Finite Group Databasebeta

← #27151974

Subgroups — #27151974

Order 12416.

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

normal subgroups only

This group has 418 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, complemented
2#20261056C212yesmaximal, complemented
3#20261032C212yesmaximal, complemented
4#20260927C212yesmaximal, complemented
5#20260868C212yesmaximal, complemented
6#20260845C212yesmaximal, complemented
7C4×#13095496C212yesmaximal, complemented
8#20260796C211yesmaximal, characteristic, complemented
9#540×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11C2×#13095496C2212yes—
12C2×#13095496C2212yes—
13C4×D388C2212yes—
14C4×D388C2212yes—
15#15690772—22no—
16#15690769C2211yescharacteristic
17C22⋊C4×D97—22no—
18#15690743C2212yescomplemented
19#15690741C2212yescomplemented
20C4⋊#11715895C2212yescomplemented
21#15690731C2212yescomplemented
22#15690731C2212yescomplemented
23#15690730C2212yes—
24#15690730C2212yes—
25C22⋊#11715895C2212yes—
26#15690721C2211yescharacteristic, complemented
27C4×#11715895C2211yescharacteristic, complemented
28#127×C97C2211yescharacteristic, complemented
29#125×C97C2211yescharacteristic, complemented
30#123×C97—22no—
31#121×C97—22no—
32#121×C97C2212yes—
33C4×D4×C97C2212yescomplemented
34#117×C97C2211yescharacteristic
35C23×D97—21no—
36C2×D388—21no—
37C2×D388—24no—
38C2×D388C2312yes—
39C2×C4×D97—22no—
40C2×C4×D97—22no—
41#13095496—24no—
42#13095496—24no—
43C2×#11715895C2312yes—
44C2×#11715895C2312yes—
45C4⋊Dic97C2312yes—
46C22⋊Dic97—21no—
47#13095488—22no—
48#13095488—22no—
49C4×Dic97—21no—
50C2×D4×C97—22no—
51C2×D4×C97—22no—
52C2×D4×C97C2312yes—
53C22×C4×C97—22no—
54C22×C4×C97C2311yescharacteristic
55C4⋊C4×C97—22no—
56C4⋊C4×C97D412yes—
57C4⋊C4×C97C2312yes—
58C22⋊C4×C97—22no—
59C22⋊C4×C97—22no—
60C22⋊C4×C97D412yes—
61C42×C97C2311yescharacteristic
62C42×C97C2311yescharacteristic
63C23×C97—22no—
64C23×C97C2×D411yescharacteristic
65C22×D97—22no—
66C22×D97—24no—
67C22×D97—82no—
68D4×C97—22no—
69D4×C97—22no—
70D4×C97—82no—
71D4×C97—82no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—82no—
75C2×C4×C97C2×D412yes—
76C2×C4×C97C2×D412yes—
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2411yescharacteristic
79C2×Dic97—22no—
80D388—24no—
81D388—22no—
82D388#4114yes—
83C4×D97—82no—
84C4×D97—82no—
85#11715895—22no—
86#11715895—22no—
87C22×C97—21no—
88C22×C97—22no—
89C22×C97—42no—
90C22×C97—42no—
91C22×C97—42no—
92C22×C97C22×D411yescharacteristic
93C2×D97—24no—
94C2×D97—42no—
95C2×D97—42no—
96C2×D97—84no—
97C4×C97—21no—
98C4×C97—22no—
99C4×C97—22no—
100C4×C97—42no—
101C4×C97C2×#4112yes—
102Dic97—42no—
103C2×C97—41no—
104C2×C97—82no—
105C2×C97#52511yescharacteristic
106C2×C97#57612yes—
107D97—44no—
108#3122—971nomaximal, Sylow
109C97#312211yescharacteristic, complemented, Sylow
110#540D9711yescharacteristic, complemented
111#518—971no—
112#482—972no—
113#478—972no—
114#459—972no—
115#447—972no—
116#442—972no—
117C4×D8—972no—
118#433—971no—
119C22×D4—971no—
120C2×D8—972no—
121C2×D8—972no—
122#127C2×D9711yescharacteristic, complemented
123#125C2×D9711yescharacteristic, complemented
124#123—22no—
125#121—22no—
126#121C2×D9712yes—
127C4×D4—972no—
128C4×D4—972no—
129C4×D4—1942no—
130C4×D4C2×D9712yescomplemented
131#117—971no—
132#117C2×D9711yescharacteristic
133C2×C22⋊C4—1942no—
134#107—972no—
135#106—972no—
136C4⋊C8—972no—
137#102—972no—
138#102—972no—
139#102—972no—
140#102—972no—
141C22⋊C8—972no—
142#97—971no—
143C4×C8—971no—
144C24—1941no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—972no—
148C2×D4—1944no—
149C2×D4—1941no—
150C2×D4C22×D9712yes—
151C22×C4—22no—
152C22×C4—1942no—
153C22×C4—1942no—
154C22×C4C22×D9711yescharacteristic
155D8—1944no—
156D8—1944no—
157C2×C8—972no—
158C2×C8—972no—
159C4⋊C4—22no—
160C4⋊C4—972no—
161C4⋊C4C22⋊D9712yes—
162C4⋊C4C22×D9712yes—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—1941no—
166C22⋊C4—1942no—
167C22⋊C4—1942no—
168C22⋊C4C22⋊D9712yes—
169C42—1941no—
170C42C22×D9711yescharacteristic
171C42C22×D9711yescharacteristic
172C23—22no—
173C23—7762no—
174C23—1944no—
175C23—1942no—
176C23C2×C22⋊D9711yescharacteristic
177D4—22no—
178D4—22no—
179D4—82no—
180D4—82no—
181D4—974no—
182D4—1944no—
183D4—1942no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—7762no—
188C2×C4—7762no—
189C2×C4—1942no—
190C2×C4C2×C22⋊D9711yescharacteristic
191C2×C4C2×C22⋊D9712yes—
192C2×C4C2×C22⋊D9712yes—
193C2×C4C23×D9711yescharacteristic
194C8—1942no—
195C8—1942no—
196C22—21no—
197C22—22no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—7764no—
202C22—3882no—
203C22—3882no—
204C22—1944no—
205C22C22×C22⋊D9711yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—3882no—
211C4#41×D9712yes—
212C2—41no—
213C2—82no—
214C2—3884no—
215C2#2026124211yescharacteristic
216C2#2026153812yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#540×C97C2
socleC22×C97C22×D4
hypercenter#540D97
hypocenterC97#3122
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #20261209C2semidirect
2. #20261056C2semidirect
3. #20261032C2semidirect
4. #20260927C2semidirect
5. #20260868C2semidirect
6. #20260845C2semidirect
7. C4×#13095496C2semidirect
8. #20260796C2semidirect
9. #540×C97C2semidirect
18. #15690743C22semidirect
19. #15690741C22semidirect
20. C4⋊#11715895C22semidirect
21. #15690731C22semidirect
22. #15690731C22semidirect
26. #15690721C22semidirect
27. C4×#11715895C22semidirect
28. #127×C97C22semidirect
29. #125×C97C22semidirect
33. C4×D4×C97C22semidirect
109. C97#3122semidirect
110. #540D97semidirect
122. #127C2×D97semidirect
123. #125C2×D97semidirect
130. C4×D4C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612092
#202610562
#202610322
#202609272
#202608682
#202608452
C4×#130954962
#202607962
#540×C972
#312297