Finite Group Databasebeta

← #6436658

Subgroups — #6436658

Order 512.

1938 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3107C212yesmaximal
2#36645C211yesmaximal, characteristic, complemented
3#34026C211yesmaximal, characteristic
4#29866C211yesmaximal, characteristic, complemented
5#29864C211yesmaximal, characteristic, complemented
6#29831C212yesmaximal, complemented
7#29829C212yesmaximal, complemented
8#29799C211yesmaximal, characteristic, complemented
9#29798C211yesmaximal, characteristic, complemented
10#29759C214yesmaximal, complemented
11#29717C214yesmaximal
12#23679C214yesmaximal, complemented
13#23677C214yesmaximal, complemented
14#21389C212yesmaximal, complemented
15#20808C211yesmaximal, characteristic, complemented
16C2×#533C2212yes—
17C2×C4×#41C2211yescharacteristic
18#3107—216no—
19C2×#485C2214yes—
20C2×#475C2212yes—
21C2×#474C2212yes—
22C2×#462C2212yes—
23C2×#461C2212yes—
24C2×#452C2214yes—
25C2×C4×Q16C2214yes—
26C2×C4×SD16C2214yes—
27C2×#431C2212yes—
28#2868C2211yescharacteristic
29#2868C2211yescharacteristic
30#2862C2212yes—
31#2859—22no—
32#2857—21no—
33#2857—21no—
34#2857C2214yes—
35C4×C4⋊Q8—21no—
36C4×C4⋊Q8—21no—
37C4×C4⋊Q8C2214yes—
38#2298—22no—
39#2297C2212yes—
40#2296C2212yes—
41#2295C2211yescharacteristic
42#2295C2212yes—
43#2295C2211yescharacteristic
44#2293C2211yescharacteristic
45#2293C2212yes—
46#2293C2211yescharacteristic
47#2290—21no—
48#2290—22no—
49#2290—21no—
50#2285—22no—
51C4×#117—21no—
52C4×#117—21no—
53C4×#117C2211yescharacteristic
54C2×C43C2211yescharacteristic
55#2084C2214yes—
56#2083C2214yes—
57#2072C2214yes—
58#2071C2214yes—
59#2068C2218yes—
60#2066C2218yes—
61#2056C2218yes—
62#2055C2218yes—
63#2043C2214yes—
64#2043C2214yes—
65#2016C2214yes—
66#2013C2214yes—
67#1988C2212yes—
68#1988C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1873C2218yes—
72#1842C2212yes—
73#1831C2211yescharacteristic
74#1831C2211yescharacteristic
75#1797C2214yes—
76C4×C4⋊C8C2214yes—
77C4×#103C2218yes—
78C22×#41C2311yescharacteristic
79C22×Q16C2314yes—
80C22×SD16C2314yes—
81#533—28no—
82#533—28no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2314yes—
87C2×C4⋊Q8C2312yes—
88C2×#125—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8C2314yes—
91C4×#41—24no—
92C4×#41D418yes—
93C2×C4×Q8—22no—
94C2×C4×Q8C2314yes—
95C2×C4×Q8C2311yescharacteristic
96C2×C4×Q8C2311yescharacteristic
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4C2311yescharacteristic
100C2×C4×D4C2311yescharacteristic
101C2×#117—22no—
102C2×#117—21no—
103C2×#117—21no—
104C2×#117—21no—
105C2×#117—22no—
106C2×#117C2311yescharacteristic
107C2×#117C2312yes—
108C22×C42—21no—
109C22×C42D411yescharacteristic
110C22×C42D412yes—
111C22×C42D411yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#485—216no—
115#475—28no—
116#474—28no—
117#462—28no—
118#461—28no—
119#452—216no—
120C4×Q16—216no—
121C4×SD16—216no—
122#431—28no—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2318yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2318yes—
132C2×#103C2318yes—
133C2×#102C2314yes—
134C2×C22⋊C8C2314yes—
135C2×C4×C8C2314yes—
136#391—22no—
137#391—22no—
138#391C2318yes—
139#391C2312yes—
140#391C2312yes—
141#390C2312yes—
142#390C2312yes—
143#389—24no—
144#389—24no—
145#389—24no—
146#389—24no—
147#389C2312yes—
148#389C2312yes—
149#389C2312yes—
150#389C2312yes—
151#388—22no—
152#388—22no—
153#388—22no—
154#388—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#386—22no—
160#386—22no—
161#385—22no—
162#385—22no—
163#384—21no—
164#384—21no—
165#384—22no—
166#384—22no—
167#384C2314yes—
168#382—24no—
169#380—24no—
170#380—24no—
171C4×C4⋊C4—24no—
172C4×C4⋊C4—24no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4C2311yescharacteristic
179C4×C4⋊C4C2311yescharacteristic
180C4×C4⋊C4C2312yes—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188#376—21no—
189#376—21no—
190#376—22no—
191#376—21no—
192C43—21no—
193C43D411yescharacteristic
194C43D411yescharacteristic
195C43D412yes—
196C43C2312yes—
197C43C2311yescharacteristic
198C43C2311yescharacteristic
199#340C2318yes—
200#340C2318yes—
201#336C2314yes—
202#336C2314yes—
203C2×#41—22no—
204C2×#41—44no—
205C2×#41C2×D414yes—
206C22×Q8—22no—
207C22×Q8C2414yes—
208C22×Q8C2411yescharacteristic
209C22×D4—21no—
210C22×D4C2411yescharacteristic
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4C2×D412yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D412yes—
