Finite Group Databasebeta

← #6526723

Subgroups — #6526723

Order 512.

2182 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic, complemented
2C2×#3110C211yesmaximal, characteristic
3C2×#2542C211yesmaximal, characteristic
4#34235C211yesmaximal, characteristic, complemented
5#28123C214yesmaximal
6#27648C214yesmaximal, complemented
7#25449C214yesmaximal, complemented
8#25320C214yesmaximal, complemented
9#24827C214yesmaximal, complemented
10#24694C214yesmaximal, complemented
11#21953C211yesmaximal, characteristic, complemented
12#21841C211yesmaximal, characteristic, complemented
13#20733C211yesmaximal, characteristic
14C2×#533C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8—22no—
17C22×C4×D4C2211yescharacteristic
18#3230—28no—
19C2×#500C2212yescomplemented
20C2×#487C2212yescomplemented
21#3110—28no—
22C2×#484C2212yes—
23C2×#484C2214yes—
24C2×#477C2214yescomplemented
25C2×#474C2212yes—
26C2×#461C2212yes—
27C2×#450C2212yes—
28C2×#440C2214yes—
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2542—21no—
32#2542—22no—
33#2542—21no—
34#2542C2214yes—
35#2542C2214yes—
36C2×#399—22no—
37#2365C2212yescomplemented
38#2365C2212yescomplemented
39#2360—22no—
40#2356C2212yescomplemented
41#2351C2212yescomplemented
42#2343—22no—
43#2336—22no—
44#2320—21no—
45C4×C22⋊Q8C2214yes—
46C2×#390C2212yes—
47C2×#384C2212yes—
48#2283C2211yescharacteristic
49#2276—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2083C2218yescomplemented
52#2072C2218yescomplemented
53#2067C2218yes—
54#2055C2218yes—
55#1993C2212yescomplemented
56#1993C2212yescomplemented
57#1988C2214yescomplemented
58#1947C2218yescomplemented
59#1940C2214yescomplemented
60#1939C2214yescomplemented
61#1925C2218yes—
62#1880C2218yes—
63#1824C2212yes—
64#1824C2214yes—
65#1804C2212yes—
66#1804C2212yes—
67#1803C2218yescomplemented
68#1797C2214yes—
69#1786C2212yes—
70C4×#102C2214yes—
71C23×Q8—21no—
72C23×D4C2311yescharacteristic
73C24×C4—21no—
74C22×SD16C2314yes—
75#533—24no—
76#533D418yes—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#125—21no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—416no—
91C2×C22⋊Q8C2312yes—
92C2×C4×Q8—22no—
93C2×C4×Q8—21no—
94C2×C4×Q8C2312yes—
95C2×C4×Q8C2314yes—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×#117—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4D412yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—24no—
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112#500—28no—
113#487—28no—
114#484—28no—
115#484—216no—
116#477—216no—
117#474—28no—
118#461—28no—
119#450—28no—
120#440—216no—
121#432—24no—
122#431—24no—
123C2×#107C2318yes—
124C2×#107C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2318yes—
128C2×#103C2318yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×#97C2312yes—
136C2×C4×C8C2312yes—
137#399—24no—
138#399—24no—
139#399—28no—
140#391—22no—
141#391C2314yes—
142#390—22no—
143#390—22no—
144#390C2318yes—
145#389—22no—
146#389—24no—
147#389C2314yes—
148#387—24no—
149#387—24no—
150#387—24no—
151#387—24no—
152#385—24no—
153#384—22no—
154#384—22no—
155#384C2318yes—
156#383—21no—
157#383—22no—
158#382—22no—
159#382C2311yescharacteristic
160#382C2311yescharacteristic
161#381—21no—
162#380C2312yes—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4D411yescharacteristic
167C4×C4⋊C4D412yes—
168C4×C4⋊C4D411yescharacteristic
169C4×C4⋊C4C2314yes—
170C4×C4⋊C4C2312yes—
171C4×C4⋊C4C2312yes—
172C4×C4⋊C4C2312yes—
173C4×C22⋊C4—24no—
174C4×C22⋊C4—21no—
175#376—21no—
176#376—22no—
177C2×#95D412yes—
178#340C2318yes—
179#340C2314yes—
180#340C2318yes—
181#340C2314yes—
182#336C2318yes—
183C25—21no—
184C22×Q8—21no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—48no—
188C22×Q8—44no—
189C22×Q8C2412yes—
190C22×D4—24no—
191C22×D4—21no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—42no—
195C22×D4C2412yes—
196C23×C4—22no—
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—48no—
201C23×C4—44no—
202C23×C4C2×D414yes—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2411yescharacteristic
206C2×SD16—28no—
207C2×SD16—28no—
208C2×SD16—44no—
209C22×C8C2414yes—
210C22×C8C2414yes—
211C4⋊Q8—24no—
212C4⋊Q8—24no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—24no—
216C4⋊Q8—24no—
217C4⋊Q8—44no—
218C4⋊Q8—44no—
219C4⋊Q8C2×D414yes—
220C4⋊Q8C2×D414yes—
221#125—28no—
222C22⋊Q8—28no—
223C22⋊Q8—24no—
224C22⋊Q8—416no—
225C22⋊Q8—432no—
226C22⋊Q8—416no—
227C22⋊Q8C2×D418yes—
228C4×Q8—28no—
229C4×Q8—24no—
230C4×Q8—48no—
231C4×Q8—48no—
232C4×Q8C2×D418yes—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—24no—
