Finite Group Databasebeta

← #6500863

Subgroups — #6500863

Order 512.

1805 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35150C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic, complemented
3#27621C212yesmaximal, complemented
4#27608C212yesmaximal, complemented
5#27595C212yesmaximal, complemented
6#26228C214yesmaximal, complemented
7#26222C214yesmaximal, complemented
8#26146C214yesmaximal, complemented
9#26144C214yesmaximal, complemented
10#24012C212yesmaximal, complemented
11#21769C212yesmaximal, complemented
12#21743C211yesmaximal, characteristic, complemented
13#20846C212yesmaximal, complemented
14C2×#537C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C2×#525C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#481C2212yescomplemented
19C2×#483C2212yescomplemented
20C2×#428C2214yes—
21C2×#407C2212yes—
22#2854—22no—
23#2617—22no—
24#2415—22no—
25#2415C2214yes—
26#2414—22no—
27#2414C2214yes—
28#2376—22no—
29#2358—22no—
30#2352C2214yes—
31D4×C4⋊C4C2214yes—
32#2329—22no—
33#2325—22no—
34#2316C2211yescharacteristic
35#2314C2212yes—
36#2311C2211yescharacteristic
37#2311C2212yes—
38#2311C2211yescharacteristic
39C2×#380C2212yes—
40#2101C2214yescomplemented
41#2095C2212yescomplemented
42#2093C2212yescomplemented
43#2084C2214yescomplemented
44#2076C2214yescomplemented
45#2075C2214yescomplemented
46#2074C2212yescomplemented
47#2073C2212yescomplemented
48#2035C2214yescomplemented
49#2034C2214yescomplemented
50#1928C2214yescomplemented
51#1925C2214yescomplemented
52#1920C2214yescomplemented
53#1919C2214yescomplemented
54#1895C2214yes—
55#1894C2214yes—
56#1875C2212yescomplemented
57#1874C2212yes—
58#1874C2212yes—
59#1873C2212yescomplemented
60#1872C2212yes—
61#1872C2212yes—
62#1870C2214yes—
63#1870C2214yes—
64#1854C2214yes—
65#1854C2214yes—
66#1853C2214yes—
67#1852C2214yes—
68#1852C2214yescomplemented
69#1825C2214yes—
70#1825C2214yescomplemented
71#1818C2212yes—
72#1788C2214yes—
73#1780C2212yes—
74#1736C2214yes—
75C22×#41C2311yescharacteristic
76C22×#41C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C22×M16C2314yes—
79#537—44no—
80#525—28no—
81#525—28no—
82C2×#123—21no—
83C2×#123D412yes—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8D412yes—
86C2×C22⋊Q8C2312yes—
87C2×C22⋊Q8C2312yes—
88C2×#121—22no—
89C2×#121—21no—
90C2×#121D412yes—
91C2×#121C2312yes—
92C2×#121C2312yes—
93C2×#120—21no—
94C2×#120D412yes—
95C2×C4×D4—22no—
96C2×C4×D4—24no—
97C2×C4×D4C2314yes—
98C2×#117—22no—
99C2×#117—44no—
100C2×#117C2311yescharacteristic
101C2×#117C2314yes—
102C2×#117C2312yes—
103C2×#117C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—22no—
111C22×C42—21no—
112#483—44no—
113#481—44no—
114#428—48no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123#407—28no—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2314yes—
130#397—24no—
131#394—24no—
132#394—24no—
133#392—24no—
134#391—22no—
135#391—21no—
136#391C2312yes—
137#390—22no—
138#390—21no—
139#390—24no—
140#390C2312yes—
141#389—22no—
142#389—24no—
143#389—24no—
144#389C2314yes—
145#388—24no—
146#388—24no—
147#388—24no—
148#388—24no—
149#387—24no—
150#387—24no—
151#386—22no—
152#386—22no—
153#386—22no—
154#386—24no—
155#386—22no—
156#385—24no—
157#385—24no—
158#384C2312yes—
159#382C2312yes—
160#380—22no—
161#380—22no—
162#380—22no—
163#380—22no—
164#380C2314yes—
165#380C2314yes—
166#380C2314yes—
167#380C2314yes—
168C4×C4⋊C4C2314yes—
169C4×C22⋊C4—24no—
170C2×#95—22no—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#336C2312yes—
180#336C2312yes—
181#336C2314yes—
182C25C2×D411yescharacteristic
183C2×#41—24no—
184C2×#41—24no—
185C2×#41—42no—
186C2×#41—42no—
187C2×#41—42no—
188C2×#41—42no—
189C22×Q8C2×D411yescharacteristic
190C22×Q8C2411yescharacteristic
191C22×D4—21no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4C2×D414yes—
195C22×D4C2×D412yes—
196C22×D4C2×D411yescharacteristic
197C22×D4C2411yescharacteristic
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—21no—
202C23×C4—22no—
203C23×C4—21no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4C2×D412yes—
207C23×C4C2×D411yescharacteristic
208C23×C4#4118yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C2×M16—28no—
213C2×M16—44no—
214C2×M16—44no—
215C22×C8C2414yes—
216C22×C8C2414yes—
217#123—28no—
218#123—44no—
219C22⋊Q8—28no—
220C22⋊Q8—24no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8—42no—
224C22⋊Q8—44no—
225#121—28no—
226#121—24no—
227#121—24no—
228#121—24no—
229#121—42no—
230#121—44no—
