Finite Group Databasebeta

← #6467167

Subgroups — #6467167

Order 512.

2054 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3126C212yesmaximal
2#34209C211yesmaximal, characteristic, complemented
3C4×#540C211yesmaximal, characteristic
4#28287C212yesmaximal, complemented
5#28252C212yesmaximal, complemented
6#28216C212yesmaximal, complemented
7#28169C212yesmaximal, complemented
8#28128C214yesmaximal, complemented
9#27646C212yesmaximal, complemented
10#24209C212yesmaximal, complemented
11#23689C214yesmaximal, complemented
12#21504C212yesmaximal, complemented
13C4×#461C212yesmaximal, complemented
14#20812C211yesmaximal, characteristic, complemented
15#20745C212yesmaximal, complemented
16C2×#540C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#519C2212yes—
19#3126—216no—
20C2×#482C2214yes—
21C2×#479C2214yes—
22C2×#460C2214yes—
23C2×#461C2212yes—
24C2×#461C2212yes—
25C2×#448C2214yes—
26C2×#440C2214yes—
27C2×C4×SD16C2214yes—
28C2×#433C2212yes—
29#2365—22no—
30#2365—22no—
31#2344—22no—
32#2344—22no—
33#2335—22no—
34#2333—22no—
35#2330C2211yescharacteristic
36#2328C2211yescharacteristic
37#2327C2212yes—
38#2327C2212yes—
39#2324C2212yes—
40#2319C2212yes—
41#2314C2211yescharacteristic
42#2310C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×#127C2211yescharacteristic
46C4×#125C2211yescharacteristic
47C4×#123—22no—
48C4×#121—22no—
49C4×#121C2212yes—
50#2285C2211yescharacteristic
51C42×D4C2212yes—
52C4×#117C2211yescharacteristic
53#2085C2214yes—
54#2073C2214yes—
55#2068C2214yes—
56#2066C2214yes—
57#2055C2214yes—
58#2055C2214yes—
59#2043C2214yes—
60#2013C2214yes—
61#1981C2212yes—
62#1981C2212yes—
63#1948C2212yes—
64#1947C2212yes—
65#1928C2214yes—
66#1927C2214yes—
67#1877C2214yes—
68#1877C2214yes—
69#1842C2212yes—
70#1832C2212yes—
71#1825C2214yes—
72#1805C2214yes—
73#1803C2214yes—
74#1802C2212yes—
75#1802C2212yes—
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—21no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8C2312yes—
93C2×C4⋊Q8C2312yes—
94C2×#127C2311yescharacteristic
95C2×#125C2311yescharacteristic
96C2×#123—22no—
97C2×#121—22no—
98C2×#121C2312yes—
99#519—28no—
100#519—28no—
101C2×C4×Q8—24no—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2314yes—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×#117—24no—
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C42—22no—
116C22×C42C2311yescharacteristic
117#482—216no—
118#479—216no—
119#461—28no—
120#461—28no—
121#460—216no—
122#448—216no—
123#440—216no—
124C4×SD16—216no—
125#433—28no—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×C22⋊C8C2314yes—
141C2×#97C2312yes—
142C2×C4×C8C2312yes—
143#391—24no—
144#391C2314yes—
145#391C2314yes—
146#390—22no—
147#390—22no—
148#390C2312yes—
149#390C2312yes—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389C2312yes—
156#389C2312yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#387—22no—
163#387—22no—
164#387—24no—
165#387—24no—
166#387—24no—
167#387—24no—
168#387—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#385—22no—
174#384C2311yescharacteristic
175#384C2312yes—
176#382C2311yescharacteristic
177#382C2312yes—
178#381—24no—
179#381—22no—
180#380—21no—
181#380—21no—
182#380—22no—
183#380C2312yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4D412yes—
202#376C2311yescharacteristic
203#376C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#336C2314yes—
211#336C2312yes—
212#336C2312yes—
213C22×Q8—28no—
214C22×Q8—22no—
215C22×Q8—24no—
216C22×Q8—24no—
217C22×Q8—44no—
218C22×Q8C2414yes—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4C2412yes—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C2×SD16—28no—
230C2×SD16—28no—
231C2×SD16—28no—
232C2×SD16—28no—
233C2×SD16—44no—
234C2×SD16—44no—
235C22×C8C2414yes—
236C22×C8C2414yes—
237C4⋊Q8—28no—
238C4⋊Q8—28no—
239C4⋊Q8—28no—
240C4⋊Q8—24no—
241C4⋊Q8#4118yes—
