Finite Group Databasebeta

← #27143737

Subgroups — #27143737

Order 12416.

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

normal subgroups only

This group has 270 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#20260912C211yesmaximal, characteristic, complemented
3#20260530C211yesmaximal, characteristic, complemented
4#20260522C212yesmaximal, complemented
5H(3,𝔽4)×C97C211yesmaximal, characteristic, complemented
6#15690855C2211yescharacteristic
7#15690841—21no—
8#15690779—22no—
9#15690738—21no—
10#15690727C2211yescharacteristic
11#15690725C2212yes—
12#15690724—22no—
13#124×C97—21no—
14#124×C97—22no—
15#124×C97—21no—
16#124×C97C2211yescharacteristic
17#120×C97—22no—
18#120×C97C2212yes—
19#13095511—21no—
20#13095510—41no—
21C2×C22⋊D97—22no—
22C2×C22⋊D97—42no—
23D4×D97—41no—
24#13095501—41no—
25#13095498—41no—
26#13095495—21no—
27C22⋊Dic97—22no—
28C22⋊Dic97—42no—
29C24×C97D412yescomplemented
30C2×Q8×C97—21no—
31C2×Q8×C97D411yescharacteristic, complemented
32C2×D4×C97—21no—
33C2×D4×C97—21no—
34C2×D4×C97—22no—
35C2×D4×C97C2311yescharacteristic
36C22⋊C4×C97—22no—
37C22⋊C4×C97—22no—
38C22⋊C4×C97—22no—
39C22⋊C4×C97—22no—
40C22⋊C4×C97D412yescomplemented
41C42×C97—21no—
42C42×C97D411yescharacteristic, complemented
43C23×C97—22no—
44C23×C97—42no—
45C23×C97—82no—
46C23×C97C2×D412yes—
47C22×D97—22no—
48Q8×C97—41no—
49Q8×C97—81no—
50D4×C97—41no—
51D4×C97—82no—
52D4×C97—81no—
53D4×C97—81no—
54C2×C4×C97—21no—
55C2×C4×C97—22no—
56C2×C4×C97—21no—
57C2×C4×C97C2×D411yescharacteristic
58C22⋊D97—42no—
59C22⋊D97—41no—
60C22⋊D97—84no—
61C2×Dic97—22no—
62C2×Dic97—42no—
63D388—21no—
64C4×D97—41no—
65Dic194—21no—
66#11715895—41no—
67C22×C97—22no—
68C22×C97—42no—
69C22×C97—42no—
70C22×C97—42no—
71C22×C97—44no—
72C22×C97—42no—
73C22×C97—82no—
74C22×C97#12011yescharacteristic
75C2×D97—41no—
76C2×D97—82no—
77C4×C97—21no—
78C4×C97—42no—
79C4×C97—41no—
80C4×C97—41no—
81Dic97—41no—
82Dic97—82no—
83C2×C97—21no—
84C2×C97—42no—
85C2×C97—82no—
86C2×C97B(4,𝔽2)11yescharacteristic
87D97—81no—
88#2199—971nomaximal, Sylow
89C97#219911yescharacteristic, complemented, Sylow
90H(3,𝔽4)D9711yescharacteristic, complemented
91B(4,𝔽2)—972no—
92#455—971no—
93#354—971no—
94AΣL(1,𝔽16)—972no—
95H(5,𝔽2)—971no—
96#137—1941no—
97#124—21no—
98#124—22no—
99#124—21no—
100#124C2×D9711yescharacteristic
101#120—22no—
102#120—1942no—
103#120C2×D9712yes—
104#104—1941no—
105#100—971no—
106#99—972no—
107#99—1942no—
108C24C22⋊D9712yescomplemented
109#41—3881no—
110#41—1941no—
111C2×Q8—21no—
112C2×Q8C22⋊D9711yescharacteristic, complemented
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—3881no—
117C2×D4—3882no—
118C2×D4—1942no—
119C2×D4C22×D9711yescharacteristic
120Q16—3881no—
121SD16—3881no—
122M16—1941no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—3882no—
128C22⋊C4—1942no—
129C22⋊C4C22⋊D9712yescomplemented
130C42—21no—
131C42C22⋊D9711yescharacteristic, complemented
132C23—22no—
133C23—42no—
134C23—82no—
135C23—1942no—
136C23C2×C22⋊D9712yes—
137Q8—41no—
138Q8—81no—
139Q8—1941no—
140D4—41no—
141D4—82no—
142D4—81no—
143D4—81no—
144D4—7764no—
145D4—3882no—
146D4—3881no—
147D4—1941no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—3881no—
152C2×C4—3882no—
153C2×C4—1942no—
154C2×C4C2×C22⋊D9711yescharacteristic
155C8—3881no—
156C22—22no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22—44no—
161C22—42no—
162C22—82no—
163C22—7762no—
164C22—3881no—
165C22#1569077911yescharacteristic
166C4—21no—
167C4—42no—
168C4—41no—
169C4—41no—
170C4—7762no—
171C4—3881no—
172C2—21no—
173C2—42no—
174C2—82no—
175C2—7761no—
176C2#2026092211yescharacteristic
subgroupG/H
centerC2#20260922
commutator, cosocleC2×D4×C97C23
FrattiniC2×D4C22×D97
FittingH(3,𝔽4)×C97C2
socleC2×C97B(4,𝔽2)
hypercenterH(3,𝔽4)D97
hypocenterC97#2199
LCSG ▷ C2×D4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #20260921C2semidirect
2. #20260912C2semidirect
3. #20260530C2semidirect
4. #20260522C2semidirect
5. H(3,𝔽4)×C97C2semidirect
29. C24×C97D4semidirect
31. C2×Q8×C97D4semidirect
40. C22⋊C4×C97D4semidirect
42. C42×C97D4semidirect
89. C97#2199semidirect
90. H(3,𝔽4)D97semidirect
108. C24C22⋊D97semidirect
112. C2×Q8C22⋊D97semidirect
129. C22⋊C4C22⋊D97semidirect
131. C42C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202609212
#202609122
#202605302
#202605222
H(3,𝔽4)×C972
#219997