Finite Group Databasebeta

← #6484772

Subgroups — #6484772

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3240C212yesmaximal, complemented
2#35343C211yesmaximal, characteristic
3#34214C211yesmaximal, characteristic, complemented
4#33117C212yesmaximal, complemented
5#33116C212yesmaximal
6#29769C214yesmaximal, complemented
7#28754C212yesmaximal, complemented
8#28692C212yesmaximal, complemented
9#28659C212yesmaximal, complemented
10#28658C212yesmaximal, complemented
11#28294C212yesmaximal, complemented
12#28292C212yesmaximal, complemented
13#28215C214yesmaximal, complemented
14#21867C212yesmaximal, complemented
15#20811C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#529C2212yes—
18#3240—216no—
19C2×#501C2212yes—
20C2×#501C2212yes—
21C2×#489C2212yes—
22C2×#489C2212yes—
23C2×#485C2214yes—
24C2×#482C2214yes—
25C2×#479C2214yes—
26C2×#478C2214yes—
27C2×#432C2212yes—
28#2861C2212yes—
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2621C2211yescharacteristic
32#2620C2212yes—
33#2619C2211yescharacteristic
34#2602—22no—
35#2602—22no—
36#2601—22no—
37#2601—22no—
38#2360—22no—
39#2360—22no—
40#2355—22no—
41Q8×C4⋊C4—22no—
42#2330C2212yes—
43#2330C2212yes—
44#2328C2211yescharacteristic
45#2327C2211yescharacteristic
46#2326C2212yes—
47#2317C2212yes—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2212yes—
50C2×#384C2211yescharacteristic
51#2274C2212yes—
52C4×#117C2211yescharacteristic
53#2084C2214yes—
54#2083C2214yes—
55#2068C2214yes—
56#2067C2214yes—
57#2066C2214yes—
58#2065C2214yes—
59#2042C2214yes—
60#2042C2214yes—
61#1992C2212yes—
62#1992C2212yes—
63#1982C2214yes—
64#1981C2214yes—
65#1959C2212yes—
66#1959C2212yes—
67#1949C2214yes—
68#1949C2214yes—
69#1928C2214yes—
70#1927C2214yes—
71#1926C2214yes—
72#1925C2214yes—
73#1874C2214yes—
74#1872C2214yes—
75#1843C2212yes—
76#1832C2212yes—
77#1803C2214yes—
78#1802C2214yes—
79#1798C2212yes—
80#1798C2212yes—
81#1767C2212yes—
82#1767C2211yescharacteristic
83#1767C2211yescharacteristic
84#533—24no—
85#533D418yescomplemented
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89#529—28no—
90#529—28no—
91C2×#126—22no—
92C2×#125C2312yes—
93C2×#125C2312yes—
94C2×#124C2312yes—
95C2×#124C2312yes—
96C2×#123—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8C2312yes—
99C2×#121C2312yes—
100C2×C4×Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8C2312yes—
104C2×C4×D4—22no—
105C2×C4×D4C2312yes—
106C2×#117—22no—
107C2×#117C2312yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C22×C4⋊C4—22no—
111C22×C4⋊C4C2312yes—
112C22×C42C2311yescharacteristic
113#501—28no—
114#501—28no—
115#489—28no—
116#489—28no—
117#485—216no—
118#482—216no—
119#479—216no—
120#478—216no—
121#432—28no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2312yes—
140C2×#97C2312yes—
141#398—22no—
142#397—22no—
143#396—22no—
144#396—22no—
145#394—22no—
146#394—22no—
147#394—22no—
148#392—22no—
149#391—22no—
150#391—22no—
151#391C2312yes—
152#391C2312yes—
153#390—22no—
154#390—22no—
155#390C2312yes—
156#390C2312yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—24no—
162#389—24no—
163#389C2314yes—
164#389C2314yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#387—22no—
172#387—22no—
173#387—24no—
174#387—24no—
175#386—22no—
176#386—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#384—22no—
182#384—22no—
183#384C2312yes—
184#384C2312yes—
185#384C2312yes—
186#382—22no—
187#382C2311yescharacteristic
188#382C2311yescharacteristic
189C4×C4⋊C4—24no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4D412yes—
194C4×C4⋊C4C2314yes—
195C4×C4⋊C4C2314yes—
196C4×C22⋊C4—24no—
197C4×C22⋊C4D412yes—
198C4×C22⋊C4D412yes—
199#376C2311yescharacteristic
200#376C2311yescharacteristic
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#336C2314yes—
208#336C2312yes—
209#336C2312yes—
210C22×Q8—22no—
211C22×Q8C2412yes—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4C2412yes—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4C2×D412yes—
218C23×C4C2411yescharacteristic
219C22×C8C2414yes—
220C22×C8C2414yes—
221C4⋊Q8—24no—
222C4⋊Q8—24no—
223C4⋊Q8C2×D418yes—
224#127—24no—
225#126—48no—
226#125—24no—
227#125—24no—
228#125—24no—
