Finite Group Databasebeta

← #27157486

Subgroups — #27157486

Order 12416.

360 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C2×#15690870C212yesmaximal
2C4×#13095520C212yesmaximal, complemented
3#20261615C212yesmaximal, complemented
4C2×#15690770C211yesmaximal, characteristic
5C2×#13095520C2214yes—
6#15690870C418yescomplemented
7#15690865—22no—
8C4×#11715897—22no—
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#13095520C2×C418yescomplemented
20C2×#11715897—24no—
21C23×D97C2×C411yescharacteristic
22C2×C22⋊D97—22no—
23C2×C22⋊D97—22no—
24C2×C22⋊D97C2×C414yes—
25C22×Dic97—21no—
26C22×Dic97—21no—
27C22×Dic97—42no—
28C22×Dic97C2×C411yescharacteristic
29C22×Dic97C2×C411yescharacteristic
30C22×Dic97C2311yescharacteristic
31C2×C4×D97—21no—
32C2×C4×D97—21no—
33C2×C4×D97—42no—
34#13095490—22no—
35#13095488—22no—
36C4×Dic97D412yes—
37C4×Dic97D412yes—
38C24×C97C2×C411yescharacteristic
39C22×C4×C97C2×C411yescharacteristic
40C22×C4×C97C2×C411yescharacteristic
41C22⋊C4×C97—22no—
42C23×C97—21no—
43C23×C97—21no—
44C23×C97—41no—
45C23×C97—41no—
46C23×C97C4212yes—
47C23×C97C22×C411yescharacteristic
48C22×D97—21no—
49C22×D97—21no—
50C22×D97—41no—
51C22×D97—41no—
52C22×D97C4212yes—
53C2×C4×C97—21no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97C22⋊C412yes—
57C22⋊D97—28no—
58C2×Dic97—22no—
59C2×Dic97—21no—
60C2×Dic97—42no—
61C2×Dic97—42no—
62C2×Dic97—41no—
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C22⋊C411yescharacteristic
65C2×Dic97C22×C412yes—
66C2×Dic97C2×D412yes—
67C2×Dic97#4112yes—
68C4×D97—42no—
69C4×D97—42no—
70#11715897—28no—
71C22×C97—21no—
72C22×C97—21no—
73C22×C97—22no—
74C22×C97—42no—
75C22×C97—42no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97C2×C4211yescharacteristic
79C22×C97C2×C22⋊C411yescharacteristic
80C22×C97#11711yescharacteristic
81C2×D97—22no—
82C2×D97—42no—
83C2×D97—41no—
84C2×D97—41no—
85C2×D97—42no—
86C4×C97—22no—
87C4×C97—41no—
88Dic97—21no—
89Dic97—41no—
90Dic97#10012yescomplemented
91Dic97C4×D414yes—
92C2×C97—21no—
93C2×C97—21no—
94C2×C97—42no—
95C2×C97C4×C22⋊C411yescharacteristic
96C2×C97C2×#10011yescharacteristic
97C2×C97#41311yescharacteristic
98D97—42no—
99C4×#100—971nomaximal, Sylow
100C97C4×#10011yescharacteristic, complemented, Sylow
101C2×C4×D4—971no—
102C2×#100—972no—
103C4×M16—972no—
104#343—972no—
105C22×D4—971no—
106C23×C4—971no—
107C23×C4—971no—
108C2×M16—974no—
109C4×D4—1942no—
110C4×D4—1942no—
111C2×C4⋊C4—971no—
112C2×C22⋊C4—971no—
113C2×C22⋊C4#8760411yescharacteristic
114C2×C42—971no—
115#100—978no—
116#97—1942no—
117C4×C8—1942no—
118C24—971no—
119C24C2×#8760411yescharacteristic
120C2×D4—974no—
121C2×D4—1942no—
122C2×D4—1942no—
123C22×C4—971no—
124C22×C4—971no—
125C22×C4—971no—
126C22×C4—3882no—
127C22×C4—3882no—
128C22×C4—1941no—
129C22×C4—1941no—
130C22×C4—1941no—
131C22×C4—1941no—
132C22×C4C2×#8760411yescharacteristic
133C22×C4C2×#8760411yescharacteristic
134M16—978no—
135M16—1944no—
136C2×C8—1944no—
137C4⋊C4—1942no—
138C22⋊C4—22no—
139C22⋊C4—1942no—
140C42—972no—
141C42—972no—
142C23—21no—
143C23—21no—
144C23—41no—
145C23—41no—
146C23—972no—
147C23—3881no—
148C23—3881no—
149C23—1941no—
150C23—1941no—
151C23C4×#8760412yes—
152C23C22×#8760411yescharacteristic
153D4—1948no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—972no—
158C2×C4—972no—
159C2×C4—971no—
160C2×C4—972no—
161C2×C4—971no—
162C2×C4—3882no—
163C2×C4—3882no—
164C2×C4—3882no—
165C2×C4—3882no—
166C2×C4—3881no—
167C2×C4—1942no—
168C2×C4—1941no—
169C2×C4C22⋊#8760412yes—
170C8—1948no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—42no—
176C22—41no—
177C22—41no—
178C22—3882no—
179C22—3882no—
180C22—3881no—
181C22—3881no—
182C22—1942no—
183C22C2×C4×#8760411yescharacteristic
184C22C2×C22⋊#8760411yescharacteristic
185C22#1569088911yescharacteristic
186C4—22no—
187C4—41no—
188C4—974no—
189C4—972no—
190C4—3881no—
191C4—1941no—
192C2—21no—
193C2—21no—
194C2—42no—
195C2—3882no—
196C2C4×C22⋊#8760411yescharacteristic
197C2D97⋊#10011yescharacteristic
198C2#2026172211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC22×C97C2×C42
FrattiniC23C22×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×C22⋊D97C22
cosocleC22×Dic97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97C4×#100
LCSG ▷ C22×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
2. C4×#13095520C2semidirect
3. #20261615C2semidirect
6. #15690870C4semidirect
19. #13095520C2×C4semidirect
90. Dic97#100semidirect
100. C97C4×#100semidirect

Maximal subgroups

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

subgroupindex
C2×#156908702
C4×#130955202
#202616152
C2×#156907702
C4×#10097