236C4×D4—24no—
237C4×D4—44no—
238C4×D4—44no—
239#117—28no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—416no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—416no—
275C2×C22⋊C4—44no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—44no—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D414yes—
286C2×C42#4114yes—
287C2×C42C2414yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290#107—28no—
291#107—216no—
292C4⋊C8—24no—
293C4⋊C8—24no—
294C4⋊C8—48no—
295#103—28no—
296#103—216no—
297#102—28no—
298#102—28no—
299#102—44no—
300C22⋊C8—24no—
301C22⋊C8—24no—
302#97—24no—
303C4×C8—24no—
304#95—21no—
305#95—22no—
306#95—21no—
307#95—24no—
308#95—24no—
309#95—21no—
310#95—21no—
311#95—24no—
312#95—22no—
313#95—24no—
314#95C2×D412yes—
315#95C2×D412yes—
316#95C2×D414yes—
317C24—24no—
318C24—22no—
319C24—44no—
320C24—42no—
321C24—42no—
322C24—44no—
323C24C22×D411yescharacteristic
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—48no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—48no—
333C2×Q8—816no—
334C2×Q8C22×D414yes—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D42−1+414yes—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—28no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—44no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—48no—
367C22×C4—816no—
368C22×C4#12114yes—
369C22×C4#12114yes—
370C22×C4C22×D414yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4C22×D414yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2511yescharacteristic
380SD16—416no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—28no—
391C4⋊C4—28no—
392C4⋊C4—28no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—416no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—416no—
407C4⋊C4—48no—
408C4⋊C4—84no—
409C4⋊C4AGL(1,ℤ/8)18yes—
410C4⋊C4C22×D414yes—
411C22⋊C4—28no—
412C22⋊C4—416no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—416no—
417C42—28no—
418C42—28no—
419C42—42no—
420C42—42no—
421C42—48no—
422C42—44no—
423C42—84no—
424C42#12114yes—
425C42#12114yes—
426C42C22×D412yes—
427C42C22×D412yes—
428C42C2×#4112yes—
429C42C2×#4112yes—
430C23—21no—
431C23—21no—
432C23—22no—
433C23—24no—
434C23—48no—
435C23—416no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—84no—
441C23—88no—
442C23—84no—
443C23C2×#12112yes—
444C23#57814yes—
445C23C23×D411yescharacteristic
446Q8—28no—
447Q8—44no—
448Q8—48no—
449Q8—88no—
450Q8—88no—
451D4—24no—
452D4—24no—
453D4—48no—
454D4—42no—
455D4—42no—
456C2×C4—24no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—28no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—416no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—48no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—88no—
487C2×C4—84no—
488C2×C4—88no—
489C2×C4—88no—
490C2×C4C2×#12112yes—
491C2×C4C2×#12112yes—
492C2×C4C2×#12112yes—
493C2×C4C2×#12112yes—
494C2×C4C2×#12112yes—
495C2×C4C2×AGL(1,ℤ/8)18yes—
496C2×C4C2×AGL(1,ℤ/8)14yes—
497C2×C4C23×D411yescharacteristic
498C2×C4C22×#4111yescharacteristic
499C2×C4C2×2−1+412yes—
500C2×C4#58512yes—
501C8—44no—
502C8—44no—
503C22—24no—
504C22—22no—
505C22—24no—
506C22—22no—
507C22—44no—
508C22—48no—
509C22—816no—
510C22—82no—
511C22—84no—
512C22—82no—
513C22#304414yes—
514C22C22×#12111yescharacteristic
515C22#348511yescharacteristic
516C22#348611yescharacteristic
517C22C22×AGL(1,ℤ/8)12yes—
518C22C2×#57812yes—
519C4—22no—
520C4—22no—
521C4—28no—
522C4—22no—
523C4—22no—
524C4—44no—
525C4—44no—
526C4—44no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—84no—
531C4#303814yes—
532C4#348912yes—
533C4#349512yes—
534C2—24no—
535C2—84no—
536C2#3563311yescharacteristic
537C2C2×#303811yescharacteristic
538C2C2×#304411yescharacteristic
539C2#6173712yes—
540C2#6183712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3230C2semidirect
4. #34235C2semidirect
6. #27648C2semidirect
7. #25449C2semidirect
8. #25320C2semidirect
9. #24827C2semidirect
10. #24694C2semidirect
11. #21953C2semidirect
12. #21841C2semidirect
19. C2×#500C22semidirect
20. C2×#487C22semidirect
24. C2×#477C22semidirect
37. #2365C22semidirect
38. #2365C22semidirect
40. #2356C22semidirect
41. #2351C22semidirect
51. #2083C22semidirect
52. #2072C22semidirect
55. #1993C22semidirect
56. #1993C22semidirect
57. #1988C22semidirect
58. #1947C22semidirect
59. #1940C22semidirect
60. #1939C22semidirect
67. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C2×#31102
C2×#25422
#342352
#281232
#276482
#254492
#253202
#248272
#246942
#219532
#218412
#207332