Finite Group Databasebeta

← #24920596

Subgroups — #24920596

Order 10624.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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#135×D83C212yesmaximal, complemented
2#19128143C212yesmaximal, complemented
3#19128140C212yesmaximal, complemented
4#19128138C212yesmaximal, complemented
5#19128135C212yesmaximal, complemented
6#19128133C211yesmaximal, characteristic, complemented
7#19128131C212yesmaximal, complemented
8C2×#14581731C211yesmaximal, characteristic, complemented
9#508×C83C211yesmaximal, characteristic, complemented
10#41×D83—22no—
11#14581815C2212yes—
12#14581814C2212yescomplemented
13#14581813—22no—
14#14581812—22no—
15#14581811C2212yes—
16Q16×D83C2212yescomplemented
17SD16×D83—22no—
18D8×D83C2212yescomplemented
19C2×C8×D83C2211yescharacteristic
20Dic664C2211yescharacteristic, complemented
21#14581738C2212yescomplemented
22#14581737C2212yescomplemented
23#14581736C2212yescomplemented
24#14581735C2212yescomplemented
25D1328C2211yescharacteristic, complemented
26#14581733C2212yescomplemented
27#14581731C2212yescomplemented
28#14581731C2212yescomplemented
29C2×C83⋊C16C2211yescharacteristic, complemented
30#135×C83C2212yescomplemented
31Q32×C83C2211yescharacteristic, complemented
32SD32×C83C2212yescomplemented
33D16×C83C2211yescharacteristic, complemented
34C2×C16×C83C2211yescharacteristic, complemented
35#13050591—22no—
36#13050590—22no—
37#13050590—42no—
38#13050589—22no—
39#13050589—42no—
40Q8×D83—22no—
41D4×D83—22no—
42D4×D83—42no—
43C2×C4×D83—42no—
44C2×C4×D83D411yescharacteristic
45Dic332C2312yes—
46#13050580—22no—
47#13050579—22no—
48#13050578—22no—
49#13050577—22no—
50D664C2312yes—
51#13050575—22no—
52C2×C83⋊C8D411yescharacteristic
53C8×D83D412yes—
54C8×D83C2312yes—
55C83⋊C16C2312yes—
56#41×C83—22no—
57Q16×C83C2312yes—
58SD16×C83—22no—
59D8×C83C2312yes—
60C2×C8×C83C2311yescharacteristic
61C16×C83C2311yescharacteristic
62C16×C83C2311yescharacteristic
63C22×D83—42no—
64C22×D83D811yescharacteristic
65Q8×C83—22no—
66D4×C83—22no—
67D4×C83—42no—
68C2×C4×C83—42no—
69C2×C4×C83C2×D411yescharacteristic
70C22⋊D83—42no—
71C22⋊D83—42no—
72C22⋊D83—42no—
73C2×Dic83—42no—
74C2×Dic83D811yescharacteristic
75D332—22no—
76D332—42no—
77C4×D83—42no—
78C4×D83—42no—
79C4×D83—42no—
80C4×D83D812yes—
81C4×D83C2×D412yes—
82Dic166—22no—
83Dic166—42no—
84C8×C83C2×D411yescharacteristic
85C8×C83C2411yescharacteristic
86C83⋊C8C2×D412yes—
87C22×C83—42no—
88C22×C83C2×D811yescharacteristic
89C2×D83—41no—
90C2×D83—42no—
91C2×D83—84no—
92C2×D83C2×D812yes—
93C4×C83—42no—
94C4×C83C2×D811yescharacteristic
95C4×C83C22×D411yescharacteristic
96Dic83—42no—
97Dic83C2×D812yes—
98C2×C83—21no—
99C2×C83—82no—
100C2×C83C22×D811yescharacteristic
101D83—22no—
102D83—82no—
103#3414—831nomaximal, Sylow
104C83#341411yescharacteristic, complemented, Sylow
105C2×#135—832no—
106#510—832no—
107#510—832no—
108#509—832no—
109#509—832no—
110#508—832no—
111#508—831no—
112#508D8311yescharacteristic, complemented
113C2×M32—831no—
