Finite Group Databasebeta

← #27157601

Subgroups — #27157601

Order 12416.

336 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 78 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#20261738C212yesmaximal, complemented
2C2×#15690870C212yesmaximal
3#20261615C212yesmaximal, complemented
4C2×#15690770C211yesmaximal, characteristic
5C2×#13095520C2214yes—
6#15690877—22no—
7C4⋊#11715897—22no—
8#15690870—24no—
9C22×C22⋊D97C2211yescharacteristic
10C23×Dic97C411yescharacteristic
11C22×C4×D97C411yescharacteristic
12#15690770—22no—
13#15690770—22no—
14C2×#13095490C2211yescharacteristic
15C2×#13095488C411yescharacteristic
16C2×C4×Dic97C2211yescharacteristic
17C2×C22⋊C4×C97C411yescharacteristic
18#13095520—24no—
19#13095520—24no—
20C2×#11715897—24no—
21C23×D97C2×C411yescharacteristic
22C2×C22⋊D97—22no—
23C2×C22⋊D97—22no—
24C2×C22⋊D97D412yes—
25C2×C22⋊D97Q812yes—
26C22×Dic97—21no—
27C22×Dic97—21no—
28C22×Dic97—42no—
29C22×Dic97C2×C411yescharacteristic
30C22×Dic97C2×C411yescharacteristic
31C22×Dic97C2311yescharacteristic
32C2×C4×D97—21no—
33C2×C4×D97—21no—
34C2×C4×D97—42no—
35#13095490—22no—
36#13095488—22no—
37C4×Dic97D412yes—
38C4×Dic97D412yes—
39C24×C97C2×C411yescharacteristic
40C22×C4×C97C2×C411yescharacteristic
41C22×C4×C97C2×C411yescharacteristic
42C22⋊C4×C97—22no—
43C23×C97—21no—
44C23×C97—21no—
45C23×C97—41no—
46C23×C97—41no—
47C23×C97C4⋊C412yes—
48C23×C97C22×C411yescharacteristic
49C22×D97—21no—
50C22×D97—21no—
51C22×D97—41no—
52C22×D97—41no—
53C22×D97C4⋊C412yes—
54C2×C4×C97—21no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97C22⋊C412yes—
58C22⋊D97—24no—
59C22⋊D97—24no—
60C2×Dic97—22no—
61C2×Dic97—21no—
62C2×Dic97—42no—
63C2×Dic97—42no—
64C2×Dic97—41no—
65C2×Dic97C22⋊C411yescharacteristic
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C2×D411yescharacteristic
68C2×Dic97C2×D412yes—
69C2×Dic97C2×Q811yescharacteristic
70C2×Dic97#4112yes—
71C4×D97—42no—
72C4×D97—42no—
73#11715897—44no—
74C22×C97—21no—
75C22×C97—21no—
76C22×C97—22no—
77C22×C97—42no—
78C22×C97—42no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97C2×C22⋊C411yescharacteristic
82C22×C97C2×C4⋊C411yescharacteristic
83C22×C97#11711yescharacteristic
84C2×D97—22no—
85C2×D97—42no—
86C2×D97—41no—
87C2×D97—41no—
88C2×D97—42no—
89C4×C97—22no—
90C4×C97—41no—
91Dic97—21no—
92Dic97—41no—
93Dic97#10012yescomplemented
94Dic97#12112yes—
95Dic97C22⋊Q812yes—
96C2×C97—21no—
97C2×C97—21no—
98C2×C97—42no—
99C2×C97#38011yescharacteristic
100C2×C97C2×#10011yescharacteristic
101C2×C97#41311yescharacteristic
102D97—42no—
103#1800—971nomaximal, Sylow
104C97#180011yescharacteristic, complemented, Sylow
105C2×C4×D4—971no—
106#423—972no—
107C2×#100—972no—
108#343—972no—
109C22×D4—971no—
110C23×C4—971no—
111C23×C4—971no—
112C2×M16—974no—
113C4×D4—1942no—
114C4×D4—1942no—
115C2×C4⋊C4—971no—
116C2×C22⋊C4—971no—
117C2×C22⋊C4#8760411yescharacteristic
118C2×C42—971no—
119C4⋊C8—1942no—
120C4⋊C8—1942no—
121#100—1944no—
122C24—971no—
123C24C2×#8760411yescharacteristic
124C2×D4—972no—
125C2×D4—972no—
126C2×D4—1942no—
127C2×D4—1942no—
128C22×C4—971no—
129C22×C4—971no—
130C22×C4—971no—
131C22×C4—3882no—
132C22×C4—3882no—
133C22×C4—1941no—
134C22×C4—1941no—
135C22×C4—1941no—
136C22×C4—1941no—
137C22×C4C2×#8760411yescharacteristic
138C22×C4C2×#8760411yescharacteristic
139M16—1944no—
140M16—1944no—
141C2×C8—1944no—
142C4⋊C4—1942no—
143C22⋊C4—22no—
144C22⋊C4—1942no—
145C42—972no—
146C42—972no—
147C23—21no—
148C23—21no—
149C23—41no—
150C23—41no—
151C23—972no—
152C23—3881no—
153C23—3881no—
154C23—1941no—
155C23—1941no—
156C23C4⋊#8760412yes—
157C23C22×#8760411yescharacteristic
158D4—1944no—
159D4—1944no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—972no—
164C2×C4—971no—
165C2×C4—971no—
166C2×C4—971no—
167C2×C4—972no—
168C2×C4—971no—
169C2×C4—3882no—
170C2×C4—3882no—
171C2×C4—3882no—
172C2×C4—3882no—
173C2×C4—3881no—
174C2×C4—1942no—
175C2×C4—1941no—
176C2×C4C22⋊#8760412yes—
177C8—3884no—
178C22—21no—
179C22—21no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22—41no—
185C22—3882no—
186C22—3882no—
187C22—3881no—
188C22—3881no—
189C22—1942no—
190C22C2×C22⋊#8760411yescharacteristic
191C22C2×C4⋊#8760411yescharacteristic
192C22#1569088911yescharacteristic
193C4—22no—
194C4—41no—
195C4—972no—
196C4—972no—
197C4—972no—
198C4—3881no—
199C4—1941no—
200C2—21no—
201C2—21no—
202C2—42no—
203C2—3882no—
204C2#2026167711yescharacteristic
205C2D97⋊#10011yescharacteristic
206C2#2026172211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×C22⋊D97C22
cosocleC22×Dic97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97#1800
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20261738C2semidirect
3. #20261615C2semidirect
93. Dic97#100semidirect
104. C97#1800semidirect

Maximal subgroups

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

subgroupindex
#202617382
C2×#156908702
#202616152
C2×#156907702
#180097