Finite Group Databasebeta

← #16856177

Subgroups — #16856177

Order 3968.

572 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C2×#11758486C212yesmaximal, complemented
2C2×#11758485C212yesmaximal, complemented
3#14521599C211yesmaximal, characteristic, complemented
4C2×#11758452C211yesmaximal, characteristic, complemented
5#14521443C212yesmaximal, complemented
6#14521438C212yesmaximal, complemented
7#14521433C214yesmaximal, complemented
8C4×M16×C31C211yesmaximal, characteristic, complemented
9C2×#92495C2212yes—
10C22×D124C2211yescharacteristic
11C22×Dic62C2211yescharacteristic
12C2×M16×C31C2212yes—
13C2×C42×C31C2211yescharacteristic
14#11758507—22no—
15C2×(C2×D31)⋊C4—22no—
16#11758486—28no—
17#11758485—28no—
18C2×Dic124C2214yescomplemented
19C2×D248C2214yes—
20C2×#92470C2214yes—
21C2×#92470C2214yescomplemented
22#11758452—22no—
23#11758452C2212yes—
24#11758452C2212yes—
25#11758452C2212yescomplemented
26#11758451C2211yescharacteristic, complemented
27C4×D124—22no—
28#11758446C2211yescharacteristic, complemented
29C31×#97C2212yescomplemented
30C4×C8×C31C2212yescomplemented
31C23×D31—21no—
32C22×C4×C31D412yes—
33C22×C4×C31C2311yescharacteristic
34C2×C22⋊D31—22no—
35C22×Dic31—21no—
36#92495—24no—
37#92495—24no—
38C2×D124—24no—
39C2×D124—22no—
40C2×D124—21no—
41C2×D124C2312yes—
42C2×D124C2312yes—
43C2×C4×D31—22no—
44C2×Dic62—24no—
45C2×Dic62—21no—
46C2×Dic62C2312yes—
47C2×Dic62C2312yes—
48M16×C31D418yescomplemented
49C2×C8×C31C2314yes—
50C42×C31D412yes—
51C42×C31C2311yescharacteristic
52C42×C31C2311yescharacteristic
53(C2×D31)⋊C4—24no—
54(C2×D31)⋊C4—24no—
55(C2×D31)⋊C4—24no—
56C4⋊Dic31—22no—
57Dic124—28no—
58D248—28no—
59#92470—28no—
60#92470—28no—
61C23×C31C2×D411yescharacteristic
62C22×D31—22no—
63C22×D31—22no—
64C22×D31—44no—
65C22×D31—42no—
66C2×C4×C31—42no—
67C2×C4×C31C2×D412yes—
68C2×C4×C31C2×D412yes—
69C2×C4×C31C2×D411yescharacteristic
70C2×C4×C31C2×D414yes—
71C2×C4×C31C2411yescharacteristic
72C22⋊D31—44no—
73C2×Dic31—22no—
74C2×Dic31—22no—
75C2×Dic31—42no—
76D124—22no—
77D124—24no—
78D124—22no—
79D124—44no—
80D124—42no—
81C4×D31—44no—
82Dic62—22no—
83Dic62—24no—
84Dic62—22no—
85Dic62—42no—
86C8×C31C2×D418yes—
87#3153—311nomaximal, Sylow
88C22×C31—22no—
89C22×C31#12712yes—
90C22×C31C22×D411yescharacteristic
91C2×D31—42no—
92C2×D31—42no—
93C2×D31—44no—
94C2×D31—44no—
95C2×D31—84no—
96C4×C31—22no—
97C4×C31—22no—
98C4×C31#12712yes—
99C4×C31C22×D412yes—
100Dic31—42no—
101Dic31—42no—
102C2×#137—312no—
103C2×AGL(1,ℤ/8)—312no—
104#532—311no—
105C2×#124—311no—
106#497—312no—
107#496—312no—
108#495—314no—
109C4×M16D3111yescharacteristic, complemented
110C2×C31—22no—
111C2×C31C2×#12711yescharacteristic
112C2×C31#57512yes—
113D31—82no—
114D31—82no—
115C2×#41—312no—
