Finite Group Databasebeta

← #27151988

Subgroups — #27151988

Order 12416.

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

normal subgroups only

This group has 394 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#20261347C211yesmaximal, characteristic, complemented
2#20261076C212yesmaximal, complemented
3#20261035C212yesmaximal, complemented
4#20260940C212yesmaximal, complemented
5#20260886C212yesmaximal
6#20260835C212yesmaximal, complemented
7C4×#13095499C212yesmaximal, complemented
8#20260796C211yesmaximal, characteristic, complemented
9#518×C97C211yesmaximal, characteristic
10C2×#13095499C2212yes—
11C2×#13095499C2212yes—
12#15690810C2211yescharacteristic, complemented
13#15690803—22no—
14#15690802C2211yescharacteristic, complemented
15#15690800—22no—
16#15690789C2212yes—
17C4×Dic194C2212yescomplemented
18#15690743C2212yescomplemented
19#15690741C2212yescomplemented
20C4⋊#11715895C2212yescomplemented
21#15690733C2212yescomplemented
22#15690733C2212yescomplemented
23#15690731C2212yes—
24#15690731C2212yes—
25C22⋊#11715895C2212yes—
26#15690721C2211yescharacteristic, complemented
27C4×#11715895C2211yescharacteristic, complemented
28C22×D4×C97C2211yescharacteristic
29C4×D4×C97—22no—
30C4×D4×C97C2212yes—
31C4×D4×C97C2212yes—
32#117×C97C2211yescharacteristic
33#117×C97C2211yescharacteristic
34C2×C22⋊C4×C97—22no—
35C2×Dic194C2312yes—
36#13095499—24no—
37#13095499—24no—
38C2×#11715895C2312yes—
39C2×#11715895C2312yes—
40C4⋊Dic97—22no—
41C4⋊Dic97C2312yes—
42#13095490—22no—
43#13095490—22no—
44#13095490—22no—
45C22⋊Dic97—22no—
46C4×Dic97—22no—
47C24×C97—21no—
48C2×D4×C97—24no—
49C2×D4×C97—21no—
50C2×D4×C97C2312yes—
51C22×C4×C97—22no—
52C22×C4×C97—22no—
53C22×C4×C97C2311yescharacteristic
54C4⋊C4×C97D412yes—
55C4⋊C4×C97C2312yes—
56C22⋊C4×C97—21no—
57C22⋊C4×C97—22no—
58C22⋊C4×C97—22no—
59C22⋊C4×C97D412yes—
60C42×C97—21no—
61C42×C97C2311yescharacteristic
62C42×C97C2311yescharacteristic
63C23×C97—24no—
64C23×C97—22no—
65C23×C97—82no—
66C23×C97C2×D411yescharacteristic
67D4×C97—24no—
68D4×C97—22no—
69D4×C97#4114yes—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97—82no—
73C2×C4×C97—82no—
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2×D412yes—
76C2×C4×C97C2×D412yes—
77C2×C4×C97C2411yescharacteristic
78C2×Dic97—22no—
79C2×Dic97—22no—
80Dic194—22no—
81Dic194—22no—
82#11715895—22no—
83#11715895—22no—
84C22×C97—21no—
85C22×C97—24no—
86C22×C97—22no—
87C22×C97—42no—
88C22×C97—42no—
89C22×C97—84no—
90C22×C97C22×D411yescharacteristic
91C4×C97—21no—
92C4×C97—22no—
93C4×C97—22no—
94C4×C97—42no—
95C4×C97C2×#4112yes—
96Dic97—42no—
97Dic97—42no—
98C2×C97—41no—
99C2×C97—44no—
100C2×C97#52511yescharacteristic
101C2×C97#57712yes—
102#3123—971nomaximal, Sylow
103C97#312311yescharacteristic, complemented, Sylow
104#541—971no—
105#518D9711yescharacteristic
106#485—972no—
107#478—972no—
108#461—972no—
109#450—972no—
110#440—972no—
111C4×SD16—972no—
112#433—971no—
113C22×D4C2×D9711yescharacteristic
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—22no—
123C4×D4C2×D9712yes—
124C4×D4C2×D9712yes—
125#117C2×D9711yescharacteristic
126#117C2×D9711yescharacteristic
127C2×C22⋊C4—22no—
128#107—972no—
129#106—972no—
130C4⋊C8—972no—
131#103—972no—
132#103—972no—
133#102—972no—
134#102—972no—
135C22⋊C8—972no—
136#97—971no—
137C4×C8—971no—
138C24—21no—
139C2×Q8—972no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4C22×D9712yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C22×D9711yescharacteristic
146SD16—1944no—
147SD16—1944no—
148C2×C8—972no—
149C2×C8—972no—
150C4⋊C4—972no—
151C4⋊C4—1942no—
152C4⋊C4—1942no—
153C4⋊C4—1942no—
154C4⋊C4—1942no—
155C4⋊C4C22⋊D9712yes—
156C4⋊C4C22×D9712yes—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—1942no—
161C22⋊C4C22⋊D9712yes—
162C42—21no—
163C42—1942no—
164C42C22×D9711yescharacteristic
165C42C22×D9711yescharacteristic
166C23—24no—
167C23—22no—
168C23—82no—
169C23C2×C22⋊D9711yescharacteristic
170Q8—1942no—
171Q8—1942no—
172D4—24no—
173D4—22no—
174D4#1309551114yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—1942no—
180C2×C4—1942no—
181C2×C4C2×C22⋊D9711yescharacteristic
182C2×C4C2×C22⋊D9712yes—
183C2×C4C2×C22⋊D9712yes—
184C2×C4C23×D9711yescharacteristic
185C8—1942no—
186C8—1942no—
187C22—21no—
188C22—24no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22C22×C22⋊D9711yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—3882no—
199C4—3882no—
200C4C2×#1309551112yes—
201C2—41no—
202C2—44no—
203C2#2026124611yescharacteristic
204C2#2026154612yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#518×C97C2
socleC22×C97C22×D4
prime radicalC22×D4×C97C22
hypercenter#518D97
hypocenterC97#3123
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #20261347C2semidirect
2. #20261076C2semidirect
3. #20261035C2semidirect
4. #20260940C2semidirect
6. #20260835C2semidirect
7. C4×#13095499C2semidirect
8. #20260796C2semidirect
12. #15690810C22semidirect
14. #15690802C22semidirect
17. C4×Dic194C22semidirect
18. #15690743C22semidirect
19. #15690741C22semidirect
20. C4⋊#11715895C22semidirect
21. #15690733C22semidirect
22. #15690733C22semidirect
26. #15690721C22semidirect
27. C4×#11715895C22semidirect
103. C97#3123semidirect

Maximal subgroups

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

subgroupindex
#202613472
#202610762
#202610352
#202609402
#202608862
#202608352
C4×#130954992
#202607962
#518×C972
#312397