Finite Group Databasebeta

← #6510726

Subgroups — #6510726

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35079C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic
3#29111C212yesmaximal, complemented
4#29106C212yesmaximal, complemented
5#27966C212yesmaximal, complemented
6#27960C212yesmaximal, complemented
7#27721C214yesmaximal, complemented
8#27715C214yesmaximal, complemented
9#27707C214yesmaximal, complemented
10#27697C214yesmaximal, complemented
11#21765C211yesmaximal, characteristic
12#21762C211yesmaximal, characteristic
13#20700C211yesmaximal, characteristic
14C2×#1823C211yesmaximal, characteristic
15#20525C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2857C2212yes—
21#2851C2212yes—
22#2667—21no—
23#2665—21no—
24#2569—21no—
25#2549—21no—
26#2502—22no—
27#2502—21no—
28#2502—21no—
29#2502C2212yes—
30#2502C2214yes—
31#2502C2212yes—
32#2442C2212yes—
33#2441C2212yes—
34#2440C2212yes—
35#2439C2212yes—
36#2437—22no—
37C2×#398—21no—
38C2×#398—21no—
39#2329—22no—
40C2×#391C2212yes—
41C2×#385—22no—
42C2×#384C2212yes—
43#2280C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#2100C2214yescomplemented
47#2100C2214yescomplemented
48#2099C2214yescomplemented
49#2099C2214yescomplemented
50#2074C2214yescomplemented
51#2072C2214yescomplemented
52#2026C2214yescomplemented
53#2024C2214yescomplemented
54#1992C2214yescomplemented
55#1990C2214yescomplemented
56#1935C2214yescomplemented
57#1935C2214yescomplemented
58#1933C2214yescomplemented
59#1933C2214yescomplemented
60#1894C2212yes—
61#1894C2212yes—
62#1892C2214yes—
63#1873C2214yes—
64#1872C2214yes—
65#1836C2214yescomplemented
66#1834C2214yescomplemented
67#1824C2214yes—
68#1823C2211yescharacteristic
69#1823C2211yescharacteristic
70#1823C2212yes—
71#1823C2212yes—
72#1823C2212yes—
73#1823C2212yes—
74#1823C2212yes—
75#1823C2212yes—
76#1823C2212yes—
77#1787C2211yescharacteristic
78#1787C2211yescharacteristic
79#1787C2212yes—
80#1753C2214yes—
81#1752C2212yes—
82C8×C22⋊C4C2212yes—
83#1749C2212yes—
84C4×C22⋊C8C2212yes—
85#1736C2214yes—
86C2×#340C2214yes—
87C23×Q8C2311yescharacteristic
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C22×M16C2312yes—
91C23×C8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×#125C2312yes—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×#117C2312yes—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—22no—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4D412yes—
119C22×C42C2312yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124#407—24no—
125C2×C22⋊C8—24no—
126C2×C22⋊C8C2311yescharacteristic
127C2×C22⋊C8C2311yescharacteristic
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×#97C2314yes—
132C2×C4×C8C2314yes—
133#400—22no—
134#399—22no—
135#398—22no—
136#398—24no—
137#398—24no—
138#398—22no—
139#398—22no—
140#398—22no—
141#398—22no—
142#398—22no—
143#396—22no—
144#396—22no—
145#395—22no—
146#393—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391C2314yes—
152#391C2314yes—
153#389—22no—
154#389—22no—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#385—22no—
164#385—22no—
165#385—22no—
166#385—22no—
167#385—24no—
168#385—24no—
169#385—24no—
170#385—24no—
171#384—22no—
172#384—22no—
173#384C2312yes—
174#384C2314yes—
175#384C2314yes—
176#382C2312yes—
177#381—21no—
178#380—22no—
179#380—21no—
180#380—21no—
181#380—22no—
182#380C2311yescharacteristic
183#380C2312yes—
184#380C2311yescharacteristic
185#380C2311yescharacteristic
186#380C2311yescharacteristic
187#380C2312yes—
188#379—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—21no—
191C4×C22⋊C4D412yes—
192C4×C22⋊C4D411yescharacteristic
193C4×C22⋊C4D411yescharacteristic
194C4×C22⋊C4C2312yes—
195C4×C22⋊C4C2311yescharacteristic
196C4×C22⋊C4C2311yescharacteristic
197C2×#95—21no—
198C2×#95—22no—
199C2×#95—22no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—44no—
222C22×Q8C2412yes—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—82no—
232C23×C4—82no—
233C23×C4C2×D412yes—
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4#4114yes—
