Finite Group Databasebeta

← #6640543

Subgroups — #6640543

Order 512.

1974 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35165C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic, complemented
3#32979C212yesmaximal, complemented
4#32260C214yesmaximal, complemented
5#32028C214yesmaximal, complemented
6#31996C218yesmaximal, complemented
7#29892C212yesmaximal, complemented
8#27611C212yesmaximal, complemented
9#24553C214yesmaximal, complemented
10#24061C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic, complemented
12C2×#535—21no—
13C2×#532C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#481C2214yes—
17C22×C22⋊C8C2212yes—
18#2847C2212yes—
19#2847C2212yes—
20#2617—21no—
21#2617—21no—
22#2547C2212yes—
23#2544—22no—
24#2541C2212yes—
25#2537C2214yes—
26#2535—22no—
27#2487—22no—
28#2487—22no—
29#2487C2214yes—
30#2487C2214yes—
31C2×#400—21no—
32C2×#399—21no—
33#2311—21no—
34C2×#389C2212yes—
35C2×#386—21no—
36C2×#385—21no—
37#2285C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40#2099C2214yescomplemented
41#2084C2214yescomplemented
42#2077C2218yescomplemented
43#2077C2218yescomplemented
44#2058C2214yescomplemented
45#2041C2214yescomplemented
46#2034C2218yescomplemented
47#2034C2218yescomplemented
48#1920C2218yescomplemented
49#1893C2218yescomplemented
50#1874C2214yes—
51#1874C2218yescomplemented
52#1869C2214yes—
53#1869C2214yes—
54#1864C2218yescomplemented
55#1853C2214yes—
56#1853C2218yescomplemented
57#1824C2214yes—
58#1817C2212yes—
59#1807C2218yescomplemented
60#1779C2212yes—
61C22×#41—21no—
62C23×D4D411yescharacteristic
63C24×C4—21no—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C23×C8C2314yes—
67#535—48no—
68#532—44no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#127—22no—
71C2×#127C2311yescharacteristic
72C2×#124—22no—
73C2×#124—21no—
74C2×#121—24no—
75C2×#121—21no—
76C2×#121C2312yes—
77C2×#121C2314yes—
78C2×#121C2314yes—
79C2×#120—22no—
80C2×#120—22no—
81C2×C4×D4—21no—
82C2×C4×D4—44no—
83C2×C4×D4D412yes—
84C2×C4×D4D412yes—
85C2×C4×D4C2312yes—
86C2×C4×D4C2312yes—
87C2×C4×D4C2312yes—
88C2×#117—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4D411yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4D411yescharacteristic
100C22×C22⋊C4D411yescharacteristic
101C22×C42C2311yescharacteristic
102#481—48no—
103C2×#106C2318yes—
104C2×#106C2318yes—
105C2×#102C2318yes—
106C2×#102C2318yes—
107C2×C22⋊C8—44no—
108C2×C22⋊C8C2314yes—
109C2×C22⋊C8C2314yes—
110C2×C22⋊C8C2318yes—
111#400—24no—
112#400—24no—
113#399—24no—
114#399—24no—
115#397—22no—
116#397—22no—
117#394—24no—
118#394—24no—
119#392—22no—
120#392—22no—
121#391—22no—
122#390—22no—
123#389—22no—
124#389—22no—
125#389C2314yes—
126#389C2314yes—
127#386—24no—
128#386—24no—
129#386—22no—
130#386—22no—
131#386—24no—
132#386—24no—
133#385—24no—
134#385—24no—
135#384C2312yes—
136#382—21no—
137#380—21no—
138#380—24no—
139#380—22no—
140#380—24no—
141#380C2314yes—
142#380C2318yes—
143#380C2314yes—
144#380C2314yes—
145#380C2312yes—
146C4×C4⋊C4—21no—
147C4×C4⋊C4C2312yes—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—22no—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#336C2314yes—
156#336C2314yes—
157C25C2×D411yescharacteristic
158C25C2×D411yescharacteristic
159C2×#41—44no—
160C2×#41—44no—
161C2×#41—44no—
162C22×Q8—21no—
163C22×D4—22no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4SD1618yes—
171C22×D4C2×D412yes—
172C22×D4C2×D412yes—
173C22×D4C2412yes—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—84no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C22×C8—48no—
194C22×C8—44no—
195C22×C8C2418yes—
196C4⋊Q8—42no—
197#127—42no—
198#127—48no—
199#124—44no—
200#124—48no—
201#121—28no—
202#121—44no—
203#121—416no—
204#121—48no—
205#121—48no—
206#120—48no—
207#120—48no—
208C4×D4—44no—
209C4×D4—44no—
210C4×D4—44no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—44no—
216#117—44no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—84no—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2414yes—
236C2×C4⋊C4C2414yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2414yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—84no—
252C2×C22⋊C4C2×D412yes—
253C2×C22⋊C4C2×D412yes—
254C2×C22⋊C4C2×D412yes—
