Finite Group Databasebeta

← #27143381

Subgroups — #27143381

Order 12416.

280 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261296C211yesmaximal, characteristic
2#100×D97C211yesmaximal, characteristic, complemented
3#99×D97C211yesmaximal, characteristic, complemented
4#20260525C211yesmaximal, characteristic, complemented
5#20260523C211yesmaximal, characteristic, complemented
6#20260521C211yesmaximal, characteristic, complemented
7#352×C97C211yesmaximal, characteristic, complemented
8C22×D388C411yescharacteristic, complemented
9C2×D4×D97C2211yescharacteristic
10M16×D97—21no—
11#15690795—21no—
12#15690793C411yescharacteristic, complemented
13#15690785—21no—
14#15690783—21no—
15#15690780—21no—
16#15690780C411yescharacteristic, complemented
17#15690778C2211yescharacteristic
18C22⋊C4×D97—21no—
19C22⋊C4×D97C411yescharacteristic, complemented
20#15690727C2211yescharacteristic
21#15690726—21no—
22#15690725C2211yescharacteristic
23#15690724—21no—
24#123×C97C2211yescharacteristic
25#100×C97C2211yescharacteristic
26#99×C97C2211yescharacteristic
27C23×D97C2×C411yescharacteristic
28C23×D97C2×C411yescharacteristic, complemented
29C23×D97D411yescharacteristic
30C2×D388—21no—
31C2×D388—21no—
32C2×D388—21no—
33C2×D388—41no—
34C2×D388—41no—
35C2×D388D411yescharacteristic
36C2×C22⋊D97—21no—
37C2×C22⋊D97C2×C411yescharacteristic
38C2×C22⋊D97D411yescharacteristic
39D4×D97—41no—
40D4×D97—41no—
41C2×C4×D97—21no—
42C2×C4×D97—41no—
43C2×C4×D97D411yescharacteristic
44#13095495—21no—
45#13095494—41no—
46C8×D97—41no—
47#13095490—21no—
48C22⋊Dic97—21no—
49C22⋊Dic97C2×C411yescharacteristic, complemented
50#13095488—21no—
51#13095488—21no—
52#13095488—21no—
53#13095488—21no—
54#13095488—41no—
55C2×D4×C97C2311yescharacteristic
56C22×C4×C97C2×C411yescharacteristic, complemented
57M16×C97—21no—
58C4⋊C4×C97—21no—
59C22⋊C4×C97—21no—
60C22⋊C4×C97—21no—
61C22⋊C4×C97C2×C411yescharacteristic, complemented
62C23×C97C22×C411yescharacteristic
63C23×C97C2×D411yescharacteristic
64C22×D97—21no—
65C22×D97—21no—
66C22×D97—21no—
67C22×D97—21no—
68C22×D97—21no—
69C22×D97—41no—
70C22×D97—81no—
71C22×D97—81no—
72C22×D97—81no—
73C22×D97C22⋊C411yescharacteristic
74C22×D97C22⋊C411yescharacteristic
75C22×D97C22⋊C411yescharacteristic
76D4×C97—41no—
77C2×C4×C97—21no—
78C2×C4×C97—21no—
79C2×C4×C97—41no—
80C2×C4×C97—41no—
81C2×C4×C97C2×D411yescharacteristic
82C22⋊D97—41no—
83C22⋊D97—41no—
84C22⋊D97—81no—
85C2×Dic97—21no—
86C2×Dic97—41no—
87C2×Dic97C22⋊C411yescharacteristic
88D388—41no—
89D388—42no—
90D388—41no—
91D388—41no—
92D388—81no—
93C4×D97—21no—
94C4×D97—21no—
95C4×D97—81no—
96C4×D97—82no—
97C8×C97—41no—
98#11715895—41no—
99C22×C97—21no—
100C22×C97—21no—
101C22×C97—41no—
102C22×C97—41no—
103C22×C97C2×C22⋊C411yescharacteristic
104C2×D97—21no—
105C2×D97—21no—
106C2×D97—41no—
107C2×D97—42no—
108C2×D97—41no—
109C2×D97—41no—
110C2×D97—41no—
111C2×D97—41no—
112C2×D97—41no—
113C2×D97—81no—
114C2×D97—81no—
115C2×D97—81no—
116C2×D97—82no—
117C2×D97#9911yescharacteristic
118C2×D97#9911yescharacteristic
119C4×C97—21no—
120C4×C97—41no—
121C4×C97—41no—
122C4×C97—81no—
123Dic97—21no—
124Dic97—41no—
125Dic97—81no—
126C2×C97—21no—
127C2×C97—41no—
128C2×C97—41no—
129C2×C97C2×#9911yescharacteristic
130D97—21no—
131D97—21no—
132D97—41no—
133D97—41no—
