Finite Group Databasebeta

← #6436721

Subgroups — #6436721

Order 512.

1912 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3126C212yesmaximal, complemented
2#35278C211yesmaximal, characteristic
3#34254C211yesmaximal, characteristic, complemented
4#29871C212yesmaximal, complemented
5#29835C214yesmaximal, complemented
6#29796C212yesmaximal, complemented
7#29765C212yesmaximal, complemented
8#29758C212yesmaximal, complemented
9#29722C212yesmaximal, complemented
10#29713C212yesmaximal, complemented
11#23690C214yesmaximal, complemented
12#23684C212yesmaximal, complemented
13#23678C212yesmaximal, complemented
14#21399C212yesmaximal, complemented
15#20813C211yesmaximal, characteristic, complemented
16C2×#540C2212yes—
17C2×#519C2211yescharacteristic
18#3126—216no—
19C2×#482C2214yes—
20C2×#479C2214yes—
21C2×#460C2214yes—
22C2×#448C2214yes—
23C2×#440C2214yes—
24C2×C4×SD16C2214yes—
25C2×#433C2212yes—
26#2867C2211yescharacteristic
27#2859C2211yescharacteristic
28#2856C2212yes—
29#2641C2212yes—
30#2619C2212yes—
31#2445—22no—
32#2445—22no—
33#2428—22no—
34#2428—22no—
35#2356—22no—
36#2355—22no—
37#2343—22no—
38#2343—22no—
39#2328C2212yes—
40#2320C2212yes—
41#2320C2212yes—
42#2317C2211yescharacteristic
43#2310C2211yescharacteristic
44C4×#127C2212yes—
45#2296C2211yescharacteristic
46#2295C2212yes—
47#2295C2212yes—
48C2×#391C2211yescharacteristic
49#2274C2211yescharacteristic
50C4×#117C2211yescharacteristic
51#2085C2214yes—
52#2084C2212yes—
53#2083C2212yes—
54#2074C2214yes—
55#2072C2212yes—
56#2072C2212yes—
57#2067C2214yes—
58#2067C2214yes—
59#2065C2214yes—
60#2065C2214yes—
61#2055C2214yes—
62#2055C2214yes—
63#2055C2214yes—
64#2055C2214yes—
65#2041C2212yes—
66#2041C2212yes—
67#2041C2214yes—
68#2014C2212yes—
69#2014C2212yes—
70#2014C2214yes—
71#1989C2214yes—
72#1982C2214yes—
73#1875C2214yes—
74#1874C2214yes—
75#1843C2212yes—
76#1831C2212yes—
77#1798C2212yes—
78#1797C2212yes—
79#1768C2212yes—
80#1767C2211yescharacteristic
81C4×C4⋊C8C2211yescharacteristic
82#1762C2214yes—
83C4×#102C2214yes—
84C23×Q8C2311yescharacteristic
85C22×SD16C2314yes—
86C22×SD16C2314yes—
87#540—28no—
88#540—28no—
89C2×#127—22no—
90C2×#127—22no—
91C2×#127C2312yes—
92C2×#127C2312yes—
93C2×#127C2312yes—
94C2×#125C2312yes—
95C2×#123—24no—
96C2×#121—22no—
97C2×#121—22no—
98C2×#121—24no—
99C2×#121C2314yes—
100#519—24no—
101#519D418yescomplemented
102C2×C4×Q8—22no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4C2314yes—
108C2×#117—22no—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#482—216no—
116#479—216no—
117#460—216no—
118#448—216no—
119#440—216no—
120C4×SD16—216no—
121#433—28no—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2312yes—
138C2×C4×C8C2312yes—
139#394—22no—
140#394—22no—
141#394—24no—
142#394—24no—
143#392—22no—
144#392—22no—
145#391—21no—
146#391—21no—
147#391—22no—
148#391C2312yes—
149#391C2312yes—
150#391C2312yes—
151#391C2312yes—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390C2314yes—
157#390C2314yes—
158#389—22no—
159#389—22no—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#386—24no—
173#386—24no—
174#385—22no—
175#385—21no—
176#385—21no—
177#384—21no—
178#384C2312yes—
179#383—22no—
180#383—22no—
181#382—21no—
182#382—22no—
183#382—22no—
184#382C2312yes—
185#381—21no—
186#381—21no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—22no—
196C4×C22⋊C4D412yes—
197C4×C22⋊C4D412yes—
198#376—22no—
199#376—22no—
200#376C2311yescharacteristic
201#376C2311yescharacteristic
202C2×#95—22no—
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2312yes—
210#336C2312yes—
211#336C2314yes—
212C22×Q8—21no—
213C22×Q8—22no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8C2×D414yes—
217C22×Q8C2412yes—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—24no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4C2414yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—24no—
228C23×C4C2×D412yes—
229C23×C4C2411yescharacteristic
230C2×SD16—28no—
231C2×SD16—28no—
232C2×SD16—44no—
233C2×SD16—44no—
234C2×SD16—44no—
235C2×SD16—44no—
236C22×C8C2414yes—
