Finite Group Databasebeta

← #6643210

Subgroups — #6643210

Order 512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2496C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic, complemented
4#33146C211yesmaximal, characteristic, complemented
5#31713C214yesmaximal, complemented
6#31711C214yesmaximal, complemented
7#31445C214yesmaximal, complemented
8#31443C214yesmaximal, complemented
9#29968C211yesmaximal, characteristic, complemented
10#29966C211yesmaximal, characteristic, complemented
11#29303C212yesmaximal, complemented
12#27966C212yesmaximal, complemented
13#27602C212yesmaximal
14#27602C212yesmaximal
15#21906C211yesmaximal, characteristic
16C22×#126—21no—
17#2857C2211yescharacteristic
18#2857C2211yescharacteristic
19#2851C2212yes—
20#2626—21no—
21#2626—21no—
22#2608—22no—
23#2607—22no—
24#2496—22no—
25#2496—22no—
26#2496C2214yes—
27#2496C2214yes—
28#2441C2212yes—
29#2441C2212yes—
30#2439C2212yes—
31#2439C2212yes—
32C2×#400—21no—
33C2×#397—21no—
34#2329—21no—
35#2329—21no—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#387—21no—
39C2×#384C2211yescharacteristic
40#2280C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#2100C2212yescomplemented
46#2100C2214yescomplemented
47#2100C2214yescomplemented
48#2099C2212yescomplemented
49#2088C2214yescomplemented
50#2086C2214yescomplemented
51#2085C2214yescomplemented
52#2077C2214yes—
53#2077C2214yes—
54#2076C2214yes—
55#2076C2214yes—
56#2059C2214yescomplemented
57#2058C2212yescomplemented
58#2057C2212yescomplemented
59#1992C2214yescomplemented
60#1988C2212yescomplemented
61#1987C2212yescomplemented
62#1948C2212yescomplemented
63#1947C2212yescomplemented
64#1944C2214yescomplemented
65#1943C2214yescomplemented
66#1924C2214yes—
67#1924C2214yes—
68#1920C2214yescomplemented
69#1918C2214yescomplemented
70#1854C2214yes—
71#1854C2214yes—
72#1853C2212yes—
73#1853C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1836C2214yescomplemented
77#1825C2214yes—
78#1825C2214yes—
79#1823C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1823C2212yes—
83#1797C2211yescharacteristic
84#1797C2211yescharacteristic
85#1779C2212yes—
86#1749C2212yes—
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×#126—24no—
91C2×#126—24no—
92C2×#126—48no—
93C2×#125C2312yes—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×#117C2312yes—
97C2×#117C2312yes—
98C22×C4⋊C4—21no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4D411yescharacteristic
109C22×C22⋊C4D411yescharacteristic
110C22×C42—21no—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2312yes—
119C2×C22⋊C8C2312yes—
120C2×C22⋊C8C2312yes—
121C2×C22⋊C8C2314yes—
122C2×C22⋊C8C2314yes—
123C2×#97C2314yes—
124#400—24no—
125#400—24no—
126#399—22no—
127#399—22no—
128#397—24no—
129#397—24no—
130#396—22no—
131#396—22no—
132#396—22no—
133#396—22no—
134#393—22no—
135#393—22no—
136#391—21no—
137#391—21no—
138#391—22no—
139#391—22no—
140#389—21no—
141#389—21no—
142#389—22no—
143#389—21no—
144#389—21no—
145#389—22no—
146#389—22no—
147#389C2312yes—
148#389C2312yes—
149#389C2314yes—
150#389C2312yes—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—24no—
161#387—24no—
162#385—22no—
163#385—22no—
164#384—22no—
165#384C2311yescharacteristic
166#384C2311yescharacteristic
167#384C2314yes—
168#382C2312yes—
169#381—22no—
170#380—21no—
171#380—21no—
172#380—22no—
173#380—22no—
174#380C2312yes—
175#380C2312yes—
176#380C2314yes—
177#380C2314yes—
178#380C2312yes—
179#380C2312yes—
180#379—22no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4D412yes—
188C4×C22⋊C4D412yes—
189C4×C22⋊C4C2312yes—
190C4×C22⋊C4C2312yes—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95D411yescharacteristic
