Finite Group Databasebeta

← #16856253

Subgroups — #16856253

Order 3968.

380 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 157 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#14521606C211yesmaximal, characteristic
2#14521455C214yesmaximal, complemented
3#14521452C212yesmaximal, complemented
4#14521445C212yesmaximal, complemented
5#14521089C214yesmaximal
6C4×#92474C211yesmaximal, characteristic
7C2×C31×C4⋊Q8C211yesmaximal, characteristic
8C22×Q8×C31—21no—
9C31×C4⋊Q8—21no—
10C31×C4⋊Q8—21no—
11C31×C4⋊Q8C2212yes—
12C31×C4⋊Q8C2212yes—
13C2×C4⋊C4×C31—21no—
14C2×C4⋊C4×C31C2212yes—
15C2×C42×C31C2211yescharacteristic
16#11758503C2212yes—
17#11758501—21no—
18#11758501—21no—
19C2×#92474C2212yes—
20#11758447C2212yes—
21#11758446C2211yescharacteristic
22#11758446C2211yescharacteristic
23C4×Dic62—21no—
24C4×Dic62—21no—
25#11758396C2218yes—
26#11758395C2218yes—
27#11758391C2212yes—
28C4×C31⋊C8C2212yes—
29C2×Q8×C31—22no—
30C2×Q8×C31—22no—
31C2×Q8×C31—44no—
32C22×C4×C31—22no—
33C22×C4×C31D411yescharacteristic
34C22×C4×C31D411yescharacteristic
35C22×C4×C31C2311yescharacteristic
36C2×Dic62—21no—
37C2×Dic62—21no—
38C4⋊C4×C31—22no—
39C4⋊C4×C31—22no—
40C4⋊C4×C31—22no—
41C4⋊C4×C31C2314yes—
42C42×C31D411yescharacteristic
43C42×C31D411yescharacteristic
44C42×C31C2311yescharacteristic
45C42×C31C2311yescharacteristic
46C22⋊Dic31—22no—
47C4⋊Dic31—21no—
48C4⋊Dic31—21no—
49C4⋊Dic31C2312yes—
50C4⋊Dic31C2312yes—
51Dic31⋊C4—22no—
52Dic31⋊C4—22no—
53C4×Dic31—22no—
54#92474Q818yescomplemented
55C2×C31⋊C8C2314yes—
56C23×C31C2×D411yescharacteristic
57Q8×C31—44no—
58Q8×C31—44no—
59C2×C4×C31—22no—
60C2×C4×C31—22no—
61C2×C4×C31—24no—
62C2×C4×C31—44no—
63C2×C4×C31C2×D411yescharacteristic
64C2×C4×C31C2×D411yescharacteristic
65C2×C4×C31C2×D411yescharacteristic
66C2×C4×C31C2×D411yescharacteristic
67C2×C4×C31C2×D411yescharacteristic
68C2×C4×C31C2×Q814yes—
69C2×C4×C31C2411yescharacteristic
70C2×Dic31—22no—
71C2×Dic31—22no—
72Dic62—42no—
73Dic62—42no—
74C31⋊C8C2×Q818yescomplemented
75#3165—311nomaximal, Sylow
76C22×C31—21no—
77C22×C31—21no—
78C22×C31C4⋊Q812yes—
79C22×C31C22×D411yescharacteristic
80C4×C31—21no—
81C4×C31—21no—
82C4×C31—44no—
83C4×C31C4⋊Q812yes—
84C4×C31AGL(1,ℤ/8)12yes—
85C4×C31#13712yes—
86C4×C31C22×Q812yes—
87Dic31—42no—
88Dic31—42no—
89#533—311no—
90C2×C4⋊Q8D3111yescharacteristic
91#501—314no—
92#500—312no—
93#498—312no—
94#428—314no—
95C4×M16—311no—
96C2×C31—22no—
97C2×C31C2×C4⋊Q811yescharacteristic
98C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
99C2×C31C2×#13711yescharacteristic
100C22×Q8—21no—
101C2×M16—312no—
102C4⋊Q8—21no—
103C4⋊Q8—21no—
104C4⋊Q8—311no—
105C4⋊Q8—311no—
106C4⋊Q8C2×D3112yes—
107C4⋊Q8C2×D3112yes—
108#125—312no—
109C22⋊Q8—621no—
110C22⋊Q8—621no—
111C4×Q8—621no—
112C4×Q8—621no—
113#117—312no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4C2×D3112yes—
116C2×C42C2×D3111yescharacteristic
117#107—318no—
118#106—318no—
119#97—312no—
120C4×C8—312no—
121C31#316511yescharacteristic, complemented, Sylow
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—44no—
125C2×Q8—621no—
126C2×Q8—621no—
127C22×C4—22no—
128C22×C4C22⋊D3111yescharacteristic
129C22×C4C22⋊D3111yescharacteristic
130C22×C4C22×D3111yescharacteristic
131M16—318no—
132C2×C8—314no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—312no—
137C4⋊C4—312no—
138C4⋊C4—622no—
139C4⋊C4—622no—
140C4⋊C4—621no—
141C4⋊C4—621no—
142C4⋊C4C22×D3114yes—
143C22⋊C4—622no—
144C42—622no—
145C42C22⋊D3111yescharacteristic
146C42C22⋊D3111yescharacteristic
147C42C22×D3111yescharacteristic
148C42C22×D3111yescharacteristic
149C23C2×C22⋊D3111yescharacteristic
150Q8—44no—
151Q8—44no—
152Q8—1242no—
153Q8—1242no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—44no—
158C2×C4—622no—
159C2×C4—622no—
160C2×C4Q8×D3114yes—
161C2×C4C2×C22⋊D3111yescharacteristic
162C2×C4C2×C22⋊D3111yescharacteristic
163C2×C4C2×C22⋊D3111yescharacteristic
164C2×C4C2×C22⋊D3111yescharacteristic
165C2×C4C2×C22⋊D3111yescharacteristic
166C2×C4C23×D3111yescharacteristic
167C8—318no—
168C22—21no—
169C22—21no—
170C22#1175852112yes—
171C22C22×C22⋊D3111yescharacteristic
172C4—21no—
173C4—21no—
174C4—44no—
175C4—1242no—
176C4—1242no—
177C4#1175850912yes—
178C4#1175851912yes—
179C4#1175852112yes—
180C4C2×Q8×D3112yes—
181C2—22no—
182C2C2×#1175852111yescharacteristic
183C2C2×#1175850911yescharacteristic
184C2C2×#1175851911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×C4⋊Q8C2
socleC22×C31C22×D4
prime radicalC23×C31C2×D4
hypercenterC2×C4⋊Q8D31
hypocenterC31#3165
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #14521455C2semidirect
3. #14521452C2semidirect
4. #14521445C2semidirect
54. #92474Q8semidirect
74. C31⋊C8C2×Q8semidirect
121. C31#3165semidirect

Maximal subgroups

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

subgroupindex
#145216062
#145214552
#145214522
#145214452
#145210892
C4×#924742
C2×C31×C4⋊Q82
#316531