Finite Group Databasebeta

← #6519080

Subgroups — #6519080

Order 512.

1906 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35514C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic, complemented
3#27609C212yesmaximal, complemented
4#27602C212yesmaximal, complemented
5#25022C214yesmaximal, complemented
6#25012C214yesmaximal, complemented
7#24982C214yesmaximal, complemented
8#24974C214yesmaximal, complemented
9#24013C212yesmaximal, complemented
10#24005C212yesmaximal, complemented
11#21990C212yesmaximal, complemented
12#21792C211yesmaximal, characteristic, complemented
13#20744C212yesmaximal, complemented
14C2×#535C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C2×#518C2211yescharacteristic
18C2×#481C2214yes—
19C2×#481C2212yescomplemented
20C2×#480C2212yescomplemented
21C2×#433C2212yes—
22#2537C2214yes—
23#2535C2214yes—
24#2428—22no—
25#2427—22no—
26#2412—22no—
27#2410—22no—
28#2376—22no—
29#2358—22no—
30#2352C2214yescomplemented
31D4×C4⋊C4C2214yescomplemented
32#2329—22no—
33#2325—22no—
34#2317C2211yescharacteristic
35#2314C2212yes—
36#2311C2212yes—
37#2311C2211yescharacteristic
38#2280C2211yescharacteristic
39C2×#380C2212yes—
40#2077C2214yescomplemented
41#2077C2214yescomplemented
42#2077C2214yescomplemented
43#2076C2214yescomplemented
44#2034C2214yes—
45#2034C2214yes—
46#2034C2214yescomplemented
47#2033C2214yescomplemented
48#1955C2214yescomplemented
49#1954C2214yescomplemented
50#1949C2214yescomplemented
51#1948C2214yescomplemented
52#1875C2212yescomplemented
53#1875C2212yescomplemented
54#1874C2214yes—
55#1854C2214yes—
56#1853C2214yes—
57#1853C2212yescomplemented
58#1853C2214yescomplemented
59#1852C2212yescomplemented
60#1847C2212yescomplemented
61#1844C2212yescomplemented
62#1836C2212yescomplemented
63#1835C2212yescomplemented
64#1825C2214yes—
65#1825C2214yescomplemented
66#1823C2212yescomplemented
67#1823C2212yescomplemented
68#1804C2214yescomplemented
69#1802C2214yescomplemented
70#1786C2214yes—
71#1780C2212yes—
72#1779C2212yes—
73#1762C2214yescomplemented
74C4×#102C2214yescomplemented
75C22×#41C2311yescharacteristic
76C23×D4—21no—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79#535—44no—
80C2×#127—21no—
81C2×#124—21no—
82#525—28no—
83C2×#123D412yes—
84C2×C22⋊Q8D412yes—
85C2×#121—22no—
86C2×#121—24no—
87C2×#121—44no—
88C2×#121D412yes—
89C2×#121C2312yes—
90C2×#121C2314yes—
91C2×#121C2312yes—
92C2×#120—22no—
93C2×#120D412yes—
94#518—28no—
95C2×C4×D4—24no—
96C2×C4×D4—22no—
97C2×C4×D4C2314yes—
98C2×C4×D4C2314yes—
99C2×#117C2311yescharacteristic
100C2×#117C2312yes—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—22no—
109C22×C22⋊C4—22no—
110#481—48no—
111#481—44no—
112#480—44no—
113#433—28no—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×C22⋊C8C2314yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2314yes—
130C2×#97C2312yes—
131C2×C4×C8C2312yes—
132#397—24no—
133#394—24no—
134#394—24no—
135#392—24no—
136#391—22no—
137#391—21no—
138#390—22no—
139#390—21no—
140#389—22no—
141#389—24no—
142#388—24no—
143#388—24no—
144#387—24no—
145#387—24no—
146#386—22no—
147#386—22no—
148#386—22no—
149#386—24no—
150#386—24no—
151#386—24no—
152#385—22no—
153#385—24no—
154#385—24no—
155#384C2312yes—
156#384C2312yes—
157#382C2312yes—
158#382C2312yes—
159#381—21no—
160#380—22no—
161#380—22no—
162#380—24no—
163#380C2314yes—
164#380C2314yes—
165#380C2314yes—
166#380C2312yes—
167#380C2314yes—
168#380C2312yes—
169#379—21no—
170C4×C4⋊C4C2314yes—
171C4×C22⋊C4—22no—
172C4×C22⋊C4—24no—
173C2×#95—22no—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182C25—21no—
183C25C2×D411yescharacteristic
184C2×#41—24no—
185C2×#41—42no—
186C2×#41—42no—
187C22×Q8C2×D411yescharacteristic
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—21no—
193C22×D4—24no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—84no—
197C22×D4C2×D414yes—
198C22×D4C2×D412yes—
199C22×D4C2×D411yescharacteristic
200C22×D4C2412yes—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—21no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C22×C8C2414yes—
215C22×C8C2414yes—
216#127—44no—
217#124—44no—
218#123—28no—
219C22⋊Q8—28no—
220#121—28no—
221#121—28no—
222#121—24no—
223#121—24no—
224#121—44no—
225#121—42no—
226#121—42no—
227#121—48no—
228#121—88no—
229#120—28no—
230#120—48no—
231C4×D4—28no—
232C4×D4—28no—
233C4×D4—28no—
234C4×D4—28no—
235C4×D4—48no—
236C4×D4—88no—
237#117—24no—
238#117—42no—
239#117—44no—
240#117#4118yes—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2414yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2411yescharacteristic
260C2×C4⋊C4C2411yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—48no—
276C2×C22⋊C4—84no—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D414yes—
