Finite Group Databasebeta

← #24917138

Subgroups — #24917138

Order 10624.

504 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581819C211yesmaximal, characteristic, complemented
2C2×#14581800C211yesmaximal, characteristic, complemented
3#19128281C211yesmaximal, characteristic, complemented
4#19128267C211yesmaximal, characteristic, complemented
5#19127954C211yesmaximal, characteristic, complemented
6#19127953C211yesmaximal, characteristic, complemented
7#19127950C211yesmaximal, characteristic, complemented
8#19127938C211yesmaximal, characteristic, complemented
9#19127935C211yesmaximal, characteristic, complemented
10#19127932C211yesmaximal, characteristic, complemented
11#19127927C211yesmaximal, characteristic, complemented
12#19127742C211yesmaximal, characteristic, complemented
13#19127696C211yesmaximal, characteristic, complemented
14#102×D83C211yesmaximal, characteristic, complemented
15#468×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic
17C2×#13050590C2211yescharacteristic
18C2×D4×D83C2211yescharacteristic
19#14581819—24no—
20C2×#13050577C2211yescharacteristic
21C2×D664C2211yescharacteristic
22C2×#13050575C2211yescharacteristic
23#14581800—24no—
24C2×#13050573C2211yescharacteristic
25C2×#13050573C2211yescharacteristic, complemented
26C22×C83⋊C8C2211yescharacteristic, complemented
27C2×C8×D83C2211yescharacteristic
28C2×C8×D83C2211yescharacteristic, complemented
29#14581788—21no—
30#14581778—21no—
31#14581770—21no—
32#14581768C2211yescharacteristic, complemented
33#14581767—21no—
34#14581767C2211yescharacteristic
35#14581767C2211yescharacteristic, complemented
36#14581766C2211yescharacteristic
37#14581766C2211yescharacteristic, complemented
38#14581765—21no—
39#14581765—21no—
40#14581763C2211yescharacteristic
41#14581762—21no—
42#14581760—21no—
43#14581750C2211yescharacteristic
44C4⋊C4×D83C2211yescharacteristic
45#14581725C2211yescharacteristic, complemented
46#14581724C2211yescharacteristic, complemented
47#14581723C2211yescharacteristic, complemented
48#14581716C2211yescharacteristic, complemented
49#14581714C2211yescharacteristic
50#14581714C2211yescharacteristic, complemented
51#14581713C2211yescharacteristic
52#14581712C2211yescharacteristic
53#14581712C2211yescharacteristic, complemented
54C2×SD16×C83C2211yescharacteristic
55C2×M16×C83C2211yescharacteristic, complemented
56C22⋊Q8×C83C2211yescharacteristic, complemented
57#121×C83C2211yescharacteristic
58#107×C83C2211yescharacteristic, complemented
59#103×C83C2211yescharacteristic, complemented
60#102×C83C2211yescharacteristic
61C23×D83—21no—
62#13050591—22no—
63#13050591—22no—
64#13050590—22no—
65#13050590—22no—
66C2×Dic166D411yescharacteristic
67C2×D332—21no—
68C2×D332—21no—
69C2×D332—21no—
70C2×D332D411yescharacteristic
71C2×D332C2311yescharacteristic
72C2×D332C2311yescharacteristic
73C2×C22⋊D83—21no—
74C2×C22⋊D83—21no—
75C2×C22⋊D83—21no—
76C2×C22⋊D83—21no—
77C2×C22⋊D83D411yescharacteristic
78C2×C22⋊D83D411yescharacteristic
79D4×D83—22no—
80D4×D83—22no—
81C2×C4×D83—21no—
82C2×C4×D83—21no—
83C2×C4×D83C2311yescharacteristic
84C2×C4×D83C2311yescharacteristic
85#13050577—22no—
86D664—22no—
87#13050575—22no—
88#13050573—22no—
89#13050573D414yescomplemented
90C2×C83⋊C8—22no—
91C2×C83⋊C8C2311yescharacteristic
92C2×C83⋊C8C2311yescharacteristic
93C8×D83—22no—
94C8×D83—22no—
95C4⋊Dic83C2311yescharacteristic
96C4⋊Dic83C2311yescharacteristic
97Dic83⋊C4—21no—
98Dic83⋊C4—21no—
99C22⋊Dic83—21no—
100(C2×D83)⋊C4—21no—
101(C2×D83)⋊C4—21no—
102(C2×D83)⋊C4—21no—
103(C2×D83)⋊C4—21no—
104(C2×D83)⋊C4—21no—
105(C2×D83)⋊C4—21no—
