Finite Group Databasebeta

← #27152133

Subgroups — #27152133

Order 12416.

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

normal subgroups only

This group has 392 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#20261284C211yesmaximal, characteristic
2#125×D97C211yesmaximal, characteristic, complemented
3#20261102C212yesmaximal, complemented
4#20261098C212yesmaximal, complemented
5#20261092C211yesmaximal, characteristic, complemented
6#20261090C211yesmaximal, characteristic, complemented
7#20261089C211yesmaximal, characteristic, complemented
8#20260735C212yesmaximal, complemented
9#20260733C212yesmaximal, complemented
10C4×#13095494C211yesmaximal, characteristic, complemented
11#488×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2211yescharacteristic
13C2×#13095510C2211yescharacteristic
14C2×#13095494C2212yes—
15#15690808C2211yescharacteristic
16#15690806C2211yescharacteristic
17#15690802C2211yescharacteristic
18#15690801C2212yes—
19#15690800—22no—
20#15690798C2211yescharacteristic
21#15690783—22no—
22#15690770—22no—
23C4⋊C4×D97—22no—
24C4⋊C4×D97C2212yes—
25C42×D97C2211yescharacteristic
26#15690735C2212yes—
27#15690734C2212yes—
28#15690733C2212yes—
29#15690732C2212yes—
30#15690731C2212yes—
31#15690730C2212yes—
32#15690720C2212yes—
33C4×#11715895C2211yescharacteristic
34#125×C97C2211yescharacteristic
35#124×C97C2211yescharacteristic
36#103×C97C2212yes—
37#102×C97C2212yes—
38C4×C8×C97C2211yescharacteristic
39#13095512—24no—
40#13095510—24no—
41C2×Dic194C2311yescharacteristic
42C2×D388—21no—
43C2×D388C2311yescharacteristic
44C2×C22⋊D97—21no—
45C2×C22⋊D97—22no—
46C22×Dic97—21no—
47C2×C4×D97—22no—
48C2×C4×D97—21no—
49C2×C4×D97D412yes—
50C2×C4×D97C2311yescharacteristic
51#13095494—24no—
52C2×#11715895C2312yes—
53C4⋊Dic97—22no—
54C4⋊Dic97C2312yes—
55#13095490—22no—
56#13095490—22no—
57#13095490—22no—
58#13095490—22no—
59#13095488—22no—
60C4×Dic97D412yes—
61C4×Dic97C2311yescharacteristic
62C2×Q8×C97C2311yescharacteristic
63C2×D4×C97C2311yescharacteristic
64C2×C8×C97C2312yes—
65C4⋊C4×C97—22no—
66C4⋊C4×C97C2312yes—
67C22⋊C4×C97—22no—
68C42×C97C2311yescharacteristic
69C23×C97—21no—
70C22×D97—21no—
71C22×D97C2×D411yescharacteristic
72Q8×C97—22no—
73D4×C97—22no—
74C2×C4×C97—21no—
75C2×C4×C97—22no—
76C2×C4×C97C2×D412yes—
77C2×C4×C97C2411yescharacteristic
78C22⋊D97—44no—
79C22⋊D97—42no—
80C2×Dic97—21no—
81C2×Dic97—22no—
82C2×Dic97—42no—
83C2×Dic97C2×D412yes—
84C2×Dic97C2×D411yescharacteristic
85D388—22no—
86D388—42no—
87C4×D97—42no—
88C4×D97—44no—
89C4×D97—42no—
90C4×D97#4114yes—
91Dic194—22no—
92C8×C97—22no—
93#11715895—22no—
94C22×C97—41no—
95C22×C97—42no—
96C22×C97C22×D411yescharacteristic
97C2×D97—22no—
98C2×D97—41no—
99C2×D97—42no—
100C2×D97#12412yes—
101C4×C97—22no—
102C4×C97—41no—
103C4×C97—42no—
104C4×C97C2×#4112yes—
105Dic97—22no—
106Dic97—41no—
107Dic97—42no—
108Dic97#12412yes—
109C2×C97—81no—
