Finite Group Databasebeta

← #16844983

Subgroups — #16844983

Order 3968.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#14521772C211yesmaximal, characteristic
2#14521447C211yesmaximal, characteristic, complemented
3#14521330C211yesmaximal, characteristic
4#14521259C211yesmaximal, characteristic, complemented
5#14520796C211yesmaximal, characteristic, complemented
6#14520778C211yesmaximal, characteristic
7C31×Q8⋊C8C211yesmaximal, characteristic, complemented
8C4×Q8×C31C2211yescharacteristic
9#11758523—21no—
10C2×#92482—21no—
11#11758475—22no—
12#11758469—21no—
13#11758452—21no—
14C4×D124C2211yescharacteristic
15#11758446C2211yescharacteristic
16#11758446C2211yescharacteristic
17C4×Dic62—21no—
18C31×C4⋊C8C2211yescharacteristic
19C4×C8×C31C2211yescharacteristic
20#11758409—21no—
21#11758406—21no—
22#11758405—22no—
23#11758398—21no—
24C4⋊(C31⋊C8)C2211yescharacteristic
25C2×Q8×C31D411yescharacteristic
26C2×D124D411yescharacteristic
27C2×C4×D31—21no—
28C2×Dic62—21no—
29C2×Dic62—21no—
30C2×C8×C31—21no—
31C2×C8×C31—21no—
32C4⋊C4×C31—21no—
33C4⋊C4×C31D411yescharacteristic
34C42×C31—21no—
35C42×C31C2311yescharacteristic
36#92482—42no—
37(C2×D31)⋊C4—22no—
38(C2×D31)⋊C4—22no—
39(C2×D31)⋊C4—21no—
40C4⋊Dic31—21no—
41C4⋊Dic31—21no—
42C4⋊Dic31—21no—
43C4⋊Dic31D412yes—
44C4⋊Dic31D411yescharacteristic
45Dic31⋊C4—22no—
46C4×Dic31—22no—
47C2×C31⋊C8—21no—
48C22×D31—21no—
49Q8×C31—21no—
50Q8×C31—21no—
51C2×C4×C31—21no—
52C2×C4×C31—21no—
53C2×C4×C31C2×D411yescharacteristic
54C2×C4×C31C2×D411yescharacteristic
55C2×C4×C31C2×D411yescharacteristic
56C2×Dic31—21no—
57C2×Dic31—22no—
58C2×Dic31—41no—
59D124—21no—
60D124—21no—
61C4×D31—41no—
62C4×D31—41no—
63Dic62—41no—
64Dic62—41no—
65Dic62—41no—
66Dic62—41no—
67C8×C31—22no—
68C8×C31—41no—
69C31⋊C8—41no—
70#1652—311nomaximal, Sylow
71C22×C31#12011yescharacteristic
72C2×D31—41no—
73C2×D31—41no—
74C2×D31—41no—
75C4×C31—21no—
76C4×C31—41no—
77C4×C31—41no—
78C4×C31#13511yescharacteristic
79C4×C31#13511yescharacteristic
80C4×C31AGL(1,ℤ/8)11yescharacteristic
81C4×C31#13711yescharacteristic
82Dic31—41no—
83Dic31—42no—
84Dic31—81no—
85#556—311no—
86#498—311no—
87#474—311no—
88#462—311no—
89#332—311no—
90Q8⋊C8D3111yescharacteristic, complemented
91D4⋊C8—311no—
92C2×C31#44911yescharacteristic
93C2×C31#45211yescharacteristic
94C2×C31#45611yescharacteristic
95D31—81no—
96C2×Q16—621no—
97C4⋊Q8—311no—
98C4⋊Q8—311no—
99#126—622no—
100#124—621no—
101C22⋊Q8—621no—
102C4×Q8—621no—
103C4×Q8C2×D3111yescharacteristic
104C4×D4—311no—
105#117—621no—
106#107—621no—
107#106—622no—
108C4⋊C8—311no—
109C4⋊C8C2×D3111yescharacteristic
110#103—621no—
111#102—621no—
112C4×C8C2×D3111yescharacteristic
113C31#165211yescharacteristic, complemented, Sylow
114C2×Q8—621no—
115C2×Q8—621no—
116C2×Q8D12411yescharacteristic
117C2×D4—311no—
118C22×C4—621no—
119Q16—1242no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—621no—
123C4⋊C4—21no—
124C4⋊C4—312no—
125C4⋊C4—311no—
126C4⋊C4—622no—
127C4⋊C4—621no—
128C4⋊C4—621no—
129C4⋊C4—621no—
130C4⋊C4D12411yescharacteristic
131C22⋊C4—622no—
132C22⋊C4—622no—
133C22⋊C4—621no—
134C42—21no—
135C42—622no—
136C42C22×D3111yescharacteristic
137C23—621no—
138Q8—21no—
139Q8—21no—
140Q8—1241no—
141Q8—1241no—
142Q8—1241no—
143Q8—1241no—
144D4—621no—
145D4—621no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—621no—
149C2×C4—622no—
150C2×C4—1241no—
151C2×C4—1241no—
152C2×C4—1241no—
153C2×C4C2×D12411yescharacteristic
154C2×C4D4×D3111yescharacteristic
155C2×C4D4×D3111yescharacteristic
156C8—22no—
157C8—41no—
158C8—1241no—
159C22—1241no—
160C22—1241no—
161C22—1241no—
162C22#1175845911yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—1241no—
167C4—1242no—
168C4—2481no—
169C4#1175848111yescharacteristic
170C4#1175848511yescharacteristic
171C4#1175849311yescharacteristic
172C4#1175849511yescharacteristic
173C2—2481no—
174C2#1452119611yescharacteristic
175C2#1452121411yescharacteristic
176C2#1452123311yescharacteristic
subgroupG/H
centerC22#11758459
commutator, cosocleC42×C31C23
FrattiniC42C22×D31
FittingC31×Q8⋊C8C2
socleC22×C31#120
prime radicalC2×D124D4
hypercenterQ8⋊C8D31
hypocenterC31#1652
LCSG ▷ C42×C31 ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #14521447C2semidirect
4. #14521259C2semidirect
5. #14520796C2semidirect
7. C31×Q8⋊C8C2semidirect
90. Q8⋊C8D31semidirect
113. C31#1652semidirect

Maximal subgroups

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

subgroupindex
#145217722
#145214472
#145213302
#145212592
#145207962
#145207782
C31×Q8⋊C82
#165231