Finite Group Databasebeta

← #27150073

Subgroups — #27150073

Order 12416.

564 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 169 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×#15690787C211yesmaximal, characteristic
2C22×C4×Dic97C211yesmaximal, characteristic
3#20260639C212yesmaximal, complemented
4#20260632C212yesmaximal, complemented
5#20260622C212yesmaximal, complemented
6#20260618C212yesmaximal, complemented
7#20260610C211yesmaximal, characteristic
8#20260607C211yesmaximal, characteristic
9#20260598C211yesmaximal, characteristic
10#20260593C211yesmaximal, characteristic
11#381×C97C211yesmaximal, characteristic
12C22×Dic194C2211yescharacteristic
13C23×Dic97C2211yescharacteristic
14C23×Dic97C2211yescharacteristic
15#15690787—24no—
16C2×C4⋊Dic97C2212yes—
17C2×C4⋊Dic97C2211yescharacteristic
18C2×#13095490C2211yescharacteristic
19C2×#13095490C2212yes—
20C2×#13095490C2211yescharacteristic
21C2×#13095490C2212yes—
22C2×C22⋊Dic97C2211yescharacteristic
23C2×C22⋊Dic97C2211yescharacteristic
24C2×C4×Dic97—24no—
25C2×C4×Dic97C2212yes—
26C2×C4×Dic97C2212yes—
27#15690717C2212yes—
28#15690717C2212yes—
29#15690717C2212yes—
30#15690717C2212yes—
31#15690717C2212yes—
32C23×C4×C97C2211yescharacteristic
33C2×C22⋊C4×C97C2211yescharacteristic
34C2×C22⋊C4×C97C2211yescharacteristic
35#95×C97C2212yes—
36#95×C97C2212yes—
37C2×Dic194—22no—
38C2×Dic194—24no—
39C22×Dic97—22no—
40C22×Dic97—24no—
41C22×Dic97—24no—
42C22×Dic97D414yes—
43C22×Dic97C2311yescharacteristic
44C22×Dic97C2312yes—
45C22×Dic97C2311yescharacteristic
46C22×Dic97C2311yescharacteristic
47C22×Dic97C2312yes—
48C22×Dic97C2311yescharacteristic
49C4⋊Dic97—22no—
50C4⋊Dic97—24no—
51#13095490—22no—
52#13095490—22no—
53#13095490—24no—
54#13095490—24no—
55C22⋊Dic97—22no—
56C22⋊Dic97—22no—
57C4×Dic97—24no—
58C4×Dic97—24no—
59C24×C97C2311yescharacteristic
60C22×C4×C97—22no—
61C22×C4×C97—22no—
62C22×C4×C97Q814yescomplemented
63C22×C4×C97C2311yescharacteristic
64C22×C4×C97C2312yes—
65C22×C4×C97C2311yescharacteristic
66C22×C4×C97C2312yes—
67C22⋊C4×C97—22no—
68C22⋊C4×C97—22no—
69C23×C97—22no—
70C23×C97—22no—
71C23×C97C2×D412yes—
72C23×C97C2×Q812yes—
73C23×C97#4112yes—
74C23×C97C2411yescharacteristic
75C2×C4×C97—28no—
76C2×C4×C97—22no—
77C2×C4×C97—22no—
78C2×C4×C97—21no—
79C2×C4×C97—21no—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97—22no—
85C2×C4×C97C2×Q814yes—
86C2×Dic97—28no—
87C2×Dic97—22no—
88C2×Dic97—22no—
89C2×Dic97—22no—
90C2×Dic97—28no—
91C2×Dic97—21no—
92C2×Dic97—21no—
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97—22no—
96C2×Dic97—22no—
97C2×Dic97—22no—
98C2×Dic97C2×D412yes—
99C2×Dic97C2×D412yes—
100C2×Dic97#4114yes—
101C2×Dic97#4114yes—
102Dic194—44no—
103C22×C97—22no—
104C22×C97—22no—
105C22×C97—24no—
106C22×C97—24no—
107C22×C97C22⋊Q814yes—
108C22×C97C22×D411yescharacteristic
109C22×C97C22×Q811yescharacteristic
110C22×C97C2×#4112yes—
111C22×C97C2×#4112yes—
112C22×C97C2×#4111yescharacteristic
113C4×C97—24no—
114C4×C97—41no—
115C4×C97—41no—
116C4×C97—42no—
117Dic97—22no—
118Dic97—24no—
119Dic97—41no—
120Dic97—41no—
121Dic97—42no—
122Dic97#12314yes—
123C2×C97—24no—
