Finite Group Databasebeta

← #27140278

Subgroups — #27140278

Order 12416.

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

normal subgroups only

This group has 286 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
1#20261248C211yesmaximal, characteristic
2#20260697C211yesmaximal, characteristic, complemented
3#20260594C211yesmaximal, characteristic, complemented
4#20260472C211yesmaximal, characteristic, complemented
5#20260470C211yesmaximal, characteristic, complemented
6#20260468C211yesmaximal, characteristic, complemented
7#328×C97C211yesmaximal, characteristic, complemented
8C2×#13095512C411yescharacteristic, complemented
9C22×C4×D97C2211yescharacteristic
10C2×#13095494—21no—
11#15690795—21no—
12#15690786C411yescharacteristic, complemented
13#15690785C411yescharacteristic, complemented
14#15690784C2211yescharacteristic
15#15690780—21no—
16#15690772—21no—
17#15690770—21no—
18#15690768C411yescharacteristic, complemented
19C2×#13095490—21no—
20C2×#13095488—21no—
21C22⋊#11715895C2211yescharacteristic
22#15690722—21no—
23#15690717C2211yescharacteristic
24C22⋊Q8×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26#95×C97C2211yescharacteristic
27C23×D97D411yescharacteristic
28#13095512—42no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388C2×C411yescharacteristic
32C2×C22⋊D97—21no—
33C22×Dic97—21no—
34C22×Dic97D411yescharacteristic
35C2×C4×D97—21no—
36C2×C4×D97—42no—
37C2×C4×D97C2×C411yescharacteristic
38C2×C4×D97C2×C411yescharacteristic
39C2×C4×D97D411yescharacteristic
40C2×C4×D97D411yescharacteristic
41#13095494—42no—
42C2×#11715895—21no—
43C4⋊Dic97C2×C411yescharacteristic
44#13095490—21no—
45#13095490—42no—
46#13095488—21no—
47#13095488—21no—
48#13095488—21no—
49#13095488—21no—
50#13095488—42no—
51C4×Dic97—21no—
52C2×Q8×C97C2×C411yescharacteristic
53C22×C4×C97—21no—
54C22×C4×C97C2311yescharacteristic
55C2×C8×C97—21no—
56C4⋊C4×C97—21no—
57C4⋊C4×C97C2×C411yescharacteristic
58C22⋊C4×C97—21no—
59C23×C97C2×D411yescharacteristic
60C22×D97—21no—
61C22×D97—21no—
62C22×D97—21no—
63C22×D97—41no—
64C22×D97—41no—
65C22×D97C22⋊C411yescharacteristic
66C22×D97C22⋊C411yescharacteristic
67Q8×C97—41no—
68C2×C4×C97—21no—
69C2×C4×C97—21no—
70C2×C4×C97—41no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97—41no—
74C2×C4×C97C22×C411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C22⋊D97—41no—
77C22⋊D97—41no—
78C2×Dic97—22no—
79C2×Dic97—21no—
80C2×Dic97—41no—
81C2×Dic97—41no—
82C2×Dic97—41no—
83C2×Dic97C22⋊C411yescharacteristic
84C2×Dic97C22⋊C411yescharacteristic
85D388—41no—
86D388—42no—
87D388—42no—
88C4×D97—22no—
89C4×D97—22no—
90C4×D97—41no—
91C4×D97—41no—
92C4×D97—42no—
93C8×C97—41no—
94#11715895—41no—
95C22×C97—21no—
96C22×C97—21no—
97C22×C97C2×C22⋊C411yescharacteristic
98C22×C97AGL(1,ℤ/8)11yescharacteristic
99C22×C97#13711yescharacteristic
100C2×D97—21no—
101C2×D97—21no—
102C2×D97—21no—
103C2×D97—21no—
104C2×D97—21no—
105C2×D97—21no—
106C2×D97—42no—
107C2×D97—42no—
108C2×D97—41no—
109C2×D97—41no—
110C2×D97—41no—
111C4×C97—21no—
112C4×C97—21no—
113C4×C97—41no—
114C4×C97—41no—
115C4×C97—81no—
116Dic97—21no—
117Dic97—41no—
118Dic97—81no—
119Dic97#9912yes—
120C2×C97—21no—
121C2×C97—21no—
