Finite Group Databasebeta

← #27145001

Subgroups — #27145001

Order 12416.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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×#15690770C211yesmaximal, characteristic
2C2×#15690768C211yesmaximal, characteristic, complemented
3#20260673C211yesmaximal, characteristic, complemented
4#20260667C211yesmaximal, characteristic, complemented
5#20260661C212yesmaximal, complemented
6#20260660C211yesmaximal, characteristic, complemented
7#20260651C212yesmaximal
8C4⋊C4×Dic97C212yesmaximal, complemented
9C22⋊C4×Dic97C212yesmaximal, complemented
10#20260582C211yesmaximal, characteristic, complemented
11#399×C97C211yesmaximal, characteristic, complemented
12C22×C22⋊D97C2211yescharacteristic
13C23×Dic97C2211yescharacteristic
14C22×C4×D97C2211yescharacteristic
15#15690770—24no—
16#15690768—24no—
17C2×C4⋊Dic97C2212yes—
18C2×C4⋊Dic97C2211yescharacteristic
19C2×#13095490C2211yescharacteristic
20C2×C22⋊Dic97C2212yes—
21C2×#13095488C2212yes—
22C2×#13095488C2212yes—
23C2×#13095488C2211yescharacteristic
24C2×C4×Dic97C2212yes—
25C2×C4×Dic97C2211yescharacteristic
26C2×C4×Dic97C2212yes—
27#15690717C2212yes—
28#15690717C2212yes—
29#15690717C2211yescharacteristic
30#15690717C2212yes—
31#15690717C2212yes—
32C2×C4⋊C4×C97C2212yes—
33C2×C4⋊C4×C97C2211yescharacteristic
34C2×C22⋊C4×C97C2212yes—
35C2×C22⋊C4×C97C2211yescharacteristic
36#95×C97C2211yescharacteristic
37C23×D97C2311yescharacteristic
38C2×C22⋊D97—22no—
39C2×C22⋊D97—22no—
40C2×C22⋊D97—22no—
41C22×Dic97—22no—
42C22×Dic97—24no—
43C22×Dic97C2312yes—
44C22×Dic97C2312yes—
45C22×Dic97C2311yescharacteristic
46C22×Dic97C2311yescharacteristic
47C22×Dic97C2311yescharacteristic
48C2×C4×D97—24no—
49C2×C4×D97D414yes—
50C4⋊Dic97—22no—
51C4⋊Dic97—24no—
52#13095490—22no—
53C22⋊Dic97—24no—
54#13095488—24no—
55#13095488—22no—
56#13095488—24no—
57C4×Dic97—24no—
58C4×Dic97—22no—
59C4×Dic97—24no—
60C24×C97C2311yescharacteristic
61C22×C4×C97C2311yescharacteristic
62C22×C4×C97C2312yes—
63C22×C4×C97C2311yescharacteristic
64C22×C4×C97C2312yes—
65C4⋊C4×C97—22no—
66C4⋊C4×C97—24no—
67C22⋊C4×C97—22no—
68C22⋊C4×C97—24no—
69C23×C97—21no—
70C23×C97—22no—
71C23×C97—22no—
72C23×C97—21no—
73C23×C97#4112yes—
74C23×C97C2411yescharacteristic
75C22×D97—22no—
76C22×D97—22no—
77C22×D97—21no—
78C22×D97—21no—
79C22×D97C2×D412yes—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—21no—
84C2×C4×C97—22no—
85C2×C4×C97—22no—
86C2×C4×C97—22no—
87C2×C4×C97—22no—
88C2×C4×C97C2×D412yes—
89C2×C4×C97#4114yes—
90C22⋊D97—28no—
91C2×Dic97—21no—
92C2×Dic97—21no—
93C2×Dic97—22no—
94C2×Dic97—21no—
95C2×Dic97—22no—
96C2×Dic97—22no—
97C2×Dic97—22no—
98C2×Dic97—22no—
99C2×Dic97—22no—
100C2×Dic97—22no—
101C2×Dic97—44no—
102C2×Dic97C2×D412yes—
103C2×Dic97#4112yes—
104C2×Dic97#4114yes—
105C4×D97—28no—
106C22×C97—22no—
107C22×C97—24no—
108C22×C97—41no—
109C22×C97—42no—
110C22×C97—41no—
111C22×C97C22×D411yescharacteristic
112C22×C97C2×#4112yes—
113C22×C97C2×#4111yescharacteristic
114C22×C97C2×#4112yes—
115C22×C97C2×#4111yescharacteristic
116C2×D97—22no—
117C2×D97—22no—
118C2×D97—22no—
119C2×D97—42no—
120C2×D97—42no—
121C4×C97—22no—
122C4×C97—24no—
123C4×C97—41no—
124C4×C97—42no—
