Finite Group Databasebeta

← #27152294

Subgroups — #27152294

Order 12416.

412 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 129 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#20261290C211yesmaximal, characteristic, complemented
2#20261283C211yesmaximal, characteristic
3#20261131C211yesmaximal, characteristic, complemented
4#20261107C211yesmaximal, characteristic, complemented
5#20261086C211yesmaximal, characteristic, complemented
6#20260963C214yesmaximal, complemented
7#20260826C214yesmaximal, complemented
8C4×C8×D97C211yesmaximal, characteristic, complemented
9#486×C97C211yesmaximal, characteristic, complemented
10C2×#13095510—21no—
11C2×#13095499C2214yes—
12C2×C8×D97C2212yes—
13#15690810C2211yescharacteristic
14#15690810C2211yescharacteristic
15#15690801—21no—
16#15690797—21no—
17#15690790C2212yes—
18#15690786—21no—
19#15690781—21no—
20#15690781C2212yes—
21Q8×Dic97—21no—
22#15690775C2212yes—
23D4×Dic97—21no—
24D4×Dic97C2212yes—
25#15690768—21no—
26C42×D97C2211yescharacteristic
27#15690741C2214yes—
28#15690735C2214yes—
29C4×#11715895C2211yescharacteristic
30C8×Dic97C2212yes—
31C4⋊Q8×C97C2211yescharacteristic
32#127×C97C2211yescharacteristic
33#102×C97C2214yes—
34C4×C8×C97C2211yescharacteristic
35#13095510—44no—
36C2×Dic194—21no—
37C2×Dic194C2312yes—
38C2×C22⋊D97—22no—
39C22×Dic97—22no—
40C2×C4×D97D411yescharacteristic
41C2×C4×D97D411yescharacteristic
42C2×C4×D97C2311yescharacteristic
43#13095499—28no—
44C2×#11715895C2312yes—
45C8×D97—24no—
46C4⋊Dic97—21no—
47C4⋊Dic97—21no—
48C4⋊Dic97—21no—
49C4⋊Dic97—22no—
50C4⋊Dic97C2312yes—
51#13095490—22no—
52C22⋊Dic97—22no—
53C22⋊Dic97—22no—
54#13095488—22no—
55C4×Dic97—22no—
56C4×Dic97D411yescharacteristic
57C4×Dic97D411yescharacteristic
58C4×Dic97C2311yescharacteristic
59C2×Q8×C97—21no—
60C2×D4×C97—21no—
61C2×D4×C97—21no—
62C2×D4×C97C2312yes—
63C2×C8×C97C2312yes—
64C4⋊C4×C97—21no—
65C4⋊C4×C97C2312yes—
66C42×C97C2311yescharacteristic
67C23×C97—22no—
68C22×D97C2×D411yescharacteristic
69Q8×C97—42no—
70D4×C97—24no—
71D4×C97—42no—
72D4×C97—42no—
73C2×C4×C97—22no—
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C2×C4×C97C2411yescharacteristic
77C22⋊D97—44no—
78C2×Dic97—22no—
79C2×Dic97—22no—
80C2×Dic97—44no—
81C2×Dic97C2×D411yescharacteristic
82C2×Dic97C2×D411yescharacteristic
83C2×Dic97C2×D411yescharacteristic
84C4×D97—22no—
85C4×D97—22no—
86C4×D97SD1614yescomplemented
87Dic194—24no—
88Dic194—42no—
89C8×C97—22no—
90#11715895#4114yescomplemented
91C22×C97—42no—
92C22×C97—42no—
93C22×C97—42no—
94C22×C97C22×D411yescharacteristic
95C2×D97—21no—
96C2×D97—21no—
97C2×D97C2×SD1612yes—
98C4×C97—42no—
99C4×C97C2×SD1612yes—
100C4×C97#13512yes—
101C4×C97C2×#4112yes—
102Dic97—21no—
103Dic97—21no—
104Dic97—42no—
105Dic97—42no—
106Dic97C2×SD1612yes—
107C2×C97—82no—
108C2×C97#53211yescharacteristic
109C2×C97C22×SD1611yescharacteristic
