Finite Group Databasebeta

← #6644670

Subgroups — #6644670

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35082C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3#33004C212yesmaximal, complemented
4#32437C214yesmaximal, complemented
5#32373C214yesmaximal, complemented
6#31665C214yesmaximal, complemented
7#31527C214yesmaximal, complemented
8#29638C212yesmaximal, complemented
9#29110C211yesmaximal, characteristic, complemented
10#28301C211yesmaximal, characteristic
11#28114C212yesmaximal, complemented
12#27964C211yesmaximal, characteristic, complemented
13#27918C211yesmaximal, characteristic, complemented
14#27651C212yesmaximal, complemented
15#21914C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#481C2212yes—
19C2×#480C2212yes—
20C2×#480C2212yes—
21C2×#447C2212yes—
22#2856C2211yescharacteristic
23#2854C2212yes—
24#2851C2211yescharacteristic
25#2651—22no—
26#2563—22no—
27#2563—22no—
28#2563C2214yes—
29#2563C2214yes—
30#2513C2212yescomplemented
31#2507C2212yescomplemented
32#2454C2212yescomplemented
33#2445C2212yescomplemented
34#2443—22no—
35#2425—22no—
36C2×#393—22no—
37#2335—22no—
38C4×#121C2212yes—
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#380—22no—
44C2×C4×C4⋊C4C2211yescharacteristic
45#2101C2214yescomplemented
46#2093C2212yes—
47#2092C2214yescomplemented
48#2085C2214yescomplemented
49#2073C2212yes—
50#2073C2214yescomplemented
51#2067C2214yes—
52#2057C2214yescomplemented
53#2057C2214yescomplemented
54#2054C2214yes—
55#2025C2212yes—
56#2023C2214yescomplemented
57#2023C2214yescomplemented
58#2011C2212yes—
59#1991C2212yescomplemented
60#1970C2214yes—
61#1961C2212yescomplemented
62#1949C2214yescomplemented
63#1936C2214yes—
64#1925C2214yes—
65#1917C2214yescomplemented
66#1916C2214yescomplemented
67#1879C2214yes—
68#1879C2214yes—
69#1879C2214yes—
70#1864C2214yes—
71#1864C2214yes—
72#1847C2212yescomplemented
73#1835C2212yescomplemented
74#1824C2212yes—
75#1823C2212yes—
76#1823C2212yes—
77#1817C2212yes—
78#1804C2214yescomplemented
79#1797C2212yes—
80#1786C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C23×Q8—21no—
84C23×D4C2311yescharacteristic
85C24×C4—21no—
86C22×D8C2314yes—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—44no—
89C2×C4⋊Q8C2314yes—
90C2×#127C2311yescharacteristic
91C2×#125C2311yescharacteristic
92C2×#123—22no—
93C2×#123—22no—
94C2×#121—22no—
95C2×#121C2312yes—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—44no—
99C2×C4×D4C2314yes—
100C2×C4×D4C2312yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4D412yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C22⋊C4—22no—
108C22×C22⋊C4—22no—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#481—44no—
113#480—44no—
114#480—44no—
115#447—44no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×#97C2312yes—
129C2×C4×C8C2312yes—
130#399—22no—
131#398—22no—
132#398—22no—
133#397—22no—
134#397—22no—
135#396—22no—
136#394—22no—
137#394—22no—
138#393—24no—
139#393—24no—
140#393—24no—
141#393—24no—
142#390—21no—
143#390—22no—
144#390—21no—
145#390—22no—
146#390C2314yes—
147#390C2312yes—
148#390C2312yes—
149#390C2312yes—
150#390C2312yes—
151#390C2312yes—
152#390C2312yes—
153#389—22no—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#387—22no—
164#387—22no—
165#386—22no—
166#386—22no—
167#386—22no—
168#385—22no—
169#385—22no—
170#384C2311yescharacteristic
171#382—21no—
172#382—22no—
173#382—21no—
174#382C2314yes—
175#382C2312yes—
176#382C2312yes—
177#382C2311yescharacteristic
178#380—24no—
179#380—24no—
180#380—24no—
181#380—24no—
182#380—22no—
183#380C2312yes—
184C4×C4⋊C4D412yes—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C22⋊C4—22no—
190C2×#95—22no—
191C2×#95—22no—
192C2×#95D412yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204C25—21no—
205C22×Q8—24no—
206C22×Q8—44no—
207C22×Q8—84no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4—22no—
212C22×D4—24no—
213C22×D4—42no—
214C22×D4C2412yes—
215C22×D4C2412yes—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—44no—
222C23×C4—48no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2411yescharacteristic
228C2×D8—28no—
