Finite Group Databasebeta

← #6647988

Subgroups — #6647988

Order 512.

1880 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic, complemented
2C2×#2548C211yesmaximal, characteristic
3#33099C212yesmaximal, complemented
4#32680C214yesmaximal, complemented
5#32647C214yesmaximal, complemented
6#32627C218yesmaximal, complemented
7#29625C212yesmaximal
8#27582C212yesmaximal, complemented
9#24507C214yesmaximal, complemented
10#23933C212yesmaximal, complemented
11#21822C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C22×#127C2211yescharacteristic
14C22×#124—21no—
15C22×C4×Q8C2211yescharacteristic
16C2×#450C2214yes—
17C2×#432C2212yes—
18#2855C2212yes—
19#2855C2212yes—
20#2627—21no—
21#2627—21no—
22#2582C2214yescomplemented
23#2581—22no—
24#2568C2212yescomplemented
25#2567C2212yescomplemented
26#2565—22no—
27#2548—22no—
28#2548—22no—
29#2548C2214yes—
30#2548C2214yes—
31C2×#394—21no—
32C2×#394—21no—
33#2317—21no—
34#2310—21no—
35C2×#390—21no—
36C2×#390C2212yes—
37C2×#390C2212yes—
38#2285C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#2093C2214yescomplemented
41#2071C2214yescomplemented
42#2067C2218yescomplemented
43#2055C2218yescomplemented
44#2041C2218yes—
45#2023C2214yescomplemented
46#2014C2218yes—
47#2013C2214yescomplemented
48#1985C2214yescomplemented
49#1985C2214yescomplemented
50#1973C2218yescomplemented
51#1936C2218yescomplemented
52#1928C2218yescomplemented
53#1925C2218yescomplemented
54#1921C2218yescomplemented
55#1880C2218yescomplemented
56#1872C2214yes—
57#1852C2214yes—
58#1817C2212yes—
59#1817C2212yes—
60#1786C2214yes—
61C23×Q8D411yescharacteristic
62C23×D4—21no—
63C23×D4—21no—
64C23×D4C2312yes—
65C22×SD16C2318yes—
66#533—44no—
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×#127—22no—
70C2×#127—44no—
71C2×#127C2314yes—
72C2×#125—21no—
73C2×#124—24no—
74C2×#124—24no—
75C2×#124—48no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8C2312yes—
78C2×C4×Q8—21no—
79C2×C4×Q8—44no—
80C2×C4×Q8D412yes—
81C2×C4×Q8D412yes—
82C2×C4×Q8C2312yes—
83C2×C4×Q8C2312yes—
84C2×C4×Q8C2312yes—
85C2×#117—21no—
86C2×#117—21no—
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4—22no—
92C22×C22⋊C4—22no—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C42—21no—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#450—48no—
100#432—44no—
101C2×#107C2318yes—
102C2×C4⋊C8C2314yes—
103C2×#103C2318yes—
104C2×#102C2318yes—
105C2×#102C2318yes—
106C2×#102C2318yes—
107C2×C22⋊C8C2314yes—
108C2×#97C2314yes—
109#400—22no—
110#400—22no—
111#398—24no—
112#398—24no—
113#395—22no—
114#395—22no—
115#394—24no—
116#394—24no—
117#394—24no—
118#394—24no—
119#391—22no—
120#391—22no—
121#391—24no—
122#391C2314yes—
123#391C2314yes—
124#390—24no—
125#390—24no—
126#390—22no—
127#390—22no—
128#390—24no—
129#390C2318yes—
130#390C2314yes—
131#390C2314yes—
132#389—24no—
133#389C2314yes—
134#389C2314yes—
135#387—24no—
136#387—24no—
137#385—22no—
138#385—22no—
139#384—22no—
140#384C2312yes—
141#382—22no—
142#382—21no—
143#381—22no—
144#381—22no—
145#380—21no—
146#380C2312yes—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4C2312yes—
150C4×C4⋊C4C2314yes—
151C2×#95—21no—
152#340C2318yes—
153#340C2318yes—
154#336C2314yes—
155#336C2318yes—
156#336C2314yes—
157C25—22no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—21no—
161C22×Q8—42no—
162C22×Q8C2×D412yes—
163C22×Q8C2×D412yes—
164C22×Q8C2412yes—
165C22×D4—24no—
166C22×D4—22no—
167C22×D4—24no—
168C22×D4—28no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—84no—
176C22×D4C2414yes—
177C23×C4—22no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C2×SD16—48no—
186C2×SD16—48no—
187C2×SD16—48no—
188C22×C8C2418yes—
189C4⋊Q8—42no—
190C4⋊Q8—42no—
191#127—44no—
192#127—44no—
193#127—48no—
194#125—44no—
195#124—416no—
196#124—416no—
197C22⋊Q8—28no—
198C22⋊Q8—44no—
199C4×Q8—44no—
200C4×Q8—44no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—44no—
206C4×Q8—44no—
