Finite Group Databasebeta

← #27157002

Subgroups — #27157002

Order 12416.

280 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 60 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, complemented
2#20261703C211yesmaximal, characteristic, complemented
3#20261580C212yesmaximal, complemented
4#20261578C212yesmaximal, complemented
5C2×#15690792C211yesmaximal, characteristic
6C2×#13095521—21no—
7C2×#13095519—21no—
8#15690867—21no—
9#15690867—21no—
10C22⋊#11715897C2212yes—
11C22⋊#11715897C2212yes—
12C22×C22⋊D97C411yescharacteristic
13C22×C4×D97C2211yescharacteristic
14#15690792—21no—
15#15690792—21no—
16#15690792—21no—
17#15690792—21no—
18C2×C4⋊Dic97C2211yescharacteristic
19C2×#13095490C411yescharacteristic
20C2×C22⋊Dic97C2211yescharacteristic
21C2×#13095488C411yescharacteristic
22C2×C22⋊C4×C97C411yescharacteristic
23#13095521—42no—
24#13095519—42no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27C23×D97C2×C411yescharacteristic
28C2×C22⋊D97—21no—
29C2×C22⋊D97—21no—
30C2×C22⋊D97—42no—
31C22×Dic97C2×C411yescharacteristic
32C22×Dic97C2×C411yescharacteristic
33C22×Dic97C2311yescharacteristic
34C2×C4×D97—42no—
35C2×C4×D97D411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39C4⋊Dic97—21no—
40C4⋊Dic97—21no—
41#13095490—21no—
42#13095490—21no—
43C22⋊Dic97—21no—
44C22⋊Dic97—21no—
45#13095488—21no—
46#13095488—21no—
47C24×C97C2×C411yescharacteristic
48C22×C4×C97C2×C411yescharacteristic
49C22×C4×C97C2×C411yescharacteristic
50C22⋊C4×C97—21no—
51C22⋊C4×C97—21no—
52C23×C97—21no—
53C23×C97—22no—
54C23×C97—41no—
55C23×C97—41no—
56C23×C97C22×C411yescharacteristic
57C22×D97—22no—
58C22×D97—22no—
59C22×D97—41no—
60C22×D97C22⋊C411yescharacteristic
61C22×D97C22⋊C411yescharacteristic
62C2×C4×C97—21no—
63C2×C4×C97—22no—
64C2×C4×C97—41no—
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22⋊C411yescharacteristic
67C22⋊D97—42no—
68C22⋊D97—42no—
69C2×Dic97—21no—
70C2×Dic97—21no—
71C2×Dic97—41no—
72C2×Dic97—41no—
73C2×Dic97—41no—
74C2×Dic97C2×D411yescharacteristic
75C2×Dic97C2×D411yescharacteristic
76C4×D97—22no—
77C4×D97—22no—
78C4×D97—41no—
79C4×D97—41no—
80#11715897—42no—
81#11715897—42no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97—41no—
85C22×C97—41no—
86C22×C97—41no—
87C22×C97—82no—
88C22×C97C2×C22⋊C411yescharacteristic
89C22×C97#13112yes—
90C2×D97—21no—
91C2×D97—21no—
92C2×D97—21no—
93C2×D97—21no—
94C2×D97—21no—
95C2×D97—44no—
96C2×D97#9912yes—
97C4×C97—21no—
98C4×C97—21no—
99C4×C97—41no—
100Dic97—21no—
101Dic97—21no—
102Dic97—41no—
103Dic97—41no—
104C2×C97—22no—
105C2×C97—81no—
106C2×C97#40811yescharacteristic
107C2×C97C2×#9911yescharacteristic
108C2×C97#41311yescharacteristic
109D97—22no—
110D97—41no—
111#1468—971nomaximal, Sylow
112C97#146811yescharacteristic, complemented, Sylow
113C2×#123—971no—
114#407—971no—
115C2×C22⋊C8—971no—
116#324—972no—
117#323—972no—
118C22×D4—971no—
119C23×C4—971no—
120C2×M16—1941no—
121C22×C8—1941no—
122#123—1941no—
123#123—1941no—
124#123—1941no—
125#123—1941no—
126C2×C4⋊C4—971no—
127C2×C4⋊C4—971no—
128C2×C22⋊C4—971no—
129C2×C22⋊C4—971no—
130C2×C22⋊C4#8760411yescharacteristic
131C22⋊C8—972no—
132C22⋊C8—972no—
133C22⋊C8—1941no—
134C22⋊C8—1941no—
135C24—971no—
136C24C2×#8760411yescharacteristic
137C2×D4—3882no—
138C2×D4—1941no—
139C2×D4—1941no—
140C22×C4—971no—
141C22×C4—971no—
142C22×C4—971no—
143C22×C4—971no—
144C22×C4—971no—
145C22×C4—971no—
146C22×C4—971no—
147C22×C4—3882no—
148C22×C4C2×#8760411yescharacteristic
149C22×C4C2×#8760411yescharacteristic
150M16—3882no—
151C2×C8—3882no—
152C2×C8—1942no—
153C2×C8—1942no—
154C4⋊C4—1941no—
155C4⋊C4—1941no—
156C4⋊C4—1941no—
157C4⋊C4—1941no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—1941no—
161C22⋊C4—1941no—
162C22⋊C4—1941no—
163C22⋊C4—1941no—
164C23—21no—
165C23—22no—
166C23—41no—
167C23—41no—
168C23—971no—
169C23—971no—
170C23—3881no—
171C23—1942no—
172C23—1942no—
173C23C22×#8760411yescharacteristic
174D4—3882no—
175D4—3882no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—971no—
180C2×C4—971no—
181C2×C4—3881no—
182C2×C4—3881no—
183C2×C4—3881no—
184C2×C4—3881no—
185C2×C4—3881no—
186C2×C4—1942no—
187C2×C4—1942no—
188C2×C4—1941no—
189C2×C4—1941no—
190C2×C4C22⋊#8760411yescharacteristic
191C2×C4C22⋊#8760411yescharacteristic
192C8—3882no—
193C8—3882no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—82no—
200C22—972no—
201C22—3884no—
202C22—1941no—
203C22—1941no—
204C22—1941no—
205C22—1941no—
206C22—1941no—
207C22C2×C22⋊#8760411yescharacteristic
208C22#1569089712yes—
209C4—21no—
210C4—21no—
211C4—41no—
212C4—3881no—
213C4—3881no—
214C4—1941no—
215C4—1941no—
216C2—22no—
217C2—81no—
218C2—3881no—
219C2—1942no—
220C2#2026170911yescharacteristic
221C2D97⋊#9911yescharacteristic
222C2#2026172111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×C22⋊D97C4
cosocleC22×Dic97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97#1468
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. #20261706C2semidirect
2. #20261703C2semidirect
3. #20261580C2semidirect
4. #20261578C2semidirect
112. C97#1468semidirect

Maximal subgroups

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

subgroupindex
#202617062
#202617032
#202615802
#202615782
C2×#156907922
#146897