Finite Group Databasebeta

← #27151989

Subgroups — #27151989

Order 12416.

394 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 121 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#20261347C211yesmaximal, characteristic, complemented
2#20261209C211yesmaximal, characteristic
3#20261077C212yesmaximal, complemented
4#20261036C212yesmaximal, complemented
5#20260941C212yesmaximal, complemented
6#20260888C212yesmaximal
7#20260837C212yesmaximal, complemented
8C4×#13095500C212yesmaximal, complemented
9#433×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11C2×#13095500C2212yes—
12C2×#13095500C2212yes—
13#15690810C2211yescharacteristic, complemented
14#15690803—22no—
15#15690802C2211yescharacteristic, complemented
16#15690800—22no—
17#15690789C2212yes—
18C4×Dic194C2212yescomplemented
19C4×D388C2212yes—
20C4×D388C2212yes—
21#15690772—22no—
22#15690769C2211yescharacteristic
23C22⋊C4×D97—22no—
24#15690744C2212yescomplemented
25#15690742C2212yescomplemented
26#15690735C2212yescomplemented
27#15690735C2212yescomplemented
28#15690732C2212yes—
29#15690732C2212yes—
30#117×C97C2211yescharacteristic
31C4⋊C8×C97C2212yescomplemented
32C22⋊C8×C97C2212yes—
33#97×C97C2211yescharacteristic, complemented
34C4×C8×C97C2211yescharacteristic, complemented
35C23×D97—21no—
36C2×Dic194C2312yes—
37C2×D388—21no—
38C2×D388—24no—
39C2×D388C2312yes—
40C2×C4×D97—22no—
41C2×C4×D97—22no—
42#13095500—24no—
43#13095500—24no—
44C4⋊Dic97—22no—
45C4⋊Dic97C2312yes—
46C4⋊Dic97C2312yes—
47#13095490—22no—
48#13095490—22no—
49#13095490—22no—
50C22⋊Dic97—21no—
51C22⋊Dic97—22no—
52#13095488—22no—
53#13095488—22no—
54C4×Dic97—21no—
55C4×Dic97—22no—
56C22×C4×C97C2311yescharacteristic
57C2×C8×C97C2312yes—
58C2×C8×C97C2312yes—
59C4⋊C4×C97D412yes—
60C22⋊C4×C97D412yes—
61C42×C97C2311yescharacteristic
62C42×C97C2311yescharacteristic
63C23×C97C2×D411yescharacteristic
64C22×D97—22no—
65C22×D97—24no—
66C22×D97—82no—
67C2×C4×C97—22no—
68C2×C4×C97C2×D412yes—
69C2×C4×C97C2×D412yes—
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2411yescharacteristic
72C2×Dic97—22no—
73C2×Dic97—22no—
74C2×Dic97—22no—
75D388—24no—
76D388—22no—
77D388#4114yes—
78C4×D97—82no—
79C4×D97—82no—
80Dic194—22no—
81Dic194—22no—
82C8×C97—22no—
83C8×C97—22no—
84C22×C97—21no—
85C22×C97—22no—
86C22×C97C22×D411yescharacteristic
87C2×D97—24no—
88C2×D97—42no—
89C2×D97—42no—
90C2×D97—84no—
91C4×C97—21no—
92C4×C97—22no—
93C4×C97—22no—
94C4×C97C2×#4112yes—
95Dic97—42no—
96Dic97—42no—
97Dic97—42no—
98C2×C97—41no—
99C2×C97#52511yescharacteristic
100C2×C97#57712yes—
101D97—44no—
102#3123—971nomaximal, Sylow
103C97#312311yescharacteristic, complemented, Sylow
104#541—971no—
105#518—971no—
106#485—972no—
107#478—972no—
108#461—972no—
109#450—972no—
110#440—972no—
111C4×SD16—972no—
112#433D9711yescharacteristic, complemented
113C22×D4—971no—
114C2×SD16—972no—
115C2×SD16—972no—
116C4⋊Q8—971no—
117#126—1942no—
118#125—971no—
119#125—1942no—
120C22⋊Q8—972no—
121C4×Q8—972no—
122C4×D4—972no—
123C4×D4—972no—
124C4×D4—1942no—
125#117—971no—
126#117C2×D9711yescharacteristic
127C2×C22⋊C4—1942no—
128#107—972no—
129#106—972no—
130C4⋊C8C2×D9712yescomplemented
131#103—972no—
132#103—972no—
133#102—972no—
134#102—972no—
135C22⋊C8C2×D9712yes—
136#97C2×D9711yescharacteristic, complemented
137C4×C8C2×D9711yescharacteristic, complemented
138C24—1941no—
139C2×Q8—972no—
140C2×D4—972no—
141C2×D4—1944no—
142C2×D4—1941no—
143C22×C4—1942no—
144C22×C4—1942no—
145C22×C4C22×D9711yescharacteristic
146SD16—1944no—
147SD16—1944no—
148C2×C8C22×D9712yes—
149C2×C8C22×D9712yes—
150C4⋊C4—972no—
151C4⋊C4—972no—
152C4⋊C4—1942no—
153C4⋊C4—1942no—
154C4⋊C4—1942no—
155C4⋊C4—1942no—
156C4⋊C4D38812yes—
157C22⋊C4—1941no—
158C22⋊C4—1942no—
159C22⋊C4—1942no—
160C22⋊C4—1942no—
161C22⋊C4D38812yes—
162C42—1941no—
163C42—1942no—
164C42C22×D9711yescharacteristic
165C42C22×D9711yescharacteristic
166C23—7762no—
167C23—1944no—
168C23—1942no—
169C23C2×D38811yescharacteristic
170Q8—1942no—
171Q8—1942no—
172D4—974no—
173D4—1944no—
174D4—1942no—
175C2×C4—22no—
176C2×C4—7762no—
177C2×C4—7762no—
178C2×C4—1942no—
179C2×C4—1942no—
180C2×C4—1942no—
181C2×C4C2×D38812yes—
182C2×C4C2×D38811yescharacteristic
183C2×C4C2×D38812yes—
184C2×C4C23×D9711yescharacteristic
185C8—22no—
186C8—22no—
187C22—21no—
188C22—22no—
189C22—7764no—
190C22—3882no—
191C22—3882no—
192C22—1944no—
193C22C22×D38811yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—3882no—
198C4—3882no—
199C4—3882no—
200C4#41×D9712yes—
201C2—41no—
202C2—3884no—
203C2#2026124711yescharacteristic
204C2#2026154712yes—
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#433×C97C2
socleC22×C97C22×D4
prime radicalC22×D388C22
hypercenter#433D97
hypocenterC97#3123
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #20261347C2semidirect
3. #20261077C2semidirect
4. #20261036C2semidirect
5. #20260941C2semidirect
7. #20260837C2semidirect
8. C4×#13095500C2semidirect
9. #433×C97C2semidirect
13. #15690810C22semidirect
15. #15690802C22semidirect
18. C4×Dic194C22semidirect
24. #15690744C22semidirect
25. #15690742C22semidirect
26. #15690735C22semidirect
27. #15690735C22semidirect
31. C4⋊C8×C97C22semidirect
33. #97×C97C22semidirect
34. C4×C8×C97C22semidirect
103. C97#3123semidirect
112. #433D97semidirect
130. C4⋊C8C2×D97semidirect
136. #97C2×D97semidirect
137. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613472
#202612092
#202610772
#202610362
#202609412
#202608882
#202608372
C4×#130955002
#433×C972
#312397