Finite Group Databasebeta

← #6434765

Subgroups — #6434765

Order 512.

1689 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#35264C211yesmaximal, characteristic
2#34217C211yesmaximal, characteristic, complemented
3#30074C212yesmaximal, complemented
4#30072C212yesmaximal, complemented
5#30071C212yesmaximal, complemented
6#30071C212yesmaximal, complemented
7#30066C214yesmaximal, complemented
8#30065C214yesmaximal, complemented
9#23731C211yesmaximal, characteristic, complemented
10#23730C211yesmaximal, characteristic, complemented
11#23557C212yesmaximal, complemented
12#23511C212yesmaximal
13#23453C211yesmaximal, characteristic, complemented
14C2×#2025C211yesmaximal, characteristic, complemented
15#21065C212yesmaximal, complemented
16#21062C212yesmaximal, complemented
17#20774C211yesmaximal, characteristic, complemented
18C2×#544C2211yescharacteristic
19C2×#535C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#417C2212yes—
24C2×#418C2212yes—
25C2×#416C2212yes—
26C22×#102C2212yes—
27#2861C2212yes—
28#2858C2212yes—
29#2641C2212yes—
30#2640C2212yes—
31#2491—22no—
32#2490—22no—
33#2472—24no—
34#2370—22no—
35#2370—22no—
36#2345—24no—
37#2330C2212yescomplemented
38#2330C2212yescomplemented
39#2316C2212yes—
40#2311C2211yescharacteristic
41#2309C2211yescharacteristic
42C4×#125C2214yescomplemented
43#2297C2211yescharacteristic
44C2×#390C2211yescharacteristic
45#2285C2211yescharacteristic
46#2274C2212yes—
47C4×#117C2211yescharacteristic
48#2097C2212yescomplemented
49#2096C2211yescharacteristic, complemented
50#2095C2211yescharacteristic, complemented
51#2094C2212yes—
52#2094C2211yescharacteristic, complemented
53#2093C2212yes—
54#2093C2211yescharacteristic, complemented
55#2092C2214yes—
56#2092C2212yescomplemented
57#2059C2214yescomplemented
58#2059C2214yescomplemented
59#2058C2214yescomplemented
60#2057C2214yescomplemented
61#2042C2214yes—
62#2042C2214yes—
63#2041C2214yes—
64#2041C2214yes—
65#2026C2212yescomplemented
66#2026C2212yescomplemented
67#2025C2214yes—
68#2025C2212yes—
69#2025C2212yes—
70#2025C2212yescomplemented
71#2025C2212yescomplemented
72#1928C2214yescomplemented
73#1927C2214yescomplemented
74#1926C2214yes—
75#1926C2214yes—
76#1926C2214yescomplemented
77#1925C2214yes—
78#1925C2214yes—
79#1925C2214yescomplemented
80#1840C2212yescomplemented
81#1840C2212yescomplemented
82#1839C2214yescomplemented
83#1736C2212yes—
84C2×#340C2212yes—
85#1729C2212yes—
86#1729C2212yes—
87#1728C2211yescharacteristic
88C2×#336C2211yescharacteristic
89C22×#41C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C22×M16C2312yes—
92C23×C8C2312yes—
93#544—44no—
94#535—44no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8C2312yes—
97C2×#127C2312yes—
98C2×#125—21no—
99C2×#125—28no—
100C2×#125—22no—
101C2×#125C2312yes—
102C2×#125C2312yes—
103C2×#125C2312yes—
104C2×#124—24no—
105C2×#124C2312yes—
106C2×C22⋊Q8—22no—
107C2×#121—21no—
108C2×#121—21no—
109C2×#120—22no—
110C2×#117C2311yescharacteristic
111C2×#117C2312yes—
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C4⋊C4—22no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4—22no—
118C22×C42C2311yescharacteristic
119#428—44no—
120#427—44no—
121C2×#107C2312yes—
122C2×#107C2314yes—
123C2×#107C2312yes—
124C2×#106C2312yes—
125C2×#106C2314yes—
126C2×#106C2312yes—
127#418—44no—
128#417—28no—
129#416—44no—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#102—28no—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141#397—24no—
142#394—24no—
143#394—24no—
144#393—24no—
145#391C2312yes—
146#391C2311yescharacteristic
147#391C2311yescharacteristic
148#390—22no—
149#390—24no—
150#390—22no—
151#390C2312yes—
152#390C2312yes—
153#390C2312yes—
154#390C2311yescharacteristic
155#390C2312yes—
156#390C2312yes—
157#390C2311yescharacteristic
158#389—22no—
159#389—22no—
160#389—24no—
161#389—24no—
162#389C2314yes—
163#389C2314yes—
164#388—24no—
165#388—24no—
166#387—24no—
167#387—24no—
168#387—24no—
169#387—24no—
170#386—22no—
171#386—22no—
172#384C2311yescharacteristic
173#384C2311yescharacteristic
174#382C2312yes—
175#380—22no—
176#380—22no—
177#380—24no—
178#379—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—24no—
182C4×C4⋊C4—24no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4D412yes—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4—24no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4D414yes—
191#376C2312yes—
192C43C2312yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2312yes—
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2314yes—
205#336C2314yes—
206C25—21no—
207C2×#41—24no—
208C2×#41—24no—
209C2×#41—22no—
210C2×#41—22no—
211C22×Q8—22no—
212C22×Q8C2411yescharacteristic
213C22×D4—21no—
214C22×D4—21no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—24no—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4C2412yes—
222C22×D4C2411yescharacteristic
223C23×C4—22no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4C2×D412yes—
227C23×C4C2411yescharacteristic
228C2×M16—24no—
229C2×M16—24no—
230C2×M16—24no—
231C22×C8—24no—
232C22×C8—24no—
233C22×C8—24no—
234C22×C8C2412yes—
235C22×C8C2412yes—
236C22×C8C2414yes—
237C4⋊Q8—44no—
