Finite Group Databasebeta

← #27151910

Subgroups — #27151910

Order 12416.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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#20261271C211yesmaximal, characteristic
2C2×#15690775C211yesmaximal, characteristic, complemented
3#20261053C211yesmaximal, characteristic
4#20261041C212yesmaximal, complemented
5#20260960C212yesmaximal, complemented
6#20260890C211yesmaximal, characteristic, complemented
7#20260876C211yesmaximal, characteristic
8#20260839C212yesmaximal, complemented
9#20260834C212yesmaximal, complemented
10#20260789C211yesmaximal, characteristic, complemented
11#484×C97C211yesmaximal, characteristic, complemented
12C22×Dic194C2211yescharacteristic
13C2×#13095501C2212yes—
14C2×#13095500C2212yes—
15#15690805—21no—
16#15690801C2211yescharacteristic
17#15690799C2211yescharacteristic
18#15690794—21no—
19#15690784C2211yescharacteristic
20Q8×Dic97C2211yescharacteristic
21#15690775—21no—
22#15690775—21no—
23#15690775C2212yes—
24#15690775C2212yes—
25#15690772C2211yescharacteristic
26#15690770—21no—
27#15690768—21no—
28C2×#13095490—21no—
29C2×C4×Dic97—21no—
30C2×C4×Dic97C2211yescharacteristic
31#15690741C2212yes—
32#15690739C2212yes—
33#15690735C2212yes—
34#15690733C2212yes—
35#15690730C2212yes—
36C22⋊#11715895C2211yescharacteristic
37#15690720C2212yes—
38C22⋊Q8×C97C2211yescharacteristic
39C2×C4⋊C4×C97C2211yescharacteristic
40#107×C97C2212yes—
41#103×C97C2212yes—
42C22⋊C8×C97C2211yescharacteristic
43C2×Dic194—21no—
44C2×Dic194—24no—
45C2×Dic194C2312yes—
46C2×D388C2311yescharacteristic
47C2×C22⋊D97—21no—
48C22×Dic97—21no—
49C22×Dic97D411yescharacteristic
50C22×Dic97D411yescharacteristic
51C2×C4×D97—21no—
52#13095501—24no—
53#13095500—24no—
54C2×#11715895C2312yes—
55C4⋊Dic97—21no—
56C4⋊Dic97—21no—
57C4⋊Dic97C2311yescharacteristic
58#13095490—22no—
59#13095490—22no—
60#13095490—21no—
61#13095488—21no—
62#13095488—21no—
63#13095488—21no—
64#13095488—21no—
65C4×Dic97—22no—
66C4×Dic97—22no—
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×C97C2311yescharacteristic
79C4⋊C4×C97C2312yes—
80C22⋊C4×C97—21no—
81C23×C97C2×D411yescharacteristic
82C22×D97—21no—
83Q8×C97—22no—
84C2×C4×C97—22no—
85C2×C4×C97—21no—
86C2×C4×C97—21no—
87C2×C4×C97—22no—
88C2×C4×C97—42no—
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97C2411yescharacteristic
91C22⋊D97—42no—
92C2×Dic97—24no—
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97—21no—
96C2×Dic97—41no—
97C2×Dic97C2×D411yescharacteristic
98C2×Dic97C2×D411yescharacteristic
99C2×Dic97C2×D411yescharacteristic
100C2×Dic97C2×D411yescharacteristic
101D388—22no—
102C4×D97—42no—
103Dic194—24no—
104Dic194—22no—
105Dic194#4114yes—
106C8×C97—22no—
107#11715895—22no—
108C22×C97—21no—
109C22×C97—21no—
110C22×C97#13712yes—
111C22×C97C22×D411yescharacteristic
112C2×D97—41no—
113C2×D97—41no—
114C2×D97—41no—
115C4×C97—21no—
116C4×C97—41no—
117C4×C97—41no—
118C4×C97—42no—
119C4×C97C2×#4112yes—
120Dic97—24no—
121Dic97—21no—
122Dic97—21no—
123Dic97—21no—
124Dic97—21no—
125Dic97—41no—
126C2×C97—22no—
127C2×C97#52511yescharacteristic
128C2×C97C2×#13711yescharacteristic
