Finite Group Databasebeta

← #27144993

Subgroups — #27144993

Order 12416.

560 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 157 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×C4×C22⋊D97C211yesmaximal, characteristic
2#20260673C211yesmaximal, characteristic, complemented
3#20260670C211yesmaximal, characteristic, complemented
4#20260666C211yesmaximal, characteristic, complemented
5#20260663C212yesmaximal, complemented
6#20260661C211yesmaximal, characteristic, complemented
7#20260650C212yesmaximal
8C4×#13095490C212yesmaximal, complemented
9C4×#13095488C212yesmaximal, complemented
10C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
11C2×#117×C97C211yesmaximal, characteristic, complemented
12C22×C22⋊D97C2211yescharacteristic
13C22×C4×D97C2211yescharacteristic
14C4×C22⋊D97—24no—
15C2×#13095490C2212yes—
16C2×#13095490C2212yes—
17C2×#13095490C2211yescharacteristic
18C2×C22⋊Dic97C2212yes—
19C2×C22⋊Dic97C2211yescharacteristic
20C2×#13095488C2212yes—
21C2×#13095488C2212yes—
22C2×#13095488C2211yescharacteristic
23C2×C4×Dic97C2211yescharacteristic
24C2×C4×Dic97C2212yes—
25#15690717C2212yes—
26#15690717C2212yes—
27#15690717C2211yescharacteristic
28#15690717C2212yes—
29#15690717C2212yes—
30#15690717C2211yescharacteristic
31C23×C4×C97C2211yescharacteristic
32#117×C97—24no—
33C2×C4⋊C4×C97C2211yescharacteristic
34C2×C4⋊C4×C97C2211yescharacteristic
35C2×C22⋊C4×C97C2212yes—
36C2×C42×C97C2212yes—
37C23×D97C2311yescharacteristic
38C2×C22⋊D97—22no—
39C2×C22⋊D97—22no—
40C2×C22⋊D97—22no—
41C22×Dic97C2311yescharacteristic
42C22×Dic97C2312yes—
43C22×Dic97C2311yescharacteristic
44C22×Dic97C2312yes—
45C22×Dic97C2311yescharacteristic
46C2×C4×D97—22no—
47C2×C4×D97—24no—
48#13095490—22no—
49#13095490—24no—
50#13095490—24no—
51C22⋊Dic97—22no—
52C22⋊Dic97—24no—
53#13095488—24no—
54#13095488—22no—
55#13095488—24no—
56C4×Dic97—22no—
57C4×Dic97—24no—
58C24×C97C2311yescharacteristic
59C22×C4×C97—24no—
60C22×C4×C97D414yes—
61C22×C4×C97C2312yes—
62C22×C4×C97C2312yes—
63C22×C4×C97C2311yescharacteristic
64C22×C4×C97C2311yescharacteristic
65C4⋊C4×C97—22no—
66C4⋊C4×C97—22no—
67C22⋊C4×C97—24no—
68C42×C97—24no—
69C23×C97—22no—
70C23×C97—22no—
71C23×C97—21no—
72C23×C97—21no—
73C23×C97C2×D412yes—
74C23×C97C2411yescharacteristic
75C22×D97—21no—
76C22×D97—22no—
77C22×D97—22no—
78C22×D97—21no—
79C22×D97#4112yes—
80C2×C4×C97—22no—
81C2×C4×C97—28no—
82C2×C4×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97—22no—
85C2×C4×C97—22no—
86C2×C4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97C2×D412yes—
89C2×C4×C97C2×D412yes—
90C2×C4×C97#4114yes—
91C22⋊D97—28no—
92C2×Dic97—21no—
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97—21no—
96C2×Dic97—21no—
97C2×Dic97—21no—
98C2×Dic97—22no—
99C2×Dic97—22no—
100C2×Dic97—22no—
101C2×Dic97—22no—
102C2×Dic97—22no—
103C2×Dic97#4114yes—
104C2×Dic97#4112yes—
105C4×D97—44no—
106C22×C97—22no—
107C22×C97—22no—
108C22×C97—22no—
109C22×C97—42no—
110C22×C97—42no—
111C22×C97C22×D411yescharacteristic
112C22×C97C2×#4111yescharacteristic
113C22×C97C2×#4111yescharacteristic
114C22×C97C2×#4112yes—
115C22×C97C2×#4112yes—
116C2×D97—22no—
117C2×D97—24no—
118C2×D97—41no—
119C2×D97—42no—
120C2×D97—41no—
121C4×C97—22no—
122C4×C97—42no—
123C4×C97—42no—
124C4×C97#12314yes—
125Dic97—22no—
126Dic97—24no—
127Dic97—41no—
128Dic97—42no—
129Dic97—41no—
130C2×C97—42no—
131C2×C97C2×#12311yescharacteristic
132C2×C97#52511yescharacteristic
