Finite Group Databasebeta

← #6355336

Subgroups — #6355336

Order 512.

1745 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60944C211yesmaximal, characteristic, complemented
2#60558C212yesmaximal, complemented
3C2×#2975C211yesmaximal, characteristic, complemented
4#19431C214yesmaximal, complemented
5#19430C214yesmaximal, complemented
6#19425C214yesmaximal, complemented
7#19424C214yesmaximal, complemented
8#19281C212yesmaximal, complemented
9#19280C212yesmaximal, complemented
10#19279C214yesmaximal, complemented
11#17815C211yesmaximal, characteristic, complemented
12#17716C212yesmaximal, complemented
13#3444C2212yes—
14C22×C4×D4C2211yescharacteristic
15#3250—22no—
16#3231C2212yescomplemented
17#3228C2212yescomplemented
18#3210C2212yescomplemented
19#3207C2212yescomplemented
20#3191—22no—
21#3170C2211yescharacteristic, complemented
22#3169C2211yescharacteristic, complemented
23#3167C2212yescomplemented
24#3114—24no—
25#3110C2214yes—
26#3109C2214yes—
27#3105—24no—
28#3102—24no—
29#3101C2214yes—
30#3100C2214yes—
31#3092—22no—
32#3077C2214yes—
33#3036C2214yes—
34#3003—22no—
35#2975—24no—
36#2975C2214yes—
37#2975C2214yes—
38C4×AGL(1,ℤ/8)—24no—
39C2×#436—22no—
40C2×#436C2212yes—
41C2×#436C2212yes—
42#2920C2212yes—
43C2×#431C2211yescharacteristic
44C2×#431C2211yescharacteristic
45C2×#407—22no—
46#1703C2218yescomplemented
47#1701C2218yescomplemented
48#1699C2218yescomplemented
49#1697C2218yescomplemented
50#1695C2214yescomplemented
51#1693C2214yescomplemented
52#1691C2218yescomplemented
53#1689C2214yescomplemented
54#1687C2214yescomplemented
55#1685C2218yescomplemented
56#1572C2214yescomplemented
57#1572C2214yescomplemented
58#1567C2212yescomplemented
59#1566C2212yescomplemented
60#1554C2214yescomplemented
61#1553C2214yescomplemented
62#1550C2214yescomplemented
63#1547C2214yescomplemented
64#1544C2214yescomplemented
65#1543C2214yescomplemented
66C23×D4D411yescharacteristic
67C24×C4—21no—
68C2×AGL(1,ℤ/8)—24no—
69C22×M16—22no—
70C22×M16—22no—
71#533C2312yes—
72#533C2312yes—
73#533C2312yes—
74#532C2312yes—
75#532C2312yes—
76#532C2312yes—
77#529—22no—
78#529—22no—
79#529—22no—
80#529—22no—
81#529—22no—
82#529—22no—
83#525—22no—
84#525—22no—
85#525—22no—
86C4×#41—22no—
87C4×#41—22no—
88C4×#41—22no—
89C2×C4×D4—21no—
90C2×C4×D4—22no—
91C2×C4×D4—22no—
92C2×C4×D4—21no—
93C2×C4×D4D414yes—
94C2×C4×D4D412yes—
95C2×C4×D4C2312yes—
96C2×C4×D4C2314yes—
97C22×C4⋊C4D411yescharacteristic
98C22×C22⋊C4—21no—
99C22×C42C2311yescharacteristic
100#500C2311yescharacteristic, complemented
101#500C2311yescharacteristic, complemented
102#498C2312yescomplemented
103#494C2311yescharacteristic, complemented
104#493C2311yescharacteristic, complemented
105#492C2312yescomplemented
106#487C2312yescomplemented
107#487C2312yescomplemented
108#486C2312yescomplemented
109#486C2312yescomplemented
110#485—24no—
111#484—24no—
112#483—24no—
113#482—24no—
114#481—24no—
115#480—24no—
116#479—24no—
117#478—28no—
118#478—24no—
119#476C2318yes—
120#474C2318yes—
121#467—22no—
122#466—22no—
123#465—24no—
124#464—24no—
125#463—28no—
126#463—24no—
127#461C2318yes—
128#459C2318yes—
129#452—24no—
130#450—24no—
131#450—24no—
132#449—24no—
133#447—24no—
134#447—24no—
135#442—28no—
136#440—28no—
137C4×Q16—22no—
138C4×SD16—28no—
139C4×SD16—24no—
140C4×D8—28no—
141C4×D8—22no—
142#436—28no—
143#436—28no—
144#436—22no—
145#436—22no—
146#436—24no—
147#436C2318yes—
148#436C2314yes—
149#436C2314yes—
150#432—22no—
151#432C2314yes—
152#431—22no—
153#431—22no—
154#431D414yes—
155#431C2314yes—
156#431C2314yes—
157#431C2312yes—