239C23×C4#4114yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×M16—24no—
244C2×M16—44no—
245C22×C8—24no—
246C22×C8—44no—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C4⋊Q8—44no—
251#125—44no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258C22⋊Q8—48no—
259C22⋊Q8—84no—
260#117—44no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—84no—
295C2×C4⋊C4—84no—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—82no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C42—22no—
319C2×C42—44no—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2412yes—
323C2×C42C2412yes—
324#103—44no—
325#103—44no—
326#103—44no—
327#103—44no—
328C22⋊C8—22no—
329C22⋊C8—22no—
330C22⋊C8—24no—
331C22⋊C8—24no—
332C22⋊C8—24no—
333#97—44no—
334C4×C8—44no—
335#95—21no—
336#95—22no—
337#95—21no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—21no—
343#95—21no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95C2×D412yes—
352#95C2×D412yes—
353#95C2×D411yescharacteristic
354#95C2×D411yescharacteristic
355#95C2×D411yescharacteristic
356#95C2×D411yescharacteristic
357C24—24no—
358C24—22no—
359C24—22no—
360C24—42no—
361C24#12414yes—
362C24C22×D411yescharacteristic
363C24C22×D411yescharacteristic
364C24C22×D411yescharacteristic
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—44no—
370C2×Q8—42no—
371C2×Q8—44no—
372C2×Q8—82no—
373C2×Q8—82no—
374C2×Q8—84no—
375C2×Q8—84no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—816no—
415C22×C4#12414yes—
416C22×C4#12414yes—
417C22×C4#12414yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2511yescharacteristic
432M16—84no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—84no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—82no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—82no—
481C22⋊C4#13718yes—
482C42—24no—
483C42—24no—
484C42—42no—
485C42—42no—
486C42—82no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—21no—
491C23—21no—
492C23—28no—
493C23—28no—
494C23—22no—
495C23—22no—
496C23—21no—
497C23—21no—
498C23—44no—
499C23—48no—
500C23—48no—
501C23C2×#12412yes—
502C23C2×#12412yes—
503C23C2×#12412yes—
504C23C2×#13714yes—
505C23#57612yes—
506C23#57712yes—
507C23C23×D411yescharacteristic
508Q8—44no—
509Q8—42no—
510Q8—42no—
511Q8—84no—
512Q8—164no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—48no—
536C2×C4—48no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—84no—
558C2×C4—88no—
559C2×C4—88no—
560C2×C4—84no—
561C2×C4—88no—
562C2×C4—164no—
563C2×C4C2×#12412yes—
564C2×C4C2×#12412yes—
565C2×C4C2×#12412yes—
566C2×C4C2×#13714yes—
567C2×C4C2×#13714yes—
568C2×C4C22×#4111yescharacteristic
569C2×C4C22×#4111yescharacteristic
570C2×C4#58514yes—
571C8—42no—
572C8—42no—
573C8—44no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—416no—
586C22—42no—
587C22—42no—
588C22#313614yes—
589C22#313914yes—
590C22C22×#12411yescharacteristic
591C22#348512yes—
592C22C22×#13712yes—
593C22C2×#57611yescharacteristic
594C22C2×#57711yescharacteristic
595C4—24no—
596C4—24no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—42no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—84no—
609C4#349112yes—
610C4#349112yes—
611C2—24no—
612C2—24no—
613C2—42no—
614C2#3545511yescharacteristic
615C2C2×#313611yescharacteristic
616C2C2×#313911yescharacteristic
617C2#6172612yes—
618C2#6173512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #29111C2semidirect
4. #29106C2semidirect
5. #27966C2semidirect
6. #27960C2semidirect
7. #27721C2semidirect
8. #27715C2semidirect
9. #27707C2semidirect
10. #27697C2semidirect
46. #2100C22semidirect
47. #2100C22semidirect
48. #2099C22semidirect
49. #2099C22semidirect
50. #2074C22semidirect
51. #2072C22semidirect
52. #2026C22semidirect
53. #2024C22semidirect
54. #1992C22semidirect
55. #1990C22semidirect
56. #1935C22semidirect
57. #1935C22semidirect
58. #1933C22semidirect
59. #1933C22semidirect
65. #1836C22semidirect
66. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#350792
C2×#25022
#291112
#291062
#279662
#279602
#277212
#277152
#277072
#276972
#217652
#217622
#207002
C2×#18232
#205252