229C2×D8—44no—
230C2×D8—44no—
231C22×C8C2414yes—
232C22×C8C2414yes—
233C4⋊Q8—48no—
234C4⋊Q8—48no—
235#127—42no—
236#125—42no—
237#123—48no—
238#123—84no—
239#121—24no—
240#121—24no—
241#121—84no—
242C4×D4—48no—
243C4×D4—48no—
244C4×D4—44no—
245C4×D4—84no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—84no—
289C2×C22⋊C4—84no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42C2×D412yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D412yes—
297C2×C42C2×D411yescharacteristic
298C2×C42C2412yes—
299C2×C42C2412yes—
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302#107—44no—
303#107—44no—
304#106—44no—
305C4⋊C8—44no—
306#102—28no—
307#102—44no—
308#102—44no—
309#102—44no—
310#102—44no—
311#102—44no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314#97—42no—
315C4×C8—42no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95C2×D412yes—
328#95C2×D412yes—
329#95C2×D412yes—
330#95C2×D412yes—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—44no—
335C24—42no—
336C24—42no—
337C24—44no—
338C24C22×D411yescharacteristic
339C2×Q8—48no—
340C2×Q8—84no—
341C2×Q8—84no—
342C2×Q8—816no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—416no—
352C2×D4—42no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—42no—
358C2×D4—82no—
359C2×D4—82no—
360C2×D42−1+414yes—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—816no—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4H(5,𝔽2)14yes—
400C22×C4C2511yescharacteristic
401D8—88no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C22⋊C4—48no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4—82no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C42—24no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—24no—
450C42—41no—
451C42—42no—
452C42—41no—
453C42—82no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—24no—
458C23—48no—
459C23—416no—
460C23—42no—
461C23—42no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—84no—
472C23—88no—
473C23—84no—
474C23#53512yes—
475C23#57614yes—
476C23C23×D411yescharacteristic
477Q8—816no—
478D4—28no—
479D4—48no—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—88no—
484D4—164no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—816no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—84no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—164no—
529C2×C4#53512yes—
530C2×C4#57514yes—
531C2×C4C2×H(5,𝔽2)12yes—
532C2×C4C2×2−1+412yes—
533C2×C4#58512yes—
534C8—44no—
535C8—44no—
536C22—24no—
537C22—22no—
538C22—24no—
539C22—22no—
540C22—48no—
541C22—44no—
542C22—816no—
543C22—84no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22#322014yes—
554C22C2×#53511yescharacteristic
555C22#348511yescharacteristic
556C22#348611yescharacteristic
557C22C2×#57512yes—
558C22C2×#57612yes—
559C4—24no—
560C4—22no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4#353514yes—
573C2—24no—
574C2—84no—
575C2—162no—
576C2#3711811yescharacteristic
577C2C2×#322011yescharacteristic
578C2#6079911yescharacteristic
579C2#6174012yes—
580C2#6183512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28301C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35082C2semidirect
2. C2×#2563C2semidirect
3. #33004C2semidirect
4. #32437C2semidirect
5. #32373C2semidirect
6. #31665C2semidirect
7. #31527C2semidirect
8. #29638C2semidirect
9. #29110C2semidirect
11. #28114C2semidirect
12. #27964C2semidirect
13. #27918C2semidirect
14. #27651C2semidirect
15. #21914C2semidirect
30. #2513C22semidirect
31. #2507C22semidirect
32. #2454C22semidirect
33. #2445C22semidirect
45. #2101C22semidirect
47. #2092C22semidirect
48. #2085C22semidirect
50. #2073C22semidirect
52. #2057C22semidirect
53. #2057C22semidirect
56. #2023C22semidirect
57. #2023C22semidirect
59. #1991C22semidirect
61. #1961C22semidirect
62. #1949C22semidirect
65. #1917C22semidirect
66. #1916C22semidirect
72. #1847C22semidirect
73. #1835C22semidirect
78. #1804C22semidirect

Maximal subgroups

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

subgroupindex
#350822
C2×#25632
#330042
#324372
#323732
#316652
#315272
#296382
#291102
#283012
#281142
#279642
#279182
#276512
#219142