Finite Group Databasebeta

← D31×#491 #16856048

Subgroups — D31×#491 #16856048

Order 3968.

384 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C4⋊Q8×D31C211yesmaximal, characteristic
2#125×D31C211yesmaximal, characteristic
3#14521419C211yesmaximal, characteristic, complemented
4#14521418C211yesmaximal, characteristic, complemented
5#14521417C211yesmaximal, characteristic, complemented
6#14521416C214yesmaximal, complemented
7#103×D31C214yesmaximal
8#97×D31C211yesmaximal, characteristic
9C31×#491C211yesmaximal, characteristic, complemented
10C2×Q8×D31C2212yes—
11C31×C4⋊Q8C2211yescharacteristic
12C31×#125C2211yescharacteristic
13#11758521C2212yes—
14C2×C8×D31C2212yes—
15C4⋊C4×D31—22no—
16C4⋊C4×D31—22no—
17C4⋊C4×D31C2212yes—
18#11758467C2212yes—
19#11758466—22no—
20#11758465—22no—
21C42×D31C2211yescharacteristic
22#11758447C2211yescharacteristic
23#11758446C2211yescharacteristic
24C31×#103C2214yes—
25C31×#97C2211yescharacteristic
26#11758423C2214yes—
27#11758404C2214yes—
28#11758403C2212yes—
29#11758398C2214yes—
30#11758391C2211yescharacteristic
31C2×Q8×C31C2312yes—
32Q8×D31—28no—
33C2×C4×D31—22no—
34C2×C4×D31—22no—
35C2×C4×D31D412yes—
36C2×C4×D31C2311yescharacteristic
37C2×Dic62—22no—
38C2×Dic62C2312yes—
39C2×C8×C31C2312yes—
40C4⋊C4×C31—22no—
41C4⋊C4×C31—22no—
42C4⋊C4×C31C2312yes—
43C42×C31C2311yescharacteristic
44C4⋊Dic31—22no—
45C4⋊Dic31—22no—
46C4⋊Dic31C2312yes—
47Dic31⋊C4—22no—
48Dic31⋊C4—24no—
49Dic31⋊C4—22no—
50Dic31⋊C4—22no—
51C4×Dic31D412yes—
52C4×Dic31C2311yescharacteristic
53C2×C31⋊C8C2312yes—
54C8×D31—24no—
55C22×D31C2×D411yescharacteristic
56Q8×C31—24no—
57C2×C4×C31—22no—
58C2×C4×C31—22no—
59C2×C4×C31C2×D412yes—
60C2×C4×C31C2411yescharacteristic
61C2×Dic31—22no—
62C2×Dic31—22no—
63C2×Dic31C2×D412yes—
64C2×Dic31C2×D411yescharacteristic
65C4×D31—24no—
66C4×D31—44no—
67C4×D31—44no—
68C4×D31#4114yes—
69Dic62—24no—
70Dic62—44no—
71C8×C31—22no—
72C31⋊C8—22no—
73C2×#491—311nomaximal, Sylow
74C22×C31C22×D411yescharacteristic
75C2×D31#12412yes—
76C2×D31#13714yes—
77C4×C31—22no—
78C4×C31—42no—
79C4×C31—42no—
80C4×C31C2×#4112yes—
81Dic31—22no—
82Dic31—42no—
83Dic31—42no—
84Dic31#12412yes—
85C2×C4⋊Q8—311no—
86C2×#125—311no—
87#491—311no—
88#491—311no—
89#491—314no—
90#491—311no—
91#491D3111yescharacteristic, direct product factor
92C2×#103—314no—
93C2×#97—311no—
94C2×C31C2×#12411yescharacteristic
95C2×C31C2×#13712yes—
96D31#49114yesdirect product factor
97C22×Q8—312no—
98C22×C8—312no—
99C4⋊Q8—312no—
100C4⋊Q8—311no—
101C4⋊Q8—622no—
102C4⋊Q8C2×D3111yescharacteristic
103#125—312no—
104#125—311no—
105#125—622no—
106#125C2×D3111yescharacteristic
107C2×C4⋊C4—312no—
108C2×C4⋊C4—622no—
109C2×C4⋊C4—622no—
110C2×C42—311no—
111#103—314no—
112#103—314no—
113#103—314no—
114#103C2×D3114yes—
115#97—312no—
116#97—311no—
117#97C2×D3111yescharacteristic
118C31C2×#49111yescharacteristic, complemented, Sylow
119C2×Q8—312no—
120C2×Q8—622no—
121C2×Q8—628no—
122C2×Q8C22×D3112yes—
123C22×C4—311no—
124C22×C4—312no—
125C22×C4—622no—
126C22×C4—622no—
127C2×C8—312no—
128C2×C8—624no—
129C2×C8C22×D3112yes—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—312no—
133C4⋊C4—622no—
134C4⋊C4—622no—
135C4⋊C4—624no—
136C4⋊C4—622no—
137C4⋊C4—622no—
138C4⋊C4—622no—
139C4⋊C4C22×D3112yes—
140C42—312no—
141C42—311no—
142C42C22×D3111yescharacteristic
143C23—311no—
144Q8—24no—
145Q8—624no—
146Q8—1244no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—311no—
150C2×C4—314no—
151C2×C4—312no—
152C2×C4—624no—
153C2×C4—622no—
154C2×C4—622no—
155C2×C4—1244no—
156C2×C4—1244no—
157C2×C4D4×D3112yes—
158C2×C4C23×D3111yescharacteristic
159C8—22no—
160C8—622no—
161C22—312no—
162C22—314no—
163C22C2×D4×D3111yescharacteristic
164C4—22no—
165C4—42no—
166C4—42no—
167C4—312no—
168C4—622no—
169C4—1242no—
170C4—1242no—
171C4#41×D3112yes—
172C2—314no—
173C2#124×D3111yescharacteristic
174C2#137×D3112yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#491C2
socleC22×C31C22×D4
prime radicalC22×D31C2×D4
hypercenter#491D31
hypocenterC31C2×#491
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
3. #14521419C2semidirect
4. #14521418C2semidirect
5. #14521417C2semidirect
6. #14521416C2semidirect
9. C31×#491C2semidirect
91. #491D31direct (N × H)
96. D31#491direct (N × H)
118. C31C2×#491semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D312
#125×D312
#145214192
#145214182
#145214172
#145214162
#103×D312
#97×D312
C31×#4912
C2×#49131