Finite Group Databasebeta

← #6850725

Subgroups — #6850725

Order 512.

1493 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36195C211yesmaximal, characteristic
2#36188C211yesmaximal, characteristic, complemented
3#33293C211yesmaximal, characteristic, complemented
4#32053C214yesmaximal, complemented
5#32049C214yesmaximal, complemented
6#29815C212yesmaximal, complemented
7#29147C211yesmaximal, characteristic, complemented
8#28519C211yesmaximal, characteristic, complemented
9#27979C212yesmaximal, complemented
10#27896C212yesmaximal, complemented
11#24699C212yesmaximal, complemented
12#24697C212yesmaximal, complemented
13#24135C212yesmaximal, complemented
14#24131C212yesmaximal
15#21989C212yesmaximal, complemented
16#21702C211yesmaximal, characteristic, complemented
17#21271C211yesmaximal, characteristic, complemented
18C2×#556C2212yes—
19C2×#497C2211yescharacteristic
20C2×#477C2212yes—
21C2×#476C2212yes—
22C2×#441C2212yes—
23C2×#440C2212yes—
24C2×#432C2212yes—
25#2868C2211yescharacteristic
26#2862C2211yescharacteristic
27#2857C2212yes—
28#2850C2212yes—
29#2572—22no—
30#2568C2212yes—
31#2547—22no—
32#2536C2211yescharacteristic
33#2535C2211yescharacteristic
34#2532—22no—
35#2499C2214yes—
36#2496—22no—
37#2496—22no—
38#2492C2212yes—
39#2491C2212yes—
40#2459—22no—
41#2372—22no—
42#2365—22no—
43#2357C2211yescharacteristic
44#2356C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#124C2211yescharacteristic
47#2285C2212yes—
48#2283C2212yes—
49C42×D4C2211yescharacteristic
50#2096C2212yes—
51#2093C2212yes—
52#2074C2214yes—
53#2073C2214yes—
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#1949C2214yes—
72#1949C2214yes—
73#1940C2212yes—
74#1928C2214yes—
75#1927C2214yes—
76#1923C2214yes—
77#1875C2214yes—
78#1873C2214yes—
79#1854C2214yes—
80#1854C2214yes—
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#556—44no—
94#556—44no—
95C2×C4⋊Q8—22no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2312yes—
99C2×#126—24no—
100C2×#124—22no—
101C2×#124—22no—
102C2×#124C2311yescharacteristic
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2314yes—
105C2×#121C2312yes—
106C2×C4×Q8—22no—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2312yes—
109C2×C4×D4D412yes—
110C2×C4×D4C2311yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C2×#117C2314yes—
113C22×C42C2312yes—
114#497—24no—
115#477—44no—
116#476—44no—
117#441—44no—
118#440—44no—
119#432—28no—
120C2×#107C2314yes—
121C2×#107C2312yes—
122C2×#106C2314yes—
123C2×#106C2312yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2311yescharacteristic
137C2×#97C2312yes—
138C2×#97C2312yes—
139C2×#97C2311yescharacteristic
140C2×C4×C8C2312yes—
141#400—22no—
142#400—22no—
143#400—22no—
144#398—21no—
145#397—21no—
146#397—24no—
147#397—24no—
148#394—21no—
149#394—22no—
150#393—22no—
151#393—22no—
152#392—21no—
153#391—24no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391C2312yes—
158#391C2314yes—
159#391C2312yes—
160#391C2311yescharacteristic
161#390C2312yes—
162#390C2311yescharacteristic
163#389—22no—
164#389—22no—
165#389—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—24no—
175#387—24no—
176#386—22no—
177#386—22no—
178#386—22no—
179#384—22no—
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#383—22no—
184#383—21no—
185#382C2314yes—
186#382C2311yescharacteristic
187#382C2311yescharacteristic
188#382C2312yes—
189#382C2311yescharacteristic
190#381—22no—
191#381—21no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380C2312yes—
197#380C2312yes—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
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—21no—
230C23×C4C2×D412yes—
231C23×C4C2×D412yes—
232C23×C4C2412yes—
233C2×Q16—24no—
234C2×Q16—42no—
235C2×Q16—42no—
236C2×SD16—24no—
237C2×SD16—42no—
238C2×SD16—42no—
239C22×C8C2412yes—
240C22×C8C2414yes—
241C22×C8C2412yes—
242C4⋊Q8—22no—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8—22no—
246C4⋊Q8—44no—
247C4⋊Q8—48no—
248#126—416no—
249#124—22no—
250#124—22no—
251#124—48no—
252#124—48no—
253C22⋊Q8—44no—