125Dic97—22no—
126Dic97—41no—
127Dic97—42no—
128Dic97—42no—
129Dic97#12314yes—
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×#117—971no—
141#400—971no—
142#399D9711yescharacteristic, complemented
143#398—971no—
144#397—972no—
145#397—971no—
146#394—972no—
147C4×C4⋊C4—972no—
148C4×C22⋊C4—972no—
149C4×C22⋊C4—971no—
150C22×D4—971no—
151C23×C4—971no—
152C23×C4—971no—
153C4×D4—1944no—
154#117—1944no—
155C2×C4⋊C4—971no—
156C2×C4⋊C4—971no—
157C2×C4⋊C4—972no—
158C2×C4⋊C4C2×D9711yescharacteristic
159C2×C4⋊C4C2×D9712yes—
160C2×C22⋊C4—972no—
161C2×C22⋊C4—972no—
162C2×C22⋊C4—971no—
163C2×C22⋊C4—972no—
164C2×C22⋊C4C2×D9711yescharacteristic
165C2×C22⋊C4C2×D9712yes—
166C2×C42—972no—
167C2×C42—971no—
168C2×C42—972no—
169#95—972no—
170#95—971no—
171#95—972no—
172#95—972no—
173#95—972no—
174#95C2×D9711yescharacteristic
175C24—971no—
176C24C22×D9711yescharacteristic
177C2×D4—1942no—
178C2×D4—1942no—
179C2×D4—1942no—
180C22×C4—971no—
181C22×C4—971no—
182C22×C4—974no—
183C22×C4—972no—
184C22×C4—971no—
185C22×C4—972no—
186C22×C4—1944no—
187C22×C4—1942no—
188C22×C4—1944no—
189C22×C4C22×D9711yescharacteristic
190C22×C4C22×D9712yes—
191C22×C4C22×D9711yescharacteristic
192C22×C4C22×D9712yes—
193C4⋊C4—22no—
194C4⋊C4—24no—
195C4⋊C4—1944no—
196C4⋊C4—1942no—
197C4⋊C4—1942no—
198C22⋊C4—22no—
199C22⋊C4—24no—
200C22⋊C4—1944no—
201C22⋊C4—1942no—
202C22⋊C4—1944no—
203C22⋊C4—1944no—
204C42—1944no—
205C42—1944no—
206C42—1942no—
207C23—21no—
208C23—22no—
209C23—21no—
210C23—22no—
211C23—972no—
212C23—1942no—
213C23—1941no—
214C23—1942no—
215C23—1941no—
216C23#1309551212yes—
217C23C23×D9711yescharacteristic
218D4—1948no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—972no—
228C2×C4—974no—
229C2×C4—972no—
230C2×C4—3884no—
231C2×C4—1948no—
232C2×C4—1942no—
233C2×C4—1941no—
234C2×C4—1942no—
235C2×C4—1942no—
236C2×C4—1942no—
237C2×C4—1942no—
238C2×C4—1942no—
239C2×C4—1942no—
240C2×C4—1941no—
241C2×C4—1941no—
242C2×C4D4×D9712yes—
243C2×C4#1309551014yes—
244C22—22no—
245C22—24no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—3882no—
250C22—3882no—
251C22—1942no—
252C22—1942no—
253C22—1942no—
254C22C2×D4×D9711yescharacteristic
255C22#41×D9712yes—
256C22#41×D9711yescharacteristic
257C22C2×#1309551012yes—
258C22C2×#1309551211yescharacteristic
259C4—22no—
260C4—24no—
261C4—41no—
262C4—42no—
263C4—974no—
264C4—3881no—
265C4—3882no—
266C4—3882no—
267C4—1942no—
268C2—42no—
269C2—3882no—
270C2#123×D9711yescharacteristic
271C2#2026124811yescharacteristic
272C2#2026126212yes—
273C2#2026127112yes—
274C2#2026128311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#399×C97C2
prime radicalC22×C22⋊D97C22
hypercenter#399D97
hypocenterC97#2395
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#15690768C2semidirect
3. #20260673C2semidirect
4. #20260667C2semidirect
5. #20260661C2semidirect
6. #20260660C2semidirect
8. C4⋊C4×Dic97C2semidirect
9. C22⋊C4×Dic97C2semidirect
10. #20260582C2semidirect
11. #399×C97C2semidirect
138. C97#2395semidirect
142. #399D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907702
C2×#156907682
#202606732
#202606672
#202606612
#202606602
#202606512
C4⋊C4×Dic972
C22⋊C4×Dic972
#202605822
#399×C972
#239597