Finite Group Databasebeta

← #27143380

Subgroups — #27143380

Order 12416.

280 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261294C211yesmaximal, characteristic
2C2×#15690727C211yesmaximal, characteristic, complemented
3#20260524C212yesmaximal, complemented
4#20260521C212yesmaximal, complemented
5C2×#99×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97C411yescharacteristic, complemented
7C2×#13095495—21no—
8#15690793—21no—
9#15690791C411yescharacteristic, complemented
10#15690791C2211yescharacteristic
11#15690782—21no—
12#15690782—21no—
13#15690779—21no—
14#15690778C411yescharacteristic, complemented
15#15690778C2211yescharacteristic
16C2×C22⋊Dic97C411yescharacteristic, complemented
17#15690727—21no—
18#15690727C2212yes—
19C22×D4×C97C2211yescharacteristic
20C2×C22⋊C4×C97—21no—
21#99×C97—21no—
22#99×C97C2212yes—
23C23×D97C2×C411yescharacteristic, complemented
24C2×C22⋊D97—21no—
25C2×C22⋊D97—21no—
26C2×C22⋊D97—21no—
27C2×C22⋊D97—21no—
28C2×C22⋊D97—41no—
29C2×C22⋊D97—41no—
30C22×Dic97—21no—
31C22×Dic97C2×C411yescharacteristic, complemented
32#13095495—22no—
33#13095495—41no—
34C2×#11715895—41no—
35#13095490—21no—
36#13095490—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39C22⋊Dic97—21no—
40C22⋊Dic97C2×C411yescharacteristic, complemented
41C22⋊Dic97C2×C411yescharacteristic, complemented
42#13095488—21no—
43#13095488—21no—
44C24×C97C2×C411yescharacteristic
45C24×C97D411yescharacteristic
46C2×D4×C97—41no—
47C2×D4×C97—41no—
48C2×D4×C97C2×C411yescharacteristic
49C2×D4×C97D411yescharacteristic
50C2×D4×C97D411yescharacteristic
51C2×D4×C97C2311yescharacteristic
52C22×C4×C97—41no—
53C22×C4×C97D411yescharacteristic
54C22⋊C4×C97—22no—
55C22⋊C4×C97—41no—
56C23×C97—21no—
57C23×C97—21no—
58C23×C97—21no—
59C23×C97—21no—
60C23×C97—81no—
61C23×C97—81no—
62C23×C97C22⋊C411yescharacteristic
63C23×C97C22⋊C411yescharacteristic
64C23×C97C22⋊C411yescharacteristic
65C23×C97C22×C411yescharacteristic
66C23×C97C2×D411yescharacteristic
67C22×D97—21no—
68C22×D97—41no—
69C22×D97—81no—
70D4×C97—41no—
71D4×C97—41no—
72D4×C97—41no—
73D4×C97—41no—
74C2×C4×C97—21no—
75C2×C4×C97—21no—
76C2×C4×C97—42no—
77C2×C4×C97—82no—
78C2×C4×C97C22⋊C411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C22⋊D97—42no—
81C22⋊D97—41no—
82C22⋊D97—41no—
83C22⋊D97—81no—
84C22⋊D97—81no—
85C2×Dic97—21no—
86C2×Dic97—21no—
87C2×Dic97—21no—
88C2×Dic97—41no—
89C2×Dic97—81no—
90#11715895—42no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97—21no—
94C22×C97—21no—
95C22×C97—41no—
96C22×C97—41no—
97C22×C97—41no—
98C22×C97—41no—
99C22×C97—41no—
100C22×C97—42no—
101C22×C97—81no—
102C22×C97—81no—
103C22×C97—81no—
104C22×C97#9911yescharacteristic
105C22×C97#9911yescharacteristic
106C22×C97C2×C22⋊C411yescharacteristic
107C2×D97—41no—
108C2×D97—41no—
109C2×D97—41no—
110C2×D97—82no—
111C4×C97—21no—
112C4×C97—21no—
113C4×C97—82no—
114Dic97—41no—
115Dic97—41no—
116Dic97—41no—
117C2×C97—21no—
118C2×C97—21no—
119C2×C97—21no—
120C2×C97—41no—
121C2×C97—41no—
122C2×C97—41no—
123C2×C97—41no—
124C2×C97C2×#9911yescharacteristic
