Finite Group Databasebeta

← #6467191

Subgroups — #6467191

Order 512.

2046 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3126C212yesmaximal
2#34280C211yesmaximal, characteristic
3#34209C211yesmaximal, characteristic, complemented
4#28294C212yesmaximal, complemented
5#28260C212yesmaximal, complemented
6#28209C212yesmaximal, complemented
7#28160C212yesmaximal, complemented
8#28128C214yesmaximal, complemented
9#27650C212yesmaximal, complemented
10#24209C212yesmaximal, complemented
11#23683C212yesmaximal, complemented
12#23677C212yesmaximal, complemented
13#21514C214yesmaximal, complemented
14#20813C211yesmaximal, characteristic, complemented
15#20745C212yesmaximal, complemented
16C2×#540C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#519C2212yes—
19#3126—216no—
20C2×#482C2214yes—
21C2×#479C2214yes—
22C2×#460C2214yes—
23C2×#461C2214yes—
24C2×#448C2214yes—
25C2×#440C2214yes—
26C2×C4×SD16C2214yes—
27C2×#433C2212yes—
28#2374—22no—
29#2365—22no—
30#2365—22no—
31#2364—22no—
32#2363—22no—
33#2359—22no—
34#2344—22no—
35#2344—22no—
36#2330C2211yescharacteristic
37#2328C2211yescharacteristic
38#2327C2212yes—
39#2327C2212yes—
40#2324C2212yes—
41#2314C2211yescharacteristic
42#2310C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×#127C2211yescharacteristic
46C4×#125C2211yescharacteristic
47C4×#121C2212yes—
48#2293C2212yes—
49#2285C2211yescharacteristic
50#2275C2212yes—
51C4×#117C2211yescharacteristic
52#2084C2212yes—
53#2083C2212yes—
54#2071C2212yes—
55#2071C2212yes—
56#2068C2214yes—
57#2066C2214yes—
58#2055C2214yes—
59#2055C2214yes—
60#2043C2212yes—
61#2043C2212yes—
62#2013C2212yes—
63#2013C2212yes—
64#1981C2214yes—
65#1949C2214yes—
66#1928C2214yes—
67#1927C2214yes—
68#1877C2214yes—
69#1877C2214yes—
70#1843C2212yes—
71#1831C2212yes—
72#1825C2214yes—
73#1805C2214yes—
74#1804C2214yes—
75#1803C2214yes—
76#1798C2212yes—
77#1797C2212yes—
78#1787C2214yes—
79#1768C2212yes—
80#1767C2211yescharacteristic
81C4×C4⋊C8C2211yescharacteristic
82#1763C2214yes—
83C4×#103C2214yes—
84C23×Q8C2312yes—
85C22×SD16C2314yes—
86C22×SD16C2314yes—
87#540—24no—
88#540D418yes—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×#127C2311yescharacteristic
94C2×#125C2311yescharacteristic
95C2×#123—22no—
96C2×#121—22no—
97C2×#121C2312yes—
98#519—28no—
99#519—28no—
100C2×C4×Q8—24no—
101C2×C4×Q8C2314yes—
102C2×C4×Q8C2314yes—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117—24no—
109C2×#117—22no—
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4—22no—
114C22×C42C2311yescharacteristic
115#482—216no—
116#479—216no—
117#461—216no—
118#460—216no—
119#448—216no—
120#440—216no—
121C4×SD16—216no—
122#433—28no—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2312yes—
139C2×C4×C8C2312yes—
140#391—22no—
141#391—22no—
142#391C2314yes—
143#391C2314yes—
144#390—22no—
145#390—22no—
146#390C2312yes—
147#390C2312yes—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389C2312yes—
154#389C2312yes—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#387—22no—
161#387—22no—
162#387—24no—
163#387—24no—
164#387—24no—
165#387—24no—
166#387—22no—
167#386—22no—
168#386—22no—
169#386—22no—
170#386—22no—
171#385—22no—
172#384C2311yescharacteristic
173#384C2312yes—
174#382C2311yescharacteristic
175#382C2312yes—
176#381—22no—
177#381—22no—
178#380—22no—
179#380C2312yes—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4D412yes—
187C4×C4⋊C4D412yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4D412yes—
197C4×C22⋊C4D412yes—
198#376C2311yescharacteristic
199#376C2311yescharacteristic
200C2×#95—22no—
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#336C2312yes—
208#336C2312yes—
209#336C2314yes—
210C22×Q8—28no—
211C22×Q8—22no—
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8—44no—
215C22×Q8C2414yes—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4C2412yes—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4C2×D412yes—
224C23×C4C2411yescharacteristic
225C2×SD16—28no—
226C2×SD16—28no—
227C2×SD16—28no—
228C2×SD16—28no—
229C2×SD16—44no—
230C2×SD16—44no—
231C22×C8C2414yes—
232C22×C8C2414yes—
233C4⋊Q8—216no—
