Finite Group Databasebeta

← #6635556

Subgroups — #6635556

Order 512.

1676 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37453C211yesmaximal, characteristic
2#35672C211yesmaximal, characteristic
3#32915C212yesmaximal, complemented
4#31951C214yesmaximal, complemented
5#31939C214yesmaximal, complemented
6#31006C214yesmaximal, complemented
7#30990C214yesmaximal, complemented
8#29766C212yesmaximal, complemented
9#28322C212yesmaximal, complemented
10#27985C212yesmaximal, complemented
11#24010C212yesmaximal
12#23921C212yesmaximal
13#21908C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2211yescharacteristic
16C2×#519C2211yescharacteristic
17C2×#485C2214yes—
18C2×#462C2212yescomplemented
19C2×#462C2212yescomplemented
20C2×#451C2214yes—
21C2×#451C2214yes—
22#2817—22no—
23#2627—22no—
24#2627—22no—
25#2621C2212yes—
26#2621C2212yes—
27#2503C2214yes—
28#2484—22no—
29#2483—22no—
30#2482—22no—
31#2434C2214yes—
32#2423—22no—
33#2423C2214yes—
34#2421—22no—
35#2421C2214yes—
36#2317C2211yescharacteristic
37#2317C2211yescharacteristic
38#2310C2211yescharacteristic
39#2280C2211yescharacteristic
40#2280C2211yescharacteristic
41#2097C2214yescomplemented
42#2085C2214yescomplemented
43#2074C2214yescomplemented
44#2072C2214yescomplemented
45#2068C2214yescomplemented
46#2066C2214yescomplemented
47#2043C2214yescomplemented
48#2036C2214yes—
49#2035C2214yes—
50#2016C2214yescomplemented
51#2001C2214yes—
52#2001C2214yes—
53#1971C2214yescomplemented
54#1969C2214yescomplemented
55#1960C2212yescomplemented
56#1956C2212yescomplemented
57#1924C2214yes—
58#1923C2214yes—
59#1878C2214yescomplemented
60#1878C2214yescomplemented
61#1873C2214yes—
62#1872C2214yes—
63#1872C2214yes—
64#1853C2212yes—
65#1852C2212yes—
66#1836C2212yescomplemented
67#1835C2212yescomplemented
68#1805C2212yescomplemented
69#1805C2212yescomplemented
70#1798C2212yes—
71#1787C2212yes—
72#1787C2212yes—
73#1779C2212yes—
74#1750C2212yes—
75C23×Q8C2311yescharacteristic
76C23×Q8C2312yes—
77C24×C4C2311yescharacteristic
78C22×Q16C2314yes—
79C22×Q16C2314yes—
80C2×C4⋊Q8C2312yes—
81C2×C4⋊Q8C2312yes—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—48no—
86C2×C22⋊Q8—44no—
87C2×C22⋊Q8C2314yes—
88C2×C22⋊Q8C2314yes—
89C2×C22⋊Q8C2314yes—
90#519—44no—
91C2×C4×Q8—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2312yes—
95C2×#117—22no—
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C22×C4⋊C4—22no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—22no—
106#485—48no—
107#462—44no—
108#462—44no—
109#451—48no—
110#451—48no—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×C4⋊C8C2312yes—
114C2×C4⋊C8C2312yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×C22⋊C8C2314yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2314yes—
127C2×#97C2312yes—
128C2×C4×C8C2312yes—
129#400—24no—
130#400—24no—
131#400—24no—
132#399—24no—
133#398—22no—
134#398—22no—
135#398—24no—
136#398—24no—
137#397—22no—
138#397—22no—
139#397—24no—
140#397—24no—
141#395—24no—
142#395—24no—
143#391—22no—
144#391C2314yes—
145#391C2314yes—
146#391C2314yes—
147#389—22no—
148#389C2314yes—
149#388—24no—
150#388—24no—
151#387—24no—
152#387—24no—
153#385—22no—
154#385—22no—
155#385—24no—
156#385—24no—
157#384—21no—
158#384C2312yes—
159#382—21no—
160#382C2312yes—
161#382C2312yes—
162#382C2312yes—
163#381—21no—
164#381—22no—
165#381—24no—
166#381—24no—
167#381D412yes—
168#380D412yes—
169#380D412yes—
170#380C2312yes—
171#380C2312yes—
172#379—21no—
173#379D412yes—
174C4×C22⋊C4—22no—
175C4×C22⋊C4C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#336C2314yes—
181#336C2314yes—
182C25C2×D411yescharacteristic
183C22×Q8—24no—
184C22×Q8—21no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—41no—
188C22×Q8—41no—
189C22×Q8—44no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8—44no—
193C22×Q8C2414yes—
194C22×Q8C2412yes—
195C23×C4—21no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—42no—
201C23×C4—41no—
202C23×C4—41no—
203C23×C4C2×D412yes—
204C23×C4C2×D411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C2×Q16—28no—
208C2×Q16—28no—
209C2×Q16—44no—
210C2×Q16—44no—
211C2×Q16—44no—
212C2×Q16—44no—
213C22×C8C2414yes—
