Finite Group Databasebeta

← #27157001

Subgroups — #27157001

Order 12416.

280 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 62 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, complemented
2C2×C22⋊#11715897C211yesmaximal, characteristic, complemented
3#20261581C212yesmaximal, complemented
4#20261578C212yesmaximal, complemented
5C2×#15690791C211yesmaximal, characteristic
6C2×#13095520—21no—
7C22×#11715897—21no—
8C22⋊#11715897—21no—
9C22⋊#11715897—21no—
10C22⋊#11715897C2212yes—
11C22⋊#11715897C2212yes—
12C23×Dic97C2211yescharacteristic
13#15690791—21no—
14#15690791—21no—
15#15690791—21no—
16#15690791—21no—
17C2×#13095490C411yescharacteristic
18C2×#13095490C2211yescharacteristic
19C2×C22⋊Dic97C411yescharacteristic
20C2×C22⋊Dic97C411yescharacteristic
21C2×C22⋊Dic97C2211yescharacteristic
22C22×D4×C97C411yescharacteristic
23#13095520—42no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×#11715897—42no—
27C22×Dic97—42no—
28C22×Dic97C2×C411yescharacteristic
29C22×Dic97C2×C411yescharacteristic
30C22×Dic97C2×C411yescharacteristic
31C22×Dic97D411yescharacteristic
32C22×Dic97D411yescharacteristic
33C22×Dic97D411yescharacteristic
34C22×Dic97D411yescharacteristic
35C22×Dic97C2311yescharacteristic
36#13095490—21no—
37#13095490—21no—
38#13095490—21no—
39#13095490—21no—
40C22⋊Dic97—21no—
41C22⋊Dic97—21no—
42C22⋊Dic97—21no—
43C22⋊Dic97—21no—
44C22⋊Dic97—21no—
45C22⋊Dic97—21no—
46C24×C97C2×C411yescharacteristic
47C24×C97C2×C411yescharacteristic
48C2×D4×C97—21no—
49C2×D4×C97—21no—
50C2×D4×C97—42no—
51C22×C4×C97C2×C411yescharacteristic
52C23×C97—22no—
53C23×C97—22no—
54C23×C97—21no—
55C23×C97—22no—
56C23×C97—41no—
57C23×C97—41no—
58C23×C97—41no—
59C23×C97C22⋊C411yescharacteristic
60C23×C97C22⋊C411yescharacteristic
61C23×C97C22×C411yescharacteristic
62D4×C97—42no—
63D4×C97—42no—
64C2×C4×C97—21no—
65C2×C4×C97—41no—
66C2×Dic97—22no—
67C2×Dic97—22no—
68C2×Dic97—21no—
69C2×Dic97—21no—
70C2×Dic97—22no—
71C2×Dic97—41no—
72C2×Dic97—41no—
73C2×Dic97—41no—
74C2×Dic97—41no—
75C2×Dic97—41no—
76C2×Dic97C22⋊C411yescharacteristic
77C2×Dic97C22⋊C411yescharacteristic
78C2×Dic97C2×D411yescharacteristic
79C2×Dic97C2×D411yescharacteristic
80#11715897—42no—
81#11715897—42no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97—21no—
88C22×C97—21no—
89C22×C97—44no—
90C22×C97—41no—
91C22×C97—41no—
92C22×C97—41no—
93C22×C97—82no—
94C22×C97#9912yes—
95C22×C97C2×C22⋊C411yescharacteristic
96C22×C97#13112yes—
97C4×C97—41no—
98Dic97—21no—
99Dic97—21no—
100Dic97—21no—
101Dic97—21no—
102Dic97—41no—
103Dic97—41no—
104C2×C97—22no—
105C2×C97—22no—
106C2×C97—41no—
107C2×C97—81no—
108C2×C97#40811yescharacteristic
109C2×C97C2×#9911yescharacteristic
110C2×C97#41311yescharacteristic
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#8760411yescharacteristic
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—971no—
131C22⋊C8—972no—
132C22⋊C8—972no—
133C22⋊C8—1941no—
134C22⋊C8—1941no—
135C24C2×#8760411yescharacteristic
136C24C2×#8760411yescharacteristic
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—42no—
140C22×C4—971no—
141C22×C4—971no—
142C22×C4—971no—
143C22×C4—971no—
144C22×C4—971no—
145C22×C4—971no—
146C22×C4—971no—
147C22×C4—971no—
148C22×C4—3882no—
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—1941no—
159C22⋊C4—1941no—
160C22⋊C4—1941no—
161C22⋊C4—1941no—
162C22⋊C4—1941no—
163C22⋊C4—1941no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23C22⋊#8760411yescharacteristic
172C23C22⋊#8760411yescharacteristic
173C23C22×#8760411yescharacteristic
174D4—42no—
175D4—42no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—971no—
179C2×C4—971no—
180C2×C4—971no—
181C2×C4—971no—
182C2×C4—3881no—
183C2×C4—3881no—
184C2×C4—3881no—
185C2×C4—3881no—
186C2×C4—3881no—
187C2×C4—1941no—
188C2×C4—1941no—
189C2×C4—1942no—
190C2×C4—1942no—
191C2×C4—1942no—
192C8—3882no—
193C8—3882no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—82no—
206C22#1569086812yes—
207C22C2×C22⋊#8760411yescharacteristic
208C22#1569089712yes—
209C4—41no—
210C4—3881no—
211C4—3881no—
212C4—1941no—
213C4—1941no—
214C4—1941no—
215C4—1941no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—81no—
220C2#2026170811yescharacteristic
221C2C2×#1569086811yescharacteristic
222C2#2026172211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
Fitting, prime radicalC22×D4×C97C4
socleC22×C97C2×C22⋊C4
cosocleC22×Dic97C23
hypercenterC22×D4#87604
hypocenterC97#1468
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20261705C2semidirect
2. C2×C22⋊#11715897C2semidirect
3. #20261581C2semidirect
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
#202617052
C2×C22⋊#117158972
#202615812
#202615782
C2×#156907912
#146897