Finite Group Databasebeta

← #16846149

Subgroups — #16846149

Order 3968.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#11758507C211yesmaximal, characteristic
2C2×#11758409C212yesmaximal, complemented
3#14520909C211yesmaximal, characteristic, complemented
4#14520812C212yesmaximal, complemented
5C22×M16×C31C211yesmaximal, characteristic, complemented
6C23×C4×C31C2211yescharacteristic
7C22×C22⋊D31—21no—
8C22×D124C2211yescharacteristic
9C2×M16×C31—22no—
10C2×M16×C31—24no—
11C22×C8×C31C2212yes—
12C2×C22⋊Dic31—21no—
13#11758507—22no—
14#11758507C414yescomplemented
15C2×(C2×D31)⋊C4—21no—
16C2×C4⋊Dic31C2212yes—
17C2×C4⋊Dic31C2211yescharacteristic
18C31⋊#95—21no—
19#11758409—24no—
20C24×C31D411yescharacteristic
21C23×D31—21no—
22C22×C4×C31—22no—
23C22×C4×C31—22no—
24C22×C4×C31C2×C412yes—
25C22×C4×C31D411yescharacteristic
26C22×C4×C31D412yes—
27C22×C4×C31C2311yescharacteristic
28C2×C22⋊D31—24no—
29C2×C22⋊D31—44no—
30C22×Dic31—21no—
31C22×Dic31—22no—
32C2×D124—22no—
33C2×D124—22no—
34C2×D124—21no—
35C2×D124C2×C412yes—
36M16×C31—28no—
37C2×C8×C31—22no—
38C2×C8×C31—22no—
39C2×C8×C31D414yescomplemented
40C22⋊Dic31—42no—
41(C2×D31)⋊C4—24no—
42C4⋊Dic31—21no—
43C4⋊Dic31—22no—
44C4⋊Dic31—22no—
45C4⋊Dic31C2×C412yes—
46C23×C31—22no—
47C23×C31—21no—
48C23×C31—22no—
49C23×C31—21no—
50C23×C31C22⋊C412yes—
51C23×C31C2×D411yescharacteristic
52C22×D31—22no—
53C22×D31—42no—
54C22×D31—42no—
55C22×D31—41no—
56C22×D31—41no—
57C2×C4×C31—22no—
58C2×C4×C31—24no—
59C2×C4×C31—22no—
60C2×C4×C31—22no—
61C2×C4×C31C22⋊C412yes—
62C2×C4×C31C22×C411yescharacteristic
63C2×C4×C31C2×D412yes—
64C2×C4×C31C2×D411yescharacteristic
65C2×C4×C31#4112yes—
66C22⋊D31—48no—
67C2×Dic31—22no—
68C2×Dic31—42no—
69C2×Dic31—42no—
70C2×Dic31—42no—
71C2×Dic31—42no—
72D124—24no—
73D124—41no—
74D124—41no—
75C8×C31—24no—
76#1895—311nomaximal, Sylow
77C22×C31—22no—
78C22×C31—22no—
79C22×C31—22no—
80C22×C31—42no—
81C22×C31—42no—
82C22×C31C2×C22⋊C411yescharacteristic
83C22×C31#12112yes—
84C22×C31AGL(1,ℤ/8)11yescharacteristic
85C22×C31AGL(1,ℤ/8)12yes—
86C22×C31#13711yescharacteristic
87C2×D31—42no—
88C2×D31—42no—
89C2×D31—42no—
90C2×D31—82no—
91C2×D31—82no—
92C4×C31—22no—
93C4×C31C4×D412yes—
94C4×C31#12011yescharacteristic
95C4×C31#12311yescharacteristic
96Dic31—42no—
97Dic31—82no—
98C22×M16D3111yescharacteristic, complemented
99C2×#121—311no—
100C2×#102—312no—
101#380—311no—
102#340—312no—
103C2×C31—42no—
104C2×C31#38611yescharacteristic
105C2×C31#41711yescharacteristic
106C2×C31#41811yescharacteristic
107C2×C31#46812yes—
108C2×C31#46912yes—
109D31—82no—
110C22×D4—311no—
111C22×D4—621no—
112C23×C4C2×D3111yescharacteristic
113C2×M16—22no—
114C2×M16—24no—
115C22×C8C2×D3112yes—
116#121—314no—
117#121—622no—
118C2×C4⋊C4—311no—
119C2×C4⋊C4—312no—
120C2×C22⋊C4—621no—
121C2×C22⋊C4—621no—
122#102—624no—
123#95—621no—
124C31#189511yescharacteristic, complemented, Sylow
125C24—621no—
126C24D12411yescharacteristic
127C2×D4—312no—
128C2×D4—624no—
129C2×D4—622no—
130C2×D4—622no—
131C2×D4—621no—
132C2×D4—1244no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—621no—
136C22×C4—622no—
137C22×C4C4×D3112yes—
138C22×C4D12411yescharacteristic
139C22×C4C22⋊D3112yes—
140C22×C4C22×D3111yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8C22⋊D3114yescomplemented
145C4⋊C4—312no—
146C4⋊C4—621no—
147C4⋊C4—622no—
148C4⋊C4—622no—
149C22⋊C4—624no—
150C22⋊C4—1242no—
151C23—21no—
152C23—22no—
153C23—21no—
154C23—22no—
155C23—622no—
156C23—1242no—
157C23—1241no—
158C23—1241no—
159C23—1242no—
160C23(C2×D31)⋊C412yes—
161C23C2×D12411yescharacteristic
162D4—624no—
163D4—1248no—
164D4—1241no—
165D4—1241no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—622no—
171C2×C4—1242no—
172C2×C4—1242no—
173C2×C4—1242no—
174C2×C4—1242no—
175C2×C4(C2×D31)⋊C412yes—
176C2×C4C2×C4×D3111yescharacteristic
177C2×C4#9249512yes—
178C2×C4C2×C22⋊D3112yes—
179C2×C4C2×C22⋊D3111yescharacteristic
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—1242no—
187C22—1242no—
188C22—1242no—
189C22—2482no—
190C22—2482no—
191C22#1175848512yes—
192C22#1175848511yescharacteristic
193C22#1175848611yescharacteristic
194C22C2×(C2×D31)⋊C411yescharacteristic
195C22#1175850712yes—
196C4—22no—
197C4—1242no—
198C4—2482no—
199C4C4×C22⋊D3112yes—
200C4#1175850611yescharacteristic
201C4#1175853011yescharacteristic
202C2—42no—
203C2—2482no—
204C2#1452092811yescharacteristic
205C2#1452105111yescharacteristic
206C2#1452105511yescharacteristic
207C2#1452129512yes—
208C2#1452130212yes—
subgroupG/H
centerC23C2×D124
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC22×M16×C31C2
socleC23×C31C2×D4
prime radicalC2×#11758507C2
cosocleC22×C4×C31C23
hypercenterC22×M16D31
hypocenterC31#1895
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#11758409C2semidirect
3. #14520909C2semidirect
4. #14520812C2semidirect
5. C22×M16×C31C2semidirect
14. #11758507C4semidirect
39. C2×C8×C31D4semidirect
98. C22×M16D31semidirect
124. C31#1895semidirect
144. C2×C8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585072
C2×#117584092
#145209092
#145208122
C22×M16×C312
#189531