Finite Group Databasebeta

← #6500830

Subgroups — #6500830

Order 512.

1792 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#35633C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic
3#27621C212yesmaximal, complemented
4#27604C212yesmaximal, complemented
5#27596C212yesmaximal, complemented
6#26217C214yesmaximal, complemented
7#26201C214yesmaximal, complemented
8#26144C214yesmaximal, complemented
9#26128C214yesmaximal, complemented
10#24003C212yesmaximal, complemented
11#21771C212yesmaximal, complemented
12#21744C211yesmaximal, characteristic
13#20848C212yesmaximal, complemented
14C2×#536C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C2×#520C2211yescharacteristic
18C2×#483C2214yes—
19C2×#428C2212yescomplemented
20C22×#106C2212yescomplemented
21C2×#408C2212yes—
22#2534—22no—
23#2486—22no—
24#2429—22no—
25#2427—22no—
26#2417C2214yes—
27#2415C2214yes—
28#2376—22no—
29#2358—22no—
30#2352C2214yes—
31D4×C4⋊C4C2214yes—
32#2329—22no—
33#2325—22no—
34#2316C2211yescharacteristic
35#2314C2212yes—
36#2311C2212yes—
37#2310C2211yescharacteristic
38#2280C2211yescharacteristic
39C2×#380C2212yes—
40#2101C2214yescomplemented
41#2096C2212yescomplemented
42#2095C2212yescomplemented
43#2084C2214yescomplemented
44#2077C2214yes—
45#2075C2214yes—
46#2074C2212yescomplemented
47#2074C2212yescomplemented
48#2035C2214yes—
49#2035C2214yes—
50#1928C2214yescomplemented
51#1927C2214yescomplemented
52#1920C2214yescomplemented
53#1919C2214yescomplemented
54#1894C2214yescomplemented
55#1892C2214yescomplemented
56#1873C2214yescomplemented
57#1872C2214yes—
58#1872C2214yescomplemented
59#1870C2214yes—
60#1869C2214yes—
61#1854C2214yes—
62#1854C2214yescomplemented
63#1853C2212yes—
64#1853C2214yes—
65#1853C2214yescomplemented
66#1852C2212yes—
67#1825C2214yes—
68#1824C2212yes—
69#1824C2212yes—
70#1818C2212yes—
71#1787C2214yes—
72#1779C2212yes—
73#1736C2212yescomplemented
74#1735C2212yescomplemented
75C22×#41C2311yescharacteristic
76C23×Q8C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C22×M16C2312yes—
79C23×C8C2312yes—
80#536—28no—
81C2×C4⋊Q8—21no—
82C2×#124—21no—
83#525—28no—
84C2×#123—22no—
85C2×#123D412yes—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—44no—
88C2×C22⋊Q8D412yes—
89C2×C22⋊Q8C2314yes—
90C2×C22⋊Q8C2314yes—
91C2×#121D412yes—
92C2×#120D412yes—
93#520—44no—
94C2×C4×Q8—21no—
95C2×C4×D4—21no—
96C2×C4×D4—22no—
97C2×C4×D4—24no—
98C2×C4×D4C2314yes—
99C2×#117C2311yescharacteristic
100C2×#117C2312yes—
101C2×#117C2311yescharacteristic
102C2×#117C2312yes—
103C2×#117C2311yescharacteristic
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—22no—
112#483—48no—
113#428—44no—
114C2×#107C2314yes—
115C2×#106—44no—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123#408—28no—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2312yes—
129#398—24no—
130#397—24no—
131#396—24no—
132#394—24no—
133#391—22no—
134#391—22no—
135#391C2312yes—
136#391C2312yes—
137#390—22no—
138#389—22no—
139#389—24no—
140#389—24no—
141#389C2314yes—
142#388—24no—
143#388—24no—
144#388—24no—
145#387—24no—
146#387—24no—
147#386—22no—
148#386—22no—
149#386—24no—
150#386—24no—
151#385—24no—
152#385—24no—
153#384C2312yes—
154#382C2312yes—
155#381—21no—
156#381—22no—
157#380—22no—
158#380—22no—
159#380—22no—
160#380—24no—
161#380C2314yes—
162#380C2314yes—
163#380C2312yes—
164#380C2314yes—
165#380C2312yes—
166#379—21no—
167C4×C4⋊C4C2314yes—
168C4×C22⋊C4—22no—
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—42no—
184C2×#41—41no—
185C2×#41—41no—
186C2×#41#4118yes—
187C22×Q8—21no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8—41no—
191C22×Q8—41no—
192C22×Q8C2×D411yescharacteristic
193C22×Q8C2412yes—
194C22×D4C2×D414yes—
195C22×D4C2×D412yes—
196C22×D4C2×D411yescharacteristic
197C23×C4—21no—
198C23×C4—24no—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—42no—
205C23×C4—41no—
206C23×C4—41no—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C2×M16—24no—
213C2×M16—42no—
214C2×M16—42no—
215C22×C8—24no—
216C22×C8—42no—
217C22×C8—42no—
218C22×C8C2414yes—
219C22×C8C2414yes—
220C4⋊Q8—44no—
221#124—44no—
222#123—28no—
223#123—48no—
224C22⋊Q8—28no—
225C22⋊Q8—28no—
226C22⋊Q8—28no—
227C22⋊Q8—28no—
228C22⋊Q8—44no—
229C22⋊Q8—48no—
230#121—28no—
231#120—28no—
232C4×Q8—44no—
233C4×D4—28no—
234C4×D4—28no—
235C4×D4—44no—
236C4×D4—48no—