133C2×C97#52912yes—
134C2×C97#52912yes—
135C2×C97#53211yescharacteristic
136D97—42no—
137#2395—971nomaximal, Sylow
138C97#239511yescharacteristic, complemented, Sylow
139C2×C4×D4—971no—
140C2×#117D9711yescharacteristic, complemented
141#400—971no—
142#399—971no—
143#398—971no—
144#397—971no—
145#397—972no—
146#394—972no—
147C4×C4⋊C4—972no—
148C4×C22⋊C4—971no—
149C4×C22⋊C4—972no—
150C22×D4—971no—
151C23×C4—971no—
152C23×C4C2×D9711yescharacteristic
153C4×D4—1944no—
154#117—24no—
155C2×C4⋊C4—971no—
156C2×C4⋊C4—972no—
157C2×C4⋊C4—972no—
158C2×C4⋊C4C2×D9711yescharacteristic
159C2×C4⋊C4C2×D9711yescharacteristic
160C2×C22⋊C4—971no—
161C2×C22⋊C4—972no—
162C2×C22⋊C4—971no—
163C2×C22⋊C4—972no—
164C2×C22⋊C4—972no—
165C2×C22⋊C4C2×D9712yes—
166C2×C42—971no—
167C2×C42—972no—
168C2×C42C2×D9712yes—
169#95—972no—
170#95—971no—
171#95—972no—
172#95—971no—
173#95—972no—
174#95—972no—
175C24—971no—
176C24C22×D9711yescharacteristic
177C2×D4—1942no—
178C2×D4—1942no—
179C2×D4—1942no—
180C22×C4—24no—
181C22×C4—972no—
182C22×C4—971no—
183C22×C4—972no—
184C22×C4—971no—
185C22×C4—971no—
186C22×C4—1942no—
187C22×C4—1944no—
188C22×C4C22⋊D9714yes—
189C22×C4C22×D9712yes—
190C22×C4C22×D9712yes—
191C22×C4C22×D9711yescharacteristic
192C22×C4C22×D9711yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—1944no—
196C4⋊C4—1944no—
197C4⋊C4—1942no—
198C22⋊C4—24no—
199C22⋊C4—1944no—
200C22⋊C4—1944no—
201C22⋊C4—1944no—
202C22⋊C4—1942no—
203C22⋊C4—1942no—
204C42—24no—
205C42—1942no—
206C42—1944no—
207C23—22no—
208C23—21no—
209C23—21no—
210C23—22no—
211C23—972no—
212C23—1941no—
213C23—1942no—
214C23—1941no—
215C23—1942no—
216C23C2×C22⋊D9712yes—
217C23C23×D9711yescharacteristic
218D4—1948no—
219C2×C4—22no—
220C2×C4—28no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—974no—
228C2×C4—972no—
229C2×C4—3884no—
230C2×C4—1941no—
231C2×C4—1941no—
232C2×C4—1942no—
233C2×C4—1942no—
234C2×C4—1942no—
235C2×C4—1942no—
236C2×C4—1942no—
237C2×C4—1941no—
238C2×C4—1942no—
239C2×C4—1941no—
240C2×C4—1942no—
241C2×C4C2×C22⋊D9712yes—
242C2×C4C2×C22⋊D9712yes—
243C2×C4#1309551114yes—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—3881no—
250C22—3882no—
251C22—3881no—
252C22—1942no—
253C22—1944no—
254C22C22×C22⋊D9711yescharacteristic
255C22#41×D9711yescharacteristic
256C22#41×D9711yescharacteristic
257C22#41×D9712yes—
258C22C2×#1309551112yes—
259C4—22no—
260C4—42no—
261C4—42no—
262C4—3881no—
263C4—3882no—
264C4—3881no—
265C4—1942no—
266C4—1944no—
267C4#1569079314yes—
268C2—42no—
269C2—3882no—
270C2C2×#1569079311yescharacteristic
271C2#2026124211yescharacteristic
272C2#2026126312yes—
273C2#2026126312yes—
274C2#2026128111yescharacteristic
subgroupG/H
centerC22×C4C22×D97
commutator, socle, cosocleC23×C97C24
FrattiniC23C23×D97
FittingC2×#117×C97C2
prime radicalC22×C22⋊D97C22
hypercenterC2×#117D97
hypocenterC97#2395
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C22×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20260673C2semidirect
3. #20260670C2semidirect
4. #20260666C2semidirect
5. #20260663C2semidirect
6. #20260661C2semidirect
8. C4×#13095490C2semidirect
9. C4×#13095488C2semidirect
10. C4×C22⋊Dic97C2semidirect
11. C2×#117×C97C2semidirect
138. C97#2395semidirect
140. C2×#117D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D972
#202606732
#202606702
#202606662
#202606632
#202606612
#202606502
C4×#130954902
C4×#130954882
C4×C22⋊Dic972
C2×#117×C972
#239597