Finite Group Databasebeta

← #6633318

Subgroups — #6633318

Order 512.

1955 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37639C211yesmaximal, characteristic, complemented
2#35541C211yesmaximal, characteristic, complemented
3#32911C212yesmaximal, complemented
4#31665C214yesmaximal, complemented
5#31645C214yesmaximal, complemented
6#30749C214yesmaximal, complemented
7#30733C214yesmaximal, complemented
8#29725C212yesmaximal, complemented
9#29340C212yesmaximal, complemented
10#25734C212yesmaximal, complemented
11#23932C212yesmaximal, complemented
12#23930C212yesmaximal, complemented
13#21918C211yesmaximal, characteristic, complemented
14C2×#535C2211yescharacteristic
15C2×#532C2212yes—
16C2×#518C2211yescharacteristic
17C2×#496C2212yes—
18C2×#493C2212yes—
19C2×#459C2214yes—
20C2×#449C2214yes—
21C2×#447C2214yes—
22#2819—22no—
23#2619C2212yes—
24#2619C2212yes—
25#2615—24no—
26#2563C2214yescomplemented
27#2541C2214yescomplemented
28#2537—22no—
29#2447C2214yes—
30#2445C2214yes—
31#2422—22no—
32#2420—22no—
33#2412—24no—
34#2316C2211yescharacteristic
35#2311C2211yescharacteristic
36#2297C2211yescharacteristic
37C2×#390C2211yescharacteristic
38#2285C2211yescharacteristic
39#2101C2214yescomplemented
40#2074C2212yescomplemented
41#2073C2212yescomplemented
42#2055C2214yescomplemented
43#2054C2214yescomplemented
44#2041C2214yescomplemented
45#2024C2212yescomplemented
46#2023C2214yes—
47#2023C2212yescomplemented
48#2014C2212yescomplemented
49#2014C2212yescomplemented
50#2012C2214yes—
51#2011C2214yes—
52#2011C2214yes—
53#2011C2212yescomplemented
54#2011C2212yescomplemented
55#1961C2212yes—
56#1958C2212yes—
57#1936C2214yescomplemented
58#1934C2214yescomplemented
59#1917C2214yescomplemented
60#1916C2214yescomplemented
61#1879C2214yes—
62#1879C2214yes—
63#1879C2214yes—
64#1879C2214yes—
65#1879C2214yescomplemented
66#1877C2214yescomplemented
67#1875C2214yes—
68#1874C2214yes—
69#1817C2212yes—
70#1804C2214yes—
71#1798C2212yes—
72#1750C2212yes—
73C22×#41—22no—
74C22×#41C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C22×SD16C2314yes—
77C22×D8C2314yes—
78C22×D8C2314yes—
79C22×D8C2314yes—
80#535—44no—
81#532—48no—
82C2×C4⋊Q8C2312yes—
83C2×#127C2312yes—
84C2×#127C2312yes—
85C2×#127C2312yes—
86C2×#127C2312yes—
87C2×#124—22no—
88C2×#124C2312yes—
89C2×#123—24no—
90C2×#121—22no—
91C2×#121—22no—
92C2×#121—24no—
93C2×#121—24no—
94C2×#121C2314yes—
95C2×#120—22no—
96#518—44no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4C2314yes—
100C2×C4×D4C2314yes—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—22no—
106C22×C22⋊C4—22no—
107C22×C42C2311yescharacteristic
108#496—44no—
109#493—44no—
110#459—48no—
111#449—48no—
112#447—48no—
113C2×C4⋊C8C2314yes—
114C2×#103C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×C22⋊C8C2314yes—
123C2×#97C2312yes—
124C2×C4×C8C2312yes—
125#397—24no—
126#397—24no—
127#394—24no—
128#394—24no—
129#394—24no—
130#394—24no—
131#393—24no—
132#393—24no—
133#392—24no—
134#392—24no—
135#392—24no—
136#392—24no—
137#391C2312yes—
138#391C2312yes—
139#390—21no—
140#390—21no—
141#390C2314yes—
142#390C2314yes—
143#390C2312yes—
144#390C2314yes—
145#390C2312yes—
146#389—22no—
147#389C2314yes—
148#386—22no—
149#386—22no—
150#386—24no—
151#385—24no—
152#384D412yes—
153#382D412yes—
154#382C2312yes—
155#382C2312yes—
156#381—24no—
157#380—24no—
158#380—24no—
159#380—22no—
160#380—22no—
161#380D414yes—
162C4×C4⋊C4C2314yes—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165#340C2314yes—
166#340C2314yes—
167#340C2314yes—
168#340C2314yes—
169#336C2312yes—
170#336C2312yes—
171#336C2314yes—
172#336C2312yes—
173#336C2312yes—
174#336C2314yes—
175C25—21no—
176C2×#41—24no—
177C2×#41—42no—
178C2×#41—42no—
179C2×#41—48no—
180C2×#41—84no—
181C2×#41—84no—
182C22×Q8—22no—
183C22×Q8C2411yescharacteristic
184C22×D4—24no—
185C22×D4—21no—
186C22×D4—24no—
187C22×D4—21no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—42no—
195C22×D4C2414yes—
196C22×D4C2411yescharacteristic
197C22×D4C2412yes—
198C23×C4—21no—
199C23×C4—24no—
200C23×C4—21no—
201C23×C4—24no—
202C23×C4C2×D412yes—
203C23×C4C2411yescharacteristic
204C2×SD16—44no—
205C2×SD16—44no—
206C2×SD16—44no—
207C2×D8—28no—
208C2×D8—28no—
209C2×D8—44no—
210C2×D8—44no—
211C2×D8—44no—
212C2×D8—44no—
213C2×D8—44no—
214C2×D8—44no—
215C2×D8—44no—
216C22×C8C2414yes—
217C22×C8C2414yes—
218C4⋊Q8—44no—
219#127—24no—
220#127—44no—
221#127—42no—
222#127—44no—
223#127—42no—
224#127—42no—
225#124—24no—
226#124—42no—
227#124—48no—
228#123—88no—
229#121—28no—
230#121—28no—
231#121—48no—
232#121—48no—
233#121—88no—
234#121—88no—
235#120—48no—
236C4×D4—28no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4—44no—
240C4×D4—48no—
241C4×D4—48no—
242#117—42no—