237C4×D4—88no—
238#117—24no—
239#117—24no—
240#117—24no—
241#117—42no—
242#117—42no—
243#117—42no—
244#117—42no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2411yescharacteristic
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2411yescharacteristic
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—84no—
284C2×C22⋊C4C2×D412yes—
285C2×C22⋊C4C2×D414yes—
286C2×C22⋊C4C2×D414yes—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42C2412yes—
293#107—44no—
294#106—28no—
295#106—28no—
296#106—44no—
297#103—44no—
298#103—44no—
299#103—44no—
300#103—44no—
301C22⋊C8—28no—
302C22⋊C8—28no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305C22⋊C8—44no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316C24—22no—
317C24—24no—
318C24—41no—
319C24—42no—
320C24—41no—
321C24C22×D411yescharacteristic
322C24C22×D412yes—
323C24C22×D412yes—
324#41—48no—
325#41—48no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—84no—
336C2×Q8—84no—
337C2×Q8#12314yes—
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—44no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4#12318yes—
346C2×D4#12314yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4C22×D414yes—
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D412yes—
392C22×C4C2×#4114yes—
393C22×C4C2×#4114yes—
394C22×C4C2×#4114yes—
395C22×C4C2511yescharacteristic
396M16—48no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—48no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—28no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—84no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—84no—
423C4⋊C4H(5,𝔽2)14yes—
424C22⋊C4—24no—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—48no—
428C22⋊C4—44no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—48no—
432C22⋊C4—44no—
433C22⋊C4—48no—
434C22⋊C4—82no—
435C22⋊C4—84no—
436C22⋊C4—82no—
437C22⋊C4—84no—
438C42—24no—
439C42—44no—
440C42—42no—
441C42—42no—
442C42—82no—
443C42—84no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23—21no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—22no—
458C23—48no—
459C23—44no—
460C23—82no—
461C23—84no—
462C23—82no—
463C23C2×#12314yes—
464C23C2×#12312yes—
465C23#57714yes—
466C23C23×D411yescharacteristic
467Q8—44no—
468Q8—44no—
469Q8—44no—
470Q8—84no—
471D4—44no—
472D4—48no—
473D4—416no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—84no—
511C2×C4—82no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4—164no—
515C2×C4—164no—
516C2×C4—164no—
517C2×C4—164no—
518C2×C4C2×#12312yes—
519C2×C4C2×#12314yes—
520C2×C4C22×#4112yes—
521C2×C4C2×H(5,𝔽2)12yes—
522C2×C4#58511yescharacteristic
523C2×C4#58511yescharacteristic
524C8—44no—
525C8—44no—
526C22—22no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—28no—
531C22—24no—
532C22—24no—
533C22—42no—
534C22—42no—
535C22—44no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22—88no—
544C22C22×#12311yescharacteristic
545C22#348411yescharacteristic
546C22#348511yescharacteristic
547C22C2×#57712yes—
548C22#358612yes—
549C4—22no—
550C4—24no—
551C4—42no—
552C4—44no—
553C4—44no—
554C4—82no—
555C4—84no—
556C4—82no—
557C4—81no—
558C4—84no—
559C4—81no—
560C4#350512yes—
561C4#350512yes—
562C2—44no—
563C2—42no—
564C2—44no—
565C2#3535111yescharacteristic
566C2#6052212yes—
567C2#6163311yescharacteristic
568C2#6164911yescharacteristic
569C2#6175512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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
3. #27621C2semidirect
4. #27604C2semidirect
5. #27596C2semidirect
6. #26217C2semidirect
7. #26201C2semidirect
8. #26144C2semidirect
9. #26128C2semidirect
10. #24003C2semidirect
11. #21771C2semidirect
13. #20848C2semidirect
19. C2×#428C22semidirect
20. C22×#106C22semidirect
40. #2101C22semidirect
41. #2096C22semidirect
42. #2095C22semidirect
43. #2084C22semidirect
46. #2074C22semidirect
47. #2074C22semidirect
50. #1928C22semidirect
51. #1927C22semidirect
52. #1920C22semidirect
53. #1919C22semidirect
54. #1894C22semidirect
55. #1892C22semidirect
56. #1873C22semidirect
58. #1872C22semidirect
62. #1854C22semidirect
65. #1853C22semidirect
73. #1736C22semidirect
74. #1735C22semidirect

Maximal subgroups

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

subgroupindex
#356332
#343602
#276212
#276042
#275962
#262172
#262012
#261442
#261282
#240032
#217712
#217442
#208482