Finite Group Databasebeta

← #24918405

Subgroups — #24918405

Order 10624.

488 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128256C211yesmaximal, characteristic, complemented
2#127×D83C211yesmaximal, characteristic
3#19128074C211yesmaximal, characteristic, complemented
4#19128073C211yesmaximal, characteristic, complemented
5#19128049C211yesmaximal, characteristic, complemented
6#19127900C214yesmaximal, complemented
7#19127853C214yesmaximal, complemented
8#19127752C211yesmaximal, characteristic, complemented
9#486×C83C211yesmaximal, characteristic, complemented
10C2×D4×D83—21no—
11C2×D4×D83—21no—
12C2×D4×D83C2212yes—
13C2×#13050579C2214yescomplemented
14C2×#13050578C2214yescomplemented
15#14581792C2211yescharacteristic, complemented
16#14581789C2211yescharacteristic
17#14581788—21no—
18#14581788—21no—
19#14581788C2212yes—
20#14581783—21no—
21#14581772C2212yes—
22#14581768—21no—
23Q8×Dic83—21no—
24#14581757C2212yescomplemented
25#14581750—21no—
26C42×D83C2211yescharacteristic
27Dic83⋊C8C2212yescomplemented
28#14581712C2214yescomplemented
29(C2×D83)⋊C8C2212yes—
30C4×C83⋊C8C2211yescharacteristic, complemented
31C4⋊Q8×C83C2211yescharacteristic, complemented
32#127×C83C2211yescharacteristic
33#102×C83C2214yescomplemented
34C4×C8×C83C2211yescharacteristic, complemented
35C23×D83—22no—
36C2×Dic166C2312yes—
37C2×D332—21no—
38C2×D332—21no—
39C2×D332C2312yes—
40C2×C22⋊D83—22no—
41C2×C22⋊D83—22no—
42C2×C22⋊D83—22no—
43D4×D83—44no—
44D4×D83—44no—
45D4×D83—44no—
46C2×C4×D83D411yescharacteristic
47C2×C4×D83D411yescharacteristic
48C2×C4×D83C2311yescharacteristic
49#13050579—44no—
50#13050578—44no—
51C2×C83⋊C8C2312yes—
52C4⋊Dic83—21no—
53C4⋊Dic83—21no—
54C4⋊Dic83—22no—
55Dic83⋊C4—22no—
56(C2×D83)⋊C4—22no—
57C4×Dic83—22no—
58C4×Dic83D411yescharacteristic
59C4×Dic83D411yescharacteristic
60C4×Dic83C2311yescharacteristic
61C2×Q8×C83—21no—
62C2×D4×C83—21no—
63C2×D4×C83—21no—
64C2×D4×C83C2312yes—
65C2×C8×C83C2312yes—
66C4⋊C4×C83—21no—
67C4⋊C4×C83C2312yes—
68C42×C83C2311yescharacteristic
69C23×C83—22no—
70C22×D83—22no—
71C22×D83—44no—
72C22×D83—44no—
73C22×D83—44no—
74C22×D83C2×D411yescharacteristic
75Q8×C83—42no—
76D4×C83—24no—
77D4×C83—42no—
78D4×C83—42no—
79C2×C4×C83—22no—
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83C2411yescharacteristic
83C22⋊D83—44no—
84C22⋊D83—44no—
85C22⋊D83—44no—
86C2×Dic83—22no—
87C2×Dic83C2×D411yescharacteristic
88C2×Dic83C2×D411yescharacteristic
89C2×Dic83C2×D411yescharacteristic
90D332—24no—
91D332—42no—
92D332—42no—
93C4×D83—22no—
94C4×D83—22no—
95C4×D83SD1614yescomplemented
96Dic166—24no—
97C8×C83—22no—
98C83⋊C8—22no—
99C22×C83—42no—
100C22×C83—42no—
101C22×C83—42no—
102C22×C83C22×D411yescharacteristic
103C2×D83—21no—
104C2×D83—21no—
105C2×D83—42no—
106C2×D83—42no—
107C2×D83—42no—
108C2×D83—88no—
109C2×D83C2×SD1612yes—
110C4×C83—42no—
111C4×C83C2×SD1612yes—
112C4×C83AGL(1,ℤ/8)12yes—
113C4×C83H(5,𝔽2)12yes—
114Dic83—21no—
115Dic83—21no—
116Dic83—42no—
117Dic83C2×SD1612yes—
118C2×C83—82no—
119C2×C83#53511yescharacteristic
120C2×C83C22×SD1611yescharacteristic
121C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
122D83—22no—
123D83—82no—
124#3208—831nomaximal, Sylow
125C83#320811yescharacteristic, complemented, Sylow