237C22×C8C2414yes—
238#127—24no—
239#127—24no—
240#127—44no—
241#127—42no—
242#127—42no—
243#127—84no—
244#127—84no—
245#125—24no—
246#125—24no—
247#123—416no—
248#121—28no—
249#121—28no—
250#121—416no—
251#121—84no—
252#121—84no—
253C4×Q8—216no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8C2×D418yes—
257C4×Q8C2×D418yes—
258C4×D4—28no—
259C4×D4—28no—
260C4×D4—84no—
261C4×D4—84no—
262#117—22no—
263#117—216no—
264#117—42no—
265#117—42no—
266#117C2×D414yes—
267#117C2×D414yes—
268#117#4114yes—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4C2×D411yescharacteristic
296C2×C22⋊C4C2×D411yescharacteristic
297C2×C22⋊C4C2×D412yes—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—42no—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316#107—28no—
317#106—28no—
318C4⋊C8—28no—
319C4⋊C8—44no—
320C4⋊C8—42no—
321C4⋊C8—42no—
322#103—28no—
323#103—28no—
324#102—28no—
325#102—28no—
326#102—44no—
327#102—44no—
328#102—44no—
329#102—44no—
330C22⋊C8—28no—
331#97—24no—
332C4×C8—24no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
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—22no—
347#95—22no—
348#95—22no—
349#95C2×D412yes—
350#95C2×D412yes—
351#95C2×D412yes—
352#95C2×D412yes—
353C24—24no—
354C24C22×D411yescharacteristic
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—48no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—48no—
363C2×Q8C22×D414yes—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—84no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—48no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—82no—
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C2×#4112yes—
415C22×C4C2511yescharacteristic
416SD16—416no—
417SD16—416no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4#12118yes—
442C4⋊C4C22×D414yes—
443C22⋊C4—24no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—42no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C22⋊C4—84no—
453C22⋊C4#12118yes—
454C42—28no—
455C42—28no—
456C42—24no—
457C42—24no—
458C42—24no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—82no—
468C42—82no—
469C42C22×D412yes—
470C42C22×D412yes—
471C42C2×#4112yes—
472C42C2×#4112yes—
473C23—22no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23C2×#12112yes—
485C23C23×D411yescharacteristic
486Q8—28no—
487Q8—48no—
488Q8—44no—
489D4—44no—
490D4—44no—
491D4—44no—
492D4—44no—
493D4—88no—
494D4—88no—
495D4—168no—
496D4—164no—
497D4—164no—
498D4—168no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—48no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—84no—
537C2×C4—168no—
538C2×C4—162no—
539C2×C4C2×#12114yes—
540C2×C4C2×#12114yes—
541C2×C4C2×#12112yes—
542C2×C4#57714yes—
543C2×C4C23×D411yescharacteristic
544C2×C4C22×#4111yescharacteristic
545C2×C4#58512yes—
546C2×C4#58512yes—
547C8—44no—
548C8—44no—
549C22—24no—
550C22—22no—
551C22—42no—
552C22—41no—
553C22—41no—
554C22—84no—
555C22—84no—
556C22—84no—
557C22—84no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22C22×#12111yescharacteristic
562C22#344911yescharacteristic
563C22#344911yescharacteristic
564C22C2×#57712yes—
565C22#358512yes—
566C4—24no—
567C4—24no—
568C4—24no—
569C4—22no—
570C4—44no—
571C4—42no—
572C4—44no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—84no—
578C4—82no—
579C4—82no—
580C4#344912yes—
581C4#345512yes—
582C2—42no—
583C2—164no—
584C2#3463311yescharacteristic
585C2#6043112yes—
586C2#6097912yes—
587C2#6101211yescharacteristic
588C2#6101711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35278C2
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×#3126C2semidirect
3. #34254C2semidirect
4. #29871C2semidirect
5. #29835C2semidirect
6. #29796C2semidirect
7. #29765C2semidirect
8. #29758C2semidirect
9. #29722C2semidirect
10. #29713C2semidirect
11. #23690C2semidirect
12. #23684C2semidirect
13. #23678C2semidirect
14. #21399C2semidirect
15. #20813C2semidirect
101. #519D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31262
#352782
#342542
#298712
#298352
#297962
#297652
#297582
#297222
#297132
#236902
#236842
#236782
#213992
#208132