Finite Group Databasebeta

← #27140084

Subgroups — #27140084

Order 12416.

280 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 81 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#20260699C211yesmaximal, characteristic, complemented
2C2×C22⋊#11715895C211yesmaximal, characteristic, complemented
3#20260461C212yesmaximal, complemented
4#20260459C212yesmaximal, complemented
5C2×#123×C97C211yesmaximal, characteristic
6C2×#13095495—21no—
7C22×#11715895—21no—
8C22⋊#11715895—21no—
9C22⋊#11715895—21no—
10C22⋊#11715895C2212yes—
11C22⋊#11715895C2212yes—
12C22×D4×C97C411yescharacteristic
13C23×C4×C97C2211yescharacteristic
14#123×C97—21no—
15#123×C97—21no—
16#123×C97—21no—
17#123×C97—21no—
18C2×C4⋊C4×C97C411yescharacteristic
19C2×C4⋊C4×C97C2211yescharacteristic
20C2×C22⋊C4×C97C411yescharacteristic
21C2×C22⋊C4×C97C411yescharacteristic
22C2×C22⋊C4×C97C2211yescharacteristic
23#13095495—42no—
24C2×#11715895—22no—
25C2×#11715895—22no—
26C2×#11715895—42no—
27C24×C97C2×C411yescharacteristic
28C24×C97C2×C411yescharacteristic
29C2×D4×C97—21no—
30C2×D4×C97—21no—
31C2×D4×C97—42no—
32C22×C4×C97—42no—
33C22×C4×C97C2×C411yescharacteristic
34C22×C4×C97C2×C411yescharacteristic
35C22×C4×C97C2×C411yescharacteristic
36C22×C4×C97C2×C411yescharacteristic
37C22×C4×C97D411yescharacteristic
38C22×C4×C97D411yescharacteristic
39C22×C4×C97D411yescharacteristic
40C22×C4×C97D411yescharacteristic
41C22×C4×C97C2311yescharacteristic
42C4⋊C4×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C22⋊C4×C97—21no—
47C22⋊C4×C97—21no—
48C22⋊C4×C97—21no—
49C22⋊C4×C97—21no—
50C22⋊C4×C97—21no—
51C22⋊C4×C97—21no—
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—22no—
65C2×C4×C97—22no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—21no—
69C2×C4×C97—22no—
70C2×C4×C97—41no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97—41no—
76C2×C4×C97C22⋊C411yescharacteristic
77C2×C4×C97C22⋊C411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80#11715895—42no—
81#11715895—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—21no—
98C4×C97—21no—
99C4×C97—21no—
100C4×C97—21no—
101C4×C97—41no—
102C4×C97—41no—
103C4×C97—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×#123D9711yescharacteristic
114#407—971no—
115C2×C22⋊C8—971no—
116#324—972no—
117#323—972no—
118C22×D4Dic9711yescharacteristic
119C23×C4C2×D9711yescharacteristic
120C2×M16—1941no—
121C22×C8—1941no—
122#123—21no—
123#123—21no—
124#123—21no—
125#123—21no—
126C2×C4⋊C4Dic9711yescharacteristic
127C2×C4⋊C4C2×D9711yescharacteristic
128C2×C22⋊C4Dic9711yescharacteristic
129C2×C22⋊C4Dic9711yescharacteristic
130C2×C22⋊C4C2×D9711yescharacteristic
131C22⋊C8—972no—
132C22⋊C8—972no—
133C22⋊C8—1941no—
134C22⋊C8—1941no—
135C24C2×Dic9711yescharacteristic
136C24C2×Dic9711yescharacteristic
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—42no—
140C22×C4—42no—
141C22×C4C2×Dic9711yescharacteristic
142C22×C4C2×Dic9711yescharacteristic
143C22×C4C2×Dic9711yescharacteristic
144C22×C4C2×Dic9711yescharacteristic
145C22×C4C22⋊D9711yescharacteristic
146C22×C4C22⋊D9711yescharacteristic
147C22×C4C22⋊D9711yescharacteristic
148C22×C4C22⋊D9711yescharacteristic
149C22×C4C22×D9711yescharacteristic
150M16—3882no—
151C2×C8—3882no—
152C2×C8—1942no—
153C2×C8—1942no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23C22⋊Dic9711yescharacteristic
172C23C22⋊Dic9711yescharacteristic
173C23C22×Dic9711yescharacteristic
174D4—42no—
175D4—42no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4C22⋊Dic9711yescharacteristic
189C2×C4C22⋊Dic9711yescharacteristic
190C2×C4C2×C22⋊D9711yescharacteristic
191C2×C4C2×C22⋊D9711yescharacteristic
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#1569072512yes—
207C22C2×C22⋊Dic9711yescharacteristic
208C22#1569081712yes—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—41no—
214C4—41no—
215C4—41no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—81no—
220C2#2026070211yescharacteristic
221C2C2×#1569072511yescharacteristic
222C2#2026071911yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#123×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D4×C97C4
cosocleC22×C4×C97C23
hypercenterC2×#123D97
hypocenterC97#1468
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #20260699C2semidirect
2. C2×C22⋊#11715895C2semidirect
3. #20260461C2semidirect
4. #20260459C2semidirect
112. C97#1468semidirect

Maximal subgroups

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

subgroupindex
#202606992
C2×C22⋊#117158952
#202604612
#202604592
C2×#123×C972
#146897