134D97—81no—
135#2121—971nomaximal, Sylow
136C97#212111yescharacteristic, complemented, Sylow
137#534—971no—
138C2×#100—971no—
139C2×#99—971no—
140#352—971no—
141#352D9711yescharacteristic, complemented
142AΣL(1,𝔽16)—971no—
143AΣL(1,𝔽16)—971no—
144C22×D4—971no—
145C22×D4—971no—
146C2×M16—1941no—
147#123—971no—
148#123—1941no—
149#123C2×D9711yescharacteristic
150#121—1941no—
151#121—1941no—
152#120—971no—
153#120—971no—
154#120—1941no—
155C2×C22⋊C4—971no—
156C2×C22⋊C4—1941no—
157#100—971no—
158#100—1941no—
159#100C2×D9711yescharacteristic
160#99—971no—
161#99—1941no—
162#99C2×D9711yescharacteristic
163C24—971no—
164C24—971no—
165C24—971no—
166C2×D4—971no—
167C2×D4—971no—
168C2×D4—971no—
169C2×D4—3881no—
170C2×D4—3881no—
171C2×D4—3881no—
172C2×D4—3881no—
173C2×D4—1941no—
174C2×D4—1941no—
175C2×D4—1941no—
176C2×D4—1941no—
177C2×D4C22×D9711yescharacteristic
178C22×C4—971no—
179C22×C4—3881no—
180C22×C4—1941no—
181C22×C4C4×D9711yescharacteristic, complemented
182M16—21no—
183M16—3881no—
184M16—1941no—
185C2×C8—3881no—
186C4⋊C4—21no—
187C4⋊C4—1941no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—971no—
191C22⋊C4—3881no—
192C22⋊C4—1941no—
193C22⋊C4—1941no—
194C22⋊C4—1941no—
195C22⋊C4—1941no—
196C22⋊C4—1941no—
197C22⋊C4C4×D9711yescharacteristic, complemented
198C23—7761no—
199C23—7761no—
200C23—7761no—
201C23—971no—
202C23—971no—
203C23—971no—
204C23—3881no—
205C23—1941no—
206C23—1941no—
207C23—1941no—
208C23—1941no—
209C23—1941no—
210C23C2×C4×D9711yescharacteristic
211C23D4×D9711yescharacteristic
212D4—41no—
213D4—7761no—
214D4—7761no—
215D4—3881no—
216D4—3881no—
217D4—3881no—
218D4—3882no—
219D4—3881no—
220D4—3881no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—7761no—
226C2×C4—7762no—
227C2×C4—971no—
228C2×C4—3881no—
229C2×C4—1941no—
230C2×C4—1941no—
231C2×C4—1941no—
232C2×C4D4×D9711yescharacteristic
233C8—41no—
234C8—3881no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—41no—
239C22—7762no—
240C22—7761no—
241C22—7761no—
242C22—7761no—
243C22—971no—
244C22—971no—
245C22—3881no—
246C22—3881no—
247C22—3881no—
248C22—3881no—
249C22—3881no—
250C22—3882no—
251C22—3881no—
252C22—1941no—
253C22—1941no—
254C22C22⋊C4×D9711yescharacteristic
255C4—21no—
256C4—41no—
257C4—41no—
258C4—81no—
259C4—7761no—
260C4—3881no—
261C4—1941no—
262C2—21no—
263C2—41no—
264C2—41no—
265C2—7761no—
266C2—3881no—
267C2—3881no—
268C2—1941no—
269C2—1941no—
270C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC23×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#352×C97C2
socleC2×C97C2×#99
prime radical#20261296C2
cosocleC2×D4×C97C23
hypercenter#352D97
hypocenterC97#2121
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
2. #100×D97C2semidirect
3. #99×D97C2semidirect
4. #20260525C2semidirect
5. #20260523C2semidirect
6. #20260521C2semidirect
7. #352×C97C2semidirect
8. C22×D388C4semidirect
12. #15690793C4semidirect
16. #15690780C4semidirect
19. C22⋊C4×D97C4semidirect
28. C23×D97C2×C4semidirect
49. C22⋊Dic97C2×C4semidirect
56. C22×C4×C97C2×C4semidirect
61. C22⋊C4×C97C2×C4semidirect
136. C97#2121semidirect
141. #352D97semidirect
181. C22×C4C4×D97semidirect
197. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
#202612962
#100×D972
#99×D972
#202605252
#202605232
#202605212
#352×C972
#212197