Finite Group Databasebeta

← #27157177

Subgroups — #27157177

Order 12416.

218 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 56 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#20261754C211yesmaximal, characteristic, complemented
2#20261680C211yesmaximal, characteristic, complemented
3#20261589C212yesmaximal, complemented
4#20261586C212yesmaximal
5#20261264C211yesmaximal, characteristic
6C4⋊#11715897—21no—
7#15690867C2212yes—
8C4×#11715897—21no—
9(C2×#87604)⋊C4—21no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4C2212yes—
12#15690803—21no—
13#15690802C411yescharacteristic, complemented
14#15690792—21no—
15#15690786C2211yescharacteristic
16#15690781C2211yescharacteristic
17Q8×Dic97C411yescharacteristic, complemented
18D4×Dic97C411yescharacteristic, complemented
19#15690770—21no—
20#15690766—21no—
21C42×D97C2211yescharacteristic
22#124×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24C2×#11715897—22no—
25C2×C22⋊D97—21no—
26C22×Dic97—21no—
27C2×C4×D97D411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C2×C4×D97C2311yescharacteristic
30C4⋊Dic97—21no—
31C4⋊Dic97C2×C411yescharacteristic
32C4⋊Dic97C2×C411yescharacteristic
33#13095490—21no—
34#13095490—21no—
35#13095490—21no—
36C22⋊Dic97—21no—
37C22⋊Dic97—21no—
38#13095488—21no—
39C4×Dic97—21no—
40C4×Dic97C2×C411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C2×Q8×C97C2×C411yescharacteristic
44C2×D4×C97C2×C411yescharacteristic
45C22⋊C4×C97—21no—
46C22⋊C4×C97—21no—
47C42×C97C2×C411yescharacteristic
48C23×C97—21no—
49C22×D97C2×D411yescharacteristic
50C2×#87604—42no—
51C2×#87604—42no—
52C2×#87604—42no—
53Q8×C97—41no—
54D4×C97—41no—
55C2×C4×C97—21no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22×C411yescharacteristic
59C22⋊D97—42no—
60C2×Dic97—21no—
61C2×Dic97—21no—
62C2×Dic97—42no—
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C22⋊C411yescharacteristic
65C2×Dic97C2×D411yescharacteristic
66C4×D97—22no—
67C4×D97—41no—
68C4×D97—41no—
69#11715897—42no—
70C22×C97—41no—
71C22×C97—41no—
72C22×C97—41no—
73C22×C97C2×C22⋊C411yescharacteristic
74C2×D97—21no—
75C2×D97—21no—
76C2×D97#13512yes—
77#87604—82no—
78C4×C97—21no—
79C4×C97—21no—
80C4×C97—21no—
81C4×C97—41no—
82Dic97—21no—
83Dic97—21no—
84Dic97—21no—
85Dic97—41no—
86Dic97—41no—
87C2×C97—81no—
88C2×C97#41011yescharacteristic
89C2×C97#41611yescharacteristic
90C2×C97#42111yescharacteristic
91D97—22no—
92#1521—971nomaximal, Sylow
93C97#152111yescharacteristic, complemented, Sylow
94#529—971no—
95#431—971no—
96#383—971no—
97#328—972no—
98#327—972no—
99#126—1941no—
100#125—971no—
101#124#8760411yescharacteristic, complemented
102#123—1941no—
103C22⋊Q8—971no—
104#121—971no—
105C4×Q8—971no—
106C4×D4—971no—
107C4×D4—1941no—
108#117—1941no—
109C2×C42—971no—
110C4⋊C8—1941no—
111C22⋊C8—972no—
112C4×C8—1941no—
113#95—972no—
114#95—1941no—
115#95—1941no—
116C2×Q8C2×#8760411yescharacteristic
117C2×D4—1941no—
118C2×D4C2×#8760411yescharacteristic
119C22×C4—971no—
120C22×C4—971no—
121C22×C4—971no—
122C22×C4—1941no—
123C22×C4—1942no—
124C2×C8—1942no—
125C4⋊C4—971no—
126C4⋊C4—971no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—1941no—
134C22⋊C4—1941no—
135C22⋊C4—1941no—
136C42—971no—
137C42—971no—
138C42—971no—
139C42—1941no—
140C42C2×#8760411yescharacteristic
141C23—21no—
142C23—971no—
143Q8—41no—
144D4—41no—
145D4—3882no—
146C2×C4—21no—
147C2×C4—971no—
148C2×C4—971no—
149C2×C4—971no—
150C2×C4—3881no—
151C2×C4—3881no—
152C2×C4—3882no—
153C2×C4—3882no—
154C2×C4—3882no—
155C2×C4—3882no—
156C2×C4—1942no—
157C2×C4—1941no—
158C2×C4—1941no—
159C2×C4C22⋊#8760411yescharacteristic
160C2×C4C22⋊#8760411yescharacteristic
161C2×C4C22×#8760411yescharacteristic
162C8—3882no—
163C22—41no—
164C22—41no—
165C22—41no—
166C22—972no—
167C22—1941no—
168C22—1941no—
169C22C2×C22⋊#8760411yescharacteristic
170C4—21no—
171C4—21no—
172C4—21no—
173C4—41no—
174C4—7762no—
175C4—3881no—
176C4—3881no—
177C4—1941no—
178C4—1941no—
179C4—1941no—
180C2—81no—
181C2—1942no—
182C2#2026171511yescharacteristic
183C2#2026172511yescharacteristic
184C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC22×C97C2×C22⋊C4
prime radical#15690781C22
cosocleC2×C4×D97C23
hypercenter#124#87604
hypocenterC97#1521
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20261754C2semidirect
2. #20261680C2semidirect
3. #20261589C2semidirect
13. #15690802C4semidirect
17. Q8×Dic97C4semidirect
18. D4×Dic97C4semidirect
22. #124×C97C4semidirect
93. C97#1521semidirect
101. #124#87604semidirect

Maximal subgroups

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

subgroupindex
#202617542
#202616802
#202615892
#202615862
#202612642
#152197