Finite Group Databasebeta

← #6477205

Subgroups — #6477205

Order 512.

1623 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35310C211yesmaximal, characteristic
2C2×#2629C211yesmaximal, characteristic
3#33214C212yesmaximal, complemented
4#33056C212yesmaximal, complemented
5#30375C214yesmaximal, complemented
6#29172C214yesmaximal, complemented
7#27966C211yesmaximal, characteristic, complemented
8#27905C211yesmaximal, characteristic, complemented
9#27884C212yesmaximal, complemented
10#25912C211yesmaximal, characteristic, complemented
11#25815C211yesmaximal, characteristic, complemented
12#25683C212yesmaximal, complemented
13#24559C212yesmaximal
14#24551C212yesmaximal
15#22084C212yesmaximal, complemented
16#22078C212yesmaximal, complemented
17#20379C211yesmaximal, characteristic
18C2×#541C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C2×#499C2212yescomplemented
22C2×#491C2212yescomplemented
23C2×#483C2212yes—
24C2×#483C2212yes—
25C2×#407C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C2×#405C2212yes—
28#2857C2211yescharacteristic
29#2851C2211yescharacteristic
30#2848C2212yes—
31#2629—22no—
32#2629—22no—
33#2629C2212yes—
34#2629C2212yes—
35#2629C2212yes—
36#2629C2212yes—
37#2629C2211yescharacteristic
38#2625C2212yes—
39#2621C2211yescharacteristic
40#2480—22no—
41#2478—22no—
42#2475—22no—
43#2473—22no—
44C2×#398—22no—
45C2×#396—22no—
46#2323C2212yes—
47#2314C2211yescharacteristic
48C4×C22⋊Q8C2212yes—
49C2×#389C2212yes—
50C2×#384C2211yescharacteristic
51C2×#380C2212yes—
52#2100C2214yescomplemented
53#2096C2214yescomplemented
54#2077C2214yes—
55#2077C2214yes—
56#2059C2214yescomplemented
57#2036C2214yes—
58#2035C2214yes—
59#2026C2214yescomplemented
60#1992C2212yescomplemented
61#1992C2212yescomplemented
62#1991C2212yescomplemented
63#1956C2212yescomplemented
64#1944C2214yescomplemented
65#1941C2214yescomplemented
66#1935C2214yescomplemented
67#1933C2214yescomplemented
68#1920C2214yescomplemented
69#1919C2214yescomplemented
70#1894C2214yes—
71#1892C2214yes—
72#1872C2212yes—
73#1872C2212yes—
74#1865C2214yescomplemented
75#1863C2214yescomplemented
76#1853C2212yes—
77#1852C2212yes—
78#1844C2212yescomplemented
79#1836C2212yescomplemented
80#1823C2212yes—
81#1823C2212yes—
82#1809C2214yes—
83#1807C2214yes—
84#1750C2212yes—
85#1750C2212yes—
86#1749C2211yescharacteristic
87#1749C2211yescharacteristic
88C4×C22⋊C8C2211yescharacteristic
89C4×C22⋊C8C2211yescharacteristic
90#1729C2212yes—
91C24×C4C2311yescharacteristic
92C22×M16C2312yes—
93C23×C8C2312yes—
94#541—44no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#126—22no—
97C2×#125—22no—
98C2×#125—22no—
99C2×#125—44no—
100C2×#125C2311yescharacteristic
101C2×#125C2312yes—
102C2×#125C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C4×Q8C2312yes—
107C2×#117—44no—
108C2×#117D412yes—
109C2×#117D412yes—
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C4⋊C4—22no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4D412yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4—22no—
121C22×C22⋊C4D412yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#499—44no—
125#491—44no—
126#483—28no—
127#483—28no—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136#407—24no—
137C2×C22⋊C8—24no—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2311yescharacteristic
141C2×C22⋊C8C2311yescharacteristic
142C2×C22⋊C8C2311yescharacteristic
143C2×C22⋊C8C2311yescharacteristic
144#405—44no—
145C2×#97C2312yes—
146C2×#97C2312yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#398—22no—
151#398—22no—
152#398—24no—
153#398—24no—
154#398—24no—
155#398—24no—
156#398—22no—
157#397—22no—
158#397—22no—
159#397—22no—
160#396—24no—
161#396—24no—
162#396—24no—
163#396—24no—
164#396—22no—
165#391C2312yes—
166#391C2312yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2312yes—
172#389C2312yes—
173#389C2314yes—
174#389C2314yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—22no—
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382C2311yescharacteristic
192#380—22no—
193#380—22no—
194#380—22no—
195#380C2312yes—
196#380C2312yes—
197#380C2314yes—
198#380C2314yes—
199#380C2312yes—
