Finite Group Databasebeta

← #27151863

Subgroups — #27151863

Order 12416.

380 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 119 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#20261287C211yesmaximal, characteristic
2C2×#15690775C211yesmaximal, characteristic
3#20261065C211yesmaximal, characteristic
4#20261019C212yesmaximal, complemented
5#20260944C212yesmaximal, complemented
6#20260893C211yesmaximal, characteristic
7#20260890C211yesmaximal, characteristic
8#20260841C212yesmaximal, complemented
9#20260839C212yesmaximal, complemented
10#20260789C211yesmaximal, characteristic
11#483×C97C211yesmaximal, characteristic
12C22×Dic194C2211yescharacteristic
13C2×Dic388C2212yes—
14C2×#13095501C2212yes—
15#15690809C2211yescharacteristic
16#15690808C2211yescharacteristic
17#15690800—21no—
18#15690789C2211yescharacteristic
19#15690787—21no—
20Q8×Dic97C2211yescharacteristic
21#15690775—21no—
22#15690775—21no—
23#15690775—21no—
24#15690775C2212yes—
25#15690775C2212yes—
26#15690775C2211yescharacteristic
27#15690766—21no—
28C2×#13095490—21no—
29C2×C4×Dic97—21no—
30C2×C4×Dic97C2211yescharacteristic
31#15690741C2212yes—
32#15690739C2212yes—
33#15690735C2212yes—
34#15690733C2212yes—
35#15690733C2212yes—
36C22⋊#11715895C2211yescharacteristic
37#15690720C2212yes—
38C22⋊Q8×C97C2211yescharacteristic
39C2×C4⋊C4×C97C2211yescharacteristic
40#106×C97C2212yes—
41#103×C97C2212yes—
42C22⋊C8×C97C2211yescharacteristic
43C2×Dic194—21no—
44C2×Dic194—24no—
45C2×Dic194—21no—
46C2×Dic194C2311yescharacteristic
47C2×Dic194C2312yes—
48C22×Dic97—21no—
49C22×Dic97D411yescharacteristic
50C22×Dic97D411yescharacteristic
51Dic388—24no—
52#13095501—24no—
53C2×#11715895C2312yes—
54C4⋊Dic97—21no—
55C4⋊Dic97C2311yescharacteristic
56#13095490—21no—
57#13095490—22no—
58#13095490—22no—
59#13095490—21no—
60#13095490—21no—
61#13095490—21no—
62#13095490—21no—
63C22⋊Dic97—21no—
64C4×Dic97—22no—
65C4×Dic97—22no—
66C4×Dic97—21no—
67C4×Dic97—21no—
68C4×Dic97D411yescharacteristic
69C4×Dic97D411yescharacteristic
70C4×Dic97C2311yescharacteristic
71C4×Dic97C2311yescharacteristic
72C2×Q8×C97C2311yescharacteristic
73C22×C4×C97—21no—
74C22×C4×C97C2311yescharacteristic
75C2×C8×C97C2312yes—
76C4⋊C4×C97—21no—
77C4⋊C4×C97—21no—
78C4⋊C4×C97C2312yes—
79C4⋊C4×C97C2311yescharacteristic
80C22⋊C4×C97—21no—
81C23×C97C2×D411yescharacteristic
82Q8×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97—21no—
85C2×C4×C97—21no—
86C2×C4×C97—22no—
87C2×C4×C97—42no—
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97C2411yescharacteristic
90C2×Dic97—24no—
91C2×Dic97—22no—
92C2×Dic97—22no—
93C2×Dic97—22no—
94C2×Dic97—21no—
95C2×Dic97—21no—
96C2×Dic97C2×D411yescharacteristic
97C2×Dic97C2×D411yescharacteristic
98C2×Dic97C2×D411yescharacteristic
99C2×Dic97C2×D411yescharacteristic
100Dic194—24no—
101Dic194—22no—
102Dic194—22no—
103Dic194—42no—
104Dic194#4114yes—
105C8×C97—22no—
106#11715895—22no—
107C22×C97—21no—
108C22×C97—21no—
109C22×C97#13712yes—
110C22×C97C22×D411yescharacteristic
111C4×C97—21no—
112C4×C97—41no—
113C4×C97—41no—
114C4×C97—42no—
115C4×C97C2×#4112yes—
