Finite Group Databasebeta

← #6478189

Subgroups — #6478189

Order 512.

1534 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic, complemented
4#33108C211yesmaximal, characteristic, complemented
5#32984C212yesmaximal
6#29969C211yesmaximal, characteristic, complemented
7#29894C212yesmaximal
8#29757C211yesmaximal, characteristic, complemented
9#27480C214yesmaximal, complemented
10#27416C214yesmaximal, complemented
11#27352C214yesmaximal, complemented
12#27184C214yesmaximal, complemented
13#22085C212yesmaximal, complemented
14#21906C211yesmaximal, characteristic
15#21579C212yesmaximal, complemented
16C22×#117C2211yescharacteristic
17C2×#484C2212yes—
18C2×#483C2212yes—
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21#2848C2212yes—
22#2646—21no—
23#2626—21no—
24#2589—22no—
25#2587—22no—
26#2497C2212yes—
27#2492C2212yes—
28#2429C2212yes—
29#2423C2212yes—
30#2346—22no—
31#2346—22no—
32#2346C2214yes—
33#2346C2214yes—
34#2329—21no—
35#2312—21no—
36C4×C22⋊Q8C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#388—21no—
40C2×#388—21no—
41C2×#382—21no—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×#376—21no—
47#2100C2212yescomplemented
48#2087C2214yescomplemented
49#2083C2212yescomplemented
50#2077C2214yes—
51#2075C2214yes—
52#2068C2214yescomplemented
53#2057C2212yescomplemented
54#2043C2212yescomplemented
55#2036C2214yes—
56#2035C2214yes—
57#1987C2212yescomplemented
58#1981C2212yescomplemented
59#1948C2212yescomplemented
60#1948C2212yescomplemented
61#1943C2214yescomplemented
62#1935C2214yescomplemented
63#1928C2214yescomplemented
64#1927C2214yescomplemented
65#1924C2214yes—
66#1924C2214yes—
67#1920C2214yescomplemented
68#1919C2214yescomplemented
69#1918C2214yescomplemented
70#1873C2214yes—
71#1873C2214yes—
72#1872C2212yes—
73#1872C2212yes—
74#1865C2214yescomplemented
75#1854C2214yes—
76#1854C2214yes—
77#1853C2212yes—
78#1852C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1779C2212yes—
84#1776C2214yescomplemented
85#1763C2214yescomplemented
86#1749C2212yes—
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8C2312yes—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C2×C4×Q8C2312yes—
95C2×#117—22no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117C2314yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4D411yescharacteristic
108C22×C22⋊C4D411yescharacteristic
109C22×C42—21no—
110C22×C42—21no—
111C22×C42—21no—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114#484—28no—
115#483—28no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130C2×#97C2314yes—
131#400—22no—
132#400—22no—
133#399—22no—
134#398—22no—
135#398—22no—
136#396—22no—
137#396—22no—
138#393—22no—
139#391—21no—
140#391—22no—
141#391C2312yes—
142#391C2312yes—
143#389—21no—
144#389—21no—
145#389—21no—
146#389—22no—
147#389—21no—
148#389—21no—
149#389—22no—
150#389C2312yes—
151#389C2312yes—
152#389C2312yes—
153#389C2314yes—
154#389C2312yes—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—24no—
160#388—24no—
161#388—24no—
162#388—24no—
163#387—22no—
164#387—22no—
165#387—22no—
166#385—22no—
167#385—22no—
168#385—22no—
169#385—22no—
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#382—24no—
173#382—24no—
174#380—21no—
175#380—21no—
176#380—21no—
177#380—22no—
178#380C2312yes—
179#380C2314yes—
180#380C2312yes—
181#380C2312yes—
182#379—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4C2314yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192C4×C22⋊C4D412yes—
193C4×C22⋊C4D412yes—
194C4×C22⋊C4C2312yes—
195C4×C22⋊C4C2312yes—
196#376—24no—
197#376—24no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95D411yescharacteristic
204C2×#95D411yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2312yes—
214#336C2312yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C22×Q8—21no—
218C22×Q8C2412yes—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C22×C8C2414yes—
236C22×C8C2414yes—
237C4⋊Q8—42no—
238C4⋊Q8—42no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—44no—
242C22⋊Q8—44no—