200#379—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204C2×#95—22no—
205C2×#95—22no—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218C25C2×D411yescharacteristic
219C22×Q8C2412yes—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—42no—
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C2×M16—24no—
235C2×M16—24no—
236C2×M16—42no—
237C22×C8—24no—
238C22×C8—24no—
239C22×C8—42no—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C4⋊Q8—24no—
245#126—48no—
246#125—24no—
247#125—24no—
248#125—24no—
249#125—44no—
250#125—48no—
251#125—48no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—28no—
255C22⋊Q8—44no—
256C4×Q8—44no—
257#117—24no—
258#117—24no—
259#117—42no—
260#117—42no—
261#117—44no—
262#117—44no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—84no—
286C2×C4⋊C4—84no—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—84no—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—42no—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320#107—44no—
321#106—28no—
322#106—28no—
323#106—44no—
324#103—28no—
325#103—28no—
326#103—44no—
327#103—44no—
328C22⋊C8—22no—
329C22⋊C8—22no—
330C22⋊C8—24no—
331C22⋊C8—24no—
332C22⋊C8#4114yes—
333C22⋊C8#4114yes—
334#97—24no—
335#97—42no—
336C4×C8—24no—
337C4×C8—42no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—41no—
357C24—42no—
358C24C22×D411yescharacteristic
359C24C22×D411yescharacteristic
360C24C22×D411yescharacteristic
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—24no—
364C2×Q8—24no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C42−1+414yes—
413C22×C4C2511yescharacteristic
414M16—48no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—48no—
426C2×C8C2×#4114yes—
427C2×C8C2×#4114yes—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—42no—
474C42—44no—
475C42—82no—
476C42H(5,𝔽2)14yes—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—21no—
482C23—21no—
483C23—22no—
484C23—22no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—44no—
490C23—48no—
491C23—44no—
492C23—82no—
493C23—84no—
494C23#53212yes—
495C23#53612yes—
496C23#57714yes—
497C23C23×D411yescharacteristic
498Q8—44no—
499Q8—44no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—44no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—84no—
547C2×C4—88no—
548C2×C4—88no—
549C2×C4#53212yes—
550C2×C4#53414yes—
551C2×C4#53612yes—
552C2×C4C22×#4111yescharacteristic
553C2×C4C22×#4111yescharacteristic
554C2×C4C2×H(5,𝔽2)12yes—
555C2×C4C2×2−1+412yes—
556C8—24no—
557C8—24no—
558C8—42no—
559C8—42no—
560C22—22no—
561C22—22no—
562C22—24no—
563C22—24no—
564C22—22no—
565C22—24no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—88no—
575C22#317414yes—
576C22C2×#53211yescharacteristic
577C22C2×#53411yescharacteristic
578C22C2×#53611yescharacteristic
579C22C2×#57712yes—
580C22#358612yes—
581C4—22no—
582C4—22no—
583C4—24no—
584C4—41no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—84no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—84no—
593C4—82no—
594C4#352012yes—
595C4#352012yes—
596C2—24no—
597C2—42no—
598C2—81no—
599C2#3508511yescharacteristic
600C2C2×#317411yescharacteristic
601C2#6070511yescharacteristic
602C2#6076312yes—
603C2#6087212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33214C2semidirect
4. #33056C2semidirect
5. #30375C2semidirect
6. #29172C2semidirect
7. #27966C2semidirect
8. #27905C2semidirect
9. #27884C2semidirect
10. #25912C2semidirect
11. #25815C2semidirect
12. #25683C2semidirect
15. #22084C2semidirect
16. #22078C2semidirect
21. C2×#499C22semidirect
22. C2×#491C22semidirect
52. #2100C22semidirect
53. #2096C22semidirect
56. #2059C22semidirect
59. #2026C22semidirect
60. #1992C22semidirect
61. #1992C22semidirect
62. #1991C22semidirect
63. #1956C22semidirect
64. #1944C22semidirect
65. #1941C22semidirect
66. #1935C22semidirect
67. #1933C22semidirect
68. #1920C22semidirect
69. #1919C22semidirect
74. #1865C22semidirect
75. #1863C22semidirect
78. #1844C22semidirect
79. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#353102
C2×#26292
#332142
#330562
#303752
#291722
#279662
#279052
#278842
#259122
#258152
#256832
#245592
#245512
#220842
#220782
#203792