Finite Group Databasebeta

← #27157015

Subgroups — #27157015

Order 12416.

270 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 64 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#20261705C211yesmaximal, characteristic
2C2×C22⋊#11715897C211yesmaximal, characteristic
3#20261581C212yesmaximal, complemented
4#20261580C212yesmaximal, complemented
5C2×#15690787C211yesmaximal, characteristic
6C2×#13095520—21no—
7C22×#11715897—21no—
8C22⋊#11715897—21no—
9C22⋊#11715897—21no—
10C22⋊#11715897C2212yes—
11C22⋊#11715897C2212yes—
12C22×Dic194C411yescharacteristic
13C23×Dic97C2211yescharacteristic
14#15690787—21no—
15#15690787—21no—
16#15690787—21no—
17#15690787—21no—
18C2×C4⋊Dic97C2211yescharacteristic
19C2×#13095490C411yescharacteristic
20C2×#13095490C2211yescharacteristic
21C2×C22⋊Dic97C411yescharacteristic
22C2×C22⋊C4×C97C411yescharacteristic
23#13095520—42no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×#11715897—42no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194—42no—
30C22×Dic97—42no—
31C22×Dic97C2×C411yescharacteristic
32C22×Dic97C2×C411yescharacteristic
33C22×Dic97C2×C411yescharacteristic
34C22×Dic97D411yescharacteristic
35C22×Dic97D411yescharacteristic
36C22×Dic97D411yescharacteristic
37C22×Dic97D411yescharacteristic
38C22×Dic97C2311yescharacteristic
39C4⋊Dic97—21no—
40C4⋊Dic97—21no—
41#13095490—21no—
42#13095490—21no—
43#13095490—21no—
44#13095490—21no—
45C22⋊Dic97—21no—
46C22⋊Dic97—21no—
47C24×C97C2×C411yescharacteristic
48C22×C4×C97C2×C411yescharacteristic
49C22×C4×C97C2×C411yescharacteristic
50C22⋊C4×C97—21no—
51C22⋊C4×C97—21no—
52C23×C97—22no—
53C23×C97—22no—
54C23×C97—41no—
55C23×C97C22⋊C411yescharacteristic
56C23×C97C22⋊C411yescharacteristic
57C23×C97C22×C411yescharacteristic
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—22no—
61C2×C4×C97—41no—
62C2×Dic97—22no—
63C2×Dic97—22no—
64C2×Dic97—22no—
65C2×Dic97—21no—
66C2×Dic97—21no—
67C2×Dic97—22no—
68C2×Dic97—41no—
69C2×Dic97—41no—
70C2×Dic97—41no—
71C2×Dic97—41no—
72C2×Dic97C22⋊C411yescharacteristic
73C2×Dic97C22⋊C411yescharacteristic
74C2×Dic97C2×D411yescharacteristic
75C2×Dic97C2×D411yescharacteristic
76Dic194—42no—
77Dic194—42no—
78#11715897—42no—
79#11715897—42no—
80C22×C97—21no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97—44no—
88C22×C97#10112yes—
89C22×C97C2×C22⋊C411yescharacteristic
90C22×C97#13112yes—
91C4×C97—41no—
92C4×C97—41no—
93Dic97—21no—
94Dic97—21no—
95Dic97—21no—
96Dic97—41no—
97Dic97—41no—
98Dic97#9912yes—
99C2×C97—22no—
100C2×C97—22no—
101C2×C97—41no—
102C2×C97#40811yescharacteristic
103C2×C97C2×#9911yescharacteristic
104C2×C97C2×#10111yescharacteristic
105#1472—971nomaximal, Sylow
106C97#147211yescharacteristic, complemented, Sylow
107C2×C22⋊Q8—971no—
108#407—971no—
109C2×C22⋊C8—971no—
110#324—972no—
111#324—972no—
112C22×Q8—971no—
113C23×C4—971no—
114C2×M16—1941no—
115C22×C8—1941no—
116C22⋊Q8—1941no—
117C22⋊Q8—1941no—
118C22⋊Q8—1941no—
119C22⋊Q8—1941no—
120C2×C4⋊C4—971no—
121C2×C4⋊C4—971no—
122C2×C4⋊C4—971no—
123C2×C22⋊C4—971no—
124C2×C22⋊C4#8760411yescharacteristic
125C22⋊C8—972no—
126C22⋊C8—972no—
127C22⋊C8—1941no—
128C22⋊C8—1941no—
129C24C2×#8760411yescharacteristic
130C2×Q8—3882no—
131C2×Q8—1941no—
132C2×Q8—1941no—
133C22×C4—971no—
134C22×C4—971no—
135C22×C4—971no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—971no—
139C22×C4—971no—
140C22×C4—971no—
141C22×C4—3882no—
142C22×C4C2×#8760411yescharacteristic
143C22×C4C2×#8760411yescharacteristic
144M16—3882no—
145C2×C8—3882no—
146C2×C8—1942no—
147C2×C8—1942no—
148C4⋊C4—1941no—
149C4⋊C4—1941no—
150C4⋊C4—1941no—
151C4⋊C4—1941no—
152C4⋊C4—1941no—
153C4⋊C4—1941no—
154C22⋊C4—21no—
155C22⋊C4—21no—
156C22⋊C4—1941no—
157C22⋊C4—1941no—
158C23—22no—
159C23—22no—
160C23—41no—
161C23C22⋊#8760411yescharacteristic
162C23C22⋊#8760411yescharacteristic
163C23C22×#8760411yescharacteristic
164Q8—3882no—
165Q8—3882no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—971no—
171C2×C4—971no—
172C2×C4—971no—
173C2×C4—971no—
174C2×C4—3881no—
175C2×C4—3881no—
176C2×C4—3881no—
177C2×C4—3881no—
178C2×C4—1942no—
179C2×C4—1942no—
180C2×C4—1942no—
181C2×C4—1941no—
182C2×C4—1941no—
183C2×C4—1942no—
184C8—3882no—
185C8—3882no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—44no—
194C22#1569087112yes—
195C22C2×C22⋊#8760411yescharacteristic
196C22#1569089712yes—
197C4—41no—
198C4—41no—
199C4—972no—
200C4—3881no—
201C4—3881no—
202C4—1941no—
203C4—1941no—
204C4—1941no—
205C2—22no—
206C2—22no—
207C2—41no—
208C2#2026170811yescharacteristic
209C2D97⋊#9911yescharacteristic
210C2C2×#1569087111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC24×C97C2×C4
cosocleC22×Dic97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97#1472
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #20261581C2semidirect
4. #20261580C2semidirect
106. C97#1472semidirect

Maximal subgroups

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

subgroupindex
#202617052
C2×C22⋊#117158972
#202615812
#202615802
C2×#156907872
#147297