116Dic97—24no—
117Dic97—21no—
118Dic97—21no—
119Dic97—21no—
120Dic97—41no—
121Dic97—41no—
122Dic97#13712yes—
123C2×C97—22no—
124C2×C97#52511yescharacteristic
125C2×C97C2×#13711yescharacteristic
126C2×C97C2×#13711yescharacteristic
127#3113—971nomaximal, Sylow
128C97#311311yescharacteristic, complemented, Sylow
129#533—971no—
130C2×C4×Q8—971no—
131#483—971no—
132#483D9711yescharacteristic
133#475—972no—
134#462—972no—
135#451—971no—
136#451—971no—
137#441—972no—
138#441—972no—
139#432—971no—
140C22×Q8—971no—
141C2×Q16—972no—
142C2×Q16—972no—
143C4⋊Q8—971no—
144C4⋊Q8—971no—
145#125—1941no—
146C22⋊Q8—971no—
147C22⋊Q8—1941no—
148C22⋊Q8C2×D9711yescharacteristic
149C4×Q8—971no—
150C4×Q8—972no—
151C4×Q8—972no—
152C4×Q8—971no—
153C4×Q8—1941no—
154C4×Q8—1941no—
155C4×Q8—1941no—
156#117—1941no—
157C2×C4⋊C4—1941no—
158C2×C4⋊C4C2×D9711yescharacteristic
159C2×C42—971no—
160C2×C42—1941no—
161#106—972no—
162#106C2×D9712yes—
163C4⋊C8—972no—
164#103—972no—
165#103—972no—
166#103—972no—
167#103C2×D9712yes—
168C22⋊C8—971no—
169C22⋊C8C2×D9711yescharacteristic
170#97—972no—
171C2×Q8—972no—
172C2×Q8—971no—
173C2×Q8—1941no—
174C2×Q8—1944no—
175C2×Q8—1941no—
176C2×Q8C22×D9711yescharacteristic
177C22×C4—21no—
178C22×C4—971no—
179C22×C4—971no—
180C22×C4—1941no—
181C22×C4C22×D9711yescharacteristic
182Q16—1944no—
183Q16—1944no—
184C2×C8—972no—
185C2×C8C22×D9712yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—971no—
189C4⋊C4—1942no—
190C4⋊C4—1942no—
191C4⋊C4—1941no—
192C4⋊C4—1941no—
193C4⋊C4—1941no—
194C4⋊C4—1941no—
195C4⋊C4—1941no—
196C4⋊C4—1941no—
197C4⋊C4C22×D9711yescharacteristic
198C4⋊C4C22×D9712yes—
199C22⋊C4—21no—
200C22⋊C4—1941no—
201C42—971no—
202C42—971no—
203C42—971no—
204C42—971no—
205C42—1942no—
206C42—1942no—
207C42—1941no—
208C42—1941no—
209C23D4×D9711yescharacteristic
210Q8—22no—
211Q8—974no—
212Q8—3882no—
213Q8—1944no—
214Q8—1942no—
215Q8—1942no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—971no—
222C2×C4—971no—
223C2×C4—971no—
224C2×C4—971no—
225C2×C4—1942no—
226C2×C4—1941no—
227C2×C4—1941no—
228C2×C4—1944no—
229C2×C4—1942no—
230C2×C4—1942no—
231C2×C4D4×D9711yescharacteristic
232C2×C4C23×D9711yescharacteristic
233C8—22no—
234C8—1942no—
235C22—21no—
236C22—21no—
237C22#1569083912yes—
238C22C2×D4×D9711yescharacteristic
239C4—21no—
240C4—41no—
241C4—41no—
242C4—42no—
243C4—972no—
244C4—3881no—
245C4—3881no—
246C4—1944no—
247C4—1941no—
248C4—1941no—
249C4—1941no—
250C4#41×D9712yes—
251C2—22no—
252C2#2026124311yescharacteristic
253C2#137×D9711yescharacteristic
254C2C2×#1569083911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#483×C97C2
socleC22×C97C22×D4
prime radicalC23×C97C2×D4
hypercenter#483D97
hypocenterC97#3113
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
4. #20261019C2semidirect
5. #20260944C2semidirect
8. #20260841C2semidirect
9. #20260839C2semidirect
128. C97#3113semidirect

Maximal subgroups

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

subgroupindex
#202612872
C2×#156907752
#202610652
#202610192
#202609442
#202608932
#202608902
#202608412
#202608392
#202607892
#483×C972
#311397