279C2×C22⋊C4C2×D412yes—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42C2412yes—
285#107—44no—
286#106—44no—
287#106—44no—
288#106—44no—
289C4⋊C8—28no—
290#102—28no—
291#102—28no—
292#102—44no—
293#102—44no—
294C22⋊C8—28no—
295C22⋊C8—42no—
296C22⋊C8—42no—
297C22⋊C8—44no—
298C22⋊C8—42no—
299C22⋊C8—42no—
300C22⋊C8—44no—
301#97—24no—
302C4×C8—24no—
303#95—24no—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312C24—24no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—42no—
317C24—41no—
318C24—41no—
319C24—42no—
320C24—42no—
321C24—82no—
322C24—82no—
323C24C22×D412yes—
324C24C22×D411yescharacteristic
325C24C22×D412yes—
326#41—88no—
327C2×Q8—22no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×D4—24no—
331C2×D4—28no—
332C2×D4—24no—
333C2×D4—28no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—416no—
339C2×D4—44no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—42no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—42no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—816no—
354C2×D4H(5,𝔽2)14yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—48no—
381C22×C4—88no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—84no—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D414yes—
395C22×C4C2×#4114yes—
396C22×C4C2511yescharacteristic
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C4⋊C4—28no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—42no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4#12318yes—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—48no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—48no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—44no—
428C22⋊C4—48no—
429C22⋊C4—82no—
430C22⋊C4—84no—
431C22⋊C4—82no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C22⋊C4#12318yes—
435C42—42no—
436C42—42no—
437C42—82no—
438C42—84no—
439C42C2×#4114yes—
440C42C2×#4114yes—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—21no—
446C23—22no—
447C23—22no—
448C23—21no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—22no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—44no—
458C23—48no—
459C23—48no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—82no—
465C23—84no—
466C23—82no—
467C23—84no—
468C23—84no—
469C23—164no—
470C23—164no—
471C23C2×#12312yes—
472C23#57714yes—
473C23C23×D411yescharacteristic
474Q8—82no—
475D4—28no—
476D4—416no—
477D4—48no—
478D4—44no—
479D4—82no—
480D4—84no—
481D4—816no—
482D4—164no—
483D4—164no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—84no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4—168no—
518C2×C4—164no—
519C2×C4—164no—
520C2×C4—164no—
521C2×C4C2×#12312yes—
522C2×C4C2×#12318yes—
523C2×C4C22×#4112yes—
524C2×C4C2×H(5,𝔽2)12yes—
525C2×C4#58512yes—
526C8—44no—
527C8—44no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—22no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—42no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—42no—
544C22—42no—
545C22—44no—
546C22—42no—
547C22—44no—
548C22—88no—
549C22—84no—
550C22—84no—
551C22—168no—
552C22C22×#12311yescharacteristic
553C22#348411yescharacteristic
554C22#348511yescharacteristic
555C22C2×#57712yes—
556C22#358512yes—
557C4—28no—
558C4—22no—
559C4—42no—
560C4—42no—
561C4—84no—
562C4—82no—
563C4—82no—
564C4—81no—
565C4—84no—
566C4—81no—
567C4#349414yes—
568C2—44no—
569C2—42no—
570C2—82no—
571C2—84no—
572C2#3554011yescharacteristic
573C2#6052512yes—
574C2#6163111yescharacteristic
575C2#6164811yescharacteristic
576C2#6175812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35514C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34360C2semidirect
3. #27609C2semidirect
4. #27602C2semidirect
5. #25022C2semidirect
6. #25012C2semidirect
7. #24982C2semidirect
8. #24974C2semidirect
9. #24013C2semidirect
10. #24005C2semidirect
11. #21990C2semidirect
12. #21792C2semidirect
13. #20744C2semidirect
19. C2×#481C22semidirect
20. C2×#480C22semidirect
30. #2352C22semidirect
31. D4×C4⋊C4C22semidirect
40. #2077C22semidirect
41. #2077C22semidirect
42. #2077C22semidirect
43. #2076C22semidirect
46. #2034C22semidirect
47. #2033C22semidirect
48. #1955C22semidirect
49. #1954C22semidirect
50. #1949C22semidirect
51. #1948C22semidirect
52. #1875C22semidirect
53. #1875C22semidirect
57. #1853C22semidirect
58. #1853C22semidirect
59. #1852C22semidirect
60. #1847C22semidirect
61. #1844C22semidirect
62. #1836C22semidirect
63. #1835C22semidirect
65. #1825C22semidirect
66. #1823C22semidirect
67. #1823C22semidirect
68. #1804C22semidirect
69. #1802C22semidirect
73. #1762C22semidirect
74. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#355142
#343602
#276092
#276022
#250222
#250122
#249822
#249742
#240132
#240052
#219902
#217922
#207442