125D97—81no—
126#2121—971nomaximal, Sylow
127C97#212111yescharacteristic, complemented, Sylow
128#534—971no—
129C2×#100—971no—
130C2×#99D9711yescharacteristic, complemented
131#352—972no—
132AΣL(1,𝔽16)—972no—
133C22×D4—971no—
134C22×D4C2×D9711yescharacteristic
135C2×M16—1941no—
136#123—971no—
137#123—971no—
138#123—1941no—
139#121—1941no—
140#121—1941no—
141#120—971no—
142#120—971no—
143#120—1941no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—971no—
146#100—972no—
147#100—1941no—
148#99—21no—
149#99C2×D9712yes—
150C24—971no—
151C24C4×D9711yescharacteristic
152C24C22⋊D9711yescharacteristic
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—3881no—
156C2×D4—3881no—
157C2×D4—1941no—
158C2×D4—1941no—
159C2×D4—1941no—
160C2×D4—1941no—
161C2×D4C4×D9711yescharacteristic
162C2×D4D38811yescharacteristic
163C2×D4C22⋊D9711yescharacteristic
164C2×D4C22×D9711yescharacteristic
165C22×C4—41no—
166C22×C4—971no—
167C22×C4—1941no—
168C22×C4D38811yescharacteristic
169M16—3881no—
170M16—1942no—
171C2×C8—3881no—
172C4⋊C4—1941no—
173C4⋊C4—1941no—
174C22⋊C4—22no—
175C22⋊C4—41no—
176C22⋊C4—971no—
177C22⋊C4—971no—
178C22⋊C4—1941no—
179C22⋊C4—1941no—
180C22⋊C4—1941no—
181C22⋊C4—1941no—
182C22⋊C4—1941no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—81no—
188C23—81no—
189C23—7761no—
190C23—3881no—
191C23—1941no—
192C23#1309548811yescharacteristic
193C23#1309548811yescharacteristic
194C23#1309548811yescharacteristic
195C23C2×C4×D9711yescharacteristic
196C23C2×C22⋊D9711yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—7761no—
202D4—7761no—
203D4—3882no—
204D4—3881no—
205D4—3881no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—82no—
210C2×C4—7761no—
211C2×C4—3881no—
212C2×C4—1941no—
213C2×C4—1941no—
214C2×C4—1941no—
215C2×C4#1309548811yescharacteristic
216C2×C4C2×D38811yescharacteristic
217C8—3882no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—7762no—
232C22—3881no—
233C22—3881no—
234C22—3881no—
235C22#1569072411yescharacteristic
236C22#1569072411yescharacteristic
237C22C2×#1309548811yescharacteristic
238C4—21no—
239C4—21no—
240C4—82no—
241C4—3881no—
242C4—3881no—
243C4—3881no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—7761no—
252C2C2×#1569072411yescharacteristic
subgroupG/H
centerC2C2×#15690724
commutatorC23×C97C22×C4
FrattiniC2×D4C22×D97
FittingC2×#99×C97C2
socleC2×C97C2×#99
prime radical#20261294C2
cosocleC2×D4×C97C23
hypercenterC2×#99D97
hypocenterC97#2121
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#15690727C2semidirect
3. #20260524C2semidirect
4. #20260521C2semidirect
5. C2×#99×C97C2semidirect
6. C22×C22⋊D97C4semidirect
9. #15690791C4semidirect
14. #15690778C4semidirect
16. C2×C22⋊Dic97C4semidirect
23. C23×D97C2×C4semidirect
31. C22×Dic97C2×C4semidirect
40. C22⋊Dic97C2×C4semidirect
41. C22⋊Dic97C2×C4semidirect
127. C97#2121semidirect
130. C2×#99D97semidirect

Maximal subgroups

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

subgroupindex
#202612942
C2×#156907272
#202605242
#202605212
C2×#99×C972
#212197