114C2×#41—1662no—
115#135—832no—
116#135—832no—
117#135—832no—
118#135—1662no—
119#135—1662no—
120#135C2×D8312yescomplemented
121C2×Q16—832no—
122C2×SD16—1662no—
123C2×D8—832no—
124C22×C8—831no—
125Q32—832no—
126Q32—831no—
127Q32C2×D8311yescharacteristic, complemented
128SD32—832no—
129SD32—832no—
130SD32—832no—
131SD32C2×D8312yescomplemented
132D16—832no—
133D16—831no—
134D16C2×D8311yescharacteristic, complemented
135M32—832no—
136M32—832no—
137C2×C16—831no—
138C2×C16C2×D8311yescharacteristic, complemented
139#41—22no—
140#41—3322no—
141#41—3322no—
142#41—1662no—
143#41—1662no—
144#41—1662no—
145C2×Q8—1662no—
146C2×D4—3322no—
147C2×D4—1662no—
148C22×C4—3322no—
149C22×C4—831no—
150Q16—832no—
151Q16—1662no—
152Q16C22×D8312yes—
153SD16—22no—
154SD16—1662no—
155SD16—1662no—
156SD16—1662no—
157D8—832no—
158D8—1662no—
159D8C22×D8312yes—
160C2×C8—831no—
161C2×C8—832no—
162C2×C8—832no—
163C2×C8C22×D8311yescharacteristic
164C16—832no—
165C16C22×D8311yescharacteristic
166C16C22×D8311yescharacteristic
167C23—3322no—
168C23—831no—
169Q8—22no—
170Q8—3322no—
171Q8—1662no—
172D4—22no—
173D4—42no—
174D4—3322no—
175D4—3322no—
176D4—3322no—
177D4—3322no—
178D4—1662no—
179C2×C4—42no—
180C2×C4—3322no—
181C2×C4—3322no—
182C2×C4—3322no—
183C2×C4—3322no—
184C2×C4—831no—
185C2×C4—832no—
186C2×C4—832no—
187C2×C4D4×D8311yescharacteristic
188C8—832no—
189C8D4×D8311yescharacteristic
190C8C23×D8311yescharacteristic
191C22—42no—
192C22—3321no—
193C22—3322no—
194C22—832no—
195C22—6644no—
196C22D8×D8311yescharacteristic
197C4—42no—
198C4—3322no—
199C4—832no—
200C4D8×D8311yescharacteristic
201C4C2×D4×D8311yescharacteristic
202C2—21no—
203C2—82no—
204C2—6642no—
205C2—1662no—
206C2C2×D8×D8311yescharacteristic
subgroupG/H
centerC4D8×D83
commutator, cosocleC8×C83C24
FrattiniC8C23×D83
Fitting#508×C83C2
socleC2×C83C22×D8
hypercenter#508D83
hypocenterC83#3414
LCSG ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D83C2semidirect
2. #19128143C2semidirect
3. #19128140C2semidirect
4. #19128138C2semidirect
5. #19128135C2semidirect
6. #19128133C2semidirect
7. #19128131C2semidirect
8. C2×#14581731C2semidirect
9. #508×C83C2semidirect
12. #14581814C22semidirect
16. Q16×D83C22semidirect
18. D8×D83C22semidirect
20. Dic664C22semidirect
21. #14581738C22semidirect
22. #14581737C22semidirect
23. #14581736C22semidirect
24. #14581735C22semidirect
25. D1328C22semidirect
26. #14581733C22semidirect
27. #14581731C22semidirect
28. #14581731C22semidirect
29. C2×C83⋊C16C22semidirect
30. #135×C83C22semidirect
31. Q32×C83C22semidirect
32. SD32×C83C22semidirect
33. D16×C83C22semidirect
34. C2×C16×C83C22semidirect
104. C83#3414semidirect
112. #508D83semidirect
120. #135C2×D83semidirect
127. Q32C2×D83semidirect
131. SD32C2×D83semidirect
134. D16C2×D83semidirect
138. C2×C16C2×D83semidirect

Maximal subgroups

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

subgroupindex
#135×D832
#191281432
#191281402
#191281382
#191281352
#191281332
#191281312
C2×#145817312
#508×C832
#341483