Finite Group Databasebeta

← #6506554

Subgroups — #6506554

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37450C211yesmaximal, characteristic
2#35151C211yesmaximal, characteristic, complemented
3#29343C212yesmaximal, complemented
4#29340C212yesmaximal, complemented
5#27986C212yesmaximal, complemented
6#27725C214yesmaximal, complemented
7#27703C214yesmaximal, complemented
8#27279C214yesmaximal, complemented
9#27273C214yesmaximal, complemented
10#25915C212yesmaximal, complemented
11#21767C212yesmaximal, complemented
12#20912C212yesmaximal, complemented
13#20529C211yesmaximal, characteristic, complemented
14C2×#544C2211yescharacteristic, complemented
15C22×#121C2212yes—
16C22×#117C2211yescharacteristic
17C2×#490C2212yescomplemented
18C2×#486C2212yescomplemented
19C2×#418C2214yescomplemented
20C2×#407C2212yes—
21#2855—22no—
22#2815—22no—
23#2626—22no—
24#2619C2212yes—
25#2619C2212yes—
26#2617—22no—
27#2617—22no—
28#2504C2214yes—
29#2441—22no—
30#2441C2214yescomplemented
31#2439—22no—
32#2439C2214yescomplemented
33#2433C2214yes—
34#2403—22no—
35#2317C2211yescharacteristic, complemented
36#2317C2211yescharacteristic, complemented
37#2316C2211yescharacteristic, complemented
38#2280C2211yescharacteristic
39#2280C2211yescharacteristic
40#2101C2214yescomplemented
41#2101C2214yescomplemented
42#2073C2214yescomplemented
43#2071C2214yescomplemented
44#2041C2214yescomplemented
45#2041C2214yescomplemented
46#2025C2214yescomplemented
47#2023C2214yescomplemented
48#1993C2212yescomplemented
49#1991C2212yescomplemented
50#1961C2212yescomplemented
51#1957C2212yescomplemented
52#1936C2214yescomplemented
53#1936C2214yescomplemented
54#1934C2214yescomplemented
55#1934C2214yescomplemented
56#1895C2214yes—
57#1895C2214yes—
58#1875C2214yes—
59#1875C2214yes—
60#1875C2214yescomplemented
61#1875C2214yescomplemented
62#1874C2214yescomplemented
63#1874C2214yescomplemented
64#1836C2212yescomplemented
65#1835C2212yescomplemented
66#1823C2212yescomplemented
67#1823C2212yescomplemented
68#1786C2212yescomplemented
69#1786C2212yescomplemented
70#1753C2214yescomplemented
71#1753C2214yescomplemented
72#1750C2212yescomplemented
73#1750C2212yescomplemented
74#1736C2214yescomplemented
75C23×D4—22no—
76C23×D4C2312yes—
77C24×C4C2311yescharacteristic
78C22×M16C2314yes—
79#544—44no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8C2312yes—
82C2×#127C2312yes—
83C2×#127C2312yes—
84C2×#125—21no—
85C2×#125C2312yes—
86C2×C22⋊Q8—22no—
87C2×#121—22no—
88C2×#121—22no—
89C2×#121—28no—
90C2×#121—28no—
91C2×#121C2314yes—
92C2×#121C2314yes—
93C2×#120—24no—
94C2×#120—24no—
95C2×#117—22no—
96C2×#117—44no—
97C2×#117C2314yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—21no—
106C22×C22⋊C4—22no—
107C22×C42—21no—
108#490—44no—
109#486—44no—
110#418—48no—
111C2×#102C2314yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119#407—28no—
120C2×C22⋊C8C2312yes—
121C2×C22⋊C8C2312yes—
122C2×C22⋊C8C2314yes—
123C2×#97C2312yes—
124C2×#97C2312yes—
125C2×C4×C8C2312yes—
126C2×C4×C8C2312yes—
127#399—24no—
128#397—24no—
129#397—24no—
130#396—24no—
131#396—24no—
132#394—22no—
133#394—22no—
134#394—24no—
135#393—24no—
136#392—22no—
137#392—22no—
138#392—24no—
139#392—24no—
140#392—24no—
141#391—24no—
142#390C2314yes—
143#390C2314yes—
144#389—22no—
145#389—22no—
146#389—24no—
147#389C2314yes—
148#389C2314yes—
149#387—24no—
150#387—24no—
151#386—24no—
152#386—24no—
153#385—22no—
154#385—22no—
155#384—21no—
156#384C2312yes—
157#382—21no—
158#382C2312yes—
159#382C2312yes—
160#382C2312yes—
161#381—21no—
162#381D412yes—
163#380—22no—
164#380D412yes—
165#380D412yes—
166#380C2312yes—
167#380C2312yes—
168#379—21no—
169#379D412yes—
170C4×C22⋊C4—22no—
171C4×C22⋊C4C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#336C2314yes—
181#336C2314yes—
182C25—22no—
183C25C2×D411yescharacteristic
184C22×Q8—22no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—48no—
196C22×D4—84no—
197C22×D4—84no—
198C22×D4C2414yes—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—21no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4C2×D412yes—
205C23×C4C2×D411yescharacteristic
206C23×C4#4118yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C2×M16—28no—
210C2×M16—44no—
211C2×M16—44no—
212C22×C8C2414yes—
213C22×C8C2414yes—
214C4⋊Q8—24no—
215C4⋊Q8—42no—
216C4⋊Q8—44no—
217#127—42no—
218#127—42no—
219#127—42no—
220#127—42no—
221#125—24no—
222#125—42no—
223#125—44no—
224C22⋊Q8—48no—
225#121—28no—
226#121—28no—
227#121—28no—
228#121—28no—
229#121—48no—
230#121—48no—
231#120—88no—
232#120—88no—
233#117—28no—
234#117—42no—
235#117—42no—
236#117—44no—
237#117—42no—
238#117—42no—
239#117—48no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2414yes—
258C2×C4⋊C4C2411yescharacteristic
259C2×C4⋊C4C2411yescharacteristic