234C4⋊Q8—28no—
235C4⋊Q8—24no—
236C4⋊Q8#4118yes—
237#127—22no—
238#127C2×D414yes—
239#125—22no—
240#125C2×D414yes—
241#123—216no—
242#121—24no—
243#121—216no—
244#121C2×D418yes—
245C4×Q8—28no—
246C4×Q8—28no—
247C4×Q8—28no—
248C4×Q8—28no—
249C4×Q8—416no—
250C4×D4—24no—
251C4×D4—28no—
252C4×D4—44no—
253C4×D4—84no—
254C4×D4—84no—
255C4×D4C2×D418yes—
256#117—24no—
257#117—24no—
258#117—42no—
259#117—416no—
260#117—84no—
261#117C2×D414yes—
262#117#4114yes—
263C2×C4⋊C4—216no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4C2×D411yescharacteristic
276C2×C4⋊C4C2×D411yescharacteristic
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2414yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4C2×D411yescharacteristic
293C2×C22⋊C4C2×D411yescharacteristic
294C2×C22⋊C4C2×D412yes—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313#107—28no—
314#106—28no—
315C4⋊C8—28no—
316C4⋊C8—28no—
317C4⋊C8—42no—
318C4⋊C8—42no—
319#103—28no—
320#103—28no—
321#103—28no—
322#103—28no—
323#103—44no—
324#103—44no—
325#102—28no—
326#102—28no—
327C22⋊C8—28no—
328#97—24no—
329C4×C8—24no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—22no—
342#95C2×D412yes—
343#95C2×D412yes—
344#95C2×D412yes—
345#95C2×D412yes—
346C24—22no—
347C24C22×D411yescharacteristic
348C2×Q8—28no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—416no—
355C2×Q8—416no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8C2×#4118yes—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—82no—
365C2×D4—82no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4C22×D414yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C2×#4112yes—
402C22×C4C2511yescharacteristic
403SD16—416no—
404SD16—416no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C4⋊C4—28no—
412C4⋊C4—28no—
413C4⋊C4—28no—
414C4⋊C4—216no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—44no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4#12118yes—
430C4⋊C4C22×D414yes—
431C22⋊C4—28no—
432C22⋊C4—28no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4—82no—
443C22⋊C4#12118yes—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—42no—
451C42—42no—
452C42—44no—
453C42—48no—
454C42—48no—
455C42—82no—
456C42—82no—
457C42—82no—
458C42C22×D412yes—
459C42C22×D412yes—
460C42C2×#4112yes—
461C42C2×#4112yes—
462C42C2×#4114yes—
463C23—22no—
464C23—21no—
465C23—21no—
466C23—22no—
467C23—24no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23C2×#12112yes—
475C23C23×D411yescharacteristic
476Q8—28no—
477Q8—28no—
478Q8—416no—
479Q8—44no—
480Q8—44no—
481D4—28no—
482D4—44no—
483D4—88no—
484D4—88no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—48no—
508C2×C4—416no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—88no—
523C2×C4—164no—
524C2×C4—164no—
525C2×C4C2×#12114yes—
526C2×C4C2×#12114yes—
527C2×C4C2×#12112yes—
528C2×C4#52514yes—
529C2×C4#57714yes—
530C2×C4C23×D411yescharacteristic
531C2×C4C22×#4112yes—
532C2×C4C22×#4111yescharacteristic
533C2×C4#58512yes—
534C8—44no—
535C8—44no—
536C22—24no—
537C22—22no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22C22×#12111yescharacteristic
549C22C2×#52511yescharacteristic
550C22#344911yescharacteristic
551C22C2×#57712yes—
552C22#358612yes—
553C4—24no—
554C4—24no—
555C4—22no—
556C4—24no—
557C4—24no—
558C4—44no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—48no—
564C4—84no—
565C4—82no—
566C4—82no—
567C4D4×#4112yes—
568C4#347512yes—
569C2—42no—
570C2—84no—
571C2#3493511yescharacteristic
572C2#6042912yes—
573C2#6058212yes—
574C2#6101211yescharacteristic
575C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
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
3. #34209C2semidirect
4. #28294C2semidirect
5. #28260C2semidirect
6. #28209C2semidirect
7. #28160C2semidirect
8. #28128C2semidirect
9. #27650C2semidirect
10. #24209C2semidirect
11. #23683C2semidirect
12. #23677C2semidirect
13. #21514C2semidirect
14. #20813C2semidirect
15. #20745C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31262
#342802
#342092
#282942
#282602
#282092
#281602
#281282
#276502
#242092
#236832
#236772
#215142
#208132
#207452