254C22⋊Q8—44no—
255C22⋊Q8—44no—
256#121—44no—
257C4×Q8—24no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—48no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—44no—
264#117—44no—
265#117—44no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4C2×D411yescharacteristic
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2411yescharacteristic
282C2×C4⋊C4C2411yescharacteristic
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4C2×D412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D412yes—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2411yescharacteristic
311C2×C42C2412yes—
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314#107—42no—
315#107—44no—
316#106—42no—
317#106—44no—
318C4⋊C8—28no—
319C4⋊C8—24no—
320#103—44no—
321#103—44no—
322#103—42no—
323#103—42no—
324#103—42no—
325#103—44no—
326#102—42no—
327#102—44no—
328C22⋊C8—28no—
329C22⋊C8—44no—
330#97—24no—
331#97—24no—
332#97—22no—
333#97#4114yes—
334C4×C8—42no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—21no—
340#95—22no—
341#95—21no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—21no—
352#95—21no—
353#95—22no—
354#95C2×D411yescharacteristic
355#95C2×D411yescharacteristic
356#95C2×D412yes—
357#95C2×D412yes—
358C24—21no—
359C24C22×D412yes—
360C2×Q8—21no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—48no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—82no—
374C2×Q8—82no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—42no—
381C2×D4—44no—
382C2×D4—82no—
383C2×D4—82no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—82no—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C2511yescharacteristic
412Q16—48no—
413SD16—48no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8C2×#4114yes—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—42no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—44no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—44no—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C42—24no—
459C42—24no—
460C42—24no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—42no—
465C42—48no—
466C42—44no—
467C42—42no—
468C42—44no—
469C42—82no—
470C42C2×#4112yes—
471C42H(5,𝔽2)14yes—
472C422−1+412yes—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—42no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23C23×D411yescharacteristic
486Q8—42no—
487Q8—42no—
488Q8—42no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—88no—
493Q8—164no—
494D4—44no—
495D4—42no—
496D4—42no—
497D4—164no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
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—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—162no—
539C2×C4#53512yes—
540C2×C4#53512yes—
541C2×C4#54714yes—
542C2×C4#57514yes—
543C2×C4C22×#4111yescharacteristic
544C2×C4C2×H(5,𝔽2)12yes—
545C2×C4C2×2−1+411yescharacteristic
546C2×C4#58512yes—
547C8—24no—
548C8—44no—
549C8—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—81no—
558C22—82no—
559C22—82no—
560C22—81no—
561C22—81no—
562C22C2×#53511yescharacteristic
563C22C2×#54711yescharacteristic
564C22#350011yescharacteristic
565C22C2×#57512yes—
566C22#358612yes—
567C4—24no—
568C4—24no—
569C4—22no—
570C4—22no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—82no—
575C4—81no—
576C4—82no—
577C4—82no—
578C4—81no—
579C4—84no—
580C4—82no—
581C4—81no—
582C4#350612yes—
583C4#354712yes—
584C2—82no—
585C2—161no—
586C2#3949111yescharacteristic
587C2#6080111yescharacteristic
588C2#6082811yescharacteristic
589C2#6125712yes—
590C2#6258512yes—
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. #36188C2semidirect
3. #33293C2semidirect
4. #32053C2semidirect
5. #32049C2semidirect
6. #29815C2semidirect
7. #29147C2semidirect
8. #28519C2semidirect
9. #27979C2semidirect
10. #27896C2semidirect
11. #24699C2semidirect
12. #24697C2semidirect
13. #24135C2semidirect
15. #21989C2semidirect
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
#361882
#332932
#320532
#320492
#298152
#291472
#285192
#279792
#278962
#246992
#246972
#241352
#241312
#219892
#217022
#212712