Finite Group Databasebeta

← #16844249

Subgroups — #16844249

Order 3968.

222 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 77 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#14521106C211yesmaximal, characteristic, complemented
2#14520915C211yesmaximal, characteristic, complemented
3#14520786C212yesmaximal, complemented
4#14520783C212yesmaximal
5C31×#529C211yesmaximal, characteristic
6C31×#126—21no—
7C31×#125C411yescharacteristic, complemented
8C31×#124C411yescharacteristic, complemented
9C31×#123—21no—
10C31×C22⋊Q8C2211yescharacteristic
11C31×#121C2211yescharacteristic
12C4×Q8×C31C411yescharacteristic, complemented
13C4×D4×C31—21no—
14C4×D4×C31C411yescharacteristic, complemented
15C31×#117—21no—
16C2×C42×C31C2211yescharacteristic
17C2×C4⋊Dic31—21no—
18C31⋊#95—21no—
19C31⋊#95C2212yes—
20C22⋊(C31⋊C8)C2212yes—
21C4⋊(C31⋊C8)—21no—
22#11758391—21no—
23C2×Q8×C31C2×C411yescharacteristic
24C2×D4×C31—21no—
25C2×D4×C31C2×C411yescharacteristic
26C22×C4×C31—21no—
27C22×C4×C31D411yescharacteristic
28C22×C4×C31D411yescharacteristic
29C22×C4×C31C2311yescharacteristic
30C22×Dic31—22no—
31C4⋊C4×C31—21no—
32C4⋊C4×C31—21no—
33C4⋊C4×C31—21no—
34C4⋊C4×C31—21no—
35C4⋊C4×C31C2×C411yescharacteristic
36C4⋊C4×C31C2×C411yescharacteristic
37C22⋊C4×C31—21no—
38C22⋊C4×C31—21no—
39C22⋊C4×C31—21no—
40C22⋊C4×C31—21no—
41C22⋊C4×C31—21no—
42C42×C31—21no—
43C42×C31C2×C411yescharacteristic
44C42×C31C2×C411yescharacteristic
45C42×C31D411yescharacteristic
46C42×C31D411yescharacteristic
47C4⋊Dic31—42no—
48C2×C31⋊C8—22no—
49C23×C31—21no—
50C23×C31C2×D411yescharacteristic
51Q8×C31—41no—
52D4×C31—41no—
53D4×C31—41no—
54D4×C31—41no—
55C2×C4×C31—21no—
56C2×C4×C31—21no—
57C2×C4×C31—21no—
58C2×C4×C31—21no—
59C2×C4×C31—21no—
60C2×C4×C31—41no—
61C2×C4×C31—41no—
62C2×C4×C31—42no—
63C2×C4×C31C22⋊C411yescharacteristic
64C2×C4×C31C22⋊C411yescharacteristic
65C2×C4×C31C22⋊C411yescharacteristic
66C2×C4×C31C22⋊C411yescharacteristic
67C2×C4×C31C22×C411yescharacteristic
68C2×C4×C31C2×D411yescharacteristic
69C2×Dic31—42no—
70C2×Dic31—42no—
71C2×Dic31—42no—
72C31⋊C8—42no—
73#1512—311nomaximal, Sylow
74C22×C31—21no—
75C22×C31—21no—
76C22×C31—41no—
77C22×C31—41no—
78C22×C31—41no—
79C22×C31C2×C22⋊C411yescharacteristic
80C22×C31AGL(1,ℤ/8)11yescharacteristic
81C22×C31#13711yescharacteristic
82C4×C31—21no—
83C4×C31—21no—
84C4×C31—21no—
85C4×C31—21no—
86C4×C31—21no—
87C4×C31—21no—
88C4×C31—41no—
89C4×C31—41no—
90C4×C31—41no—
91Dic31—82no—
92#529D3111yescharacteristic
93#432—311no—
94#382—311no—
95#328—312no—
96#327—312no—
97C2×C31—21no—
98C2×C31—21no—
99C2×C31—81no—
100C2×C31#41011yescharacteristic
101C2×C31#41711yescharacteristic
102C2×C31#42111yescharacteristic
103#126—21no—
104#125Dic3111yescharacteristic, complemented
105#124Dic3111yescharacteristic, complemented
106#123—21no—
107C22⋊Q8C2×D3111yescharacteristic
108#121C2×D3111yescharacteristic
109C4×Q8Dic3111yescharacteristic, complemented
110C4×D4—21no—
111C4×D4Dic3111yescharacteristic, complemented
112#117—21no—
113C2×C4⋊C4—621no—
114C2×C42C2×D3111yescharacteristic
115C4⋊C8—621no—
116C22⋊C8—312no—
117#97—621no—
118#95—312no—
119#95—621no—
120C31#151211yescharacteristic, complemented, Sylow
121C2×Q8C2×Dic3111yescharacteristic
122C2×D4—21no—
123C2×D4C2×Dic3111yescharacteristic
124C22×C4—21no—
125C22×C4—622no—
126C22×C4C22⋊D3111yescharacteristic
127C22×C4C22⋊D3111yescharacteristic
128C22×C4C22×D3111yescharacteristic
129C2×C8—622no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—1242no—
135C4⋊C4C2×Dic3111yescharacteristic
136C4⋊C4C2×Dic3111yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C2×Dic3111yescharacteristic
144C42C2×Dic3111yescharacteristic
145C42C22⋊D3111yescharacteristic
146C42C22⋊D3111yescharacteristic
147C23—21no—
148C23C2×C22⋊D3111yescharacteristic
149Q8—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—1242no—
162C2×C4—1242no—
163C2×C4—1242no—
164C2×C4C22⋊Dic3111yescharacteristic
165C2×C4C22⋊Dic3111yescharacteristic
166C2×C4C22⋊Dic3111yescharacteristic
167C2×C4C22⋊Dic3111yescharacteristic
168C2×C4C22×Dic3111yescharacteristic
169C2×C4C2×C22⋊D3111yescharacteristic
170C8—1242no—
171C22—21no—
172C22—21no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22#1175852611yescharacteristic
177C22#1175852811yescharacteristic
178C22C2×C22⋊Dic3111yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—2482no—
189C2—21no—
190C2—21no—
191C2—81no—
192C2#1452102311yescharacteristic
193C2#1452105011yescharacteristic
194C2#1452106611yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic31
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC31×#529C2
socleC22×C31C2×C22⋊C4
prime radicalC31×#121C22
cosocleC22×C4×C31C23
hypercenter#529D31
hypocenterC31#1512
LCSG ▷ C2×C4×C31 ▷ C22×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #14521106C2semidirect
2. #14520915C2semidirect
3. #14520786C2semidirect
7. C31×#125C4semidirect
8. C31×#124C4semidirect
12. C4×Q8×C31C4semidirect
14. C4×D4×C31C4semidirect
104. #125Dic31semidirect
105. #124Dic31semidirect
109. C4×Q8Dic31semidirect
111. C4×D4Dic31semidirect
120. C31#1512semidirect

Maximal subgroups

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

subgroupindex
#145211062
#145209152
#145207862
#145207832
C31×#5292
#151231