116C22×Q8—311no—
117C22×D4—311no—
118#137—628no—
119AGL(1,ℤ/8)—628no—
120C2×Q16—314no—
121C2×SD16—314no—
122C2×SD16—314no—
123C2×D8—314no—
124C2×M16C2×D3112yes—
125C4⋊Q8—311no—
126#127—311no—
127#124—312no—
128#124—312no—
129#124—312no—
130#124—622no—
131#121—622no—
132C4×D4—622no—
133C2×C22⋊C4—622no—
134C2×C42C2×D3111yescharacteristic
135#97C2×D3112yescomplemented
136C4×C8C2×D3112yescomplemented
137C31#315311yescharacteristic, complemented, Sylow
138C24—621no—
139#41—624no—
140#41—624no—
141C2×Q8—312no—
142C2×Q8—312no—
143C2×Q8—621no—
144C2×Q8—624no—
145C2×D4—312no—
146C2×D4—312no—
147C2×D4—622no—
148C2×D4—624no—
149C2×D4—622no—
150C2×D4—621no—
151C22×C4—621no—
152C22×C4—622no—
153C22×C4D12412yes—
154C22×C4C22×D3111yescharacteristic
155Q16—628no—
156SD16—628no—
157SD16—628no—
158D8—628no—
159M16D12418yescomplemented
160C2×C8C22×D3114yes—
161C4⋊C4—622no—
162C22⋊C4—624no—
163C22⋊C4—624no—
164C22⋊C4—624no—
165C42D12412yes—
166C42C22×D3111yescharacteristic
167C42C22×D3111yescharacteristic
168C23—622no—
169C23—622no—
170C23—1244no—
171C23—1242no—
172C23C2×D12411yescharacteristic
173Q8—622no—
174Q8—624no—
175Q8—622no—
176Q8—1242no—
177D4—622no—
178D4—624no—
179D4—622no—
180D4—1244no—
181D4—1244no—
182D4—1242no—
183C2×C4—42no—
184C2×C4—622no—
185C2×C4—622no—
186C2×C4—1242no—
187C2×C4—1244no—
188C2×C4C2×D12412yes—
189C2×C4C2×D12412yes—
190C2×C4C2×D12411yescharacteristic
191C2×C4C2×D12414yes—
192C2×C4C23×D3111yescharacteristic
193C8C2×D12418yes—
194C22—22no—
195C22—1242no—
196C22—1242no—
197C22—1244no—
198C22—1244no—
199C22—2484no—
200C22#1175845112yes—
201C22C22×D12411yescharacteristic
202C4—22no—
203C4—22no—
204C4—1242no—
205C4—1242no—
206C4#1175845112yes—
207C4C22×D12412yes—
208C2—22no—
209C2—2482no—
210C2—2482no—
211C2C2×#1175845111yescharacteristic
212C2#1452184912yes—
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC4×M16×C31C2
socleC22×C31C22×D4
hypercenterC4×M16D31
hypocenterC31#3153
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#11758486C2semidirect
2. C2×#11758485C2semidirect
3. #14521599C2semidirect
4. C2×#11758452C2semidirect
5. #14521443C2semidirect
6. #14521438C2semidirect
7. #14521433C2semidirect
8. C4×M16×C31C2semidirect
18. C2×Dic124C22semidirect
21. C2×#92470C22semidirect
25. #11758452C22semidirect
26. #11758451C22semidirect
28. #11758446C22semidirect
29. C31×#97C22semidirect
30. C4×C8×C31C22semidirect
48. M16×C31D4semidirect
109. C4×M16D31semidirect
135. #97C2×D31semidirect
136. C4×C8C2×D31semidirect
137. C31#3153semidirect
159. M16D124semidirect

Maximal subgroups

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

subgroupindex
C2×#117584862
C2×#117584852
#145215992
C2×#117584522
#145214432
#145214382
#145214332
C4×M16×C312
#315331