Finite Group Databasebeta

← #16846179

Subgroups — #16846179

Order 3968.

338 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 97 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×#11758501C211yesmaximal, characteristic
2C2×#11758412C212yesmaximal, complemented
3#14521013C211yesmaximal, characteristic
4#14520817C212yesmaximal, complemented
5C22×M16×C31C211yesmaximal, characteristic
6C23×C4×C31C2211yescharacteristic
7C22×Dic62C2211yescharacteristic
8C2×M16×C31—21no—
9C2×M16×C31—24no—
10C2×M16×C31C2212yes—
11C22×C8×C31—21no—
12C2×C22⋊Dic31C412yes—
13C2×C4⋊Dic31C2211yescharacteristic
14#11758501—21no—
15#11758501—21no—
16#11758501—21no—
17#11758501—21no—
18C2×Dic31⋊C4C412yes—
19C2×#92474C2212yes—
20C22⋊(C31⋊C8)—21no—
21C22⋊(C31⋊C8)—21no—
22#11758412—24no—
23C24×C31C2×C411yescharacteristic
24C22×C4×C31—22no—
25C22×C4×C31—22no—
26C22×C4×C31C2×C411yescharacteristic
27C22×C4×C31D411yescharacteristic
28C22×C4×C31D411yescharacteristic
29C22×C4×C31D411yescharacteristic
30C22×C4×C31D411yescharacteristic
31C22×C4×C31C2311yescharacteristic
32C22×Dic31C2×C412yes—
33C22×Dic31C2×C412yes—
34C2×Dic62—22no—
35C2×Dic62—21no—
36C2×Dic62—21no—
37C2×Dic62—22no—
38M16×C31—24no—
39M16×C31—22no—
40M16×C31D414yes—
41C2×C8×C31—24no—
42C2×C8×C31—22no—
43C22⋊Dic31—22no—
44C22⋊Dic31—22no—
45C4⋊Dic31—21no—
46C4⋊Dic31—21no—
47Dic31⋊C4—22no—
48Dic31⋊C4—22no—
49#92474—22no—
50#92474—22no—
51C2×C31⋊C8—22no—
52C23×C31—22no—
53C23×C31—22no—
54C23×C31—41no—
55C23×C31C22⋊C411yescharacteristic
56C23×C31C22⋊C411yescharacteristic
57C23×C31C22×C411yescharacteristic
58C2×C4×C31—22no—
59C2×C4×C31—21no—
60C2×C4×C31—21no—
61C2×C4×C31—22no—
62C2×C4×C31—22no—
63C2×C4×C31—21no—
64C2×C4×C31—21no—
65C2×C4×C31C22⋊C411yescharacteristic
66C2×C4×C31C22⋊C411yescharacteristic
67C2×C4×C31C2×D411yescharacteristic
68C2×C4×C31C2×D412yes—
69C2×C4×C31C2×D411yescharacteristic
70C2×C4×C31#4112yes—
71C2×Dic31—22no—
72C2×Dic31—22no—
73C2×Dic31—24no—
74C2×Dic31—42no—
75Dic62—41no—
76Dic62—42no—
77Dic62—41no—
78C8×C31—24no—
79C31⋊C8—42no—
80#1901—311nomaximal, Sylow
81C22×C31—21no—
82C22×C31—21no—
83C22×C31—21no—
84C22×C31—21no—
85C22×C31—21no—
86C22×C31—21no—
87C22×C31—21no—
88C22×C31—44no—
89C22×C31#10112yescomplemented
90C22×C31C2×C22⋊C411yescharacteristic
91C22×C31C4×D412yes—
92C4×C31—21no—
93C4×C31—21no—
94C4×C31#12011yescharacteristic
95C4×C31#12112yes—
96C4×C31#12311yescharacteristic
97Dic31—42no—
98Dic31—42no—
99C22×M16D3111yescharacteristic
100C2×C22⋊Q8—311no—
101C2×#101—312no—
102#407—311no—
103#343—312no—
104C2×C31—22no—
105C2×C31—22no—
106C2×C31—41no—
107C2×C31#38611yescharacteristic
