Finite Group Databasebeta

← #27151983

Subgroups — #27151983

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#20261209C211yesmaximal, characteristic
2#20261073C212yesmaximal, complemented
3#20261032C212yesmaximal, complemented
4#20260937C212yesmaximal, complemented
5#20260885C212yesmaximal
6#20260833C212yesmaximal, complemented
7C4×#13095498C212yesmaximal, complemented
8#20260796C211yesmaximal, characteristic, complemented
9#541×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11C2×#13095498C2212yes—
12C2×#13095498C2212yes—
13C4×D388C2212yes—
14C4×D388C2212yes—
15#15690772—22no—
16#15690769C2211yescharacteristic
17C22⋊C4×D97—22no—
18#15690743C2212yescomplemented
19#15690741C2212yescomplemented
20C4⋊#11715895C2212yescomplemented
21#15690734C2212yescomplemented
22#15690734C2212yescomplemented
23#15690730C2212yes—
24#15690730C2212yes—
25C22⋊#11715895C2212yes—
26#15690721C2211yescharacteristic, complemented
27C4×#11715895C2211yescharacteristic, complemented
28C4⋊Q8×C97C2211yescharacteristic, complemented
29#126×C97—22no—
30#125×C97—22no—
31#125×C97C2211yescharacteristic, complemented
32C22⋊Q8×C97C2212yes—
33C4×Q8×C97C2212yescomplemented
34#117×C97C2211yescharacteristic
35C23×D97—21no—
36C2×D388—21no—
37C2×D388—24no—
38C2×D388C2312yes—
39C2×C4×D97—22no—
40C2×C4×D97—22no—
41#13095498—24no—
42#13095498—24no—
43C2×#11715895C2312yes—
44C2×#11715895C2312yes—
45C4⋊Dic97C2312yes—
46C22⋊Dic97—21no—
47#13095488—22no—
48#13095488—22no—
49C4×Dic97—21no—
50C2×Q8×C97C2312yes—
51C22×C4×C97C2311yescharacteristic
52C4⋊C4×C97—22no—
53C4⋊C4×C97—22no—
54C4⋊C4×C97—22no—
55C4⋊C4×C97—22no—
56C4⋊C4×C97D412yes—
57C4⋊C4×C97C2312yes—
58C22⋊C4×C97—22no—
59C22⋊C4×C97D412yes—
60C42×C97—22no—
61C42×C97C2311yescharacteristic
62C42×C97C2311yescharacteristic
63C23×C97C2×D411yescharacteristic
64C22×D97—22no—
65C22×D97—24no—
66C22×D97—82no—
67Q8×C97—22no—
68Q8×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97C2×D412yes—
73C2×C4×C97C2×D412yes—
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2411yescharacteristic
76C2×Dic97—22no—
77D388—24no—
78D388—22no—
79D388#4114yes—
80C4×D97—82no—
81C4×D97—82no—
82#11715895—22no—
83#11715895—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×C97—42no—
95C4×C97—42no—
96C4×C97C2×#4112yes—
97Dic97—42no—
98C2×C97—41no—
99C2×C97#52511yescharacteristic
100C2×C97#57712yes—
101D97—44no—
102#3123—971nomaximal, Sylow
103C97#312311yescharacteristic, complemented, Sylow
104#541D9711yescharacteristic, complemented
105#518—971no—
106#485—972no—
107#478—972no—
108#461—972no—
109#450—972no—
110#440—972no—
111C4×SD16—972no—
112#433—971no—
113C22×D4—971no—
114C2×SD16—972no—
115C2×SD16—972no—
116C4⋊Q8C2×D9711yescharacteristic, complemented
117#126—22no—
118#125—22no—
119#125C2×D9711yescharacteristic, complemented
120C22⋊Q8C2×D9712yes—
121C4×Q8C2×D9712yescomplemented
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⋊C8—972no—
131#103—972no—
132#103—972no—
133#102—972no—
134#102—972no—
135C22⋊C8—972no—
136#97—971no—
137C4×C8—971no—
138C24—1941no—
139C2×Q8C22×D9712yes—
140C2×D4—972no—
141C2×D4—1944no—
142C2×D4—1941no—
143C22×C4—1942no—
144C22×C4—1942no—
145C22×C4C22×D9711yescharacteristic
146SD16—1944no—
147SD16—1944no—
148C2×C8—972no—
149C2×C8—972no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—972no—
155C4⋊C4C22⋊D9712yes—
156C4⋊C4C22×D9712yes—
157C22⋊C4—22no—
158C22⋊C4—1941no—
159C22⋊C4—1942no—
160C22⋊C4—1942no—
161C22⋊C4C22⋊D9712yes—
162C42—22no—
163C42—1941no—
164C42C22×D9711yescharacteristic
165C42C22×D9711yescharacteristic
166C23—7762no—
167C23—1944no—
168C23—1942no—
169C23C2×C22⋊D9711yescharacteristic
170Q8—22no—
171Q8—22no—
172D4—974no—
173D4—1944no—
174D4—1942no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—7762no—
179C2×C4—7762no—
180C2×C4—1942no—
181C2×C4C2×C22⋊D9712yes—
182C2×C4C2×C22⋊D9711yescharacteristic
183C2×C4C2×C22⋊D9712yes—
184C2×C4C23×D9711yescharacteristic
185C8—1942no—
186C8—1942no—
187C22—21no—
188C22—22no—
189C22—7764no—
190C22—3882no—
191C22—3882no—
192C22—1944no—
193C22C22×C22⋊D9711yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—3882no—
200C4#41×D9712yes—
201C2—41no—
202C2—3884no—
203C2#2026124211yescharacteristic
204C2#2026154312yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#541×C97C2
socleC22×C97C22×D4
prime radicalC22×D388C22
hypercenter#541D97
hypocenterC97#3123
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #20261073C2semidirect
3. #20261032C2semidirect
4. #20260937C2semidirect
6. #20260833C2semidirect
7. C4×#13095498C2semidirect
8. #20260796C2semidirect
9. #541×C97C2semidirect
18. #15690743C22semidirect
19. #15690741C22semidirect
20. C4⋊#11715895C22semidirect
21. #15690734C22semidirect
22. #15690734C22semidirect
26. #15690721C22semidirect
27. C4×#11715895C22semidirect
28. C4⋊Q8×C97C22semidirect
31. #125×C97C22semidirect
33. C4×Q8×C97C22semidirect
103. C97#3123semidirect
104. #541D97semidirect
116. C4⋊Q8C2×D97semidirect
119. #125C2×D97semidirect
121. C4×Q8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612092
#202610732
#202610322
#202609372
#202608852
#202608332
C4×#130954982
#202607962
#541×C972
#312397