122C2×C97C2×#9911yescharacteristic
123C2×C97#41711yescharacteristic
124C2×C97#42111yescharacteristic
125D97—41no—
126D97—41no—
127D97—81no—
128#1513—971nomaximal, Sylow
129C97#151311yescharacteristic, complemented, Sylow
130#525—971no—
131#407—971no—
132#380—971no—
133#328—971no—
134#328D9711yescharacteristic, complemented
135#327—971no—
136#327—971no—
137C2×#41—971no—
138C23×C4—971no—
139C2×M16—1941no—
140#123—1941no—
141C22⋊Q8—971no—
142C22⋊Q8C2×D9711yescharacteristic
143#121—971no—
144#121—971no—
145#120—1941no—
146C4×D4—1941no—
147C4×D4—1941no—
148#117—971no—
149C2×C4⋊C4—1941no—
150C2×C22⋊C4—1941no—
151C22⋊C8—971no—
152C22⋊C8—1941no—
153C22⋊C8C2×D9711yescharacteristic
154#95—971no—
155#95C2×D9711yescharacteristic
156C24—971no—
157#41—3882no—
158C2×Q8C4×D9711yescharacteristic
159C2×D4—971no—
160C2×D4—1941no—
161C2×D4—1941no—
162C2×D4—1941no—
163C22×C4—21no—
164C22×C4—971no—
165C22×C4—971no—
166C22×C4—971no—
167C22×C4—971no—
168C22×C4—971no—
169C22×C4—3882no—
170C22×C4—1941no—
171C22×C4—1941no—
172C22×C4C22×D9711yescharacteristic
173M16—3882no—
174C2×C8—21no—
175C2×C8—1941no—
176C4⋊C4—21no—
177C4⋊C4—971no—
178C4⋊C4—3882no—
179C4⋊C4—1941no—
180C4⋊C4C4×D9711yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—3882no—
183C22⋊C4—1941no—
184C22⋊C4—1941no—
185C22⋊C4—1941no—
186C22⋊C4—1941no—
187C42—1941no—
188C23—971no—
189C23—971no—
190C23—3881no—
191C23—3881no—
192C23—1941no—
193C23—1941no—
194C23—1941no—
195C23D4×D9711yescharacteristic
196Q8—41no—
197D4—3881no—
198D4—3882no—
199D4—3882no—
200D4—3881no—
201D4—3881no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—971no—
209C2×C4—971no—
210C2×C4—3881no—
211C2×C4—3881no—
212C2×C4—3882no—
213C2×C4—3881no—
214C2×C4—3881no—
215C2×C4—3881no—
216C2×C4—1942no—
217C2×C4—1942no—
218C2×C4—1942no—
219C2×C4—1941no—
220C2×C4C2×C4×D9711yescharacteristic
221C2×C4D4×D9711yescharacteristic
222C8—41no—
223C8—3881no—
224C22—21no—
225C22—21no—
226C22—3882no—
227C22—3882no—
228C22—3881no—
229C22—3881no—
230C22—3881no—
231C22—1941no—
232C22—1941no—
233C22—1941no—
234C22—1941no—
235C22—1941no—
236C22—1941no—
237C22C22⋊C4×D9711yescharacteristic
238C22#1569083711yescharacteristic
239C22#1569084211yescharacteristic
240C4—21no—
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—7761no—
246C4—972no—
247C4—3881no—
248C4—1941no—
249C2—21no—
250C2—21no—
251C2—7761no—
252C2—3881no—
253C2—3881no—
254C2#99×D9711yescharacteristic
255C2#2026073511yescharacteristic
256C2#2026075011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#328×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261248C2
cosocleC22×C4×C97C23
hypercenter#328D97
hypocenterC97#1513
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #20260697C2semidirect
3. #20260594C2semidirect
4. #20260472C2semidirect
5. #20260470C2semidirect
6. #20260468C2semidirect
7. #328×C97C2semidirect
8. C2×#13095512C4semidirect
12. #15690786C4semidirect
13. #15690785C4semidirect
18. #15690768C4semidirect
129. C97#1513semidirect
134. #328D97semidirect

Maximal subgroups

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

subgroupindex
#202612482
#202606972
#202605942
#202604722
#202604702
#202604682
#328×C972
#151397