238C4⋊Q8—44no—
239#127—44no—
240#125—216no—
241#125—28no—
242#125—28no—
243#125—44no—
244#125—44no—
245#124—44no—
246#124—88no—
247C22⋊Q8—48no—
248#121—44no—
249#121—44no—
250#120—48no—
251#117—24no—
252#117—22no—
253#117—22no—
254#117—24no—
255#117—24no—
256#117—24no—
257C2×C4⋊C4—216no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4C2×D411yescharacteristic
272C2×C4⋊C4C2×D411yescharacteristic
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4#4118yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2411yescharacteristic
277C2×C4⋊C4C2411yescharacteristic
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2414yes—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—84no—
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D412yes—
295C2×C42—24no—
296C2×C42—21no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42C2×D412yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2412yes—
304C2×C42C2412yes—
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307#107—42no—
308#107—42no—
309#107—44no—
310#106—42no—
311#106—42no—
312#106—44no—
313#103—28no—
314#103—42no—
315#103—42no—
316#102—28no—
317#102—28no—
318#102—28no—
319#102—42no—
320#102—42no—
321#102—44no—
322#102—44no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95C2×D414yes—
333#95C2×D414yes—
334C24—22no—
335C24—21no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24—82no—
340C24—82no—
341C24C22×D411yescharacteristic
342#41—48no—
343#41—48no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—44no—
349C2×Q8—84no—
350C2×D4—24no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—84no—
369C2×D4—84no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—28no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4C22×D414yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4114yes—
395C22×C4H(5,𝔽2)14yes—
396C22×C42−1+412yes—
397C22×C42−1+412yes—
398C22×C4C2511yescharacteristic
399M16—48no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—48no—
410C4⋊C4—216no—
411C4⋊C4—216no—
412C4⋊C4—28no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—84no—
429C4⋊C4—82no—
430C4⋊C4—84no—
431C4⋊C4—82no—
432C4⋊C4C2×#4118yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—48no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—42no—
441C22⋊C4—84no—
442C22⋊C4—82no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—24no—
450C42—22no—
451C42—22no—
452C42—44no—
453C42—44no—
454C42—44no—
455C42—42no—
456C42—84no—
457C42—84no—
458C23—22no—
459C23—22no—
460C23—42no—
461C23—44no—
462C23—42no—
463C23—44no—
464C23—41no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—88no—
471C23—84no—
472C23—162no—
473C23—162no—
474C23—164no—
475C23#52512yes—
476C23#53512yes—
477C23#53612yes—
478C23C23×D411yescharacteristic
479Q8—42no—
480Q8—42no—
481Q8—164no—
482D4—44no—
483D4—44no—
484D4—42no—
485D4—42no—
486D4—84no—
487D4—84no—
488D4—164no—
489D4—164no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—28no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—84no—
522C2×C4—88no—
523C2×C4—84no—
524C2×C4—162no—
525C2×C4#52512yes—
526C2×C4#53512yes—
527C2×C4#53612yes—
528C2×C4#54218yes—
529C2×C4#54218yes—
530C2×C4C22×#4112yes—
531C2×C4C2×H(5,𝔽2)12yes—
532C2×C4C2×2−1+411yescharacteristic
533C2×C4C2×2−1+411yescharacteristic
534C8—42no—
535C8—42no—
536C8—44no—
537C22—22no—
538C22—22no—
539C22—24no—
540C22—24no—
541C22—81no—
542C22—82no—
543C22—81no—
544C22—82no—
545C22—81no—
546C22—82no—
547C22—82no—
548C22—84no—
549C22—82no—
550C22—84no—
551C22—168no—
552C22#263214yes—
553C22C2×#52511yescharacteristic
554C22C2×#53511yescharacteristic
555C22C2×#53611yescharacteristic
556C22#358412yes—
557C22#358512yes—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—48no—
563C4—44no—
564C4—81no—
565C4—82no—
566C4—81no—
567C4—82no—
568C4—82no—
569C4—81no—
570C4#263214yes—
571C4C2×#54212yes—
572C4C2×#54212yes—
573C2—24no—
574C2—161no—
575C2—162no—
576C2C2×#263211yescharacteristic
577C2#6058012yes—
578C2#6082311yescharacteristic
579C2#6083011yescharacteristic
580C2#6089012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34217C2semidirect
3. #30074C2semidirect
4. #30072C2semidirect
5. #30071C2semidirect
6. #30071C2semidirect
7. #30066C2semidirect
8. #30065C2semidirect
9. #23731C2semidirect
10. #23730C2semidirect
11. #23557C2semidirect
13. #23453C2semidirect
14. C2×#2025C2semidirect
15. #21065C2semidirect
16. #21062C2semidirect
17. #20774C2semidirect
37. #2330C22semidirect
38. #2330C22semidirect
42. C4×#125C22semidirect
48. #2097C22semidirect
49. #2096C22semidirect
50. #2095C22semidirect
52. #2094C22semidirect
54. #2093C22semidirect
56. #2092C22semidirect
57. #2059C22semidirect
58. #2059C22semidirect
59. #2058C22semidirect
60. #2057C22semidirect
65. #2026C22semidirect
66. #2026C22semidirect
70. #2025C22semidirect
71. #2025C22semidirect
72. #1928C22semidirect
73. #1927C22semidirect
76. #1926C22semidirect
79. #1925C22semidirect
80. #1840C22semidirect
81. #1840C22semidirect
82. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#352642
#342172
#300742
#300722
#300712
#300662
#300652
#237312
#237302
#235572
#235112
#234532
C2×#20252
#210652
#210622
#207742