Finite Group Databasebeta

← #6497516

Subgroups — #6497516

Order 512.

1846 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3233C212yesmaximal
2#36645C211yesmaximal, characteristic, complemented
3C4×#529C211yesmaximal, characteristic
4#33198C211yesmaximal, characteristic, complemented
5#33197C211yesmaximal, characteristic, complemented
6#33196C211yesmaximal, characteristic, complemented
7#33195C211yesmaximal, characteristic, complemented
8#29867C212yesmaximal, complemented
9#29865C212yesmaximal, complemented
10#28721C214yesmaximal, complemented
11#28691C214yesmaximal, complemented
12#27649C214yesmaximal, complemented
13#27646C214yesmaximal
14#21837C212yesmaximal, complemented
15#20808C211yesmaximal, characteristic, complemented
16C2×#533C2212yes—
17C2×#529C2211yescharacteristic
18#3233—216no—
19C2×#499C2214yes—
20C2×#488C2214yes—
21C2×#485C2214yes—
22C2×#482C2214yes—
23C2×#477C2212yes—
24C2×#475C2212yes—
25C2×#476C2212yes—
26C2×#474C2212yes—
27C2×#431C2212yes—
28#2868C2211yescharacteristic
29#2868C2211yescharacteristic
30#2862C2212yes—
31#2859C2214yes—
32#2857—21no—
33#2857—21no—
34#2857—22no—
35C4×C4⋊Q8—22no—
36C4×C4⋊Q8—21no—
37C4×C4⋊Q8—21no—
38C4×#125C2212yes—
39C4×#126—22no—
40C4×#124C2212yes—
41C4×C22⋊Q8C2212yes—
42C4×#123—22no—
43C4×#121C2212yes—
44#2298—21no—
45#2295C2211yescharacteristic
46#2293C2211yescharacteristic
47#2290—21no—
48#2285C2214yes—
49#2283C2211yescharacteristic
50C42×Q8C2211yescharacteristic
51C42×D4—21no—
52C42×D4C2211yescharacteristic
53C4×#117—21no—
54C2×C43C2211yescharacteristic
55#2085C2218yes—
56#2085C2218yes—
57#2074C2218yes—
58#2073C2218yes—
59#1991C2218yes—
60#1988C2212yes—
61#1988C2212yes—
62#1987C2212yes—
63#1987C2212yes—
64#1948C2214yes—
65#1948C2214yes—
66#1947C2214yes—
67#1947C2214yes—
68#1842C2211yescharacteristic
69#1842C2211yescharacteristic
70#1831C2212yes—
71#1825C2218yes—
72#1803C2214yes—
73#1803C2214yes—
74#1802C2214yes—
75#1802C2214yes—
76#1797C2214yes—
77C4×C4⋊C8C2214yes—
78#533—28no—
79#533—28no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—22no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85#529—24no—
86#529D418yes—
87C2×#126—21no—
88C2×#125C2314yes—
89C2×#125C2311yescharacteristic
90C2×#124C2311yescharacteristic
91C2×#123—21no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8C2311yescharacteristic
95C2×#121C2311yescharacteristic
96C2×C4×Q8—22no—
97C2×C4×Q8—22no—
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2311yescharacteristic
100C2×C4×D4—21no—
101C2×C4×D4—22no—
102C2×C4×D4C2312yes—
103C2×C4×D4C2311yescharacteristic
104C2×#117—21no—
105C2×#117—21no—
106C2×#117C2314yes—
107C22×C42—21no—
108C22×C42D411yescharacteristic
109C22×C42D411yescharacteristic
110C22×C42D412yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113#499—216no—
114#488—216no—
115#485—216no—
116#482—216no—
117#477—28no—
118#476—28no—
119#475—28no—
120#474—28no—
121#431—28no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×C22⋊C8C2314yes—
135C2×C4×C8C2314yes—
136#391—22no—
137#391—22no—
138#391—24no—
139#391C2312yes—
140#390C2312yes—
141#389—24no—
142#389—24no—
143#389—24no—
144#389—24no—
145#389C2312yes—
146#389C2312yes—
147#388—22no—
148#388—22no—
149#388—22no—
150#388—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#386—22no—
156#385—22no—
157#384—22no—
158#384—21no—
159#384—21no—
160#384C2314yes—
161#384C2314yes—
162#383—21no—
163#382C2312yes—
164#382C2318yes—
165#381—21no—
166#380—24no—
167#380—24no—
168#380C2312yes—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—24no—
174C4×C4⋊C4—24no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2311yescharacteristic
180C4×C4⋊C4C2311yescharacteristic
181C4×C22⋊C4—21no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189#376—21no—
190#376—21no—
191C43—21no—
192C43D412yes—
193C43D411yescharacteristic
194C43D411yescharacteristic
195C43C2312yes—
196C43C2311yescharacteristic
197C43C2311yescharacteristic
198#340C2318yes—
199#340C2318yes—
200#340C2318yes—
201#340C2318yes—
202#336C2318yes—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8C2411yescharacteristic
206C22×D4—21no—
207C22×D4C2411yescharacteristic
208C23×C4—21no—
209C23×C4—22no—
210C23×C4C2×D412yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2411yescharacteristic
215C22×C8C2414yes—
216C22×C8C2414yes—
217C4⋊Q8—24no—
218C4⋊Q8—24no—
219C4⋊Q8—24no—
220C4⋊Q8—24no—
221C4⋊Q8—44no—
222C4⋊Q8—48no—
223C4⋊Q8—44no—
224#126—28no—
225#125—28no—
226#125—22no—
227#125—28no—
228#125C2×D414yes—