106C4×Dic83—21no—
107C2×Q8×C83C2311yescharacteristic
108C2×D4×C83—21no—
109C2×D4×C83C2311yescharacteristic
110C22×C4×C83C2311yescharacteristic
111SD16×C83—22no—
112M16×C83—22no—
113C2×C8×C83C2311yescharacteristic
114C2×C8×C83C2311yescharacteristic
115C4⋊C4×C83—21no—
116C4⋊C4×C83C2311yescharacteristic
117C4⋊C4×C83C2311yescharacteristic
118C22⋊C4×C83—21no—
119C22⋊C4×C83—21no—
120C23×C83—21no—
121C23×C83C2×D411yescharacteristic
122C22×D83—21no—
123C22×D83—21no—
124C22×D83—42no—
125C22×D83—81no—
126C22×D83—81no—
127C22×D83C2×D411yescharacteristic
128C22×D83C2×D411yescharacteristic
129Q8×C83—21no—
130Q8×C83—21no—
131D4×C83—21no—
132D4×C83—21no—
133D4×C83—81no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83C2×D411yescharacteristic
140C2×C4×C83C2411yescharacteristic
141C22⋊D83—22no—
142C22⋊D83—22no—
143C22⋊D83—42no—
144C22⋊D83—81no—
145C22⋊D83—81no—
146C22⋊D83—81no—
147C2×Dic83—21no—
148C2×Dic83—21no—
149C2×Dic83C2×D411yescharacteristic
150C2×Dic83C2×D411yescharacteristic
151D332—22no—
152D332—21no—
153D332—21no—
154D332—21no—
155D332—21no—
156D332—42no—
157D332—81no—
158D332—81no—
159C4×D83—21no—
160C4×D83—21no—
161C4×D83—42no—
162C4×D83—81no—
163C4×D83C2×D412yes—
164C4×D83#4112yes—
165Dic166—22no—
166C8×C83—21no—
167C8×C83C2×D412yes—
168C83⋊C8—21no—
169C83⋊C8C2×D412yes—
170C22×C83—21no—
171C22×C83—21no—
172C22×C83—21no—
173C22×C83—41no—
174C22×C83—41no—
175C22×C83—41no—
176C22×C83C22×D411yescharacteristic
177C2×D83—21no—
178C2×D83—21no—
179C2×D83—21no—
180C2×D83—21no—
181C2×D83—21no—
182C2×D83—41no—
183C2×D83—41no—
184C2×D83—41no—
185C2×D83—41no—
186C2×D83—41no—
187C2×D83—41no—
188C2×D83—84no—
189C2×D83#12112yes—
190C4×C83—21no—
191C4×C83—41no—
192C4×C83—41no—
193C4×C83—41no—
194C4×C83C22×D411yescharacteristic
195C4×C83C2×#4111yescharacteristic
196Dic83—21no—
197Dic83—41no—
198Dic83—41no—
199Dic83#12112yes—
200C2×C83—41no—
201C2×C83—81no—
202C2×C83C2×#12111yescharacteristic
203C2×C83#57611yescharacteristic
204C2×C83#57711yescharacteristic
205D83—22no—
206D83—41no—
207D83—81no—
208D83—81no—
209#3062—831nomaximal, Sylow
210C83#306211yescharacteristic, complemented, Sylow
211C2×AGL(1,ℤ/8)—831no—
212C2×#131—831no—
213#537—831no—
214#535—831no—
215#469—831no—
216#469—831no—
217#468—831no—
218#468D8311yescharacteristic, complemented
219#466—831no—
220#466—831no—
221#465—831no—
222#465—831no—
223#428—831no—
224#416—831no—
225C2×#102—831no—
226C2×#41—831no—
227C2×#41—831no—
228C22×D4—831no—
229AGL(1,ℤ/8)—1664no—
230C2×SD16—831no—
231C2×SD16C2×D8311yescharacteristic
232C2×D8—831no—
233C2×D8—831no—
234#131—1664no—
235C2×M16—831no—
236C2×M16—831no—
237C2×M16C2×D8311yescharacteristic, complemented
238C22×C8—831no—
239C22×C8—831no—
240C22×C8—831no—
241#127—1661no—
242#124—1661no—
243C22⋊Q8—831no—
244C22⋊Q8—1661no—
245C22⋊Q8C2×D8311yescharacteristic, complemented
246#121—831no—
247#121—831no—
248#121—831no—
249#121—831no—
250#121—831no—
251#121—1661no—
252#121—1661no—
253#121—1661no—
254#121C2×D8311yescharacteristic
255#120—1661no—
256#120—1661no—
257#117—831no—
258C2×C4⋊C4—831no—
259#107—831no—
260#107C2×D8311yescharacteristic, complemented
261#106—831no—
262#106—831no—
263#103—831no—
264#103C2×D8311yescharacteristic, complemented
265#102—831no—
266#102—831no—
267#102—831no—
268#102—831no—
269#102—831no—
270#102C2×D8311yescharacteristic
271C24—1661no—
272#41—1662no—
273#41—1662no—
274#41—1662no—
275#41—1662no—
276C2×Q8—831no—
277C2×Q8C22×D8311yescharacteristic
278C2×D4—21no—
279C2×D4—831no—
280C2×D4—831no—
281C2×D4—831no—
282C2×D4—831no—
283C2×D4—831no—
284C2×D4—1661no—
285C2×D4—1661no—
286C2×D4—1661no—
