Finite Group Databasebeta

← #27140081

Subgroups — #27140081

Order 12416.

280 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 63 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×#15690792C211yesmaximal, characteristic
2#20260697C211yesmaximal, characteristic, complemented
3C22⋊C8×D97C211yesmaximal, characteristic, complemented
4#20260460C211yesmaximal, characteristic, complemented
5#20260460C211yesmaximal, characteristic, complemented
6#20260459C211yesmaximal, characteristic, complemented
7#323×C97C211yesmaximal, characteristic, complemented
8C22×C22⋊D97C411yescharacteristic
9C22×C4×D97C2211yescharacteristic
10C2×#13095494—21no—
11C2×C8×D97—21no—
12#15690792—21no—
13#15690792—21no—
14#15690792—21no—
15#15690792—21no—
16C2×C4⋊Dic97C411yescharacteristic
17C2×#13095490C2211yescharacteristic
18C2×C22⋊Dic97C411yescharacteristic
19C2×#13095488C411yescharacteristic
20C22⋊#11715895C2211yescharacteristic
21C22⋊#11715895C2211yescharacteristic
22#15690722—21no—
23#15690722—21no—
24C2×C22⋊C4×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26C22⋊C8×C97C2211yescharacteristic
27C23×D97C2×C411yescharacteristic
28C2×C22⋊D97—21no—
29C2×C22⋊D97—21no—
30C2×C22⋊D97—42no—
31C22×Dic97C2×C411yescharacteristic
32C22×Dic97C2×C411yescharacteristic
33C22×Dic97C2×C411yescharacteristic
34C2×C4×D97—42no—
35C2×C4×D97D411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39#13095494—42no—
40C2×#11715895—21no—
41C2×#11715895—21no—
42C8×D97—42no—
43C4⋊Dic97—21no—
44C4⋊Dic97—21no—
45#13095490—21no—
46#13095490—21no—
47C22⋊Dic97—21no—
48C22⋊Dic97—21no—
49#13095488—21no—
50#13095488—21no—
51C24×C97C2×C411yescharacteristic
52C22×C4×C97C2×C411yescharacteristic
53C22×C4×C97C2311yescharacteristic
54C2×C8×C97—21no—
55C2×C8×C97—21no—
56C22⋊C4×C97—21no—
57C22⋊C4×C97—21no—
58C23×C97—21no—
59C23×C97—21no—
60C23×C97—21no—
61C23×C97—41no—
62C23×C97—41no—
63C23×C97C22×C411yescharacteristic
64C22×D97—21no—
65C22×D97—21no—
66C22×D97—22no—
67C22×D97—41no—
68C22×D97C22⋊C411yescharacteristic
69C22×D97C22⋊C411yescharacteristic
70C2×C4×C97—21no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2×D411yescharacteristic
75C22⋊D97—42no—
76C22⋊D97—42no—
77C2×Dic97—21no—
78C2×Dic97—21no—
79C2×Dic97—21no—
80C2×Dic97—21no—
81C2×Dic97—41no—
82C2×Dic97—41no—
83C2×Dic97C22⋊C411yescharacteristic
84C2×Dic97C22⋊C411yescharacteristic
85C4×D97—22no—
86C4×D97—22no—
87C4×D97—41no—
88C4×D97—41no—
89C8×C97—41no—
90C8×C97—41no—
91#11715895—41no—
92#11715895—41no—
93C22×C97—21no—
94C22×C97—21no—
95C22×C97—41no—
96C22×C97—41no—
97C22×C97—41no—
98C22×C97—81no—
99C22×C97—81no—
100C22×C97C2×C22⋊C411yescharacteristic
101C22×C97#13111yescharacteristic
102C22×C97#13111yescharacteristic
103C2×D97—21no—
104C2×D97—21no—
105C2×D97—21no—
106C2×D97—21no—
107C2×D97—21no—
108C2×D97—42no—
109C2×D97—42no—
110C2×D97#9912yes—
111C4×C97—21no—
112C4×C97—21no—
113C4×C97—41no—
114Dic97—21no—
115Dic97—21no—
116Dic97—41no—
117Dic97—41no—
118C2×C97—21no—
119C2×C97—21no—
120C2×C97—81no—