214C22×C8C2414yes—
215C4⋊Q8—42no—
216C4⋊Q8—42no—
217C4⋊Q8—42no—
218C4⋊Q8—42no—
219C22⋊Q8—28no—
220C22⋊Q8—28no—
221C22⋊Q8—28no—
222C22⋊Q8—44no—
223C22⋊Q8—44no—
224C22⋊Q8—44no—
225C22⋊Q8—48no—
226C22⋊Q8—48no—
227C22⋊Q8—48no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—42no—
231C4×Q8—42no—
232C4×Q8—44no—
233C4×Q8—44no—
234#117—24no—
235#117—41no—
236#117—41no—
237#117—42no—
238#117—42no—
239#117—48no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2411yescharacteristic
262C2×C4⋊C4C2411yescharacteristic
263C2×C4⋊C4C2412yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4C2×D414yes—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D412yes—
276C2×C22⋊C4C2×D414yes—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42C2412yes—
282C2×C42C2412yes—
283#107—44no—
284#106—44no—
285C4⋊C8—42no—
286C4⋊C8—42no—
287#103—44no—
288#103—44no—
289#103—44no—
290#103—44no—
291#103—44no—
292#103—44no—
293#103—44no—
294#103—44no—
295C22⋊C8—28no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298C22⋊C8—44no—
299#97—42no—
300C4×C8—42no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95C2×D414yes—
315#95C2×D414yes—
316C24—22no—
317C24—24no—
318C24—41no—
319C24—42no—
320C24—41no—
321C24C22×D411yescharacteristic
322C24C22×D412yes—
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—42no—
338C2×Q8—44no—
339C2×Q8—42no—
340C2×Q8—82no—
341C2×Q8—82no—
342C2×Q8—88no—
343C2×Q8—88no—
344C2×Q8H(5,𝔽2)14yes—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4C22×D414yes—
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4H(5,𝔽2)14yes—
384C22×C4C2511yescharacteristic
385Q16—88no—
386Q16—88no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—48no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4—84no—
414C22⋊C4—28no—
415C22⋊C4—44no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—42no—
421C22⋊C4—48no—
422C22⋊C4—42no—
423C22⋊C4—44no—
424C22⋊C4—82no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—44no—
430C42—44no—
431C42—42no—
432C42—42no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—21no—
438C23—22no—
439C23—21no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—48no—
444C23—44no—
445C23—82no—
446C23—84no—
447C23—82no—
448C23#53512yes—
449C23#57814yes—
450C23C23×D411yescharacteristic
451Q8—24no—
452Q8—24no—
453Q8—48no—
454Q8—42no—
455Q8—42no—
456Q8—44no—
457Q8—44no—
458Q8—44no—
459Q8—44no—
460Q8—88no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—48no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4—88no—
499C2×C4—84no—
500C2×C4—88no—
501C2×C4#53512yes—
502C2×C4C2×H(5,𝔽2)12yes—
503C2×C4C2×H(5,𝔽2)12yes—
504C2×C4#58511yescharacteristic
505C2×C4#58511yescharacteristic
506C8—44no—
507C8—44no—
508C22—24no—
509C22—22no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22—88no—
521C22C2×#53511yescharacteristic
522C22#348411yescharacteristic
523C22#348511yescharacteristic
524C22C2×#57812yes—
525C22#358612yes—
526C4—22no—
527C4—44no—
528C4—42no—
529C4—44no—
530C4—42no—
531C4—42no—
532C4—84no—
533C4—82no—
534C4—84no—
535C4—81no—
536C4—81no—
537C4—82no—
538C4#353412yes—
539C4#353412yes—
540C2—44no—
541C2—42no—
542C2#3672111yescharacteristic
543C2#6081412yes—
544C2#6163411yescharacteristic
545C2#6165211yescharacteristic
546C2#6175712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32915C2semidirect
4. #31951C2semidirect
5. #31939C2semidirect
6. #31006C2semidirect
7. #30990C2semidirect
8. #29766C2semidirect
9. #28322C2semidirect
10. #27985C2semidirect
18. C2×#462C22semidirect
19. C2×#462C22semidirect
41. #2097C22semidirect
42. #2085C22semidirect
43. #2074C22semidirect
44. #2072C22semidirect
45. #2068C22semidirect
46. #2066C22semidirect
47. #2043C22semidirect
50. #2016C22semidirect
53. #1971C22semidirect
54. #1969C22semidirect
55. #1960C22semidirect
56. #1956C22semidirect
59. #1878C22semidirect
60. #1878C22semidirect
66. #1836C22semidirect
67. #1835C22semidirect
68. #1805C22semidirect
69. #1805C22semidirect

Maximal subgroups

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

subgroupindex
#374532
#356722
#329152
#319512
#319392
#310062
#309902
#297662
#283222
#279852
#240102
#239212
#219082