158#431C2312yes—
159#428—24no—
160#423C2314yes—
161C2×C4⋊C8D412yes—
162C2×C4⋊C8C2312yes—
163#418—24no—
164#417—24no—
165#407—22no—
166#407—28no—
167#407—28no—
168C2×C22⋊C8—21no—
169C2×C22⋊C8—22no—
170C2×C22⋊C8—22no—
171C2×C22⋊C8D412yes—
172C4×M16—22no—
173C2×C4×C8—21no—
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176#335C2318yescomplemented
177#334C2314yescomplemented
178#334C2314yescomplemented
179#333C2318yescomplemented
180#332C2318yescomplemented
181#332C2318yescomplemented
182#331C2318yescomplemented
183C25—21no—
184C2×#41—22no—
185C2×#41—22no—
186C2×#41—22no—
187C22×D4—22no—
188C22×D4—21no—
189C22×D4—44no—
190C22×D4C2×D412yes—
191C22×D4C2×D414yes—
192C23×C4—21no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—84no—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4#4111yescharacteristic
201AGL(1,ℤ/8)—416no—
202C2×Q16—22no—
203C2×SD16—28no—
204C2×SD16—24no—
205C2×D8—28no—
206C2×D8—22no—
207C2×M16—24no—
208C2×M16—24no—
209C2×M16—28no—
210C2×M16—24no—
211C2×M16—84no—
212C2×M16—84no—
213C22×C8—22no—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C4⋊Q8C2412yes—
217C4⋊Q8C2412yes—
218C4⋊Q8C2412yes—
219#127C2412yes—
220#126—42no—
221#126—42no—
222#125—22no—
223#125—22no—
224#125—22no—
225#124—22no—
226#124—22no—
227#124—22no—
228#123—42no—
229#123—42no—
230#123—42no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8—22no—
237#121—22no—
238#121—22no—
239#121—22no—
240#121—22no—
241#121—22no—
242#121—22no—
243#120—42no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—22no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—24no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—42no—
264C4×D4—42no—
265C4×D4—42no—
266C4×D4—42no—
267C4×D4—42no—
268C4×D4—42no—
269C4×D4C2×D418yes—
270C4×D4C2×D414yes—
271C4×D4C2414yes—
272#117—22no—
273#117—22no—
274#117—22no—
275#117—42no—
276#117—42no—
277#117—42no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2×D414yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C42—42no—
287C2×C42C2×D412yes—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D412yes—
291C2×C42#4112yes—
292C2×C42#4111yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2412yes—
295#107—22no—
296#107—22no—
297#107—28no—
298#106—24no—
299#106—28no—
300C4⋊C8—22no—
301C4⋊C8—24no—
302C4⋊C8C2×D414yes—
303C4⋊C8C2×D414yes—
304C4⋊C8C2414yes—
305C4⋊C8C2418yes—
306#103—24no—
307#103—24no—
308#103—28no—
309#102—28no—
310#102—28no—
311#102—24no—
312#102—24no—
313#102—28no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317C22⋊C8—24no—
318C22⋊C8—28no—
319C22⋊C8—24no—
320C22⋊C8C2×D414yes—
321C22⋊C8C2×D414yes—
322#97—24no—
323C4×C8—24no—
324C4×C8—24no—
325C4×C8C2×D412yes—
326C4×C8C2×D412yes—
327C4×C8C2411yescharacteristic
328C4×C8C2411yescharacteristic
329C4×C8C2412yes—
330C24—22no—
331C24—24no—
332C24—44no—
333C24—82no—
334C24—82no—
335C24#12111yescharacteristic
336#41—48no—
337#41—84no—
338#41—84no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—28no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—48no—
349C2×D4—44no—
350C2×D4—48no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4C22×D414yes—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—28no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—88no—
368C22×C4—81no—
369C22×C4—88no—
370C22×C4—84no—
371C22×C4—82no—
372C22×C4#12112yes—
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12112yes—
376C22×C4#12111yescharacteristic
377C22×C4#12112yes—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C2×#4112yes—
383C22×C4C2×#4111yescharacteristic
384Q16—82no—