287C2×D4—1662no—
288C2×D4—1662no—
289C2×D4—1661no—
290C2×D4—1661no—
291C2×D4—1661no—
292C2×D4—1661no—
293C2×D4C22×D8311yescharacteristic
294C22×C4—831no—
295C22×C4—831no—
296C22×C4—1661no—
297C22×C4—1661no—
298C22×C4C22×D8311yescharacteristic
299SD16—22no—
300SD16—1662no—
301D8—1662no—
302D8—1662no—
303M16—22no—
304M16—834no—
305M16—1662no—
306C2×C8—831no—
307C2×C8—831no—
308C2×C8—1662no—
309C2×C8—1662no—
310C2×C8—1662no—
311C2×C8C22×D8311yescharacteristic
312C2×C8C22×D8311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—831no—
315C4⋊C4—831no—
316C4⋊C4—1661no—
317C4⋊C4—1661no—
318C4⋊C4C22×D8311yescharacteristic
319C4⋊C4C22×D8311yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—1661no—
323C22⋊C4—1661no—
324C22⋊C4—1661no—
325C22⋊C4—1661no—
326C22⋊C4—1661no—
327C22⋊C4—1661no—
328C22⋊C4—1661no—
329C42—1661no—
330C23—21no—
331C23—3322no—
332C23—831no—
333C23—831no—
334C23—6641no—
335C23—6641no—
336C23—1661no—
337C23—1661no—
338C23D4×D8311yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—1662no—
342D4—21no—
343D4—21no—
344D4—81no—
345D4—3322no—
346D4—3322no—
347D4—6641no—
348D4—6641no—
349D4—6641no—
350D4—6641no—
351D4—6641no—
352D4—1662no—
353D4—1662no—
354D4—1662no—
355D4—1661no—
356D4—1661no—
357D4—1661no—
358D4—1661no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—3322no—
365C2×C4—831no—
366C2×C4—832no—
367C2×C4—832no—
368C2×C4—831no—
369C2×C4—6641no—
370C2×C4—1661no—
371C2×C4—1661no—
372C2×C4—1661no—
373C2×C4—1661no—
374C2×C4D4×D8311yescharacteristic
375C2×C4C23×D8311yescharacteristic
376C8—21no—
377C8—832no—
378C8—1661no—
379C8D4×D8312yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—3321no—
387C22—3321no—
388C22—3321no—
389C22—3321no—
390C22—3321no—
391C22—3321no—
392C22—832no—
393C22—6644no—
394C22—1661no—
395C22—1661no—
396C22—1661no—
397C22—1661no—
398C22—1661no—
399C22C2×D4×D8311yescharacteristic
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—3321no—
405C4—3321no—
406C4—832no—
407C4—1661no—
408C4C2×D4×D8311yescharacteristic
409C4#41×D8311yescharacteristic
410C2—41no—
411C2—81no—
412C2—3321no—
413C2—6641no—
414C2—6641no—
415C2—1662no—
416C2#121×D8311yescharacteristic
417C2#1912850511yescharacteristic
418C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#468×C83C2
socleC22×C83C22×D4
hypercenter#468D83
hypocenterC83#3062
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#14581819C2semidirect
2. C2×#14581800C2semidirect
3. #19128281C2semidirect
4. #19128267C2semidirect
5. #19127954C2semidirect
6. #19127953C2semidirect
7. #19127950C2semidirect
8. #19127938C2semidirect
9. #19127935C2semidirect
10. #19127932C2semidirect
11. #19127927C2semidirect
12. #19127742C2semidirect
13. #19127696C2semidirect
14. #102×D83C2semidirect
15. #468×C83C2semidirect
25. C2×#13050573C22semidirect
26. C22×C83⋊C8C22semidirect
28. C2×C8×D83C22semidirect
32. #14581768C22semidirect
35. #14581767C22semidirect
37. #14581766C22semidirect
45. #14581725C22semidirect
46. #14581724C22semidirect
47. #14581723C22semidirect
48. #14581716C22semidirect
50. #14581714C22semidirect
53. #14581712C22semidirect
55. C2×M16×C83C22semidirect
56. C22⋊Q8×C83C22semidirect
58. #107×C83C22semidirect
59. #103×C83C22semidirect
89. #13050573D4semidirect
210. C83#3062semidirect
218. #468D83semidirect
237. C2×M16C2×D83semidirect
245. C22⋊Q8C2×D83semidirect
260. #107C2×D83semidirect
264. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818192
C2×#145818002
#191282812
#191282672
#191279542
#191279532
#191279502
#191279382
#191279352
#191279322
#191279272
#191277422
#191276962
#102×D832
#468×C832
#306283