Finite Group Databasebeta

← #27151346

Subgroups — #27151346

Order 12416.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#15690832C211yesmaximal, characteristic, complemented
2C22×#13095494C211yesmaximal, characteristic, complemented
3#20261247C212yesmaximal, complemented
4#20260994C212yesmaximal, complemented
5#20260989C212yesmaximal, complemented
6#20260982C214yesmaximal, complemented
7#20260729C212yesmaximal, complemented
8#427×C97C211yesmaximal, characteristic, complemented
9C2×#13095511C2212yes—
10C22×C4×D97C2211yescharacteristic
11#15690832—24no—
12C2×Dic388C2211yescharacteristic, complemented
13C2×#13095500C2212yescomplemented
14C2×D776C2211yescharacteristic, complemented
15C2×#13095494—24no—
16C2×#13095494C2212yes—
17C2×#13095494C2212yes—
18C22×#11715895C2211yescharacteristic
19#15690794—22no—
20#15690790C2214yes—
21#15690785C2214yes—
22#15690780—22no—
23C4×D388—24no—
24#15690733C2214yescomplemented
25#15690730C2214yescomplemented
26C22×C8×C97C2211yescharacteristic
27#117×C97C2212yes—
28#107×C97C2212yescomplemented
29#106×C97C2212yescomplemented
30C23×D97D411yescharacteristic
31#13095511—24no—
32#13095511—24no—
33C2×Dic194C2312yes—
34C2×D388—24no—
35C2×D388C2312yes—
36C2×C22⋊D97—22no—
37C22×Dic97D411yescharacteristic
38C2×C4×D97—22no—
39C2×C4×D97—22no—
40C2×C4×D97—22no—
41C2×C4×D97D412yes—
42C2×C4×D97C2312yes—
43Dic388—22no—
44#13095500—24no—
45D776—22no—
46#13095494—24no—
47#13095494—24no—
48C2×#11715895—22no—
49C2×#11715895C2312yes—
50C4⋊Dic97—24no—
51#13095488—24no—
52#13095488—24no—
53C22×C4×C97C2311yescharacteristic
54C2×C8×C97D414yescomplemented
55C2×C8×C97C2311yescharacteristic
56C2×C8×C97C2311yescharacteristic
57C4⋊C4×C97C2314yes—
58C22⋊C4×C97—22no—
59C42×C97—22no—
60C23×C97C2×D411yescharacteristic
61C22×D97—22no—
62C22×D97—22no—
63C22×D97—42no—
64C22×D97C2×D412yes—
65C2×C4×C97—24no—
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×D412yes—
68C2×C4×C97#4112yes—
69C2×C4×C97C2411yescharacteristic
70C22⋊D97—44no—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97C2×D412yes—
74D388—22no—
75D388—22no—
76D388—48no—
77C4×D97—24no—
78C4×D97—22no—
79C4×D97—22no—
80C4×D97—44no—
81Dic194—22no—
82Dic194—22no—
83C8×C97C2×D412yes—
84C8×C97C2×D412yes—
85#11715895—22no—
86C22×C97—22no—
87C22×C97#12112yes—
88C22×C97C22×D411yescharacteristic
89C2×D97—22no—
90C2×D97—24no—
91C2×D97—44no—
92C2×D97—42no—
93C2×D97—44no—
94C4×C97—44no—
95C4×C97#12112yes—
96C4×C97C22×D411yescharacteristic
97C4×C97C2×#4111yescharacteristic
98Dic97—22no—
99Dic97—42no—
100C2×C97—22no—
101C2×C97C2×#12111yescharacteristic
102C2×C97#57512yes—
103D97—42no—
104D97—82no—
105#3054—971nomaximal, Sylow
106C97#305411yescharacteristic, complemented, Sylow
107C2×#135—971no—
108C22×M16—971no—
109#525—972no—
110#470—972no—
111#469—972no—
