Finite Group Databasebeta

← #6477788

Subgroups — #6477788

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3#33151C212yesmaximal, complemented
4#33148C211yesmaximal, characteristic, complemented
5#33146C211yesmaximal, characteristic, complemented
6#29972C212yesmaximal, complemented
7#29968C211yesmaximal, characteristic, complemented
8#29966C211yesmaximal, characteristic, complemented
9#26014C214yesmaximal, complemented
10#26012C214yesmaximal, complemented
11#26006C214yesmaximal, complemented
12#26004C214yesmaximal, complemented
13#21944C212yesmaximal
14#21934C211yesmaximal, characteristic
15#20700C212yesmaximal
16C22×#117—21no—
17C23×C4⋊C4C2211yescharacteristic
18C2×#407C2212yes—
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21#2851C2212yes—
22#2626—21no—
23#2626—21no—
24#2608—22no—
25#2607—22no—
26#2441C2212yes—
27#2441C2212yes—
28#2439C2212yes—
29#2439C2212yes—
30#2344—22no—
31#2344—22no—
32#2344C2214yes—
33#2344C2214yes—
34#2329—21no—
35#2329—21no—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#387—21no—
39C2×#387—21no—
40#2280C2212yes—
41C2×#381—21no—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#2101C2214yescomplemented
47#2100C2212yescomplemented
48#2099C2212yescomplemented
49#2088C2214yescomplemented
50#2088C2214yescomplemented
51#2087C2214yescomplemented
52#2086C2214yescomplemented
53#2059C2214yescomplemented
54#2058C2212yescomplemented
55#2057C2212yescomplemented
56#1989C2214yescomplemented
57#1988C2212yescomplemented
58#1987C2212yescomplemented
59#1949C2214yescomplemented
60#1948C2212yescomplemented
61#1947C2212yescomplemented
62#1921C2214yescomplemented
63#1920C2214yescomplemented
64#1919C2214yescomplemented
65#1918C2214yescomplemented
66#1854C2214yes—
67#1854C2214yes—
68#1853C2212yes—
69#1853C2212yes—
70#1852C2212yes—
71#1852C2212yes—
72#1851C2214yes—
73#1851C2214yes—
74#1824C2214yes—
75#1824C2214yes—
76#1823C2212yes—
77#1823C2212yes—
78#1823C2212yes—
79#1823C2212yes—
80#1809C2214yes—
81#1809C2214yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1797C2212yes—
85#1736C2214yes—
86#1736C2214yes—
87C24×C4—21no—
88C24×C4C2311yescharacteristic
89C22×M16C2314yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125C2312yes—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—48no—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4Q818yescomplemented
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4D411yescharacteristic
113C22×C22⋊C4D411yescharacteristic
114C22×C42—21no—
115C22×C42—21no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2314yes—
124C2×C4⋊C8C2312yes—
125#407—28no—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2314yes—
129#399—22no—
130#399—22no—
131#396—22no—
132#396—22no—
133#396—22no—
134#396—22no—
135#393—22no—
136#393—22no—
137#391—21no—
138#391—21no—
139#391—22no—
140#391—22no—
141#389—22no—
142#389—21no—
143#389—21no—
144#389—21no—
145#389—21no—
146#389—22no—
147#389—22no—
148#389C2312yes—
149#389C2312yes—
150#389C2314yes—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#389C2312yes—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—24no—
162#387—24no—
163#387—24no—
164#387—24no—
165#385—22no—
166#385—22no—
167#384C2311yescharacteristic
168#384C2311yescharacteristic
169#382C2312yes—
170#381—22no—
171#381—24no—
172#381—24no—
173#380—21no—
174#380—21no—
175#380—22no—
176#380C2312yes—
177#380C2312yes—
178#380C2314yes—
179#380C2312yes—
180#380C2312yes—
181#379—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4D412yes—
191C4×C22⋊C4D412yes—
192C4×C22⋊C4C2312yes—
193C4×C22⋊C4C2312yes—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95D411yescharacteristic
200C2×#95D411yescharacteristic
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2312yes—
214#336C2312yes—
215#336C2314yes—
216C25C2×D411yescharacteristic
217C22×Q8—21no—
218C22×Q8—21no—
219C23×C4—28no—
220C23×C4—21no—
221C23×C4—24no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D412yes—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×Q814yes—
236C23×C4#4114yes—
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C2×M16—28no—
240C2×M16—44no—
241C2×M16—44no—
242C22×C8C2414yes—
243C22×C8C2414yes—
244C4⋊Q8—42no—
245C4⋊Q8—42no—
246#125—44no—
247C22⋊Q8—44no—