385SD16—416no—
386SD16—84no—
387D8—416no—
388D8—82no—
389M16—48no—
390M16—84no—
391M16—88no—
392M16—84no—
393C2×C8—24no—
394C2×C8—28no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—44no—
398C2×C8C22×D412yes—
399C2×C8C22×D414yes—
400C2×C8C22×D412yes—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4C22×D414yes—
416C22⋊C4—28no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42#12112yes—
432C42#12112yes—
433C42#12112yes—
434C42C22×D412yes—
435C42C22×D412yes—
436C42C22×D411yescharacteristic
437C42C22×D411yescharacteristic
438C42C2×#4111yescharacteristic
439C42C2×#4112yes—
440C42C2511yescharacteristic
441C23—22no—
442C23—24no—
443C23—41no—
444C23—41no—
445C23—44no—
446C23—48no—
447C23—44no—
448C23—44no—
449C23—42no—
450C23—84no—
451C23—84no—
452C23—816no—
453C23C2×#12112yes—
454C23C2×#12111yescharacteristic
455Q8—44no—
456Q8—44no—
457Q8—82no—
458D4—28no—
459D4—48no—
460D4—44no—
461D4—44no—
462D4—44no—
463D4—82no—
464D4—84no—
465D4—84no—
466D4—84no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—48no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—88no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—84no—
491C2×C4C2×#12112yes—
492C2×C4C2×#12112yes—
493C2×C4C2×#12111yescharacteristic
494C2×C4C2×#12112yes—
495C2×C4C2×#12111yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4#57514yes—
498C2×C4C23×D411yescharacteristic
499C2×C4C23×D411yescharacteristic
500C2×C4C22×#4111yescharacteristic
501C8—22no—
502C8—24no—
503C8—22no—
504C8—44no—
505C8—48no—
506C22—21no—
507C22—21no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—88no—
516C22—816no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22#305412yes—
521C22C22×#12111yescharacteristic
522C4—21no—
523C4—22no—
524C4—22no—
525C4—22no—
526C4—21no—
527C4—21no—
528C4—44no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4#305412yes—
533C4C2×#57512yes—
534C4#358512yes—
535C2—22no—
536C2—42no—
537C2—84no—
538C2—162no—
539C2#6042311yescharacteristic
540C2C2×#305411yescharacteristic
541C2#6046911yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60944C2semidirect
2. #60558C2semidirect
3. C2×#2975C2semidirect
4. #19431C2semidirect
5. #19430C2semidirect
6. #19425C2semidirect
7. #19424C2semidirect
8. #19281C2semidirect
9. #19280C2semidirect
10. #19279C2semidirect
11. #17815C2semidirect
12. #17716C2semidirect
16. #3231C22semidirect
17. #3228C22semidirect
18. #3210C22semidirect
19. #3207C22semidirect
21. #3170C22semidirect
22. #3169C22semidirect
23. #3167C22semidirect
46. #1703C22semidirect
47. #1701C22semidirect
48. #1699C22semidirect
49. #1697C22semidirect
50. #1695C22semidirect
51. #1693C22semidirect
52. #1691C22semidirect
53. #1689C22semidirect
54. #1687C22semidirect
55. #1685C22semidirect
56. #1572C22semidirect
57. #1572C22semidirect
58. #1567C22semidirect
59. #1566C22semidirect
60. #1554C22semidirect
61. #1553C22semidirect
62. #1550C22semidirect
63. #1547C22semidirect
64. #1544C22semidirect
65. #1543C22semidirect
100. #500C23semidirect
101. #500C23semidirect
102. #498C23semidirect
103. #494C23semidirect
104. #493C23semidirect
105. #492C23semidirect
106. #487C23semidirect
107. #487C23semidirect
108. #486C23semidirect
109. #486C23semidirect
176. #335C23semidirect
177. #334C23semidirect
178. #334C23semidirect
179. #333C23semidirect
180. #332C23semidirect
181. #332C23semidirect
182. #331C23semidirect

Maximal subgroups

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

subgroupindex
#609442
#605582
C2×#29752
#194312
#194302
#194252
#194242
#192812
#192802
#192792
#178152
#177162