197C2×#95D411yescharacteristic
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#336C2312yes—
211#336C2312yes—
212#336C2314yes—
213C25C2×D411yescharacteristic
214C22×Q8—21no—
215C22×Q8—21no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C22×C8C2414yes—
233C22×C8C2414yes—
234C4⋊Q8—42no—
235C4⋊Q8—42no—
236#126—416no—
237#126—416no—
238#125—44no—
239C22⋊Q8—44no—
240C22⋊Q8—44no—
241#117—44no—
242#117—44no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4C2×D412yes—
295C2×C22⋊C4C2×D412yes—
296C2×C22⋊C4C2×D412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—44no—
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D412yes—
308C2×C42C2412yes—
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311#107—44no—
312#107—44no—
313#106—44no—
314#106—44no—
315C4⋊C8—42no—
316C4⋊C8—42no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319C22⋊C8—44no—
320C22⋊C8—44no—
321#97—44no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95C2×D412yes—
340#95C2×D412yes—
341#95C2×D412yes—
342#95C2×D412yes—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—41no—
350C24C22×D412yes—
351C24C22×D411yescharacteristic
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—81no—
355C2×Q8—81no—
356C2×Q8—81no—
357C2×Q8—81no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4AGL(1,ℤ/8)14yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C42−1+412yes—
413C22×C42−1+412yes—
414C22×C4C2511yescharacteristic
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C42—24no—
466C42—22no—
467C42—22no—
468C42—41no—
469C42—41no—
470C42—42no—
471C42—44no—
472C42—44no—
473C42—82no—
474C42—82no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23#53612yes—
494C23C2×AGL(1,ℤ/8)12yes—
495C23#57512yes—
496C23C23×D411yescharacteristic
497Q8—162no—
498Q8—162no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4#53612yes—
559C2×C4C2×AGL(1,ℤ/8)12yes—
560C2×C4C2×AGL(1,ℤ/8)12yes—
561C2×C4C2×2−1+411yescharacteristic
562C2×C4C2×2−1+411yescharacteristic
563C2×C4#58512yes—
564C2×C4#58512yes—
565C8—44no—
566C8—44no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—88no—
581C22#323514yes—
582C22C2×#53611yescharacteristic
583C22#348511yescharacteristic
584C22#348511yescharacteristic
585C22C22×AGL(1,ℤ/8)11yescharacteristic
586C22C2×#57511yescharacteristic
587C22#358611yescharacteristic
588C22#358611yescharacteristic
589C4—22no—
590C4—22no—
591C4—22no—
592C4—42no—
593C4—41no—
594C4—41no—
595C4—42no—
596C4—42no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4#356412yes—
606C4#356412yes—
607C2—24no—
608C2—82no—
609C2#3697011yescharacteristic
610C2C2×#323511yescharacteristic
611C2#6086911yescharacteristic
612C2#6170611yescharacteristic
613C2#6171111yescharacteristic
614C2#6171211yescharacteristic
615C2#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. #33148C2semidirect
4. #33146C2semidirect
5. #31713C2semidirect
6. #31711C2semidirect
7. #31445C2semidirect
8. #31443C2semidirect
9. #29968C2semidirect
10. #29966C2semidirect
11. #29303C2semidirect
12. #27966C2semidirect
45. #2100C22semidirect
46. #2100C22semidirect
47. #2100C22semidirect
48. #2099C22semidirect
49. #2088C22semidirect
50. #2086C22semidirect
51. #2085C22semidirect
56. #2059C22semidirect
57. #2058C22semidirect
58. #2057C22semidirect
59. #1992C22semidirect
60. #1988C22semidirect
61. #1987C22semidirect
62. #1948C22semidirect
63. #1947C22semidirect
64. #1944C22semidirect
65. #1943C22semidirect
68. #1920C22semidirect
69. #1918C22semidirect
76. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#24962
#331482
#331462
#317132
#317112
#314452
#314432
#299682
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