Finite Group Databasebeta

← #27152055

Subgroups — #27152055

Order 12416.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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#20261088C214yesmaximal, complemented
4#20261087C211yesmaximal, characteristic, complemented
5#20261086C211yesmaximal, characteristic, complemented
6#20261081C211yesmaximal, characteristic, complemented
7#20260728C214yesmaximal, complemented
8C4×C8×D97C211yesmaximal, characteristic, complemented
9#486×C97C211yesmaximal, characteristic, complemented
10C2×#13095510C2212yes—
11C2×C8×D97C2212yes—
12#15690810C2211yescharacteristic
13#15690810C2211yescharacteristic
14#15690801C2212yes—
15#15690797C2212yes—
16#15690790—21no—
17#15690786—21no—
18#15690781—21no—
19#15690781—21no—
20Q8×Dic97—21no—
21#15690775—21no—
22D4×Dic97—21no—
23D4×Dic97—21no—
24#15690768C2212yes—
25C42×D97C2211yescharacteristic
26#15690735C2214yes—
27#15690733C2214yes—
28#15690731C2214yes—
29C4×#11715895C2211yescharacteristic
30C8×Dic97C2212yes—
31C4⋊Q8×C97C2211yescharacteristic
32#127×C97C2211yescharacteristic
33#102×C97C2214yes—
34C4×C8×C97C2211yescharacteristic
35#13095510—28no—
36C2×Dic194—21no—
37C2×Dic194C2312yes—
38C2×C22⋊D97—22no—
39C22×Dic97—22no—
40C2×C4×D97D411yescharacteristic
41C2×C4×D97D411yescharacteristic
42C2×C4×D97C2311yescharacteristic
43C2×#11715895C2312yes—
44C8×D97—24no—
45C4⋊Dic97—21no—
46C4⋊Dic97—21no—
47C4⋊Dic97—21no—
48C4⋊Dic97—22no—
49C4⋊Dic97C2312yes—
50#13095490—22no—
51C22⋊Dic97—22no—
52C22⋊Dic97—22no—
53#13095488—22no—
54C4×Dic97—22no—
55C4×Dic97D411yescharacteristic
56C4×Dic97D411yescharacteristic
57C4×Dic97C2311yescharacteristic
58C2×Q8×C97—21no—
59C2×D4×C97—21no—
60C2×D4×C97—21no—
61C2×D4×C97C2312yes—
62C2×C8×C97C2312yes—
63C4⋊C4×C97—21no—
64C4⋊C4×C97C2312yes—
65C42×C97C2311yescharacteristic
66C23×C97—22no—
67C22×D97C2×D411yescharacteristic
68Q8×C97—42no—
69D4×C97—24no—
70D4×C97—42no—
71D4×C97—42no—
72C2×C4×C97—22no—
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2411yescharacteristic
76C22⋊D97—44no—
77C2×Dic97—22no—
78C2×Dic97—22no—
79C2×Dic97—44no—
80C2×Dic97C2×D411yescharacteristic
81C2×Dic97C2×D411yescharacteristic
82C2×Dic97C2×D411yescharacteristic
83C4×D97—22no—
84C4×D97—22no—
85C4×D97#4114yes—
86Dic194—24no—
87Dic194—42no—
88C8×C97—22no—
89#11715895—22no—
90C22×C97—42no—
91C22×C97—42no—
92C22×C97—42no—
93C22×C97C22×D411yescharacteristic
94C2×D97—21no—
95C2×D97—21no—
96C2×D97#12412yes—
97C4×C97—42no—
98C4×C97#13512yes—
99C4×C97#13512yes—
100C4×C97C2×#4112yes—
101Dic97—21no—
102Dic97—21no—
103Dic97—42no—
104Dic97—42no—
105Dic97#12412yes—
106C2×C97—82no—
107C2×C97C2×#12411yescharacteristic
108C2×C97C2×#13511yescharacteristic
109C2×C97C2×#13511yescharacteristic
110D97—22no—
111#3131—971nomaximal, Sylow
112C97#313111yescharacteristic, complemented, Sylow
113#533—971no—
114#532—971no—
115#488—974no—
116#487—971no—
117#487—971no—
118#486—971no—
119#486D9711yescharacteristic, complemented
120#416—974no—
121C2×C4×C8—971no—
122C2×#41—972no—
123C22×C8—972no—
124C4⋊Q8—971no—
125C4⋊Q8—971no—
126C4⋊Q8C2×D9711yescharacteristic
127#127C2×D9711yescharacteristic
128#125—972no—
129#124—972no—
130C22⋊Q8—1941no—
131C22⋊Q8—1941no—
132#121—1941no—
133#121—1941no—
134C4×Q8—1941no—
135C4×Q8—1941no—
136C4×D4—1941no—
137C4×D4—1941no—
138#117—972no—
139C2×C42—971no—
140#103—974no—
141#103—974no—
142#102—974no—
143#102C2×D9714yes—
144C4×C8—972no—
145C4×C8—971no—
146C4×C8C2×D9711yescharacteristic
147#41—1948no—
148C2×Q8—21no—
149C2×Q8—972no—
150C2×Q8—1941no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—1942no—
154C2×D4C22×D9712yes—
155C22×C4—971no—
156C22×C4—971no—
157C22×C4—971no—
158C22×C4—1942no—
159C2×C8—972no—
160C2×C8—1944no—
161C2×C8C22×D9712yes—
162C4⋊C4—21no—
163C4⋊C4—972no—
164C4⋊C4—1942no—
165C4⋊C4—1942no—
166C4⋊C4—1941no—
167C4⋊C4—1941no—
168C4⋊C4—1941no—
169C4⋊C4C22×D9712yes—
170C22⋊C4—1942no—
171C22⋊C4—1942no—
172C22⋊C4—1942no—
173C42—971no—
174C42—971no—
175C42—971no—
176C42—1942no—
177C42C22×D9711yescharacteristic
178C23—22no—
179C23—971no—
180Q8—42no—
181Q8—3882no—
182Q8—1944no—
183D4—24no—
184D4—42no—
185D4—42no—
186D4—3884no—
187C2×C4—22no—
188C2×C4—971no—
189C2×C4—974no—
190C2×C4—971no—
191C2×C4—971no—
192C2×C4—3884no—
193C2×C4—1942no—
194C2×C4—1942no—
195C2×C4—1942no—
196C2×C4—1942no—
197C2×C4D4×D9711yescharacteristic
198C2×C4D4×D9711yescharacteristic
199C2×C4C23×D9711yescharacteristic
200C8—22no—
201C8—1942no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—972no—
206C22—1941no—
207C22—1941no—
208C22C2×D4×D9711yescharacteristic
209C4—42no—
210C4—972no—
211C4—3882no—
212C4—3882no—
213C4—1941no—
214C4—1941no—
215C4#1569082912yes—
216C4#1569083112yes—
217C4#41×D9712yes—
218C2—82no—
219C2—1942no—
220C2#124×D9711yescharacteristic
221C2C2×#1569082911yescharacteristic
222C2C2×#1569083111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#486×C97C2
socleC22×C97C22×D4
prime radical#20261283C2
hypercenter#486D97
hypocenterC97#3131
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. #20261088C2semidirect
4. #20261087C2semidirect
5. #20261086C2semidirect
6. #20261081C2semidirect
7. #20260728C2semidirect
8. C4×C8×D97C2semidirect
9. #486×C97C2semidirect
112. C97#3131semidirect
119. #486D97semidirect

Maximal subgroups

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

subgroupindex
#202612902
#202612832
#202610882
#202610872
#202610862
#202610812
#202607282
C4×C8×D972
#486×C972
#313197