108C2×C31C2×#10111yescharacteristic
109C2×C31#41311yescharacteristic
110C22×Q8—311no—
111C23×C4C2×D3111yescharacteristic
112C2×M16—21no—
113C2×M16—24no—
114C2×M16—312no—
115C2×M16C2×D3112yes—
116C22×C8—21no—
117C22⋊Q8—621no—
118C22⋊Q8—621no—
119C22⋊Q8—621no—
120C22⋊Q8—621no—
121C2×C4⋊C4—312no—
122C2×C4⋊C4—311no—
123C2×C22⋊C4—312no—
124#101—624no—
125C22⋊C8—621no—
126C22⋊C8—621no—
127C31#190111yescharacteristic, complemented, Sylow
128C24C4×D3111yescharacteristic
129C2×Q8—621no—
130C2×Q8—622no—
131C2×Q8—622no—
132C2×Q8—621no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—312no—
136C22×C4—312no—
137C22×C4C4×D3111yescharacteristic
138C22×C4D12411yescharacteristic
139C22×C4D12411yescharacteristic
140C22×C4C22⋊D3111yescharacteristic
141C22×C4C22⋊D3111yescharacteristic
142C22×C4C22×D3111yescharacteristic
143M16—24no—
144M16—22no—
145M16—622no—
146M16—622no—
147M16C22⋊D3114yes—
148C2×C8—24no—
149C2×C8—22no—
150C2×C8—622no—
151C4⋊C4—622no—
152C4⋊C4—622no—
153C4⋊C4—621no—
154C4⋊C4—621no—
155C22⋊C4—622no—
156C22⋊C4—622no—
157C23—22no—
158C23—22no—
159C23—41no—
160C23(C2×D31)⋊C411yescharacteristic
161C23(C2×D31)⋊C411yescharacteristic
162C23C2×C4×D3111yescharacteristic
163Q8—1241no—
164Q8—1242no—
165Q8—1241no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—622no—
174C2×C4—622no—
175C2×C4—624no—
176C2×C4—1242no—
177C2×C4(C2×D31)⋊C411yescharacteristic
178C2×C4(C2×D31)⋊C411yescharacteristic
179C2×C4C2×D12411yescharacteristic
180C2×C4#9249512yes—
181C2×C4C2×C22⋊D3111yescharacteristic
182C2×C4C2×C22⋊D3112yes—
183C8—24no—
184C8—1242no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—44no—
193C22#1175841212yescomplemented
194C22C2×(C2×D31)⋊C411yescharacteristic
195C22C4×C22⋊D3112yes—
196C4—21no—
197C4—21no—
198C4—1242no—
199C4—1242no—
200C4#1175850611yescharacteristic
201C4#1175850712yes—
202C4#1175853011yescharacteristic
203C2—22no—
204C2—22no—
205C2—41no—
206C2#1452092811yescharacteristic
207C2C2×#1175841211yescharacteristic
208C2#1452103111yescharacteristic
subgroupG/H
centerC22C2×(C2×D31)⋊C4
commutatorC23×C31C22×C4
FrattiniC22×C4C22×D31
FittingC22×M16×C31C2
socleC22×C31C2×C22⋊C4
prime radicalC24×C31C2×C4
cosocleC22×C4×C31C23
hypercenterC22×M16D31
hypocenterC31#1901
LCSG ▷ C23×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#11758412C2semidirect
4. #14520817C2semidirect
89. C22×C31#101semidirect
127. C31#1901semidirect
193. C22#11758412semidirect

Maximal subgroups

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

subgroupindex
C2×#117585012
C2×#117584122
#145210132
#145208172
C22×M16×C312
#190131