255C2×C22⋊C4C2×D412yes—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2411yescharacteristic
261C2×C42C2411yescharacteristic
262#106—48no—
263#106—48no—
264#102—48no—
265#102—48no—
266C22⋊C8—28no—
267C22⋊C8—44no—
268C22⋊C8—48no—
269#95—28no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—22no—
274#95—22no—
275#95—24no—
276#95—24no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24C2×SD1614yes—
287C24C2×SD1614yes—
288C24C22×D412yes—
289C24C22×D412yes—
290C24C22×D411yescharacteristic
291#41—816no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×D4—24no—
296C2×D4—28no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—416no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—42no—
311C2×D4—82no—
312C2×D4—84no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4C2×SD1618yes—
317C2×D4C2×SD1618yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—416no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—48no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—88no—
353C22×C4—88no—
354C22×C4—816no—
355C22×C4C2×SD1614yes—
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4H(5,𝔽2)14yes—
364C22×C4C2511yescharacteristic
365C2×C8—28no—
366C2×C8—28no—
367C2×C8—28no—
368C2×C8—88no—
369C4⋊C4—28no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—28no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—48no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—82no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—44no—
389C22⋊C4—48no—
390C22⋊C4—44no—
391C22⋊C4—48no—
392C22⋊C4—48no—
393C22⋊C4—44no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C22⋊C4—88no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—44no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—24no—
406C23—28no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—28no—
414C23—48no—
415C23—48no—
416C23—44no—
417C23—44no—
418C23—48no—
419C23—44no—
420C23—42no—
421C23—44no—
422C23—42no—
423C23—42no—
424C23—82no—
425C23—84no—
426C23—82no—
427C23#53412yes—
428C23C22×SD1614yes—
429C23C22×SD1614yes—
430C23C22×SD1612yes—
431C23#57512yes—
432C23C23×D411yescharacteristic
433Q8—84no—
434D4—48no—
435D4—48no—
436D4—48no—
437D4—84no—
438D4—84no—
439D4—88no—
440D4—168no—
441D4—168no—
442C2×C4—28no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—28no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—48no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—84no—
476C2×C4—88no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—88no—
480C2×C4—88no—
481C2×C4—816no—
482C2×C4#53412yes—
483C2×C4C22×SD1612yes—
484C2×C4C22×SD1612yes—
485C2×C4C2×H(5,𝔽2)12yes—
486C2×C4#58514yes—
487C8—48no—
488C22—24no—
489C22—24no—
490C22—22no—
491C22—22no—
492C22—28no—
493C22—28no—
494C22—24no—
495C22—24no—
496C22—416no—
497C22—44no—
498C22—44no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—88no—
504C22—84no—
505C22—84no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22#318714yes—
510C22C2×#53411yescharacteristic
511C22#348512yes—
512C22C23×SD1611yescharacteristic
513C22C2×#57511yescharacteristic
514C22#358512yes—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—41no—
519C4—41no—
520C4—42no—
521C4—42no—
522C4—84no—
523C4—84no—
524C4—82no—
525C4—84no—
526C4#353314yes—
527C2—24no—
528C2—44no—
529C2—44no—
530C2—162no—
531C2#3695511yescharacteristic
532C2C2×#318711yescharacteristic
533C2#6075311yescharacteristic
534C2#6169712yes—
535C2#6171312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35165C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2487C2semidirect
3. #32979C2semidirect
4. #32260C2semidirect
5. #32028C2semidirect
6. #31996C2semidirect
7. #29892C2semidirect
8. #27611C2semidirect
9. #24553C2semidirect
10. #24061C2semidirect
11. #21738C2semidirect
40. #2099C22semidirect
41. #2084C22semidirect
42. #2077C22semidirect
43. #2077C22semidirect
44. #2058C22semidirect
45. #2041C22semidirect
46. #2034C22semidirect
47. #2034C22semidirect
48. #1920C22semidirect
49. #1893C22semidirect
51. #1874C22semidirect
54. #1864C22semidirect
56. #1853C22semidirect
59. #1807C22semidirect

Maximal subgroups

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

subgroupindex
#351652
C2×#24872
#329792
#322602
#320282
#319962
#298922
#276112
#245532
#240612
#217382