121C2×C97#40811yescharacteristic
122C2×C97C2×#9911yescharacteristic
123C2×C97#41311yescharacteristic
124D97—22no—
125D97—41no—
126#1468—971nomaximal, Sylow
127C97#146811yescharacteristic, complemented, Sylow
128C2×#123—971no—
129#407—971no—
130C2×C22⋊C8—971no—
131#324—971no—
132#324—971no—
133#323—971no—
134#323D9711yescharacteristic, complemented
135C22×D4—971no—
136C23×C4—971no—
137C2×M16—1941no—
138C22×C8—1941no—
139#123—1941no—
140#123—1941no—
141#123—1941no—
142#123—1941no—
143C2×C4⋊C4—971no—
144C2×C4⋊C4—971no—
145C2×C22⋊C4—971no—
146C2×C22⋊C4—971no—
147C2×C22⋊C4C2×D9711yescharacteristic
148C22⋊C8—971no—
149C22⋊C8—971no—
150C22⋊C8—1941no—
151C22⋊C8—1941no—
152C22⋊C8C2×D9711yescharacteristic
153C22⋊C8C2×D9711yescharacteristic
154C24—971no—
155C24C4×D9711yescharacteristic
156C2×D4—3882no—
157C2×D4—1941no—
158C2×D4—1941no—
159C22×C4—971no—
160C22×C4—971no—
161C22×C4—971no—
162C22×C4—971no—
163C22×C4—971no—
164C22×C4—971no—
165C22×C4—971no—
166C22×C4—3882no—
167C22×C4C4×D9711yescharacteristic
168C22×C4C22×D9711yescharacteristic
169M16—3882no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—3882no—
173C2×C8—1941no—
174C2×C8—1941no—
175C4⋊C4—1941no—
176C4⋊C4—1941no—
177C4⋊C4—1941no—
178C4⋊C4—1941no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—1941no—
182C22⋊C4—1941no—
183C22⋊C4—1941no—
184C22⋊C4—1941no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—41no—
190C23—971no—
191C23—971no—
192C23—3881no—
193C23—1942no—
194C23—1941no—
195C23—1941no—
196C23C2×C4×D9711yescharacteristic
197D4—3882no—
198D4—3882no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—971no—
203C2×C4—971no—
204C2×C4—3881no—
205C2×C4—3881no—
206C2×C4—3881no—
207C2×C4—3881no—
208C2×C4—1942no—
209C2×C4—1942no—
210C2×C4—1941no—
211C2×C4—1941no—
212C2×C4—1941no—
213C2×C4—1941no—
214C2×C4D4×D9711yescharacteristic
215C2×C4D4×D9711yescharacteristic
216C8—41no—
217C8—41no—
218C8—3881no—
219C8—3881no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—81no—
226C22—81no—
227C22—972no—
228C22—3882no—
229C22—3882no—
230C22—1941no—
231C22—1941no—
232C22—1941no—
233C22—1941no—
234C22—1941no—
235C22C22⋊C4×D9711yescharacteristic
236C22#1569081611yescharacteristic
237C22#1569081811yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—3881no—
242C4—3881no—
243C4—1941no—
244C4—1941no—
245C2—21no—
246C2—21no—
247C2—81no—
248C2—3881no—
249C2—1942no—
250C2#2026070011yescharacteristic
251C2#99×D9711yescharacteristic
252C2#2026071611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#323×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×C22⋊D97C4
cosocleC22×C4×C97C23
hypercenter#323D97
hypocenterC97#1468
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20260697C2semidirect
3. C22⋊C8×D97C2semidirect
4. #20260460C2semidirect
5. #20260460C2semidirect
6. #20260459C2semidirect
7. #323×C97C2semidirect
127. C97#1468semidirect
134. #323D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907922
#202606972
C22⋊C8×D972
#202604602
#202604592
#323×C972
#146897