Finite Group Databasebeta

← #27151659

Subgroups — #27151659

Order 12416.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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#20261246C211yesmaximal, characteristic
2#20261070C212yesmaximal, complemented
3#20261065C212yesmaximal, complemented
4#20261053C212yesmaximal, complemented
5#20261048C212yesmaximal, complemented
6#20260775C212yesmaximal, complemented
7#20260734C212yesmaximal, complemented
8#20260699C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11C2×#13095495C2212yes—
12#15690793—21no—
13#15690789C2211yescharacteristic
14#15690789C2211yescharacteristic
15#15690784C2211yescharacteristic
16#15690784C2211yescharacteristic
17#15690779—21no—
18C4×C22⋊D97—21no—
19C4×C22⋊D97—21no—
20#15690769C2211yescharacteristic
21#15690743C2214yescomplemented
22#15690741C2214yescomplemented
23#15690733C2214yescomplemented
24#15690730C2214yescomplemented
25C22⋊#11715895C2212yes—
26C22⋊#11715895C2212yes—
27C23×C4×C97—21no—
28C23×C4×C97C2211yescharacteristic
29C2×C4⋊C4×C97—21no—
30C2×C4⋊C4×C97—21no—
31C2×C4⋊C4×C97—21no—
32C2×C4⋊C4×C97C2212yes—
33C2×C4⋊C4×C97C2212yes—
34C2×C4⋊C4×C97C2212yes—
35#13095511—24no—
36C2×Dic194C2311yescharacteristic
37C2×D388C2311yescharacteristic
38C2×C22⋊D97—21no—
39C2×C22⋊D97—21no—
40C2×C22⋊D97—21no—
41C2×C4×D97—21no—
42#13095495—24no—
43C2×#11715895C2314yes—
44C4⋊Dic97C2311yescharacteristic
45C4⋊Dic97C2311yescharacteristic
46#13095490—21no—
47#13095490—21no—
48C22⋊Dic97—21no—
49C22⋊Dic97—21no—
50C22⋊Dic97—21no—
51#13095488—21no—
52#13095488—21no—
53C4×Dic97—21no—
54C24×C97D411yescharacteristic
55C22×C4×C97—24no—
56C22×C4×C97—22no—
57C22×C4×C97—22no—
58C22×C4×C97—22no—
59C22×C4×C97—44no—
60C22×C4×C97D411yescharacteristic
61C22×C4×C97D411yescharacteristic
62C22×C4×C97D411yescharacteristic
63C22×C4×C97C2311yescharacteristic
64C22×C4×C97C2311yescharacteristic
65C22×C4×C97C2311yescharacteristic
66C4⋊C4×C97—22no—
67C4⋊C4×C97—22no—
68C4⋊C4×C97—22no—
69C4⋊C4×C97C2314yes—
70C23×C97—22no—
71C23×C97—22no—
72C23×C97—22no—
73C23×C97C2×D411yescharacteristic
74C23×C97C2×D411yescharacteristic
75C23×C97C2×D411yescharacteristic
76C22×D97—21no—
77C2×C4×C97—22no—
78C2×C4×C97—22no—
79C2×C4×C97—22no—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—24no—
83C2×C4×C97—44no—
84C2×C4×C97—44no—
85C2×C4×C97—44no—
86C2×C4×C97C2×D411yescharacteristic
87C2×C4×C97C2×D411yescharacteristic
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97#4114yes—
90C2×C4×C97C2411yescharacteristic
91C22⋊D97—42no—
92C22⋊D97—42no—
93C22⋊D97—42no—
94C2×Dic97—21no—
95C2×Dic97—21no—
96C2×Dic97—21no—
97D388—22no—
98C4×D97—42no—
99Dic194—22no—
100#11715895—24no—
101C22×C97—28no—
102C22×C97—21no—
103C22×C97—21no—
104C22×C97—21no—
105C22×C97—21no—
106C22×C97—21no—
107C22×C97—21no—
108C22×C97#12312yes—
109C22×C97AGL(1,ℤ/8)12yes—
110C22×C97#13712yes—
111C22×C97C22×D411yescharacteristic
112C2×D97—41no—
113C2×D97—41no—
114C2×D97—41no—
115C4×C97—21no—
116C4×C97—21no—
117C4×C97—44no—
118C4×C97#12312yes—
119C4×C97C2×#4112yes—
120Dic97—41no—
121Dic97—41no—
122Dic97—41no—
123C2×C97—22no—
124C2×C97—22no—
125C2×C97—22no—
126C2×C97C2×#12311yescharacteristic
127C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C97C2×#13711yescharacteristic