229#124—22no—
230#124C2×D414yes—
231#123—28no—
232C22⋊Q8—22no—
233C22⋊Q8—48no—
234C22⋊Q8—48no—
235C22⋊Q8C2×D414yes—
236#121—22no—
237#121C2×D414yes—
238C4×Q8—22no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—48no—
242C4×Q8—48no—
243C4×Q8C2×D414yes—
244C4×D4—28no—
245C4×D4—22no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—48no—
249C4×D4C2×D414yes—
250#117—28no—
251#117—28no—
252#117—28no—
253#117—44no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2411yescharacteristic
274C2×C4⋊C4C2411yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C42—21no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—21no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—22no—
295C2×C42—44no—
296C2×C42—42no—
297C2×C42—44no—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D412yes—
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D414yes—
312C2×C42C2×D411yescharacteristic
313C2×C42#4114yes—
314C2×C42C2412yes—
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317C2×C42C2412yes—
318#107—28no—
319#107—28no—
320#106—28no—
321#106—28no—
322C4⋊C8—24no—
323C4⋊C8—24no—
324C4⋊C8—28no—
325C4⋊C8—28no—
326#103—28no—
327#103—28no—
328#102—28no—
329#102—28no—
330C22⋊C8—28no—
331C4×C8—28no—
332#95—22no—
333#95—22no—
334#95—22no—
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—28no—
344#95—28no—
345C24—21no—
346C24C22×D411yescharacteristic
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8C22×D412yes—
356C2×D4—24no—
357C2×D4—21no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—44no—
361C2×D4C22×D412yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4#12114yes—
387C22×C4#12114yes—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C2×#4112yes—
399C22×C4C2511yescharacteristic
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—28no—
403C2×C8—28no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4C22×D412yes—
430C4⋊C4C22×D412yes—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—48no—
440C22⋊C4—44no—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—24no—
445C42—22no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—24no—
450C42—22no—
451C42—22no—
452C42—42no—
453C42—42no—
454C42—42no—
455C42—48no—
456C42—44no—
457C42—44no—
458C42—42no—
459C42—42no—
460C42—44no—
461C42#12114yes—
462C42#12114yes—
463C42#13514yes—
464C42C22×D412yes—
465C42C22×D412yes—
466C42C2×#4112yes—
467C42C2×#4112yes—
468C422−1+414yes—
469C422−1+414yes—
470C23—22no—
471C23—21no—
472C23—22no—
473C23—21no—
474C23—22no—
475C23—42no—
476C23—41no—
477C23—41no—
478C23—42no—
479C23C2×#12112yes—
480C23C23×D411yescharacteristic
481Q8—22no—
482Q8—22no—
483Q8—42no—
484Q8—88no—
485Q8—88no—
486D4—22no—
487D4—22no—
488D4—48no—
489D4—42no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—28no—
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—44no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—48no—
529C2×C4—88no—
530C2×C4C2×#12112yes—
531C2×C4C2×#12112yes—
532C2×C4C2×#12112yes—
533C2×C4C2×#12112yes—
534C2×C4C2×#12112yes—
535C2×C4#53614yes—
536C2×C4#53614yes—
537C2×C4C2×#13514yes—
538C2×C4C2×#13512yes—
539C2×C4#57514yes—
540C2×C4#57512yes—
541C2×C4C23×D411yescharacteristic
542C2×C4C22×#4111yescharacteristic
543C2×C4C2×2−1+412yes—
544C2×C4C2×2−1+412yes—
545C8—44no—
546C8—44no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—82no—
556C22—82no—
557C22C22×#12111yescharacteristic
558C22C2×#53612yes—
559C22C22×#13511yescharacteristic
560C22C2×#57512yes—
561C22C2×#57511yescharacteristic
562C4—24no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—22no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—82no—
573C4—82no—
574C4—84no—
575C4—84no—
576C4#303514yes—
577C4#323318yescomplemented
578C4#349911yescharacteristic
579C4#349911yescharacteristic
580C4#350912yes—
581C2—42no—
582C2—82no—
583C2#3531611yescharacteristic
584C2C2×#303511yescharacteristic
585C2#6041511yescharacteristic
586C2C2×#323312yes—
587C2#6086312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36645C2semidirect
4. #33198C2semidirect
5. #33197C2semidirect
6. #33196C2semidirect
7. #33195C2semidirect
8. #29867C2semidirect
9. #29865C2semidirect
10. #28721C2semidirect
11. #28691C2semidirect
12. #27649C2semidirect
14. #21837C2semidirect
15. #20808C2semidirect
577. C4#3233semidirect

Maximal subgroups

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

subgroupindex
C2×#32332
#366452
C4×#5292
#331982
#331972
#331962
#331952
#298672
#298652
#287212
#286912
#276492
#276462
#218372
#208082