243#117—42no—
244#117—42no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C2×D414yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2414yes—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—84no—
272C2×C22⋊C4C2×D412yes—
273C2×C22⋊C4C2×D412yes—
274C2×C42—24no—
275C2×C42—21no—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2412yes—
279C2×C42C2412yes—
280C2×C42C2412yes—
281C4⋊C8—44no—
282#103—44no—
283#102—44no—
284#102—44no—
285#102—44no—
286#102—44no—
287#102—44no—
288#102—44no—
289#102—44no—
290C22⋊C8—28no—
291#97—42no—
292C4×C8—42no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95C2×D414yes—
300#95C2×D414yes—
301C24—24no—
302C24—22no—
303C24—21no—
304C24—24no—
305C24—42no—
306C24—42no—
307C24—82no—
308C24—82no—
309C24C22×D411yescharacteristic
310#41—88no—
311#41—832no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—44no—
316C2×Q8—82no—
317C2×Q8—82no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—416no—
332C2×D4—42no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—48no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—48no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—44no—
343C2×D4—82no—
344C2×D4—82no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4H(5,𝔽2)14yes—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—42no—
365C22×C4—48no—
366C22×C4—42no—
367C22×C4—48no—
368C22×C4—42no—
369C22×C4—48no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—48no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—82no—
376C22×C4C22×D414yes—
377C22×C4C22×D414yes—
378C22×C4C22×D412yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D412yes—
381C22×C4H(5,𝔽2)14yes—
382C22×C4C2511yescharacteristic
383SD16—88no—
384D8—88no—
385D8—88no—
386D8—88no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C4⋊C4—28no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—84no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—42no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—44no—
413C22⋊C4—82no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C22⋊C4—84no—
419C22⋊C4—84no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—24no—
425C42—24no—
426C42—42no—
427C23—22no—
428C23—21no—
429C23—21no—
430C23—22no—
431C23—24no—
432C23—48no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—44no—
438C23—42no—
439C23—44no—
440C23—44no—
441C23—41no—
442C23—41no—
443C23—44no—
444C23—41no—
445C23—44no—
446C23—42no—
447C23—44no—
448C23—44no—
449C23—88no—
450C23—164no—
451C23—162no—
452C23—162no—
453C23#53512yes—
454C23C23×D411yescharacteristic
455Q8—44no—
456Q8—88no—
457D4—28no—
458D4—48no—
459D4—44no—
460D4—44no—
461D4—44no—
462D4—44no—
463D4—44no—
464D4—44no—
465D4—88no—
466D4—88no—
467D4—84no—
468D4—816no—
469D4—168no—
470D4—168no—
471D4—168no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—84no—
499C2×C4—816no—
500C2×C4—168no—
501C2×C4—162no—
502C2×C4#53512yes—
503C2×C4#57614yes—
504C2×C4C2×H(5,𝔽2)12yes—
505C2×C4C2×H(5,𝔽2)12yes—
506C2×C4#58512yes—
507C8—44no—
508C8—44no—
509C22—22no—
510C22—24no—
511C22—41no—
512C22—42no—
513C22—41no—
514C22—44no—
515C22—48no—
516C22—84no—
517C22—82no—
518C22—82no—
519C22—81no—
520C22—84no—
521C22—84no—
522C22—84no—
523C22—82no—
524C22—81no—
525C22—84no—
526C22—82no—
527C22—84no—
528C22—84no—
529C22—84no—
530C22—81no—
531C22—168no—
532C22C2×#53511yescharacteristic
533C22#348411yescharacteristic
534C22#348711yescharacteristic
535C22C2×#57612yes—
536C22#358512yes—
537C4—22no—
538C4—24no—
539C4—42no—
540C4—44no—
541C4—44no—
542C4—81no—
543C4—84no—
544C4—82no—
545C4—82no—
546C4#353414yes—
547C2—42no—
548C2—84no—
549C2—161no—
550C2—164no—
551C2#3671011yescharacteristic
552C2#6081112yes—
553C2#6167912yes—
554C2#6193211yescharacteristic
555C2#6194211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37639C2semidirect
2. #35541C2semidirect
3. #32911C2semidirect
4. #31665C2semidirect
5. #31645C2semidirect
6. #30749C2semidirect
7. #30733C2semidirect
8. #29725C2semidirect
9. #29340C2semidirect
10. #25734C2semidirect
11. #23932C2semidirect
12. #23930C2semidirect
13. #21918C2semidirect
26. #2563C22semidirect
27. #2541C22semidirect
39. #2101C22semidirect
40. #2074C22semidirect
41. #2073C22semidirect
42. #2055C22semidirect
43. #2054C22semidirect
44. #2041C22semidirect
45. #2024C22semidirect
47. #2023C22semidirect
48. #2014C22semidirect
49. #2014C22semidirect
53. #2011C22semidirect
54. #2011C22semidirect
57. #1936C22semidirect
58. #1934C22semidirect
59. #1917C22semidirect
60. #1916C22semidirect
65. #1879C22semidirect
66. #1877C22semidirect

Maximal subgroups

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

subgroupindex
#376392
#355412
#329112
#316652
#316452
#307492
#307332
#297252
#293402
#257342
#239322
#239302
#219182