Finite Group Databasebeta

← #6479539

Subgroups — #6479539

Order 512.

1917 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#33098C212yesmaximal, complemented
4#33094C212yesmaximal, complemented
5#29909C212yesmaximal, complemented
6#29907C212yesmaximal, complemented
7#27864C214yesmaximal, complemented
8#27863C214yesmaximal, complemented
9#27612C214yesmaximal, complemented
10#27584C214yesmaximal, complemented
11#21944C212yesmaximal, complemented
12#21769C212yesmaximal
13#20682C211yesmaximal, characteristic, complemented
14C2×#525C2211yescharacteristic
15C22×#117—21no—
16C23×C42C2211yescharacteristic
17C2×#407C2212yes—
18#2855C2211yescharacteristic
19#2855C2211yescharacteristic
20#2855C2212yes—
21#2854C2211yescharacteristic
22#2854C2211yescharacteristic
23#2854C2212yes—
24#2853—22no—
25#2849—21no—
26#2849—21no—
27C4×#123—21no—
28C4×#123—21no—
29C4×#123—22no—
30#2297C2212yes—
31#2296C2212yes—
32#2285—22no—
33#2285—22no—
34#2285C2214yes—
35#2285C2214yes—
36C2×#384C2212yes—
37C2×#382—21no—
38#2280C2211yescharacteristic
39C2×#380—21no—
40C2×#380C2212yes—
41C2×C4×C4⋊C4—21no—
42C2×C4×C4⋊C4C2212yes—
43#2100C2214yescomplemented
44#2100C2214yescomplemented
45#2095C2214yescomplemented
46#2094C2214yescomplemented
47#2084C2214yescomplemented
48#2083C2214yescomplemented
49#2072C2214yescomplemented
50#2071C2214yescomplemented
51#1927C2218yescomplemented
52#1925C2218yescomplemented
53#1920C2218yescomplemented
54#1918C2218yescomplemented
55#1895C2214yes—
56#1894C2214yes—
57#1874C2214yes—
58#1872C2214yes—
59#1854C2218yes—
60#1853C2214yes—
61#1852C2214yes—
62#1825C2214yes—
63#1817C2212yes—
64#1817C2214yes—
65#1817C2212yes—
66#1809C2218yes—
67#1788C2214yes—
68#1728C2214yes—
69C22×#41C2311yescharacteristic
70C24×C4D411yescharacteristic
71C24×C4D411yescharacteristic
72C24×C4C2311yescharacteristic
73C22×M16C2314yes—
74#525—28no—
75C2×#123—22no—
76C2×#123—21no—
77C2×#123—21no—
78C2×#123—21no—
79C2×C22⋊Q8C2311yescharacteristic
80C2×C22⋊Q8C2311yescharacteristic
81C2×#121C2311yescharacteristic
82C2×#121C2311yescharacteristic
83C2×#120—21no—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—48no—
89C2×#117C2311yescharacteristic
90C2×#117C2312yes—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4C2314yes—
94C22×C22⋊C4—22no—
95C22×C42—28no—
96C22×C42—22no—
97C22×C42D412yes—
98C22×C42D411yescharacteristic
99C22×C42D411yescharacteristic
100C22×C42D412yes—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2312yes—
105C2×#107C2318yes—
106C2×#106C2318yes—
107C2×#103C2318yes—
108C2×#102C2318yes—
109#407—28no—
110C2×C22⋊C8C2314yes—
111C2×C22⋊C8C2314yes—
112#391C2312yes—
113#391C2312yes—
114#390C2312yes—
115#390C2312yes—
116#389C2312yes—
117#389C2312yes—
118#389C2312yes—
119#389C2312yes—
120#388—22no—
121#388—22no—
122#388—22no—
123#388—22no—
124#387—22no—
125#387—22no—
126#387—22no—
127#387—22no—
128#386—22no—
129#386—22no—
130#386—21no—
131#386—22no—
132#386—21no—
133#385—22no—
134#385—22no—
135#385—21no—
136#385—22no—
137#385—21no—
138#384—22no—
139#384—22no—
140#384C2314yes—
141#384C2314yes—
142#382—24no—
143#382—24no—
144#381—22no—
145#381—21no—
146#381—21no—
147#381—21no—
148#380—24no—
149#380—24no—
150#380—22no—
151#380—22no—
152#380C2312yes—
153#380C2314yes—
154#380C2314yes—
155#380C2311yescharacteristic
156#380C2311yescharacteristic
157#379—21no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—24no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4C2318yes—
164C4×C22⋊C4—21no—
165C4×C22⋊C4—21no—
166C4×C22⋊C4—22no—
167C4×C22⋊C4—22no—
168C2×#95—22no—
169C2×#95—22no—
170#340C2318yes—
171#340C2318yes—
172#340C2318yes—
173#340C2318yes—
174#336C2314yes—
175#336C2314yes—
176#336C2318yes—
177C25C2×D411yescharacteristic
178C2×#41—24no—
179C2×#41—44no—
180C22×Q8C2411yescharacteristic
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C2411yescharacteristic
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—21no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4#4118yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C2×M16—28no—
209C2×M16—48no—
210C22×C8C2418yes—
211#123—48no—
212#123—44no—
213#123—44no—
214#123—44no—
215C22⋊Q8—24no—
216C22⋊Q8—24no—
217#121—24no—
218#121—24no—
219#120—44no—
220C4×D4—44no—
221C4×D4—44no—
222#117—24no—
223#117—44no—
224#117—416no—
