Finite Group Databasebeta

← #27157016

Subgroups — #27157016

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#20261706C211yesmaximal, characteristic
2#20261703C211yesmaximal, characteristic
3#20261581C214yesmaximal, complemented
4C2×#15690786C211yesmaximal, characteristic
5C2×#13095521—21no—
6C2×#13095519—21no—
7#15690867—21no—
8#15690867—21no—
9C22⋊#11715897C2212yes—
10C22⋊#11715897C2212yes—
11C22×C4×D97C2211yescharacteristic
12#15690786—21no—
13#15690786—21no—
14#15690786—21no—
15#15690786—21no—
16C2×C4⋊Dic97C411yescharacteristic
17C2×#13095490C2212yes—
18C2×#13095488C412yes—
19C22×Q8×C97C411yescharacteristic
20#13095521—42no—
21#13095519—42no—
22C2×#11715897—22no—
23C2×#11715897—22no—
24C23×D97C2×C411yescharacteristic
25C22×Dic97C2×C412yes—
26C22×Dic97C2311yescharacteristic
27C2×C4×D97—42no—
28C2×C4×D97D411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97D411yescharacteristic
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34#13095490—22no—
35#13095490—22no—
36#13095488—22no—
37#13095488—22no—
38C2×Q8×C97—21no—
39C2×Q8×C97—21no—
40C2×Q8×C97—42no—
41C22×C4×C97C2×C412yes—
42C22×C4×C97C2×C411yescharacteristic
43C23×C97C22×C411yescharacteristic
44C22×D97—22no—
45C22×D97—22no—
46C22×D97—41no—
47C22×D97C22⋊C411yescharacteristic
48C22×D97C22⋊C411yescharacteristic
49Q8×C97—42no—
50Q8×C97—42no—
51C2×C4×C97—22no—
52C2×C4×C97—22no—
53C2×C4×C97—42no—
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22⋊C411yescharacteristic
56C2×Dic97—22no—
57C2×Dic97—24no—
58C2×Dic97—41no—
59C2×Dic97C2×D411yescharacteristic
60C2×Dic97C2×D411yescharacteristic
61C4×D97—22no—
62C4×D97—22no—
63C4×D97—41no—
64C4×D97—41no—
65#11715897—42no—
66#11715897—42no—
67C22×C97—21no—
68C22×C97—21no—
69C22×C97C2×C22⋊C411yescharacteristic
70C22×C97#13112yes—
71C2×D97—21no—
72C2×D97—21no—
73C2×D97—21no—
74C2×D97—21no—
75C2×D97—21no—
76C2×D97—44no—
77C2×D97#10112yes—
78C4×C97—21no—
79C4×C97—42no—
80C4×C97#9912yes—
81Dic97—21no—
82Dic97—21no—
83Dic97—42no—
84C2×C97—22no—
85C2×C97#40811yescharacteristic
86C2×C97C2×#9911yescharacteristic
87C2×C97C2×#10111yescharacteristic
88D97—22no—
89D97—41no—
90#1472—971nomaximal, Sylow
91C97#147211yescharacteristic, complemented, Sylow
92C2×C22⋊Q8—971no—
93#407—971no—
94C2×C22⋊C8—971no—
95#324—974no—
96C22×Q8#8760411yescharacteristic
97C23×C4—971no—
98C2×M16—1941no—
99C22×C8—1941no—
100C22⋊Q8—1941no—
101C22⋊Q8—1941no—
102C22⋊Q8—1941no—
103C22⋊Q8—1941no—
104C2×C4⋊C4—972no—
105C2×C4⋊C4—971no—
106C2×C22⋊C4—972no—
107C22⋊C8—972no—
108C22⋊C8—972no—
109C22⋊C8—1941no—
110C22⋊C8—1941no—
111C24—971no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—42no—
115C22×C4—971no—
116C22×C4—971no—
117C22×C4—971no—
118C22×C4—971no—
119C22×C4—971no—
120C22×C4—972no—
121C22×C4—3882no—
122C22×C4C2×#8760412yes—
123C22×C4C2×#8760411yescharacteristic
124M16—3882no—
125C2×C8—3882no—
126C2×C8—1942no—
127C2×C8—1942no—
128C4⋊C4—1942no—
129C4⋊C4—1942no—
130C4⋊C4—1941no—
131C4⋊C4—1941no—
132C22⋊C4—1942no—
133C22⋊C4—1942no—
134C23—971no—
135C23—971no—
136C23—3881no—
137C23—1942no—
138C23—1942no—
139C23C22×#8760411yescharacteristic
140Q8—42no—
141Q8—42no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—971no—
146C2×C4—971no—
147C2×C4—3881no—
148C2×C4—3881no—
149C2×C4—3881no—
150C2×C4—1942no—
151C2×C4—1944no—
152C2×C4—1942no—
153C2×C4—1942no—
154C2×C4C22⋊#8760411yescharacteristic
155C2×C4C22⋊#8760411yescharacteristic
156C8—3882no—
157C8—3882no—
158C22—21no—
159C22—21no—
160C22—972no—
161C22—3884no—
162C22—1941no—
163C22—1941no—
164C22—1941no—
165C22—1941no—
166C22—1941no—
167C22C2×C22⋊#8760411yescharacteristic
168C22#1569089812yes—
169C4—21no—
170C4—42no—
171C4—3882no—
172C4—1941no—
173C4—1941no—
174C4#1569086812yes—
175C2—22no—
176C2—3881no—
177C2—1942no—
178C2#2026171011yescharacteristic
179C2C2×#1569086811yescharacteristic
180C2D97⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC22×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97C2×C4
cosocleC22×Dic97C23
hypercenterC22×Q8#87604
hypocenterC97#1472
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #20261581C2semidirect
91. C97#1472semidirect

Maximal subgroups

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

subgroupindex
#202617062
#202617032
#202615812
C2×#156907862
#147297