110C2×C97C2×#13511yescharacteristic
111D97—22no—
112#3167—971nomaximal, Sylow
113C97#316711yescharacteristic, complemented, Sylow
114#533—971no—
115#532—971no—
116#498—971no—
117#492—971no—
118#487—971no—
119#486D9711yescharacteristic, complemented
120#465—974no—
121C4×SD16—974no—
122C2×C4×C8—971no—
123C2×#41—1941no—
124C2×SD16—974no—
125C22×C8—972no—
126C4⋊Q8—971no—
127C4⋊Q8—971no—
128C4⋊Q8C2×D9711yescharacteristic
129#127C2×D9711yescharacteristic
130#125—1941no—
131#124—1941no—
132C22⋊Q8—972no—
133C22⋊Q8—1941no—
134#121—972no—
135#121—1941no—
136C4×Q8—972no—
137C4×Q8—1941no—
138C4×D4—972no—
139C4×D4—1941no—
140#117—1941no—
141C2×C42—971no—
142#106—974no—
143#103—974no—
144#102C2×D9714yes—
145C4×C8—972no—
146C4×C8—971no—
147C4×C8C2×D9711yescharacteristic
148#41—3884no—
149C2×Q8—21no—
150C2×Q8—972no—
151C2×Q8—1941no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—1942no—
155C2×D4C22×D9712yes—
156C22×C4—971no—
157C22×C4—971no—
158C22×C4—971no—
159C22×C4—1942no—
160SD16—1948no—
161C2×C8—972no—
162C2×C8—1944no—
163C2×C8C22×D9712yes—
164C4⋊C4—21no—
165C4⋊C4—972no—
166C4⋊C4—1942no—
167C4⋊C4—1942no—
168C4⋊C4—1941no—
169C4⋊C4—1941no—
170C4⋊C4—1941no—
171C4⋊C4C22×D9712yes—
172C22⋊C4—1942no—
173C22⋊C4—1942no—
174C22⋊C4—1942no—
175C42—971no—
176C42—971no—
177C42—971no—
178C42—1942no—
179C42C22×D9711yescharacteristic
180C23—22no—
181C23—971no—
182Q8—42no—
183Q8—3882no—
184Q8—1944no—
185D4—24no—
186D4—42no—
187D4—42no—
188D4—3884no—
189C2×C4—22no—
190C2×C4—971no—
191C2×C4—974no—
192C2×C4—971no—
193C2×C4—971no—
194C2×C4—3884no—
195C2×C4—1942no—
196C2×C4—1942no—
197C2×C4—1942no—
198C2×C4—1942no—
199C2×C4D4×D9711yescharacteristic
200C2×C4D4×D9711yescharacteristic
201C2×C4C23×D9711yescharacteristic
202C8—22no—
203C8—974no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—972no—
208C22—1941no—
209C22—1941no—
210C22C2×D4×D9711yescharacteristic
211C4—42no—
212C4—972no—
213C4—3882no—
214C4—3882no—
215C4—1941no—
216C4—1941no—
217C4SD16×D9712yes—
218C4#1569082912yes—
219C4#41×D9712yes—
220C2—82no—
221C2—1942no—
222C2#2026128411yescharacteristic
223C2C2×SD16×D9711yescharacteristic
224C2C2×#1569082911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#486×C97C2
socleC22×C97C22×D4
prime radical#20261283C2
hypercenter#486D97
hypocenterC97#3167
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #20261290C2semidirect
3. #20261131C2semidirect
4. #20261107C2semidirect
5. #20261086C2semidirect
6. #20260963C2semidirect
7. #20260826C2semidirect
8. C4×C8×D97C2semidirect
9. #486×C97C2semidirect
86. C4×D97SD16semidirect
90. #11715895#41semidirect
113. C97#3167semidirect
119. #486D97semidirect

Maximal subgroups

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

subgroupindex
#202612902
#202612832
#202611312
#202611072
#202610862
#202609632
#202608262
C4×C8×D972
#486×C972
#316797