110C2×C97C2×#12411yescharacteristic
111C2×C97#57512yes—
112D97—22no—
113D97—81no—
114#3142—971nomaximal, Sylow
115C97#314211yescharacteristic, complemented, Sylow
116#532—971no—
117C2×#125—971no—
118#491—972no—
119#490—972no—
120#488—971no—
121#488—971no—
122#488—971no—
123#488D9711yescharacteristic, complemented
124#417—972no—
125#417—972no—
126C4×M16—971no—
127C2×#41—971no—
128C2×#41—971no—
129C2×M16—972no—
130C4⋊Q8—971no—
131#127—971no—
132#125—972no—
133#125—971no—
134#125—1942no—
135#125C2×D9711yescharacteristic
136#124—971no—
137#124C2×D9711yescharacteristic
138#121—1942no—
139C4×D4—1942no—
140C2×C4⋊C4—972no—
141C2×C4⋊C4—1942no—
142C2×C42—971no—
143#103—972no—
144#103—972no—
145#103—972no—
146#103C2×D9712yes—
147#102—972no—
148#102—972no—
149#102—972no—
150#102C2×D9712yes—
151#97—972no—
152C4×C8—971no—
153C4×C8C2×D9711yescharacteristic
154#41—1944no—
155#41—1944no—
156C2×Q8—971no—
157C2×Q8C22×D9711yescharacteristic
158C2×D4—971no—
159C2×D4—1941no—
160C2×D4—1942no—
161C2×D4—1941no—
162C2×D4C22×D9711yescharacteristic
163C22×C4—971no—
164C22×C4—972no—
165C22×C4—1941no—
166C22×C4—1942no—
167C22×C4—1941no—
168M16—1944no—
169C2×C8—972no—
170C2×C8C22×D9712yes—
171C4⋊C4—22no—
172C4⋊C4—972no—
173C4⋊C4—1942no—
174C4⋊C4—1942no—
175C4⋊C4—1942no—
176C4⋊C4—1942no—
177C4⋊C4—1942no—
178C4⋊C4C22×D9712yes—
179C22⋊C4—22no—
180C22⋊C4—1942no—
181C42—972no—
182C42—971no—
183C42C22×D9711yescharacteristic
184C23—21no—
185C23—971no—
186C23—1941no—
187Q8—22no—
188Q8—1942no—
189D4—22no—
190D4—3882no—
191D4—3882no—
192D4—3884no—
193D4—1942no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—971no—
197C2×C4—974no—
198C2×C4—972no—
199C2×C4—3882no—
200C2×C4—3882no—
201C2×C4—3884no—
202C2×C4—3882no—
203C2×C4—1941no—
204C2×C4—1942no—
205C2×C4D4×D9712yes—
206C2×C4C23×D9711yescharacteristic
207C8—22no—
208C8—1942no—
209C22—41no—
210C22—42no—
211C22—972no—
212C22—3881no—
213C22—3882no—
214C22—1942no—
215C22C2×D4×D9711yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—972no—
220C4—3881no—
221C4—3882no—
222C4—1942no—
223C4#41×D9712yes—
224C2—81no—
225C2—7761no—
226C2—1942no—
227C2#124×D9711yescharacteristic
228C2#2026153212yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#488×C97C2
socleC22×C97C22×D4
prime radical#20261284C2
hypercenter#488D97
hypocenterC97#3142
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
2. #125×D97C2semidirect
3. #20261102C2semidirect
4. #20261098C2semidirect
5. #20261092C2semidirect
6. #20261090C2semidirect
7. #20261089C2semidirect
8. #20260735C2semidirect
9. #20260733C2semidirect
10. C4×#13095494C2semidirect
11. #488×C97C2semidirect
115. C97#3142semidirect
123. #488D97semidirect

Maximal subgroups

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

subgroupindex
#202612842
#125×D972
#202611022
#202610982
#202610922
#202610902
#202610892
#202607352
#202607332
C4×#130954942
#488×C972
#314297