242#127—22no—
243#127C2×D414yes—
244#125—22no—
245#125C2×D414yes—
246#123—216no—
247#121—24no—
248#121—216no—
249#121C2×D418yes—
250C4×Q8—28no—
251C4×Q8—28no—
252C4×Q8—28no—
253C4×Q8—28no—
254C4×Q8—416no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—42no—
259C4×D4—42no—
260C4×D4—84no—
261C4×D4—84no—
262C4×D4C2×D418yes—
263#117—24no—
264#117—24no—
265#117—42no—
266#117—416no—
267#117C2×D414yes—
268#117#4114yes—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
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⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—84no—
309C2×C42C2×D412yes—
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321#107—28no—
322#106—28no—
323C4⋊C8—28no—
324C4⋊C8—24no—
325C4⋊C8—24no—
326C4⋊C8—44no—
327#103—28no—
328#103—28no—
329#103—28no—
330#103—28no—
331#103—44no—
332#103—44no—
333#102—28no—
334#102—28no—
335C22⋊C8—28no—
336#97—24no—
337C4×C8—24no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95C2×D412yes—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D412yes—
355C24—22no—
356C24C22×D411yescharacteristic
357C2×Q8—28no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—416no—
364C2×Q8—416no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8C2×#4118yes—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4C22×D414yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4112yes—
414C22×C4C2511yescharacteristic
415SD16—416no—
416SD16—416no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4#12118yes—
444C4⋊C4C22×D414yes—
445C22⋊C4—28no—
446C22⋊C4—28no—
447C22⋊C4—42no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—82no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4#12118yes—
457C42—22no—
458C42—22no—
459C42—24no—
460C42—24no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—42no—
465C42—42no—
466C42—44no—
467C42—48no—
468C42—48no—
469C42—44no—
470C42—82no—
471C42—82no—
472C42C22×D412yes—
473C42C22×D412yes—
474C42C2×#4112yes—
475C42C2×#4112yes—
476C42C2×#4114yes—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23C2×#12112yes—
489C23C23×D411yescharacteristic
490Q8—28no—
491Q8—28no—
492Q8—416no—
493Q8—44no—
494Q8—44no—
495D4—24no—
496D4—24no—
497D4—42no—
498D4—42no—
499D4—88no—
500D4—88no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—44no—
523C2×C4—416no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—88no—
539C2×C4—88no—
540C2×C4—88no—
541C2×C4C2×#12114yes—
542C2×C4C2×#12114yes—
543C2×C4C2×#12112yes—
544C2×C4#52512yes—
545C2×C4#52512yes—
546C2×C4#57714yes—
547C2×C4C23×D411yescharacteristic
548C2×C4C22×#4112yes—
549C2×C4C22×#4111yescharacteristic
550C2×C4#58512yes—
551C8—44no—
552C8—44no—
553C22—24no—
554C22—22no—
555C22—42no—
556C22—41no—
557C22—41no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22C22×#12111yescharacteristic
566C22C2×#52511yescharacteristic
567C22#344911yescharacteristic
568C22C2×#57712yes—
569C22#358612yes—
570C4—24no—
571C4—24no—
572C4—22no—
573C4—22no—
574C4—44no—
575C4—42no—
576C4—42no—
577C4—48no—
578C4—41no—
579C4—41no—
580C4—84no—
581C4—82no—
582C4—82no—
583C4#312614yescomplemented
584C4D4×#4112yes—
585C4#347512yes—
586C2—42no—
587C2—84no—
588C2#3493511yescharacteristic
589C2#6042912yes—
590C2C2×#312611yescharacteristic
591C2#6056611yescharacteristic
592C2#6099112yes—
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
2. #34209C2semidirect
4. #28287C2semidirect
5. #28252C2semidirect
6. #28216C2semidirect
7. #28169C2semidirect
8. #28128C2semidirect
9. #27646C2semidirect
10. #24209C2semidirect
11. #23689C2semidirect
12. #21504C2semidirect
13. C4×#461C2semidirect
14. #20812C2semidirect
15. #20745C2semidirect
583. C4#3126semidirect

Maximal subgroups

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

subgroupindex
C2×#31262
#342092
C4×#5402
#282872
#282522
#282162
#281692
#281282
#276462
#242092
#236892
#215042
C4×#4612
#208122
#207452