112#468—974no—
113#427D9711yescharacteristic, complemented
114#416—972no—
115C2×#41—972no—
116C23×C4—971no—
117#135—1944no—
118C2×Q16—971no—
119C2×SD16—972no—
120C2×D8—971no—
121C2×M16—972no—
122C2×M16—972no—
123C2×M16—1944no—
124C22×C8—971no—
125C22×C8C2×D9711yescharacteristic
126#123—1942no—
127C22⋊Q8—974no—
128#121—974no—
129#120—1942no—
130C4×D4—1944no—
131#117C2×D9712yes—
132#107C2×D9712yescomplemented
133#106C2×D9712yescomplemented
134#103—974no—
135#102—974no—
136C24—971no—
137#41—1944no—
138#41—1944no—
139C2×Q8—972no—
140C2×D4—972no—
141C2×D4—1944no—
142C2×D4—1942no—
143C22×C4—971no—
144C22×C4—972no—
145C22×C4—972no—
146C22×C4—1942no—
147C22×C4—1942no—
148C22×C4—1942no—
149C22×C4C22×D9711yescharacteristic
150Q16—1942no—
151SD16—1944no—
152D8—1942no—
153M16—1944no—
154M16—1944no—
155C2×C8—972no—
156C2×C8—1942no—
157C2×C8D38814yescomplemented
158C2×C8C22×D9711yescharacteristic
159C2×C8C22×D9711yescharacteristic
160C4⋊C4—1944no—
161C4⋊C4C22×D9714yes—
162C22⋊C4—22no—
163C22⋊C4—1944no—
164C22⋊C4—1944no—
165C42—22no—
166C23—972no—
167C23—3882no—
168C23—1942no—
169C23—1942no—
170C23D4×D9711yescharacteristic
171Q8—1942no—
172Q8—1942no—
173D4—3888no—
174D4—3884no—
175D4—1942no—
176D4—1942no—
177C2×C4—24no—
178C2×C4—972no—
179C2×C4—3884no—
180C2×C4—1942no—
181C2×C4—1942no—
182C2×C4—1944no—
183C2×C4—1942no—
184C2×C4—1942no—
185C2×C4D4×D9711yescharacteristic
186C2×C4C2×D38812yes—
187C2×C4#1309551212yes—
188C2×C4C23×D9711yescharacteristic
189C8—1942no—
190C8C2×D38812yes—
191C8C2×D38812yes—
192C22—22no—
193C22—3884no—
194C22—3882no—
195C22—3884no—
196C22—1942no—
197C22—1944no—
198C22#1569078512yes—
199C22C2×D4×D9711yescharacteristic
200C4—44no—
201C4—3882no—
202C4—1942no—
203C4#1569078512yes—
204C4C22×D38811yescharacteristic
205C4C2×#1309551211yescharacteristic
206C2—22no—
207C2—7762no—
208C2—3882no—
209C2C2×#1569078511yescharacteristic
210C2#2026153212yes—
subgroupG/H
centerC2×C4D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#427×C97C2
socleC22×C97C22×D4
hypercenter#427D97
hypocenterC97#3054
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. C2×#15690832C2semidirect
2. C22×#13095494C2semidirect
3. #20261247C2semidirect
4. #20260994C2semidirect
5. #20260989C2semidirect
6. #20260982C2semidirect
7. #20260729C2semidirect
8. #427×C97C2semidirect
12. C2×Dic388C22semidirect
13. C2×#13095500C22semidirect
14. C2×D776C22semidirect
24. #15690733C22semidirect
25. #15690730C22semidirect
28. #107×C97C22semidirect
29. #106×C97C22semidirect
54. C2×C8×C97D4semidirect
106. C97#3054semidirect
113. #427D97semidirect
132. #107C2×D97semidirect
133. #106C2×D97semidirect
157. C2×C8D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156908322
C22×#130954942
#202612472
#202609942
#202609892
#202609822
#202607292
#427×C972
#305497