124C2×C97C2×C22⋊Q811yescharacteristic
125C2×C97C2×#12311yescharacteristic
126C2×C97#52511yescharacteristic
127C2×C97#52912yes—
128C2×C97#53312yes—
129#2855—971nomaximal, Sylow
130C97#285511yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—971no—
132C22×C42—971no—
133#391—972no—
134#389—972no—
135#387—972no—
136#387—972no—
137#385—971no—
138#385—971no—
139#381—971no—
140#381D9711yescharacteristic
141#380—971no—
142C22×Q8—971no—
143C23×C4—971no—
144C23×C4—971no—
145C23×C4C2×D9711yescharacteristic
146C22⋊Q8—1944no—
147C2×C4⋊C4—971no—
148C2×C4⋊C4—971no—
149C2×C4⋊C4—971no—
150C2×C4⋊C4—972no—
151C2×C4⋊C4—972no—
152C2×C4⋊C4—972no—
153C2×C22⋊C4—971no—
154C2×C22⋊C4—971no—
155C2×C22⋊C4C2×D9711yescharacteristic
156C2×C22⋊C4C2×D9711yescharacteristic
157C2×C42—972no—
158C2×C42—972no—
159C2×C42—1944no—
160#95—972no—
161#95—972no—
162#95—972no—
163#95—972no—
164#95—972no—
165#95C2×D9712yes—
166#95C2×D9712yes—
167C24C22×D9711yescharacteristic
168C2×Q8—1942no—
169C2×Q8—1944no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—971no—
173C22×C4—974no—
174C22×C4—971no—
175C22×C4—972no—
176C22×C4—972no—
177C22×C4—971no—
178C22×C4—971no—
179C22×C4—1944no—
180C22×C4—1944no—
181C22×C4—1942no—
182C22×C4Dic19414yescomplemented
183C22×C4C22×D9712yes—
184C22×C4C22×D9711yescharacteristic
185C22×C4C22×D9711yescharacteristic
186C22×C4C22×D9712yes—
187C4⋊C4—1944no—
188C4⋊C4—1944no—
189C4⋊C4—1942no—
190C4⋊C4—1942no—
191C4⋊C4—1944no—
192C4⋊C4—1942no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—1942no—
196C22⋊C4—1942no—
197C42—1944no—
198C42—1944no—
199C23—22no—
200C23—22no—
201C23D4×D9712yes—
202C23C2×Dic19412yes—
203C23#1309551012yes—
204C23C23×D9711yescharacteristic
205Q8—3884no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—972no—
217C2×C4—974no—
218C2×C4—972no—
219C2×C4—974no—
220C2×C4—1948no—
221C2×C4—1941no—
222C2×C4—1941no—
223C2×C4—1942no—
224C2×C4—1942no—
225C2×C4—1942no—
226C2×C4—1942no—
227C2×C4—1942no—
228C2×C4—1948no—
229C2×C4—1942no—
230C2×C4—1942no—
231C2×C4—1942no—
232C2×C4C2×Dic19414yes—
233C22—22no—
234C22—22no—
235C22—24no—
236C22—24no—
237C22#1569078714yes—
238C22C2×D4×D9711yescharacteristic
239C22C22×Dic19411yescharacteristic
240C22#41×D9712yes—
241C22#41×D9712yes—
242C22C2×#1309551011yescharacteristic
243C4—24no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—974no—
248C4—3881no—
249C4—3881no—
250C4—3882no—
251C4—1942no—
252C4—1944no—
253C2—24no—
254C2C2×#1569078711yescharacteristic
255C2#123×D9711yescharacteristic
256C2#2026124311yescharacteristic
257C2#2026126212yes—
258C2#2026128812yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#381×C97C2
prime radicalC24×C97C23
hypercenter#381D97
hypocenterC97#2855
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #20260639C2semidirect
4. #20260632C2semidirect
5. #20260622C2semidirect
6. #20260618C2semidirect
62. C22×C4×C97Q8semidirect
130. C97#2855semidirect
182. C22×C4Dic194semidirect

Maximal subgroups

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

subgroupindex
C2×#156907872
C22×C4×Dic972
#202606392
#202606322
#202606222
#202606182
#202606102
#202606072
#202605982
#202605932
#381×C972
#285597