243C22⋊Q8—44no—
244C4×Q8—44no—
245#117—216no—
246#117—28no—
247#117—28no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—82no—
273C2×C4⋊C4—82no—
274C2×C4⋊C4#4114yes—
275C2×C4⋊C4#4114yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—82no—
295C2×C22⋊C4C2×D412yes—
296C2×C22⋊C4C2×D412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D412yes—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42C2×D412yes—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2412yes—
314C2×C42C2411yescharacteristic
315C2×C42C2411yescharacteristic
316#107—28no—
317#107—44no—
318#106—28no—
319#106—44no—
320C4⋊C8—42no—
321C4⋊C8—42no—
322#103—28no—
323#103—28no—
324#103—44no—
325#103—44no—
326C22⋊C8—24no—
327C22⋊C8—24no—
328C22⋊C8—44no—
329C22⋊C8—44no—
330#97—44no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—24no—
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#95C2×D412yes—
349#95C2×D412yes—
350#95C2×D412yes—
351#95C2×D412yes—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—41no—
356C24—41no—
357C24—41no—
358C24—41no—
359C24C22×D412yes—
360C24C22×D411yescharacteristic
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—42no—
366C2×Q8—81no—
367C2×Q8—81no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4#13714yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2511yescharacteristic
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—82no—
451C4⋊C4—84no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4C2×#4114yes—
461C4⋊C4C2×#4114yes—
462C22⋊C4—216no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—82no—
470C22⋊C4—82no—
471C22⋊C4—84no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C42—216no—
475C42—24no—
476C42—22no—
477C42—22no—
478C42—41no—
479C42—41no—
480C42—42no—
481C42—48no—
482C42—44no—
483C42—44no—
484C42—82no—
485C42—82no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—82no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23#52512yes—
505C23C2×#13712yes—
506C23#57512yes—
507C23C23×D411yescharacteristic
508Q8—44no—
509Q8—44no—
510Q8—162no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—44no—
537C2×C4—48no—
538C2×C4—44no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4#52512yes—
571C2×C4C2×#13712yes—
572C2×C4C2×#13712yes—
573C2×C4C22×#4111yescharacteristic
574C2×C4C22×#4111yescharacteristic
575C2×C4#58512yes—
576C2×C4#58512yes—
577C8—44no—
578C8—44no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—88no—
593C22#311614yes—
594C22C2×#52511yescharacteristic
595C22#348511yescharacteristic
596C22#348511yescharacteristic
597C22C22×#13711yescharacteristic
598C22C2×#57511yescharacteristic
599C22#358511yescharacteristic
600C22#358611yescharacteristic
601C4—22no—
602C4—22no—
603C4—22no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—44no—
610C4—44no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4#345512yes—
618C4#349312yes—
619C2—24no—
620C2—82no—
621C2#3509311yescharacteristic
622C2C2×#311611yescharacteristic
623C2#6056511yescharacteristic
624C2#6170511yescharacteristic
625C2#6171011yescharacteristic
626C2#6171111yescharacteristic
627C2#6172111yescharacteristic
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. #33148C2semidirect
4. #33108C2semidirect
6. #29969C2semidirect
8. #29757C2semidirect
9. #27480C2semidirect
10. #27416C2semidirect
11. #27352C2semidirect
12. #27184C2semidirect
13. #22085C2semidirect
15. #21579C2semidirect
47. #2100C22semidirect
48. #2087C22semidirect
49. #2083C22semidirect
52. #2068C22semidirect
53. #2057C22semidirect
54. #2043C22semidirect
57. #1987C22semidirect
58. #1981C22semidirect
59. #1948C22semidirect
60. #1948C22semidirect
61. #1943C22semidirect
62. #1935C22semidirect
63. #1928C22semidirect
64. #1927C22semidirect
67. #1920C22semidirect
68. #1919C22semidirect
69. #1918C22semidirect
74. #1865C22semidirect
84. #1776C22semidirect
85. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#23462
#331482
#331082
#329842
#299692
#298942
#297572
#274802
#274162
#273522
#271842
#220852
#219062
#215792