207#117—44no—
208#117—44no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2414yes—
230C2×C4⋊C4C2412yes—
231C2×C4⋊C4C2414yes—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4—48no—
242C2×C22⋊C4—84no—
243C2×C22⋊C4—84no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—44no—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2411yescharacteristic
256C2×C42C2412yes—
257C2×C42C2411yescharacteristic
258C2×C42C2412yes—
259#107—48no—
260C4⋊C8—44no—
261#103—48no—
262#102—48no—
263#102—48no—
264#102—48no—
265C22⋊C8—28no—
266#97—44no—
267#95—24no—
268#95—22no—
269#95—22no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276C24—24no—
277C24—44no—
278C24—44no—
279C24—44no—
280C24—44no—
281C24—44no—
282C24—44no—
283C24—44no—
284C24C22×D411yescharacteristic
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—28no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—416no—
294C2×Q8—42no—
295C2×Q8—82no—
296C2×D4—28no—
297C2×D4—24no—
298C2×D4—28no—
299C2×D4—24no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—48no—
307C2×D4—816no—
308C2×D4—88no—
309C2×D4—816no—
310C2×D4—816no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4AGL(1,ℤ/8)14yes—
333C22×C4#13714yes—
334C22×C4C22×D412yes—
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4H(5,𝔽2)14yes—
344C22×C4C2511yescharacteristic
345SD16—816no—
346C2×C8—28no—
347C2×C8—28no—
348C2×C8—28no—
349C4⋊C4—24no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—82no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C22⋊C4—416no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C22⋊C4—48no—
372C22⋊C4—48no—
373C22⋊C4—88no—
374C22⋊C4—84no—
375C22⋊C4—84no—
376C42—24no—
377C42—22no—
378C42—22no—
379C42—24no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—48no—
387C42—44no—
388C42—44no—
389C23—22no—
390C23—22no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—88no—
399C23—88no—
400C23—88no—
401C23—88no—
402C23—88no—
403C23—88no—
404C23—88no—
405C23#53412yes—
406C23C2×AGL(1,ℤ/8)12yes—
407C23C2×#13712yes—
408C23C23×D411yescharacteristic
409Q8—48no—
410Q8—48no—
411Q8—48no—
412Q8—84no—
413D4—48no—
414D4—48no—
415D4—816no—
416D4—88no—
417D4—816no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—28no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—88no—
454C2×C4#53412yes—
455C2×C4C2×AGL(1,ℤ/8)12yes—
456C2×C4C2×AGL(1,ℤ/8)12yes—
457C2×C4C2×#13714yes—
458C2×C4C2×H(5,𝔽2)12yes—
459C2×C4#58514yes—
460C8—48no—
461C22—22no—
462C22—22no—
463C22—24no—
464C22—24no—
465C22—84no—
466C22—84no—
467C22—84no—
468C22—84no—
469C22—84no—
470C22—84no—
471C22—84no—
472C22—1616no—
473C22#319314yes—
474C22C2×#53411yescharacteristic
475C22#348712yes—
476C22C22×AGL(1,ℤ/8)11yescharacteristic
477C22C22×#13711yescharacteristic
478C22#358512yes—
479C4—22no—
480C4—22no—
481C4—24no—
482C4—22no—
483C4—42no—
484C4—42no—
485C4—41no—
486C4—42no—
487C4—41no—
488C4—82no—
489C4—82no—
490C4—84no—
491C4—84no—
492C4#354514yes—
493C2—24no—
494C2—164no—
495C2#3718711yescharacteristic
496C2C2×#319311yescharacteristic
497C2#6075311yescharacteristic
498C2#6189012yes—
499C2#6189212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35179C2semidirect
3. #33099C2semidirect
4. #32680C2semidirect
5. #32647C2semidirect
6. #32627C2semidirect
8. #27582C2semidirect
9. #24507C2semidirect
10. #23933C2semidirect
11. #21822C2semidirect
22. #2582C22semidirect
24. #2568C22semidirect
25. #2567C22semidirect
40. #2093C22semidirect
41. #2071C22semidirect
42. #2067C22semidirect
43. #2055C22semidirect
45. #2023C22semidirect
47. #2013C22semidirect
48. #1985C22semidirect
49. #1985C22semidirect
50. #1973C22semidirect
51. #1936C22semidirect
52. #1928C22semidirect
53. #1925C22semidirect
54. #1921C22semidirect
55. #1880C22semidirect

Maximal subgroups

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

subgroupindex
#351792
C2×#25482
#330992
#326802
#326472
#326272
#296252
#275822
#245072
#239332
#218222