C4—42no—
580C4—42no—
581C4—82no—
582C4—81no—
583C4—82no—
584C4—82no—
585C4—81no—
586C4—81no—
587C4—82no—
588C4—84no—
589C4#350612yes—
590C4#354712yes—
591C2—44no—
592C2—161no—
593C2#3949111yescharacteristic
594C2#6080011yescharacteristic
595C2#6082711yescharacteristic
596C2#6125412yes—
597C2#6258212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36180C2semidirect
3. #33293C2semidirect
4. #32052C2semidirect
5. #32048C2semidirect
6. #29815C2semidirect
7. #29147C2semidirect
8. #28520C2semidirect
9. #27978C2semidirect
10. #27896C2semidirect
11. #24698C2semidirect
12. #24696C2semidirect
13. #24134C2semidirect
15. #21987C2semidirect
16. #21702C2semidirect
17. #21271C2semidirect

Maximal subgroups

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

subgroupindex
#361952
#361802
#332932
#320522
#320482
#298152
#291472
#285202
#279782
#278962
#246982
#246962
#241342
#241302
#219872
#217022
#212712