229#125C2×D418yes—
230#124—24no—
231#124—24no—
232#124—44no—
233#123—48no—
234C22⋊Q8—216no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237#121—24no—
238#121—24no—
239C4×Q8—216no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—84no—
243C4×Q8—84no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—48no—
247#117—24no—
248#117—24no—
249#117—42no—
250#117—48no—
251#117C2×D418yes—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—84no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2×D411yescharacteristic
272C2×C4⋊C4C2414yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2414yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
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⋊C4C2×D412yes—
289C2×C22⋊C4C2×D411yescharacteristic
290C2×C22⋊C4C2×D411yescharacteristic
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42C2×D414yes—
301C2×C42#4114yes—
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308#107—28no—
309#107—28no—
310#106—28no—
311#106—28no—
312C4⋊C8—28no—
313C4⋊C8—28no—
314C4⋊C8—42no—
315C4⋊C8—42no—
316#103—28no—
317#103—28no—
318#103—44no—
319#103—44no—
320#102—28no—
321#102—28no—
322#102—44no—
323#102—44no—
324C22⋊C8—28no—
325#97—24no—
326#97—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95C2×D412yes—
338#95C2×D412yes—
339#95C2×D412yes—
340#95C2×D412yes—
341C24—22no—
342C24C22×D411yescharacteristic
343C2×Q8—28no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—82no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4—82no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—84no—
374C22×C4#12118yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D412yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D412yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D412yes—
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C2×#4112yes—
386C22×C4C2511yescharacteristic
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C4⋊C4—216no—
394C4⋊C4—28no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—84no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4C22×D418yes—
419C22⋊C4—28no—
420C22⋊C4—24no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C42—28no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—42no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—44no—
443C42—84no—
444C42—84no—
445C42#12118yes—
446C42C22×D412yes—
447C42C22×D412yes—
448C42C2×#4112yes—
449C42C2×#4112yes—
450C422−1+414yes—
451C23—22no—
452C23—22no—
453C23—21no—
454C23—21no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23C2×#12112yes—
463C23C23×D411yescharacteristic
464Q8—416no—
465Q8—44no—
466Q8—44no—
467D4—44no—
468D4—44no—
469D4—88no—
470D4—164no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—48no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—88no—
509C2×C4—164no—
510C2×C4C2×#12112yes—
511C2×C4C2×#12114yes—
512C2×C4C2×#12114yes—
513C2×C4#53614yes—
514C2×C4C23×D411yescharacteristic
515C2×C4C22×#4111yescharacteristic
516C2×C4C2×2−1+412yes—
517C2×C4#58512yes—
518C8—44no—
519C8—44no—
520C22—24no—
521C22—22no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22C22×#12111yescharacteristic
533C22C2×#53611yescharacteristic
534C22#344911yescharacteristic
535C22#358512yes—
536C22#358612yes—
537C4—24no—
538C4—24no—
539C4—22no—
540C4—24no—
541C4—48no—
542C4—41no—
543C4—41no—
544C4—41no—
545C4—41no—
546C4—42no—
547C4—82no—
548C4—82no—
549C4—84no—
550C4—82no—
551C4#349712yes—
552C4#350712yes—
553C2—42no—
554C2—162no—
555C2#3517811yescharacteristic
556C2#6042512yes—
557C2#6088512yes—
558C2#6101911yescharacteristic
559C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2619C22
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
1. C2×#3240C2semidirect
3. #34214C2semidirect
4. #33117C2semidirect
6. #29769C2semidirect
7. #28754C2semidirect
8. #28692C2semidirect
9. #28659C2semidirect
10. #28658C2semidirect
11. #28294C2semidirect
12. #28292C2semidirect
13. #28215C2semidirect
14. #21867C2semidirect
15. #20811C2semidirect
85. #533D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32402
#353432
#342142
#331172
#331162
#297692
#287542
#286922
#286592
#286582
#282942
#282922
#282152
#218672
#208112