Finite Group Databasebeta

← #6499415

Subgroups — #6499415

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35338C211yesmaximal, characteristic
2#35294C211yesmaximal, characteristic, complemented
3#33282C212yesmaximal, complemented
4#33282C212yesmaximal, complemented
5#30217C212yesmaximal, complemented
6#30216C212yesmaximal, complemented
7#29363C214yesmaximal, complemented
8#29362C214yesmaximal, complemented
9#29118C212yesmaximal, complemented
10#29117C212yesmaximal, complemented
11#29116C212yesmaximal
12#29115C212yesmaximal
13#21839C212yesmaximal, complemented
14#21808C212yesmaximal
15#20467C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C2×#432C2212yes—
19#2868C2211yescharacteristic, complemented
20#2861C2212yes—
21#2861C2212yes—
22#2859C2212yescomplemented
23#2859C2211yescharacteristic, complemented
24#2640C2212yes—
25#2640C2212yes—
26#2629C2212yes—
27#2628C2212yes—
28#2592—22no—
29#2592—22no—
30#2570—22no—
31#2561—22no—
32#2561—22no—
33#2550—22no—
34#2441—22no—
35#2439—22no—
36#2331C2212yescomplemented
37#2330C2212yescomplemented
38#2310C2211yescharacteristic
39#2310C2211yescharacteristic
40#2309C2211yescharacteristic
41#2309C2211yescharacteristic
42#2285C2212yes—
43C4×#117C2211yescharacteristic
44#2074C2214yescomplemented
45#2073C2214yes—
46#2072C2214yescomplemented
47#2071C2214yes—
48#2026C2214yescomplemented
49#2025C2214yes—
50#2024C2214yescomplemented
51#2023C2214yes—
52#1992C2212yescomplemented
53#1992C2212yescomplemented
54#1992C2212yescomplemented
55#1992C2212yescomplemented
56#1988C2214yescomplemented
57#1987C2214yescomplemented
58#1983C2214yes—
59#1983C2214yes—
60#1959C2212yes—
61#1959C2212yes—
62#1959C2212yes—
63#1959C2212yes—
64#1936C2214yes—
65#1936C2214yes—
66#1935C2214yescomplemented
67#1935C2214yescomplemented
68#1934C2214yes—
69#1934C2214yes—
70#1933C2214yescomplemented
71#1933C2214yescomplemented
72#1825C2214yes—
73#1825C2214yes—
74#1787C2214yes—
75#1786C2214yes—
76#1768C2212yescomplemented
77#1768C2212yescomplemented
78#1750C2212yes—
79#1750C2212yes—
80#1745C2212yescomplemented
81C4×#97C2211yescharacteristic, complemented
82C4×#97C2211yescharacteristic, complemented
83C23×Q8C2311yescharacteristic
84C23×D4C2311yescharacteristic
85#533—28no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×#125—22no—
90C2×#125C2312yes—
91C2×#125C2312yes—
92C2×#124—22no—
93C2×#124—28no—
94C2×#124C2312yes—
95C2×#124C2312yes—
96C2×C22⋊Q8—22no—
97C2×C4×Q8—22no—
98C2×#117—22no—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C22×C22⋊C4—22no—
104C22×C42C2311yescharacteristic
105#432—28no—
106C2×C4⋊C8C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×#103C2314yes—
111C2×#102C2314yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×C22⋊C8C2314yes—
116C2×#97C2312yes—
117C2×#97C2312yes—
118C2×#97C2312yes—
119C2×#97C2312yes—
120C2×C4×C8C2312yes—
121C2×C4×C8C2312yes—
122#400—22no—
123#399—22no—
124#398—24no—
125#397—22no—
126#397—22no—
127#397—22no—
128#397—22no—
129#396—24no—
130#395—22no—
131#393—22no—
132#393—22no—
133#393—22no—
134#393—22no—
135#393—22no—
136#391—21no—
137#391—22no—
138#391—21no—
139#391—24no—
140#391C2312yes—
141#391C2312yes—
142#391C2312yes—
143#391C2312yes—
144#390—21no—
145#390—22no—
146#390—21no—
147#390C2312yes—
148#390C2312yes—
149#390C2312yes—
150#390C2312yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—24no—
156#389C2314yes—
157#389C2314yes—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#387—24no—
167#387—24no—
168#384—22no—
169#384C2312yes—
170#384C2312yes—
171#384C2312yes—
172#382—22no—
173#382—22no—
174#382C2312yes—
175#381—21no—
176#381—22no—
177#381—21no—
178#380—22no—
179#380C2314yes—
180#379—21no—
181#379—22no—
182#379—21no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4D412yes—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4C2314yes—
187C4×C22⋊C4D412yes—
188C4×C22⋊C4D412yes—
189#376C2311yescharacteristic
190#376C2311yescharacteristic
191C43C2311yescharacteristic
192C43C2311yescharacteristic
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25—21no—
204C22×Q8—21no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—24no—
208C22×Q8—42no—
209C22×Q8—42no—
