Finite Group Databasebeta

← #6478175

Subgroups — #6478175

Order 512.

1622 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic
2C2×#2345C211yesmaximal, characteristic
3#33149C211yesmaximal, characteristic, complemented
4#33109C211yesmaximal, characteristic, complemented
5#32983C212yesmaximal
6#29969C211yesmaximal, characteristic, complemented
7#29895C212yesmaximal
8#29757C211yesmaximal, characteristic, complemented
9#27460C214yesmaximal, complemented
10#27396C214yesmaximal, complemented
11#27332C214yesmaximal, complemented
12#27164C214yesmaximal, complemented
13#22079C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic
15C4×#484C212yesmaximal, complemented
16C23×C4⋊C4C2211yescharacteristic
17C2×#484C2212yes—
18C2×#484C2212yes—
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#2345—22no—
31#2345—22no—
32#2345C2214yes—
33#2345C2214yes—
34#2329—21no—
35#2312—21no—
36C4×C22⋊Q8C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#387—21no—
40C2×#387—21no—
41C2×#380—21no—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4—21no—
46C2×C4×C22⋊C4C2211yescharacteristic
47#2100C2212yescomplemented
48#2087C2214yescomplemented
49#2083C2212yescomplemented
50#2076C2214yes—
51#2076C2214yes—
52#2068C2214yescomplemented
53#2057C2212yescomplemented
54#2043C2212yescomplemented
55#2036C2214yes—
56#2036C2214yes—
57#1987C2212yescomplemented
58#1981C2212yescomplemented
59#1947C2212yescomplemented
60#1947C2212yescomplemented
61#1942C2214yescomplemented
62#1933C2214yescomplemented
63#1928C2214yescomplemented
64#1928C2214yescomplemented
65#1923C2214yes—
66#1923C2214yes—
67#1921C2214yescomplemented
68#1921C2214yescomplemented
69#1921C2214yescomplemented
70#1872C2214yes—
71#1872C2214yes—
72#1872C2212yes—
73#1872C2212yes—
74#1863C2214yescomplemented
75#1852C2212yes—
76#1852C2214yes—
77#1852C2212yes—
78#1852C2214yes—
79#1824C2212yes—
80#1824C2212yes—
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1779C2212yes—
84C4×#107C2214yescomplemented
85C4×#103C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C24×C4—21no—
88C24×C4C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C2×C22⋊Q8C2312yes—
95C2×C4×Q8C2312yes—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4D411yescharacteristic
109C22×C22⋊C4D411yescharacteristic
110C22×C42—21no—
111C22×C42—21no—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114#484—28no—
115#484—28no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#107C2314yes—
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×C4×C8C2314yes—
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#387—22no—
160#387—22no—
161#387—22no—
162#387—24no—
163#387—24no—
164#387—24no—
165#387—24no—
166#385—22no—
167#385—22no—
168#385—22no—
169#385—22no—
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#380—21no—
173#380—21no—
174#380—21no—
175#380—22no—
176#380—24no—
177#380—24no—
178#380C2312yes—
179#380C2312yes—
180#380C2314yes—
181#380C2312yes—
182#379—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—24no—
193C4×C22⋊C4—24no—
194C4×C22⋊C4D412yes—
195C4×C22⋊C4D412yes—
196C4×C22⋊C4C2312yes—
197C4×C22⋊C4C2312yes—
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—28no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—82no—
230C23×C4—82no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—42no—
242C4⋊Q8—42no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—44no—
246C22⋊Q8—44no—
247C22⋊Q8—44no—
248C4×Q8—44no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—82no—
279C2×C4⋊C4—82no—
280C2×C4⋊C4#4114yes—
281C2×C4⋊C4#4114yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C22⋊C4—24no—
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—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4Q1618yes—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D412yes—
302C2×C22⋊C4C2×D412yes—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—82no—
311C2×C42—82no—
312C2×C42C2×D412yes—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318#107—28no—
319#107—28no—
320#107—44no—
321#107—44no—
322C4⋊C8—42no—
323C4⋊C8—42no—
324#103—28no—
325#103—28no—
326#103—44no—
327#103—44no—
328C22⋊C8—24no—
329C22⋊C8—24no—
330C22⋊C8—44no—
331C22⋊C8—44no—
332C4×C8—44no—
333#95—24no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—24no—
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#95—22no—
350#95C2×D412yes—
351#95C2×D412yes—
352#95C2×D412yes—
353#95C2×D412yes—
354C24—22no—
355C24—22no—
356C24—41no—
357C24—41no—
358C24—41no—
359C24—41no—
360C24C2×Q1614yes—
361C24C22×D412yes—
362C24C22×D411yescharacteristic
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—42no—
368C2×Q8—81no—
369C2×Q8—81no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—216no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—48no—
404C22×C4—48no—
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—42no—
414C22×C4—42no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4—816no—
418C22×C4C2×Q1614yes—
419C22×C4C2×Q1614yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2511yescharacteristic
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C4⋊C4—216no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—82no—
458C4⋊C4—84no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4C2×#4114yes—
468C4⋊C4C2×#4114yes—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4C2×Q1618yes—
479C22⋊C4C2×Q1618yes—
480C42—24no—
481C42—22no—
482C42—22no—
483C42—42no—
484C42—41no—
485C42—41no—
486C42—82no—
487C42—84no—
488C42—82no—
489C42—82no—
490C42—82no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—28no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23#52512yes—
509C23C22×Q1614yes—
510C23C22×Q1612yes—
511C23#57512yes—
512C23C23×D411yescharacteristic
513Q8—44no—
514Q8—44no—
515Q8—162no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
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—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—816no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4#52512yes—
576C2×C4C22×Q1614yes—
577C2×C4C22×Q1612yes—
578C2×C4C22×Q1612yes—
579C2×C4C22×#4111yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4#58512yes—
582C2×C4#58512yes—
583C8—44no—
584C8—44no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—416no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22#310414yes—
600C22C2×#52511yescharacteristic
601C22#348511yescharacteristic
602C22#348511yescharacteristic
603C22C23×Q1611yescharacteristic
604C22C2×#57511yescharacteristic
605C22#358511yescharacteristic
606C22#358511yescharacteristic
607C4—24no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—42no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—44no—
616C4—44no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4#345512yes—
624C4#349312yes—
625C2—24no—
626C2—44no—
627C2#3509311yescharacteristic
628C2C2×#310411yescharacteristic
629C2#6056411yescharacteristic
630C2#6169811yescharacteristic
631C2#6169811yescharacteristic
632C2#6171311yescharacteristic
633C2#6171811yescharacteristic
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. #33149C2semidirect
4. #33109C2semidirect
6. #29969C2semidirect
8. #29757C2semidirect
9. #27460C2semidirect
10. #27396C2semidirect
11. #27332C2semidirect
12. #27164C2semidirect
13. #22079C2semidirect
15. C4×#484C2semidirect
47. #2100C22semidirect
48. #2087C22semidirect
49. #2083C22semidirect
52. #2068C22semidirect
53. #2057C22semidirect
54. #2043C22semidirect
57. #1987C22semidirect
58. #1981C22semidirect
59. #1947C22semidirect
60. #1947C22semidirect
61. #1942C22semidirect
62. #1933C22semidirect
63. #1928C22semidirect
64. #1928C22semidirect
67. #1921C22semidirect
68. #1921C22semidirect
69. #1921C22semidirect
74. #1863C22semidirect
84. C4×#107C22semidirect
85. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#23452
#331492
#331092
#329832
#299692
#298952
#297572
#274602
#273962
#273322
#271642
#220792
#219022
C4×#4842