Finite Group Databasebeta

← #6526721

Subgroups — #6526721

Order 512.

2310 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic, complemented
2C2×#3109C211yesmaximal, characteristic
3C2×#2542C211yesmaximal, characteristic, complemented
4#34236C211yesmaximal, characteristic, complemented
5#28117C214yesmaximal, complemented
6#27645C214yesmaximal, complemented
7#25451C214yesmaximal, complemented
8#25318C214yesmaximal, complemented
9#24829C214yesmaximal, complemented
10#24692C214yesmaximal, complemented
11#21953C211yesmaximal, characteristic, complemented
12#21841C211yesmaximal, characteristic, complemented
13#20733C211yesmaximal, characteristic, complemented
14C2×#532C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8—22no—
17C22×C4×D4C2211yescharacteristic
18#3227—28no—
19C2×#498C2212yescomplemented
20C2×#486C2212yescomplemented
21#3109—28no—
22C2×#481C2212yes—
23C2×#481C2214yes—
24C2×#476C2212yes—
25C2×#476C2214yescomplemented
26C2×#459C2212yes—
27C2×#447C2212yes—
28C2×#442C2214yes—
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2542—21no—
32#2542—22no—
33#2542—21no—
34#2542C2214yes—
35#2542C2214yes—
36C2×#399—22no—
37#2369C2212yescomplemented
38#2366C2212yescomplemented
39#2361—22no—
40#2357C2212yescomplemented
41#2349C2212yescomplemented
42#2339—22no—
43#2337—22no—
44#2319—21no—
45C4×#121C2214yes—
46C2×#390C2212yes—
47C2×#384C2212yes—
48#2283C2211yescharacteristic
49#2275—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2084C2218yescomplemented
52#2071C2218yescomplemented
53#2065C2218yes—
54#2054C2218yes—
55#1993C2212yescomplemented
56#1993C2212yescomplemented
57#1988C2214yescomplemented
58#1948C2218yescomplemented
59#1940C2214yescomplemented
60#1939C2214yescomplemented
61#1926C2218yes—
62#1879C2218yes—
63#1824C2214yes—
64#1824C2212yes—
65#1804C2212yes—
66#1804C2212yes—
67#1802C2218yescomplemented
68#1797C2214yes—
69#1786C2212yes—
70C4×#102C2214yes—
71C22×#41—21no—
72C23×Q8—21no—
73C23×D4C2311yescharacteristic
74C24×C4—21no—
75C22×D8C2314yes—
76#532—24no—
77#532D418yes—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×#127C2311yescharacteristic
86C2×#124—21no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—416no—
91C2×#121—21no—
92C2×#121C2312yes—
93C2×C4×D4—24no—
94C2×C4×D4—22no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×C4×D4C2312yes—
103C2×C4×D4C2314yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4D412yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—24no—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113#498—28no—
114#486—28no—
115#481—28no—
116#481—216no—
117#476—28no—
118#476—216no—
119#459—28no—
120#447—28no—
121#442—216no—
122#432—24no—
123#431—24no—
124C2×#106C2318yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2318yes—
129C2×#102C2314yes—
130C2×#102C2318yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×#97C2312yes—
137C2×C4×C8C2312yes—
138#399—24no—
139#399—24no—
140#399—28no—
141#391—24no—
142#390—22no—
143#390—22no—
144#390—22no—
145#390C2314yes—
146#390C2318yes—
147#389—22no—
148#389C2314yes—
149#388—24no—
150#388—24no—
151#386—24no—
152#386—24no—
153#385—24no—
154#384—22no—
155#384—22no—
156#384C2318yes—
157#383—21no—
158#382C2311yescharacteristic
159#382C2311yescharacteristic
160#381—21no—
161#381—22no—
162#380—22no—
163#380C2312yes—
164C4×C4⋊C4—22no—
165C4×C4⋊C4D411yescharacteristic
166C4×C4⋊C4D412yes—
167C4×C4⋊C4D411yescharacteristic
168C4×C4⋊C4C2314yes—
169C4×C4⋊C4C2312yes—
170C4×C4⋊C4C2312yes—
171C4×C4⋊C4C2312yes—
172C4×C22⋊C4—21no—
173C4×C22⋊C4—24no—
174C4×C22⋊C4—22no—
175#376—21no—
176C2×#95—22no—
177C2×#95D412yes—
178#340C2318yes—
179#340C2314yes—
180#340C2318yes—
181#340C2314yes—
182#336C2318yes—
183C25—21no—
184C2×#41—28no—
185C2×#41—84no—
186C22×Q8—21no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—48no—
190C22×Q8—44no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—42no—
199C22×D4C2412yes—
200C22×D4C2412yes—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—21no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—48no—
208C23×C4—44no—
209C23×C4C2×D414yes—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2411yescharacteristic
213C2×D8—28no—
214C2×D8—28no—
215C2×D8—44no—
216C22×C8C2414yes—
217C22×C8C2414yes—
218C4⋊Q8—24no—
219C4⋊Q8—24no—
220C4⋊Q8—22no—
221C4⋊Q8—24no—
222C4⋊Q8—24no—
223C4⋊Q8—44no—
224C4⋊Q8—44no—
225C4⋊Q8C2×D414yes—
226#127—22no—
227#127C2×D414yes—
228#124—28no—
229C22⋊Q8—416no—
230C22⋊Q8—432no—
231C22⋊Q8—416no—
232#121—28no—
233#121—24no—
234#121C2×D418yes—
235C4×D4—28no—
236C4×D4—24no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—48no—
242C4×D4—48no—
243C4×D4—44no—
244C4×D4—44no—
245C4×D4—88no—
246C4×D4C2×D418yes—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—416no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—416no—
