Finite Group Databasebeta

← #6508977

Subgroups — #6508977

Order 512.

1666 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34809C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27618C214yesmaximal, complemented
4#27604C214yesmaximal, complemented
5#24952C214yesmaximal, complemented
6#24950C214yesmaximal, complemented
7#24936C214yesmaximal, complemented
8#24934C214yesmaximal, complemented
9#21943C211yesmaximal, characteristic, complemented
10#21939C211yesmaximal, characteristic, complemented
11#21740C211yesmaximal, characteristic
12#20701C211yesmaximal, characteristic, complemented
13#20699C211yesmaximal, characteristic, complemented
14C22×C4×D4C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#407C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#2487C2212yes—
19#2486C2212yes—
20#2485C2214yes—
21#2484—21no—
22#2483—21no—
23#2482—21no—
24#2405—21no—
25#2395—22no—
26#2353—22no—
27#2353—22no—
28#2352C2214yes—
29#2352C2214yes—
30#2313—21no—
31#2312—21no—
32#2308—21no—
33C4×#123—22no—
34#2296C2212yes—
35#2283C2212yes—
36#2280C2211yescharacteristic
37#2280C2211yescharacteristic
38#2280C2211yescharacteristic
39#2280C2212yes—
40C2×#380C2212yes—
41#2273—21no—
42#2101C2214yescomplemented
43#2101C2214yescomplemented
44#2085C2214yescomplemented
45#2085C2214yescomplemented
46#2077C2214yescomplemented
47#2077C2214yescomplemented
48#2075C2214yescomplemented
49#2075C2214yescomplemented
50#1921C2214yescomplemented
51#1920C2214yescomplemented
52#1919C2214yescomplemented
53#1918C2214yescomplemented
54#1870C2214yes—
55#1869C2212yes—
56#1869C2212yes—
57#1854C2214yes—
58#1854C2214yes—
59#1854C2214yescomplemented
60#1854C2214yescomplemented
61#1853C2214yescomplemented
62#1852C2214yescomplemented
63#1825C2214yes—
64#1825C2214yes—
65#1825C2214yescomplemented
66#1824C2214yescomplemented
67#1824C2214yescomplemented
68#1824C2212yescomplemented
69#1823C2212yescomplemented
70#1817C2212yes—
71#1809C2214yescomplemented
72#1808C2212yescomplemented
73#1807C2212yescomplemented
74#1779C2212yes—
75#1736C2212yescomplemented
76#1736C2212yescomplemented
77#1735C2212yescomplemented
78#1735C2212yescomplemented
79C23×D4D411yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×M16C2312yes—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×#123—22no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4D412yes—
89C2×C4×D4D412yes—
90C2×C4×D4C2312yes—
91C2×C4×D4C2312yes—
92C2×#117—21no—
93C2×#117—21no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117C2312yes—
97C2×#117C2312yes—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C2×#117C2312yes—
101C22×C4⋊C4D411yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4D411yescharacteristic
108C22×C22⋊C4D411yescharacteristic
109C22×C42—21no—
110C22×C42C2311yescharacteristic
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×#106C2314yes—
115#407—24no—
116C2×C22⋊C8—24no—
117C2×C22⋊C8C2312yes—
118C2×C22⋊C8C2312yes—
119C2×C22⋊C8C2312yes—
120C2×C22⋊C8C2312yes—
121C2×C22⋊C8C2314yes—
122C2×C22⋊C8C2314yes—
123#400—22no—
124#399—22no—
125#398—22no—
126#397—22no—
127#397—24no—
128#394—24no—
129#389C2314yes—
130#389C2314yes—
131#388—24no—
132#388—24no—
133#388—24no—
134#387—24no—
135#387—24no—
136#387—24no—
137#386—24no—
138#386—24no—
139#386—22no—
140#385—22no—
141#385—22no—
142#385—22no—
143#383—22no—
144#382C2314yes—
145#381—22no—
146#381—21no—
147#381—21no—
148#381—21no—
149#381—22no—
150#381—22no—
151#380—22no—
152#380—22no—
153#380C2312yes—
154#380C2312yes—
155#380C2312yes—
156#380C2312yes—
157#380C2314yes—
158#380C2314yes—
159#380C2314yes—
160#380C2312yes—
161#380C2312yes—
162#379—21no—
163#379—21no—
164#379—22no—
165#379—21no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4C2314yes—
168C4×C22⋊C4—22no—
169C4×C22⋊C4—21no—
170C4×C22⋊C4—21no—
171C4×C22⋊C4—21no—
172C4×C22⋊C4—21no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—24no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—24no—
177C4×C22⋊C4—24no—
178#376—22no—
179C2×#95—22no—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25C2×D411yescharacteristic
195C25C2×D411yescharacteristic
196C22×D4—24no—
197C22×D4—44no—
198C22×D4C2×D412yes—
199C22×D4C2×D412yes—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—21no—
203C23×C4—24no—
204C23×C4—21no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4#4114yes—