231#121—48no—
232#120—28no—
233#120—44no—
234C4×D4—28no—
235C4×D4—28no—
236C4×D4—48no—
237C4×D4—88no—
238#117—24no—
239#117—24no—
240#117—28no—
241#117—44no—
242#117—48no—
243#117—44no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—84no—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D414yes—
282C2×C22⋊C4C2×D414yes—
283C2×C42—24no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—44no—
288C2×C42C2412yes—
289#107—44no—
290#107—44no—
291#106—28no—
292#106—28no—
293#103—44no—
294#103—44no—
295#102—44no—
296#102—44no—
297C22⋊C8—28no—
298C22⋊C8—24no—
299C22⋊C8—24no—
300C22⋊C8—44no—
301C22⋊C8—42no—
302C22⋊C8—42no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311C24—21no—
312C24—41no—
313C24—42no—
314C24—41no—
315C24—42no—
316C24#12314yes—
317C24C22×D412yes—
318C24C22×D412yes—
319C24C22×D411yescharacteristic
320#41—88no—
321#41—88no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×D4—28no—
329C2×D4—24no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—41no—
339C2×D4—42no—
340C2×D4—44no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—44no—
344C2×D4—42no—
345C2×D4—82no—
346C2×D4—82no—
347C2×D4—84no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—416no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—84no—
382C22×C4#12318yes—
383C22×C4#12314yes—
384C22×C4C22×D414yes—
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C2×#4114yes—
389C22×C4C2×#4114yes—
390C22×C4C2×#4114yes—
391C22×C4C2511yescharacteristic
392M16—416no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—28no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—84no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—84no—
418C4⋊C4H(5,𝔽2)14yes—
419C22⋊C4—24no—
420C22⋊C4—28no—
421C22⋊C4—28no—
422C22⋊C4—44no—
423C22⋊C4—48no—
424C22⋊C4—44no—
425C22⋊C4—48no—
426C22⋊C4—42no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C22⋊C4—84no—
432C22⋊C4—82no—
433C22⋊C4—84no—
434C42—24no—
435C42—48no—
436C42—42no—
437C42—82no—
438C42—84no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—21no—
447C23—22no—
448C23—21no—
449C23—24no—
450C23—24no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—44no—
455C23—48no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—84no—
462C23—82no—
463C23—82no—
464C23C2×#12312yes—
465C23C2×#12314yes—
466C23C23×D411yescharacteristic
467Q8—44no—
468Q8—82no—
469D4—416no—
470D4—44no—
471D4—82no—
472D4—84no—
473D4—84no—
474D4—88no—
475D4—164no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—48no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4—164no—
513C2×C4—164no—
514C2×C4—164no—
515C2×C4C2×#12312yes—
516C2×C4C2×#12314yes—
517C2×C4C22×#4112yes—
518C2×C4C2×H(5,𝔽2)12yes—
519C2×C4#58512yes—
520C8—44no—
521C8—44no—
522C22—22no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—41no—
531C22—42no—
532C22—41no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—44no—
537C22—42no—
538C22—42no—
539C22—88no—
540C22—81no—
541C22—82no—
542C22—82no—
543C22—81no—
544C22—81no—
545C22C22×#12311yescharacteristic
546C22#348411yescharacteristic
547C22#348511yescharacteristic
548C22#358512yes—
549C22#358612yes—
550C4—22no—
551C4—24no—
552C4—42no—
553C4—44no—
554C4—44no—
555C4—81no—
556C4—84no—
557C4—82no—
558C4—82no—
559C4—84no—
560C4#350514yes—
561C2—42no—
562C2—44no—
563C2—82no—
564C2—161no—
565C2#3535111yescharacteristic
566C2#6051712yes—
567C2#6164911yescharacteristic
568C2#6165111yescharacteristic
569C2#6177612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35150C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34360C2semidirect
3. #27621C2semidirect
4. #27608C2semidirect
5. #27595C2semidirect
6. #26228C2semidirect
7. #26222C2semidirect
8. #26146C2semidirect
9. #26144C2semidirect
10. #24012C2semidirect
11. #21769C2semidirect
12. #21743C2semidirect
13. #20846C2semidirect
18. C2×#481C22semidirect
19. C2×#483C22semidirect
40. #2101C22semidirect
41. #2095C22semidirect
42. #2093C22semidirect
43. #2084C22semidirect
44. #2076C22semidirect
45. #2075C22semidirect
46. #2074C22semidirect
47. #2073C22semidirect
48. #2035C22semidirect
49. #2034C22semidirect
50. #1928C22semidirect
51. #1925C22semidirect
52. #1920C22semidirect
53. #1919C22semidirect
56. #1875C22semidirect
59. #1873C22semidirect
68. #1852C22semidirect
70. #1825C22semidirect

Maximal subgroups

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

subgroupindex
#351502
#343602
#276212
#276082
#275952
#262282
#262222
#261462
#261442
#240122
#217692
#217432
#208462