248C22⋊Q8—44no—
249#117—416no—
250#117—416no—
251#117—44no—
252#117—44no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×Q814yes—
283C2×C4⋊C4C2×Q814yes—
284C2×C4⋊C4C2×Q8116yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4C2×D412yes—
304C2×C22⋊C4C2×D412yes—
305C2×C22⋊C4C2×D412yes—
306C2×C22⋊C4C2×D412yes—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42C2×D412yes—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2412yes—
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322#107—44no—
323#107—44no—
324#106—44no—
325#106—44no—
326C4⋊C8—44no—
327C4⋊C8—42no—
328C4⋊C8—42no—
329C22⋊C8—24no—
330C22⋊C8—24no—
331C22⋊C8—28no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95C2×D412yes—
350#95C2×D412yes—
351#95C2×D412yes—
352#95C2×D412yes—
353C24—22no—
354C24—22no—
355C24—41no—
356C24—41no—
357C24—41no—
358C24—41no—
359C24C22⋊Q814yes—
360C24C22×D412yes—
361C24C22×D411yescharacteristic
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—81no—
365C2×Q8—81no—
366C2×Q8—81no—
367C2×Q8—81no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—216no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—48no—
401C22×C4—432no—
402C22×C4—48no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4C22⋊Q818yes—
414C22×C4C22⋊Q814yes—
415C22×C4AGL(1,ℤ/8)14yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×Q814yes—
425C22×C4C22×Q812yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4114yes—
428C22×C4C2511yescharacteristic
429M16—88no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C4⋊C4—24no—
437C4⋊C4—216no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—48no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4C22×Q814yes—
466C4⋊C4C22×Q814yes—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C42—24no—
479C42—22no—
480C42—22no—
481C42—41no—
482C42—41no—
483C42—42no—
484C42—48no—
485C42—44no—
486C42—44no—
487C42—82no—
488C42—82no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—22no—
494C23—22no—
495C23—28no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23C2×C22⋊Q814yes—
507C23C2×C22⋊Q812yes—
508C23C2×AGL(1,ℤ/8)12yes—
509C23#57512yes—
510C23C23×D411yescharacteristic
511Q8—162no—
512Q8—162no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—432no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4C2×C22⋊Q814yes—
573C2×C4C2×C22⋊Q812yes—
574C2×C4C2×AGL(1,ℤ/8)12yes—
575C2×C4C2×AGL(1,ℤ/8)12yes—
576C2×C4C23×Q811yescharacteristic
577C2×C4C22×#4111yescharacteristic
578C2×C4#58512yes—
579C2×C4#58512yes—
580C8—44no—
581C8—44no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—416no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22#307714yes—
597C22C22×C22⋊Q811yescharacteristic
598C22#348511yescharacteristic
599C22#348511yescharacteristic
600C22C22×AGL(1,ℤ/8)11yescharacteristic
601C22C2×#57511yescharacteristic
602C22#358611yescharacteristic
603C22#358611yescharacteristic
604C4—22no—
605C4—22no—
606C4—22no—
607C4—24no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—42no—
612C4—44no—
613C4—44no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4#345812yes—
621C4#349112yes—
622C2—24no—
623C2—44no—
624C2#3509011yescharacteristic
625C2C2×#307711yescharacteristic
626C2#6049111yescharacteristic
627C2#6170611yescharacteristic
628C2#6171111yescharacteristic
629C2#6171611yescharacteristic
630C2#6171711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33151C2semidirect
4. #33148C2semidirect
5. #33146C2semidirect
6. #29972C2semidirect
7. #29968C2semidirect
8. #29966C2semidirect
9. #26014C2semidirect
10. #26012C2semidirect
11. #26006C2semidirect
12. #26004C2semidirect
46. #2101C22semidirect
47. #2100C22semidirect
48. #2099C22semidirect
49. #2088C22semidirect
50. #2088C22semidirect
51. #2087C22semidirect
52. #2086C22semidirect
53. #2059C22semidirect
54. #2058C22semidirect
55. #2057C22semidirect
56. #1989C22semidirect
57. #1988C22semidirect
58. #1987C22semidirect
59. #1949C22semidirect
60. #1948C22semidirect
61. #1947C22semidirect
62. #1921C22semidirect
63. #1920C22semidirect
64. #1919C22semidirect
65. #1918C22semidirect
105. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#23442
#331512
#331482
#331462
#299722
#299682
#299662
#260142
#260122
#260062
#260042
#219442
#219342
#207002