126#533—831no—
127C2×#127—831no—
128#492—831no—
129#492—831no—
130#486—831no—
131#486D8311yescharacteristic, complemented
132#460—834no—
133#450—834no—
134#431—831no—
135C22×D4—832no—
136C22×D4—1661no—
137C22×D4—1661no—
138C2×SD16—834no—
139C2×SD16—834no—
140C4⋊Q8—831no—
141C4⋊Q8C2×D8311yescharacteristic, complemented
142#127—832no—
143#127—831no—
144#127—1661no—
145#127—1661no—
146#127C2×D8311yescharacteristic
147#125—1661no—
148C22⋊Q8—832no—
149C22⋊Q8—1661no—
150C4×Q8—832no—
151C4×Q8—1661no—
152#117—1661no—
153C2×C42—831no—
154C4⋊C8—832no—
155#102—834no—
156#102C2×D8314yescomplemented
157C22⋊C8—832no—
158C4×C8—831no—
159C4×C8C2×D8311yescharacteristic, complemented
160C24—1662no—
161C2×Q8—21no—
162C2×Q8—832no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—3324no—
166C2×D4—3324no—
167C2×D4—3324no—
168C2×D4—832no—
169C2×D4—1662no—
170C2×D4—1662no—
171C2×D4—1662no—
172C2×D4—1661no—
173C2×D4—1661no—
174C2×D4C22×D8312yes—
175C22×C4—831no—
176C22×C4—831no—
177C22×C4—831no—
178SD16—3324no—
179SD16—3324no—
180C2×C8—832no—
181C2×C8C22×D8312yes—
182C4⋊C4—21no—
183C4⋊C4—1662no—
184C4⋊C4—1662no—
185C4⋊C4—1661no—
186C4⋊C4—1661no—
187C4⋊C4C22×D8312yes—
188C22⋊C4—1662no—
189C42—831no—
190C42—831no—
191C42—831no—
192C42—1662no—
193C42C22×D8311yescharacteristic
194C23—22no—
195C23—3324no—
196C23—3324no—
197C23—3324no—
198C23—831no—
199C23—1662no—
200Q8—42no—
201Q8—1664no—
202D4—24no—
203D4—42no—
204D4—42no—
205D4—3324no—
206D4—3324no—
207D4—3324no—
208D4—3322no—
209D4—3322no—
210D4—1664no—
211C2×C4—22no—
212C2×C4—831no—
213C2×C4—834no—
214C2×C4—831no—
215C2×C4—831no—
216C2×C4—1662no—
217C2×C4—1662no—
218C2×C4—1662no—
219C2×C4D4×D8311yescharacteristic
220C2×C4D4×D8311yescharacteristic
221C2×C4C23×D8311yescharacteristic
222C8—22no—
223C8—1662no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—3322no—
228C22—3322no—
229C22—3322no—
230C22—832no—
231C22—6648no—
232C22—1661no—
233C22—1661no—
234C22C2×D4×D8311yescharacteristic
235C4—42no—
236C4—3322no—
237C4—832no—
238C4—1661no—
239C4—1661no—
240C4SD16×D8312yes—
241C4#1458181612yes—
242C4#1458183812yes—
243C2—82no—
244C2—6642no—
245C2—1662no—
246C2#1912826911yescharacteristic
247C2C2×SD16×D8311yescharacteristic
248C2C2×#1458181611yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#486×C83C2
socleC22×C83C22×D4
prime radical#127×D83C2
hypercenter#486D83
hypocenterC83#3208
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #19128256C2semidirect
3. #19128074C2semidirect
4. #19128073C2semidirect
5. #19128049C2semidirect
6. #19127900C2semidirect
7. #19127853C2semidirect
8. #19127752C2semidirect
9. #486×C83C2semidirect
13. C2×#13050579C22semidirect
14. C2×#13050578C22semidirect
15. #14581792C22semidirect
24. #14581757C22semidirect
27. Dic83⋊C8C22semidirect
28. #14581712C22semidirect
30. C4×C83⋊C8C22semidirect
31. C4⋊Q8×C83C22semidirect
33. #102×C83C22semidirect
34. C4×C8×C83C22semidirect
95. C4×D83SD16semidirect
125. C83#3208semidirect
131. #486D83semidirect
141. C4⋊Q8C2×D83semidirect
156. #102C2×D83semidirect
159. C4×C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282562
#127×D832
#191280742
#191280732
#191280492
#191279002
#191278532
#191277522
#486×C832
#320883