281C2×C22⋊C4—44no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—44no—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D414yes—
290C2×C42#4114yes—
291C2×C42C2414yes—
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294#106—28no—
295#106—216no—
296C4⋊C8—24no—
297C4⋊C8—24no—
298C4⋊C8—48no—
299#102—28no—
300#102—216no—
301#102—28no—
302#102—28no—
303#102—44no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306#97—24no—
307C4×C8—24no—
308#95—21no—
309#95—24no—
310#95—22no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—22no—
315#95—24no—
316#95C2×D412yes—
317#95C2×D412yes—
318#95C2×D414yes—
319C24—24no—
320C24—22no—
321C24—22no—
322C24—44no—
323C24—42no—
324C24—42no—
325C24—44no—
326C24C22×D411yescharacteristic
327#41—816no—
328C2×Q8—24no—
329C2×Q8—48no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—48no—
333C2×Q8—82no—
334C2×Q8—816no—
335C2×D4—24no—
336C2×D4—28no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—24no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—22no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—42no—
350C2×D4—48no—
351C2×D4—42no—
352C2×D4—44no—
353C2×D4—82no—
354C2×D4C22×D414yes—
355C2×D42−1+414yes—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—22no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—44no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4—816no—
382C22×C4#12114yes—
383C22×C4#12114yes—
384C22×C4C22×D414yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D412yes—
390C22×C4C22×D414yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4112yes—
393C22×C4C2511yescharacteristic
394D8—416no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C4⋊C4—24no—
402C4⋊C4—28no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—416no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—416no—
417C4⋊C4—84no—
418C4⋊C4AGL(1,ℤ/8)18yes—
419C4⋊C4C22×D414yes—
420C22⋊C4—28no—
421C22⋊C4—28no—
422C22⋊C4—416no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—416no—
427C22⋊C4—48no—
428C22⋊C4—84no—
429C22⋊C4—84no—
430C42—28no—
431C42—42no—
432C42—42no—
433C42—44no—
434C42—84no—
435C42#12114yes—
436C42#12114yes—
437C42C22×D412yes—
438C42C22×D412yes—
439C42C2×#4112yes—
440C42C2×#4112yes—
441C23—21no—
442C23—21no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—48no—
447C23—416no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—84no—
457C23—88no—
458C23—84no—
459C23C2×#12112yes—
460C23#57714yes—
461C23C23×D411yescharacteristic
462Q8—84no—
463Q8—88no—
464Q8—88no—
465D4—24no—
466D4—28no—
467D4—24no—
468D4—48no—
469D4—416no—
470D4—42no—
471D4—44no—
472D4—42no—
473D4—48no—
474D4—84no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—28no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—416no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—88no—
503C2×C4—88no—
504C2×C4—84no—
505C2×C4—88no—
506C2×C4—88no—
507C2×C4—168no—
508C2×C4C2×#12112yes—
509C2×C4C2×#12112yes—
510C2×C4C2×#12112yes—
511C2×C4C2×#12112yes—
512C2×C4C2×#12112yes—
513C2×C4C2×AGL(1,ℤ/8)18yes—
514C2×C4C2×AGL(1,ℤ/8)14yes—
515C2×C4C23×D411yescharacteristic
516C2×C4C22×#4111yescharacteristic
517C2×C4C2×2−1+412yes—
518C2×C4#58512yes—
519C8—44no—
520C8—44no—
521C22—24no—
522C22—22no—
523C22—24no—
524C22—22no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—48no—
529C22—44no—
530C22—816no—
531C22—82no—
532C22—84no—
533C22—84no—
534C22—84no—
535C22—82no—
536C22#304214yes—
537C22C22×#12111yescharacteristic
538C22#348511yescharacteristic
539C22#348611yescharacteristic
540C22C22×AGL(1,ℤ/8)12yes—
541C22C2×#57712yes—
542C4—22no—
543C4—22no—
544C4—28no—
545C4—22no—
546C4—22no—
547C4—44no—
548C4—44no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—84no—
553C4#303814yes—
554C4#348912yes—
555C4#349512yes—
556C2—24no—
557C2—84no—
558C2—84no—
559C2#3563311yescharacteristic
560C2C2×#303811yescharacteristic
561C2C2×#304211yescharacteristic
562C2#6172812yes—
563C2#6184312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
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×#3227C2semidirect
3. C2×#2542C2semidirect
4. #34236C2semidirect
5. #28117C2semidirect
6. #27645C2semidirect
7. #25451C2semidirect
8. #25318C2semidirect
9. #24829C2semidirect
10. #24692C2semidirect
11. #21953C2semidirect
12. #21841C2semidirect
13. #20733C2semidirect
19. C2×#498C22semidirect
20. C2×#486C22semidirect
25. C2×#476C22semidirect
37. #2369C22semidirect
38. #2366C22semidirect
40. #2357C22semidirect
41. #2349C22semidirect
51. #2084C22semidirect
52. #2071C22semidirect
55. #1993C22semidirect
56. #1993C22semidirect
57. #1988C22semidirect
58. #1948C22semidirect
59. #1940C22semidirect
60. #1939C22semidirect
67. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
C2×#31092
C2×#25422
#342362
#281172
#276452
#254512
#253182
#248292
#246922
#219532
#218412
#207332