218C23×C4C2411yescharacteristic
219C2×Q16—28no—
220C2×Q16—48no—
221C2×SD16—28no—
222C2×SD16—48no—
223C22×C8C2414yes—
224C22×C8C2414yes—
225C4⋊Q8—24no—
226C4⋊Q8—24no—
227C4⋊Q8—24no—
228C4⋊Q8—24no—
229C4⋊Q8—48no—
230C4⋊Q8—44no—
231C4⋊Q8—44no—
232#125—48no—
233C22⋊Q8—28no—
234C22⋊Q8—28no—
235C22⋊Q8—48no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8—28no—
239C4×Q8—28no—
240C4×Q8—48no—
241C4×Q8C2×D414yes—
242C4×Q8C2×D414yes—
243C4×D4—28no—
244C4×D4—28no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4C2×D414yes—
248C4×D4C2×D414yes—
249#117—22no—
250#117—28no—
251#117—44no—
252#117—44no—
253#117—48no—
254#117—44no—
255#117—48no—
256#117C2×D414yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2411yescharacteristic
276C2×C4⋊C4C2411yescharacteristic
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×C22⋊C4—22no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D414yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D412yes—
314C2×C42#4114yes—
315C2×C42C2412yes—
316C2×C42C2412yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319#107—28no—
320#106—28no—
321C4⋊C8—24no—
322C4⋊C8—24no—
323C4⋊C8—48no—
324#103—28no—
325#103—28no—
326#103—28no—
327#103—48no—
328#103—48no—
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—22no—
340#95—21no—
341#95—21no—
342#95—22no—
343#95—28no—
344#95—28no—
345C24—21no—
346C24C22×D411yescharacteristic
347#41—48no—
348#41—48no—
349C2×Q8—21no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—28no—
354C2×Q8—28no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8C22×D412yes—
359C2×D4—24no—
360C2×D4—21no—
361C2×D4—24no—
362C2×D4—44no—
363C2×D4C22×D412yes—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4#12114yes—
389C22×C4#12114yes—
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C2×#4112yes—
401C22×C4C2511yescharacteristic
402Q16—416no—
403SD16—416no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—28no—
407C2×C8—28no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4C22×D412yes—
430C4⋊C4C22×D412yes—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C42—24no—
441C42—24no—
442C42—24no—
443C42—24no—
444C42—22no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—22no—
449C42—22no—
450C42—24no—
451C42—24no—
452C42—42no—
453C42—42no—
454C42—42no—
455C42—44no—
456C42—48no—
457C42—42no—
458C42—44no—
459C42—42no—
460C42#12114yes—
461C42#12114yes—
462C42#13514yes—
463C42#13718yes—
464C42C22×D412yes—
465C42C22×D412yes—
466C42C2×#4112yes—
467C42C2×#4112yes—
468C23—22no—
469C23—21no—
470C23—22no—
471C23—21no—
472C23—22no—
473C23—42no—
474C23—41no—
475C23—42no—
476C23—41no—
477C23C2×#12112yes—
478C23C23×D411yescharacteristic
479Q8—24no—
480Q8—42no—
481Q8—44no—
482Q8—44no—
483Q8—48no—
484Q8—88no—
485D4—24no—
486D4—48no—
487D4—42no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—48no—
516C2×C4—48no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—48no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4C2×#12112yes—
529C2×C4C2×#12112yes—
530C2×C4C2×#12112yes—
531C2×C4C2×#12112yes—
532C2×C4C2×#12112yes—
533C2×C4C2×#13514yes—
534C2×C4C2×#13512yes—
535C2×C4C2×#13714yes—
536C2×C4C2×#13714yes—
537C2×C4C2×#13714yes—
538C2×C4#57512yes—
539C2×C4C23×D411yescharacteristic
540C2×C4C22×#4111yescharacteristic
541C2×C4#58514yes—
542C8—44no—
543C8—44no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—82no—
553C22—82no—
554C22C22×#12111yescharacteristic
555C22#344912yes—
556C22C22×#13511yescharacteristic
557C22C22×#13712yes—
558C22C2×#57511yescharacteristic
559C4—24no—
560C4—24no—
561C4—24no—
562C4—22no—
563C4—22no—
564C4—22no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—82no—
571C4—84no—
572C4—84no—
573C4#303514yes—
574C4#303914yes—
575C4#344912yes—
576C4#345512yes—
577C2—42no—
578C2—82no—
579C2#3463311yescharacteristic
580C2C2×#303511yescharacteristic
581C2C2×#303911yescharacteristic
582C2#6094712yes—
583C2#6095212yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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. #29866C2semidirect
5. #29864C2semidirect
6. #29831C2semidirect
7. #29829C2semidirect
8. #29799C2semidirect
9. #29798C2semidirect
10. #29759C2semidirect
12. #23679C2semidirect
13. #23677C2semidirect
14. #21389C2semidirect
15. #20808C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31072
#366452
#340262
#298662
#298642
#298312
#298292
#297992
#297982
#297592
#297172
#236792
#236772
#213892
#208082