Finite Group Databasebeta

← #27140082

Subgroups — #27140082

Order 12416.

280 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 73 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×#15690793C211yesmaximal, characteristic
2#20260699C211yesmaximal, characteristic, complemented
3#20260460C212yesmaximal, complemented
4#20260458C212yesmaximal, complemented
5C2×C22⋊C8×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97C411yescharacteristic
7C2×#13095495—21no—
8#15690793—21no—
9#15690793—21no—
10#15690793—21no—
11#15690793—21no—
12C2×#13095490C411yescharacteristic
13C2×#13095490C2211yescharacteristic
14C2×C22⋊Dic97C411yescharacteristic
15C2×#13095488C411yescharacteristic
16C2×#13095488C2211yescharacteristic
17C22⋊#11715895—21no—
18C22⋊#11715895C2212yes—
19C23×C4×C97C2211yescharacteristic
20C22×C8×C97—21no—
21C22⋊C8×C97—21no—
22C22⋊C8×C97C2212yes—
23C23×D97C2×C411yescharacteristic
24C2×C22⋊D97—21no—
25C2×C22⋊D97—21no—
26C2×C22⋊D97—42no—
27C22×Dic97C2×C411yescharacteristic
28C22×Dic97C2×C411yescharacteristic
29C22×Dic97C2×C411yescharacteristic
30#13095495—42no—
31C2×#11715895—22no—
32#13095490—21no—
33#13095490—21no—
34#13095490—21no—
35#13095490—21no—
36C22⋊Dic97—21no—
37C22⋊Dic97—21no—
38#13095488—21no—
39#13095488—21no—
40#13095488—21no—
41#13095488—21no—
42C24×C97C2×C411yescharacteristic
43C22×C4×C97—42no—
44C22×C4×C97C2×C411yescharacteristic
45C22×C4×C97D411yescharacteristic
46C22×C4×C97D411yescharacteristic
47C22×C4×C97D411yescharacteristic
48C22×C4×C97D411yescharacteristic
49C22×C4×C97C2311yescharacteristic
50C2×C8×C97—22no—
51C2×C8×C97—42no—
52C23×C97—22no—
53C23×C97—22no—
54C23×C97—41no—
55C23×C97C22⋊C411yescharacteristic
56C23×C97C22⋊C411yescharacteristic
57C23×C97C22×C411yescharacteristic
58C22×D97—21no—
59C22×D97—22no—
60C22×D97—41no—
61C22×D97—41no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97—41no—
68C2×C4×C97C22⋊C411yescharacteristic
69C2×C4×C97C22⋊C411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C22⋊D97—42no—
73C22⋊D97—42no—
74C2×Dic97—21no—
75C2×Dic97—21no—
76C2×Dic97—21no—
77C2×Dic97—22no—
78C2×Dic97—41no—
79C2×Dic97—41no—
80C8×C97—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#9912yes—
91C22×C97C2×C22⋊C411yescharacteristic
92C22×C97#13112yes—
93C2×D97—41no—
94C2×D97—41no—
95C2×D97—41no—
96C2×D97—82no—
97C4×C97—21no—
98C4×C97—21no—
99C4×C97—21no—
100C4×C97—21no—
101Dic97—41no—
102Dic97—41no—
103Dic97—41no—
104C2×C97—22no—
105C2×C97—22no—
106C2×C97—41no—
107C2×C97#40811yescharacteristic
108C2×C97C2×#9911yescharacteristic
109C2×C97#41311yescharacteristic
110D97—81no—
111#1468—971nomaximal, Sylow
112C97#146811yescharacteristic, complemented, Sylow
113C2×#123—971no—
114#407—971no—
115C2×C22⋊C8D9711yescharacteristic, complemented
116#324—972no—
117#323—972no—
118C22×D4—971no—
119C23×C4C2×D9711yescharacteristic
120C2×M16—1941no—
121C22×C8—21no—
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—21no—
132C22⋊C8—972no—
133C22⋊C8—1941no—
134C22⋊C8C2×D9712yes—
135C24—971no—
136C24C4×D9711yescharacteristic
137C2×D4—3882no—
138C2×D4—1941no—
139C2×D4—1941no—
140C22×C4—42no—
141C22×C4—971no—
142C22×C4—971no—
143C22×C4—971no—
144C22×C4C4×D9711yescharacteristic
145C22×C4D38811yescharacteristic
146C22×C4D38811yescharacteristic
147C22×C4C22⋊D9711yescharacteristic
148C22×C4C22⋊D9711yescharacteristic
149C22×C4C22×D9711yescharacteristic
150M16—3882no—
151C2×C8—22no—
152C2×C8—42no—
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—41no—
167C23—3881no—
168C23—3881no—
169C23—1942no—
170C23—1941no—
171C23#1309548811yescharacteristic
172C23#1309548811yescharacteristic
173C23C2×C4×D9711yescharacteristic
174D4—3882no—
175D4—3882no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—3881no—
183C2×C4—3881no—
184C2×C4—1941no—
185C2×C4—1941no—
186C2×C4—1941no—
187C2×C4—1942no—
188C2×C4#1309548811yescharacteristic
189C2×C4#1309548811yescharacteristic
190C2×C4C2×C22⋊D9711yescharacteristic
191C2×C4C2×D38811yescharacteristic
192C8—42no—
193C8—3882no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—7762no—
203C22—3881no—
204C22—3881no—
205C22—3881no—
206C22#1569072412yes—
207C22C2×#1309548811yescharacteristic
208C22#1569081612yes—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—3881no—
214C4—3881no—
215C4—3881no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—7761no—
220C2#2026070111yescharacteristic
221C2C2×#1569072411yescharacteristic
222C2#2026071711yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×C22⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×C22⋊D97C4
cosocleC22×C4×C97C23
hypercenterC2×C22⋊C8D97
hypocenterC97#1468
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #20260699C2semidirect
3. #20260460C2semidirect
4. #20260458C2semidirect
5. C2×C22⋊C8×C97C2semidirect
112. C97#1468semidirect
115. C2×C22⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907932
#202606992
#202604602
#202604582
C2×C22⋊C8×C972
#146897