225#117—416no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—48no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4C2418yes—
241C2×C4⋊C4C2411yescharacteristic
242C2×C4⋊C4C2411yescharacteristic
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—48no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—28no—
264C2×C42—28no—
265C2×C42—24no—
266C2×C42—28no—
267C2×C42—24no—
268C2×C42—28no—
269C2×C42—28no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—48no—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2412yes—
280C2×C42C2412yes—
281#107—48no—
282#106—48no—
283#103—48no—
284#102—48no—
285C22⋊C8—28no—
286C22⋊C8—28no—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—22no—
291#95—21no—
292#95—21no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301C24—22no—
302C24—22no—
303C24—21no—
304C24—42no—
305C24—42no—
306C24#12314yes—
307C24C22×D411yescharacteristic
308C24C22×D411yescharacteristic
309C24C22×D411yescharacteristic
310#41—88no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×D4—22no—
314C2×D4—24no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C22×C4—232no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—28no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—416no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—416no—
357C22×C4—44no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—42no—
362C22×C4#12314yes—
363C22×C4#12314yes—
364C22×C4#12314yes—
365C22×C4AGL(1,ℤ/8)14yes—
366C22×C4#13714yes—
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D412yes—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C2×#4114yes—
375C22×C4C2×#4114yes—
376C22×C4C2×#4114yes—
377C22×C4C2511yescharacteristic
378M16—88no—
379C2×C8—28no—
380C2×C8—28no—
381C2×C8—28no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C42—24no—
415C42—24no—
416C42—24no—
417C42—24no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—24no—
422C42—44no—
423C42—42no—
424C42—416no—
425C23—22no—
426C23—22no—
427C23—28no—
428C23—22no—
429C23—22no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—22no—
435C23—22no—
436C23—44no—
437C23—44no—
438C23—48no—
439C23—48no—
440C23—41no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23C2×#12312yes—
446C23C2×#12312yes—
447C23C2×#12312yes—
448C23C2×AGL(1,ℤ/8)12yes—
449C23C2×#13712yes—
450C23C23×D411yescharacteristic
451Q8—44no—
452D4—44no—
453D4—84no—
454D4—84no—
455D4—84no—
456C2×C4—216no—
457C2×C4—216no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—48no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—48no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—48no—
497C2×C4—42no—
498C2×C4—84no—
499C2×C4—816no—
500C2×C4C2×#12312yes—
501C2×C4C2×#12312yes—
502C2×C4C2×#12312yes—
503C2×C4C2×AGL(1,ℤ/8)14yes—
504C2×C4C2×#13714yes—
505C2×C4#57514yes—
506C2×C4C22×#4112yes—
507C2×C4#58514yes—
508C8—48no—
509C22—24no—
510C22—22no—
511C22—24no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—416no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—82no—
525C22—81no—
526C22—82no—
527C22—81no—
528C22—81no—
529C22#309214yes—
530C22C22×#12311yescharacteristic
531C22#344912yes—
532C22C22×AGL(1,ℤ/8)11yescharacteristic
533C22C22×#13711yescharacteristic
534C22C2×#57512yes—
535C4—22no—
536C4—22no—
537C4—22no—
538C4—24no—
539C4—24no—
540C4—24no—
541C4—42no—
542C4—82no—
543C4—88no—
544C4—81no—
545C4—82no—
546C4—81no—
547C4—81no—
548C4#349114yes—
549C2—24no—
550C2—42no—
551C2—42no—
552C2—161no—
553C2#3509711yescharacteristic
554C2C2×#309211yescharacteristic
555C2#6051111yescharacteristic
556C2#6095112yes—
557C2#6096612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2285C2semidirect
3. #33098C2semidirect
4. #33094C2semidirect
5. #29909C2semidirect
6. #29907C2semidirect
7. #27864C2semidirect
8. #27863C2semidirect
9. #27612C2semidirect
10. #27584C2semidirect
11. #21944C2semidirect
13. #20682C2semidirect
43. #2100C22semidirect
44. #2100C22semidirect
45. #2095C22semidirect
46. #2094C22semidirect
47. #2084C22semidirect
48. #2083C22semidirect
49. #2072C22semidirect
50. #2071C22semidirect
51. #1927C22semidirect
52. #1925C22semidirect
53. #1920C22semidirect
54. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#366262
C2×#22852
#330982
#330942
#299092
#299072
#278642
#278632
#276122
#275842
#219442
#217692
#206822