210C22×Q8C2412yes—
211C22×D4—21no—
212C22×D4—24no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4C2412yes—
216C23×C4—21no—
217C23×C4C2×D412yes—
218C23×C4C2411yescharacteristic
219C22×C8C2414yes—
220C22×C8C2414yes—
221C4⋊Q8—28no—
222C4⋊Q8—42no—
223C4⋊Q8—42no—
224C4⋊Q8—84no—
225#125—28no—
226#125—42no—
227#125—42no—
228#125—84no—
229#124—28no—
230#124—28no—
231#124—48no—
232C22⋊Q8—48no—
233C4×Q8—48no—
234#117—24no—
235#117—24no—
236#117—42no—
237#117—42no—
238#117—84no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2414yes—
257C2×C4⋊C4C2414yes—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4C2×D411yescharacteristic
275C2×C22⋊C4C2×D412yes—
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—24no—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42#4118yes—
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C4⋊C8—28no—
293#103—44no—
294#103—44no—
295#103—44no—
296#103—44no—
297#102—44no—
298#102—44no—
299#102—44no—
300#102—44no—
301C22⋊C8—28no—
302#97—24no—
303#97—24no—
304#97—42no—
305#97—42no—
306C4×C8—42no—
307C4×C8—42no—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95C2×D412yes—
323#95C2×D412yes—
324#95C2×D412yes—
325#95C2×D412yes—
326C24—22no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24—82no—
331C24—82no—
332C24C22×D411yescharacteristic
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—42no—
341C2×Q8—82no—
342C2×Q8—82no—
343C2×Q8—84no—
344C2×Q8—88no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—88no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4#12418yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D412yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C2×#4114yes—
375C22×C4C2511yescharacteristic
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C22⋊C4—42no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C22⋊C4—84no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—84no—
432C42#12418yes—
433C42C2×#4114yes—
434C42C2×#4114yes—
435C23—22no—
436C23—22no—
437C23—21no—
438C23—21no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—84no—
447C23—88no—
448C23—164no—
449C23—162no—
450C23—162no—
451C23C2×#12412yes—
452C23C23×D411yescharacteristic
453Q8—48no—
454Q8—84no—
455Q8—88no—
456Q8—164no—
457D4—48no—
458D4—84no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—28no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—84no—
497C2×C4C2×#12412yes—
498C2×C4C2×#12414yes—
499C2×C4C2×#12414yes—
500C2×C4C22×#4112yes—
501C2×C4#58512yes—
502C2×C4#58512yes—
503C8—44no—
504C8—44no—
505C22—22no—
506C22—24no—
507C22—42no—
508C22—41no—
509C22—41no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—168no—
518C22C22×#12411yescharacteristic
519C22#344911yescharacteristic
520C22#344911yescharacteristic
521C22#358512yes—
522C22#358612yes—
523C4—22no—
524C4—24no—
525C4—24no—
526C4—42no—
527C4—42no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—82no—
533C4—84no—
534C4—84no—
535C4—82no—
536C4—82no—
537C4#351014yes—
538C2—42no—
539C2—162no—
540C2#3533811yescharacteristic
541C2#6062112yes—
542C2#6101612yes—
543C2#6101711yescharacteristic
544C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #35294C2semidirect
3. #33282C2semidirect
4. #33282C2semidirect
5. #30217C2semidirect
6. #30216C2semidirect
7. #29363C2semidirect
8. #29362C2semidirect
9. #29118C2semidirect
10. #29117C2semidirect
13. #21839C2semidirect
15. #20467C2semidirect
19. #2868C22semidirect
22. #2859C22semidirect
23. #2859C22semidirect
36. #2331C22semidirect
37. #2330C22semidirect
44. #2074C22semidirect
46. #2072C22semidirect
48. #2026C22semidirect
50. #2024C22semidirect
52. #1992C22semidirect
53. #1992C22semidirect
54. #1992C22semidirect
55. #1992C22semidirect
56. #1988C22semidirect
57. #1987C22semidirect
66. #1935C22semidirect
67. #1935C22semidirect
70. #1933C22semidirect
71. #1933C22semidirect
76. #1768C22semidirect
77. #1768C22semidirect
80. #1745C22semidirect
81. C4×#97C22semidirect
82. C4×#97C22semidirect

Maximal subgroups

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

subgroupindex
#353382
#352942
#332822
#302172
#302162
#293632
#293622
#291182
#291172
#291162
#291152
#218392
#218082
#204672