217C23×C4#4114yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C2×M16—24no—
223C2×M16—42no—
224C2×M16—42no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—42no—
228C22×C8—42no—
229C22×C8C2412yes—
230C22×C8C2412yes—
231C22×C8C2414yes—
232#123—48no—
233C4×D4—28no—
234C4×D4—28no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—24no—
239#117—24no—
240#117—24no—
241#117—24no—
242#117—24no—
243#117—42no—
244#117—42no—
245#117—44no—
246#117—44no—
247#117—42no—
248#117—42no—
249#117—42no—
250#117—42no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2414yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C22⋊C4C2×D412yes—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—44no—
301C2×C42C2×D412yes—
302C2×C42C2412yes—
303#107—44no—
304#107—44no—
305#106—44no—
306#106—44no—
307C22⋊C8—24no—
308C22⋊C8—24no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311C22⋊C8—44no—
312C22⋊C8—44no—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327C24—22no—
328C24—24no—
329C24—42no—
330C24—42no—
331C24—42no—
332C24—42no—
333C24—42no—
334C24#12314yes—
335C24C22×D411yescharacteristic
336C24C22×D411yescharacteristic
337C24C22×D411yescharacteristic
338C24C22×D412yes—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—416no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—48no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—48no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—84no—
382C22×C4—82no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4#12314yes—
386C22×C4#12314yes—
387C22×C4#12314yes—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4112yes—
400C22×C4C2511yescharacteristic
401M16—84no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—82no—
410C2×C8—82no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—48no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—48no—
439C22⋊C4—48no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4—82no—
443C22⋊C4—84no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—44no—
450C42—44no—
451C42—44no—
452C42—84no—
453C42—82no—
454C42—82no—
455C23—28no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—82no—
473C23—88no—
474C23—82no—
475C23—88no—
476C23C2×#12312yes—
477C23C2×#12312yes—
478C23C2×#12312yes—
479C23#57514yes—
480C23C23×D411yescharacteristic
481D4—48no—
482D4—48no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—84no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—84no—
526C2×C4—84no—
527C2×C4—168no—
528C2×C4C2×#12312yes—
529C2×C4C2×#12312yes—
530C2×C4C2×#12312yes—
531C2×C4C22×#4111yescharacteristic
532C2×C4C22×#4111yescharacteristic
533C2×C4#58512yes—
534C2×C4#58512yes—
535C8—42no—
536C8—42no—
537C8—44no—
538C22—28no—
539C22—22no—
540C22—22no—
541C22—24no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22—88no—
556C22—88no—
557C22C22×#12311yescharacteristic
558C22#348512yes—
559C22C2×#57512yes—
560C22#358412yes—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—84no—
568C4—84no—
569C4—84no—
570C4—82no—
571C4—82no—
572C4#349111yescharacteristic
573C4#349111yescharacteristic
574C4#349112yes—
575C2—44no—
576C2—42no—
577C2—42no—
578C2—42no—
579C2#3544811yescharacteristic
580C2#6051111yescharacteristic
581C2#6051611yescharacteristic
582C2#6174114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27618C2semidirect
4. #27604C2semidirect
5. #24952C2semidirect
6. #24950C2semidirect
7. #24936C2semidirect
8. #24934C2semidirect
9. #21943C2semidirect
10. #21939C2semidirect
12. #20701C2semidirect
13. #20699C2semidirect
42. #2101C22semidirect
43. #2101C22semidirect
44. #2085C22semidirect
45. #2085C22semidirect
46. #2077C22semidirect
47. #2077C22semidirect
48. #2075C22semidirect
49. #2075C22semidirect
50. #1921C22semidirect
51. #1920C22semidirect
52. #1919C22semidirect
53. #1918C22semidirect
59. #1854C22semidirect
60. #1854C22semidirect
61. #1853C22semidirect
62. #1852C22semidirect
65. #1825C22semidirect
66. #1824C22semidirect
67. #1824C22semidirect
68. #1824C22semidirect
69. #1823C22semidirect
71. #1809C22semidirect
72. #1808C22semidirect
73. #1807C22semidirect
75. #1736C22semidirect
76. #1736C22semidirect
77. #1735C22semidirect
78. #1735C22semidirect

Maximal subgroups

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

subgroupindex
#348092
#341852
#276182
#276042
#249522
#249502
#249362
#249342
#219432
#219392
#217402
#207012
#206992