129D97—81no—
130#3092—971nomaximal, Sylow
131C97#309211yescharacteristic, complemented, Sylow
132#525—971no—
133C22×C4⋊C4D9711yescharacteristic, complemented
134#484—972no—
135#483—972no—
136#481—972no—
137#480—972no—
138#428—972no—
139#417—972no—
140#407—971no—
141C2×#41—971no—
142C23×C4—21no—
143C23×C4C2×D9711yescharacteristic
144C2×M16—972no—
145#123—1941no—
146C22⋊Q8—971no—
147C22⋊Q8—971no—
148#121—971no—
149#121—971no—
150#120—1941no—
151C4×D4—1941no—
152C4×D4—1941no—
153#117—971no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D9712yes—
158C2×C4⋊C4C2×D9712yes—
159C2×C4⋊C4C2×D9712yes—
160#107—974no—
161#106—974no—
162#103—974no—
163#102—974no—
164C22⋊C8—972no—
165C22⋊C8—972no—
166C24C22⋊D9711yescharacteristic
167#41—1944no—
168C2×Q8—971no—
169C2×D4—971no—
170C2×D4—1941no—
171C2×D4—1941no—
172C2×D4—1941no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—1941no—
179C22×C4C22⋊D9711yescharacteristic
180C22×C4C22⋊D9711yescharacteristic
181C22×C4C22⋊D9711yescharacteristic
182C22×C4C22×D9711yescharacteristic
183C22×C4C22×D9711yescharacteristic
184C22×C4C22×D9711yescharacteristic
185M16—1944no—
186C2×C8—974no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—971no—
191C4⋊C4—971no—
192C4⋊C4—1941no—
193C4⋊C4—1941no—
194C4⋊C4C22×D9714yes—
195C22⋊C4—1941no—
196C22⋊C4—1941no—
197C22⋊C4—1941no—
198C22⋊C4—1941no—
199C22⋊C4—1941no—
200C42—1941no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—1941no—
205C23C2×C22⋊D9711yescharacteristic
206C23C2×C22⋊D9711yescharacteristic
207C23C2×C22⋊D9711yescharacteristic
208Q8—1942no—
209D4—3882no—
210D4—3882no—
211D4—3882no—
212D4—1942no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—3882no—
223C2×C4—1941no—
224C2×C4—1941no—
225C2×C4—1941no—
226C2×C4C2×C22⋊D9711yescharacteristic
227C2×C4C2×C22⋊D9711yescharacteristic
228C2×C4C2×C22⋊D9711yescharacteristic
229C2×C4#1309551114yes—
230C2×C4C23×D9711yescharacteristic
231C8—1944no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—3881no—
240C22—3881no—
241C22—3881no—
242C22#1569079312yes—
243C22#1569083612yes—
244C22#1569084112yes—
245C22C22×C22⋊D9711yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—3881no—
250C4—3881no—
251C4—3881no—
252C4#1569079312yes—
253C4C2×#1309551112yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—7761no—
258C2C2×#1569079311yescharacteristic
259C2C2×#1569083611yescharacteristic
260C2C2×#1569084111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×C4⋊C4×C97C2
socleC22×C97C22×D4
prime radical#20261246C2
hypercenterC22×C4⋊C4D97
hypocenterC97#3092
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20261070C2semidirect
3. #20261065C2semidirect
4. #20261053C2semidirect
5. #20261048C2semidirect
6. #20260775C2semidirect
7. #20260734C2semidirect
8. #20260699C2semidirect
9. C22×C4⋊C4×C97C2semidirect
21. #15690743C22semidirect
22. #15690741C22semidirect
23. #15690733C22semidirect
24. #15690730C22semidirect
131. C97#3092semidirect
133. C22×C4⋊C4D97semidirect

Maximal subgroups

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

subgroupindex
#202612462
#202610702
#202610652
#202610532
#202610482
#202607752
#202607342
#202606992
C22×C4⋊C4×C972
#309297