129C2×C97#57511yescharacteristic
130D97—81no—
131#3117—971nomaximal, Sylow
132C97#311711yescharacteristic, complemented, Sylow
133#529—971no—
134C2×C4×Q8—971no—
135#484D9711yescharacteristic, complemented
136#481—971no—
137#479—972no—
138#464—972no—
139#451—971no—
140#448—971no—
141#441—972no—
142#440—972no—
143#432—971no—
144C22×Q8—971no—
145C2×Q16—972no—
146C2×SD16—972no—
147#126—1941no—
148#125—971no—
149#124—971no—
150#123—1941no—
151C22⋊Q8C2×D9711yescharacteristic
152#121—971no—
153C4×Q8—971no—
154C4×Q8—972no—
155C4×Q8—972no—
156C4×Q8—1941no—
157C4×Q8—1941no—
158C4×D4—971no—
159C4×D4—1941no—
160#117—1941no—
161C2×C4⋊C4—1941no—
162C2×C4⋊C4C2×D9711yescharacteristic
163C2×C42—971no—
164C2×C42—1941no—
165#107C2×D9712yes—
166#106—972no—
167C4⋊C8—972no—
168#103—972no—
169#103—972no—
170#103C2×D9712yes—
171#102—972no—
172C22⋊C8—971no—
173C22⋊C8C2×D9711yescharacteristic
174#97—972no—
175C2×Q8—972no—
176C2×Q8—1944no—
177C2×Q8—1941no—
178C2×Q8C22×D9711yescharacteristic
179C2×D4—971no—
180C2×D4—1941no—
181C22×C4—21no—
182C22×C4—971no—
183C22×C4—971no—
184C22×C4—1941no—
185C22×C4—1941no—
186C22×C4C22×D9711yescharacteristic
187Q16—1944no—
188SD16—1944no—
189C2×C8—972no—
190C2×C8C22×D9712yes—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—971no—
194C4⋊C4—1942no—
195C4⋊C4—1942no—
196C4⋊C4—1941no—
197C4⋊C4—1941no—
198C4⋊C4—1941no—
199C4⋊C4C22×D9712yes—
200C4⋊C4C22×D9711yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—1941no—
203C22⋊C4—1941no—
204C22⋊C4—1941no—
205C22⋊C4—1941no—
206C42—971no—
207C42—971no—
208C42—971no—
209C42—971no—
210C42—1942no—
211C42—1942no—
212C42—1941no—
213C23—1941no—
214C23D4×D9711yescharacteristic
215Q8—22no—
216Q8—974no—
217Q8—1944no—
218Q8—1942no—
219D4—3882no—
220D4—1942no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—971no—
227C2×C4—971no—
228C2×C4—971no—
229C2×C4—971no—
230C2×C4—3882no—
231C2×C4—3881no—
232C2×C4—1941no—
233C2×C4—1944no—
234C2×C4—1942no—
235C2×C4—1942no—
236C2×C4D4×D9711yescharacteristic
237C2×C4C23×D9711yescharacteristic
238C8—22no—
239C8—1942no—
240C22—21no—
241C22—21no—
242C22—3881no—
243C22—3881no—
244C22—3881no—
245C22#1569084212yes—
246C22C2×D4×D9711yescharacteristic
247C4—21no—
248C4—41no—
249C4—42no—
250C4—41no—
251C4—3881no—
252C4—1944no—
253C4—1941no—
254C4—1941no—
255C4—1941no—
256C4—1941no—
257C4#41×D9712yes—
258C2—22no—
259C2—7761no—
260C2#2026124311yescharacteristic
261C2C2×#1569084211yescharacteristic
262C2#2026153111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#484×C97C2
socleC22×C97C22×D4
prime radical#15690784C22
hypercenter#484D97
hypocenterC97#3117
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#15690775C2semidirect
4. #20261041C2semidirect
5. #20260960C2semidirect
6. #20260890C2semidirect
8. #20260839C2semidirect
9. #20260834C2semidirect
10. #20260789C2semidirect
11. #484×C97C2semidirect
132. C97#3117semidirect
135. #484D97semidirect

Maximal subgroups

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

subgroupindex
#202612712
C2×#156907752
#202610532
#202610412
#202609602
#202608902
#202608762
#202608392
#202608342
#202607892
#484×C972
#311797