Finite Group Databasebeta

← #27143720

Subgroups — #27143720

Order 12416.

354 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 45 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#20260921C212yesmaximal, complemented
2#20260902C211yesmaximal, characteristic, complemented
3#20260528C211yesmaximal, characteristic, complemented
4#20260522C212yesmaximal, complemented
5D42×C97C211yesmaximal, characteristic, complemented
6#15690855C2211yescharacteristic, complemented
7#15690836—21no—
8#15690779—22no—
9#15690738—21no—
10#15690727C2211yescharacteristic, complemented
11#15690725C2212yescomplemented
12#15690724—22no—
13C22×D4×C97—22no—
14#127×C97C2211yescharacteristic
15#121×C97—22no—
16#120×C97—21no—
17#120×C97C2212yes—
18C4×D4×C97—21no—
19#13095511—21no—
20#13095510—41no—
21C2×C22⋊D97—22no—
22C2×C22⋊D97—42no—
23D4×D97—41no—
24#13095499—41no—
25#13095496—41no—
26#13095495—21no—
27C22⋊Dic97—22no—
28C22⋊Dic97—42no—
29C24×C97—21no—
30C24×C97D412yescomplemented
31C2×D4×C97—21no—
32C2×D4×C97—22no—
33C2×D4×C97—22no—
34C2×D4×C97—22no—
35C2×D4×C97—48no—
36C2×D4×C97D411yescharacteristic, complemented
37C2×D4×C97C2311yescharacteristic
38C22×C4×C97—22no—
39C4⋊C4×C97—21no—
40C22⋊C4×C97—21no—
41C22⋊C4×C97—22no—
42C22⋊C4×C97D412yescomplemented
43C42×C97D411yescharacteristic, complemented
44C23×C97—22no—
45C23×C97—21no—
46C23×C97—44no—
47C23×C97—42no—
48C23×C97—44no—
49C23×C97—81no—
50C23×C97C2×D412yes—
51C22×D97—22no—
52D4×C97—24no—
53D4×C97—41no—
54D4×C97—41no—
55D4×C97—44no—
56D4×C97—44no—
57D4×C97—44no—
58C2×C4×C97—21no—
59C2×C4×C97—22no—
60C2×C4×C97—44no—
61C2×C4×C97C2×D411yescharacteristic
62C22⋊D97—42no—
63C22⋊D97—41no—
64C22⋊D97—84no—
65C2×Dic97—22no—
66C2×Dic97—42no—
67D388—21no—
68C4×D97—41no—
69Dic194—21no—
70#11715895—41no—
71C22×C97—24no—
72C22×C97—22no—
73C22×C97—42no—
74C22×C97—42no—
75C22×C97—42no—
76C22×C97—44no—
77C22×C97—42no—
78C22×C97—41no—
79C22×C97—82no—
80C22×C97—84no—
81C22×C97#12011yescharacteristic
82C2×D97—41no—
83C2×D97—82no—
84C4×C97—22no—
85C4×C97—21no—
86C4×C97—42no—
87Dic97—41no—
88Dic97—82no—
89C2×C97—21no—
90C2×C97—42no—
91C2×C97—44no—
92C2×C97—81no—
93C2×C97B(4,𝔽2)11yescharacteristic
94D97—81no—
95#2196—971nomaximal, Sylow
96C97#219611yescharacteristic, complemented, Sylow
97D42D9711yescharacteristic, complemented
98B(4,𝔽2)—972no—
99#453—971no—
100#353—971no—
101AΣL(1,𝔽16)—972no—
102H(5,𝔽2)—971no—
103C22×D4—22no—
104AGL(1,ℤ/8)—1941no—
105#127C2×D9711yescharacteristic
106#121—22no—
107#120—21no—
108#120—1942no—
109#120C2×D9712yes—
110C4×D4—21no—
111#104—1941no—
112#100—971no—
113#99—972no—
114#99—1942no—
115C24—21no—
116C24C22⋊D9712yescomplemented
117#41—3881no—
118#41—1941no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—48no—
124C2×D4—3881no—
125C2×D4—3882no—
126C2×D4—1942no—
127C2×D4C22⋊D9711yescharacteristic, complemented
128C2×D4C22×D9711yescharacteristic
129C22×C4—22no—
130SD16—3881no—
131D8—3881no—
132M16—1941no—
133C4⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—22no—
136C22⋊C4—3882no—
137C22⋊C4—1942no—
138C22⋊C4C22⋊D9712yescomplemented
139C42C22⋊D9711yescharacteristic, complemented
140C23—22no—
141C23—21no—
142C23—42no—
143C23—44no—
144C23—44no—
145C23—81no—
146C23—1942no—
147C23C2×C22⋊D9712yes—
148Q8—1941no—
149D4—24no—
150D4—41no—
151D4—41no—
152D4—44no—
153D4—44no—
154D4—44no—
155D4—7764no—
156D4—3882no—
157D4—3881no—
158D4—1941no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—44no—
162C2×C4—3881no—
163C2×C4—3882no—
164C2×C4—1942no—
165C2×C4C2×C22⋊D9711yescharacteristic
166C8—3881no—
167C22—24no—
168C22—22no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—44no—
173C22—42no—
174C22—41no—
175C22—82no—
176C22—84no—
177C22—7762no—
178C22—3881no—
179C22#1569077911yescharacteristic
180C4—22no—
181C4—21no—
182C4—42no—
183C4—7762no—
184C4—3881no—
185C2—21no—
186C2—42no—
187C2—44no—
188C2—81no—
189C2—7761no—
190C2#2026092211yescharacteristic
subgroupG/H
centerC2#20260922
commutator, cosocleC2×D4×C97C23
FrattiniC2×D4C22×D97
FittingD42×C97C2
socleC2×C97B(4,𝔽2)
hypercenterD42D97
hypocenterC97#2196
LCSG ▷ C2×D4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #20260921C2semidirect
2. #20260902C2semidirect
3. #20260528C2semidirect
4. #20260522C2semidirect
5. D42×C97C2semidirect
6. #15690855C22semidirect
10. #15690727C22semidirect
11. #15690725C22semidirect
30. C24×C97D4semidirect
36. C2×D4×C97D4semidirect
42. C22⋊C4×C97D4semidirect
43. C42×C97D4semidirect
96. C97#2196semidirect
97. D42D97semidirect
116. C24C22⋊D97semidirect
127. C2×D4C22⋊D97semidirect
138. C22⋊C4C22⋊D97semidirect
139. C42C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202609212
#202609022
#202605282
#202605222
D42×C972
#219697