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4C2×D412yes—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D414yes—
279C2×C42—24no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—44no—
283C2×C42C2412yes—
284C2×C42C2412yes—
285#102—28no—
286#102—28no—
287#102—44no—
288#102—44no—
289#102—44no—
290#102—44no—
291#102—44no—
292#102—44no—
293C22⋊C8—24no—
294C22⋊C8—24no—
295C22⋊C8—28no—
296#97—42no—
297#97—42no—
298C4×C8—42no—
299C4×C8—42no—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—22no—
308#95C2×D414yes—
309#95C2×D414yes—
310C24—24no—
311C24—41no—
312C24—41no—
313C24—42no—
314C24—42no—
315C24—44no—
316C24—44no—
317C24—44no—
318C24—82no—
319C24—82no—
320C24—82no—
321C24—82no—
322C24#12414yes—
323C24C22×D412yes—
324C24C22×D411yescharacteristic
325C2×Q8—44no—
326C2×Q8—84no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—416no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—44no—
339C2×D4—88no—
340C2×D4—82no—
341C2×D4—82no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—82no—
347C2×D4—82no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—84no—
351C2×D4—832no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—416no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4#12418yes—
375C22×C4#12414yes—
376C22×C4C22×D414yes—
377C22×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4114yes—
382C22×C4C2×#4114yes—
383C22×C4C2×#4114yes—
384C22×C4C2511yescharacteristic
385M16—88no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4H(5,𝔽2)14yes—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—48no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—84no—
422C22⋊C4—84no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—48no—
432C42—42no—
433C42—42no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—21no—
441C23—22no—
442C23—21no—
443C23—22no—
444C23—44no—
445C23—48no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—88no—
454C23—88no—
455C23—88no—
456C23—84no—
457C23—82no—
458C23—82no—
459C23—164no—
460C23—164no—
461C23—164no—
462C23—164no—
463C23C2×#12414yes—
464C23C2×#12412yes—
465C23C23×D411yescharacteristic
466Q8—164no—
467D4—44no—
468D4—44no—
469D4—44no—
470D4—44no—
471D4—88no—
472D4—816no—
473D4—816no—
474D4—168no—
475D4—164no—
476D4—164no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—48no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—84no—
508C2×C4—88no—
509C2×C4—84no—
510C2×C4C2×#12412yes—
511C2×C4C2×#12414yes—
512C2×C4C22×#4112yes—
513C2×C4C2×H(5,𝔽2)12yes—
514C2×C4#58512yes—
515C8—44no—
516C8—44no—
517C22—24no—
518C22—22no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—41no—
526C22—42no—
527C22—41no—
528C22—42no—
529C22—88no—
530C22—84no—
531C22—84no—
532C22—84no—
533C22—84no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22—1616no—
538C22C22×#12411yescharacteristic
539C22#348411yescharacteristic
540C22#348511yescharacteristic
541C22#358512yes—
542C22#358512yes—
543C4—22no—
544C4—24no—
545C4—44no—
546C4—42no—
547C4—44no—
548C4—42no—
549C4—82no—
550C4—84no—
551C4—81no—
552C4—81no—
553C4—82no—
554C4#350514yes—
555C2—42no—
556C2—44no—
557C2—164no—
558C2#3537711yescharacteristic
559C2#6061912yes—
560C2#6164811yescharacteristic
561C2#6165111yescharacteristic
562C2#6177812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37450C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35151C2semidirect
3. #29343C2semidirect
4. #29340C2semidirect
5. #27986C2semidirect
6. #27725C2semidirect
7. #27703C2semidirect
8. #27279C2semidirect
9. #27273C2semidirect
10. #25915C2semidirect
11. #21767C2semidirect
12. #20912C2semidirect
13. #20529C2semidirect
14. C2×#544C22semidirect
17. C2×#490C22semidirect
18. C2×#486C22semidirect
19. C2×#418C22semidirect
30. #2441C22semidirect
32. #2439C22semidirect
35. #2317C22semidirect
36. #2317C22semidirect
37. #2316C22semidirect
40. #2101C22semidirect
41. #2101C22semidirect
42. #2073C22semidirect
43. #2071C22semidirect
44. #2041C22semidirect
45. #2041C22semidirect
46. #2025C22semidirect
47. #2023C22semidirect
48. #1993C22semidirect
49. #1991C22semidirect
50. #1961C22semidirect
51. #1957C22semidirect
52. #1936C22semidirect
53. #1936C22semidirect
54. #1934C22semidirect
55. #1934C22semidirect
60. #1875C22semidirect
61. #1875C22semidirect
62. #1874C22semidirect
63. #1874C22semidirect
64. #1836C22semidirect
65. #1835C22semidirect
66. #1823C22semidirect
67. #1823C22semidirect
68. #1786C22semidirect
69. #1786C22semidirect
70. #1753C22semidirect
71. #1753C22semidirect
72. #1750C22semidirect
73. #1750C22semidirect
74. #1736C22semidirect

Maximal subgroups

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

subgroupindex
#374502
#351512
#293432
#293402
#279862
#277252
#277032
#272792
#272732
#259152
#217672
#209122
#205292