Finite Group Databasebeta

← #27151980

Subgroups — #27151980

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#20261340C211yesmaximal, characteristic, complemented
2#20261209C211yesmaximal, characteristic, complemented
3#20261060C212yesmaximal, complemented
4#20261036C212yesmaximal, complemented
5#20260931C212yesmaximal, complemented
6#20260871C212yesmaximal, complemented
7#20260848C212yesmaximal, complemented
8C4×D776C212yesmaximal, complemented
9#433×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11C2×D776C2212yes—
12C2×D776C2212yes—
13#15690807C2211yescharacteristic, complemented
14#15690802C2211yescharacteristic, complemented
15#15690795—22no—
16#15690784C2212yes—
17#15690783—22no—
18C4×D388C2212yes—
19C4×D388C2212yes—
20C4×D388C2212yescomplemented
21#15690772—22no—
22#15690769C2211yescharacteristic
23C22⋊C4×D97—22no—
24#15690744C2212yescomplemented
25#15690742C2212yescomplemented
26#15690732C2212yes—
27#15690732C2212yes—
28#15690732C2212yescomplemented
29#15690732C2212yescomplemented
30#117×C97C2211yescharacteristic
31C4⋊C8×C97C2212yescomplemented
32C22⋊C8×C97C2212yes—
33#97×C97C2211yescharacteristic, complemented
34C4×C8×C97C2211yescharacteristic, complemented
35C23×D97—21no—
36C2×D388—21no—
37C2×D388—24no—
38C2×D388—22no—
39C2×D388C2312yes—
40C2×D388C2312yes—
41C2×C22⋊D97—22no—
42C2×C4×D97—22no—
43C2×C4×D97—22no—
44C2×C4×D97—22no—
45D776—24no—
46D776—24no—
47C4⋊Dic97C2312yes—
48C4⋊Dic97C2312yes—
49#13095490—22no—
50C22⋊Dic97—21no—
51#13095488—22no—
52#13095488—22no—
53#13095488—22no—
54#13095488—22no—
55C4×Dic97—21no—
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—22no—
67C22×D97—82no—
68C2×C4×C97—22no—
69C2×C4×C97C2×D412yes—
70C2×C4×C97C2×D412yes—
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2411yescharacteristic
73C22⋊D97—82no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76D388—24no—
77D388—22no—
78D388—22no—
79D388—22no—
80D388—82no—
81D388#4114yes—
82C4×D97—82no—
83C4×D97—82no—
84C4×D97—82no—
85C8×C97—22no—
86C8×C97—22no—
87C22×C97—21no—
88C22×C97—22no—
89C22×C97C22×D411yescharacteristic
90C2×D97—24no—
91C2×D97—42no—
92C2×D97—42no—
93C2×D97—42no—
94C2×D97—42no—
95C2×D97—42no—
96C2×D97—84no—
97C4×C97—21no—
98C4×C97—22no—
99C4×C97—22no—
100C4×C97C2×#4112yes—
101Dic97—42no—
102Dic97—42no—
103C2×C97—41no—
104C2×C97#52511yescharacteristic
105C2×C97#57612yes—
106D97—44no—
107D97—82no—
108#3122—971nomaximal, Sylow
109C97#312211yescharacteristic, complemented, Sylow
110#540—971no—
111#518—971no—
112#482—972no—
113#478—972no—
114#459—972no—
115#447—972no—
116#442—972no—
117C4×D8—972no—
118#433D9711yescharacteristic, complemented
119C22×D4—971no—
120C2×D8—972no—
121C2×D8—972no—
122#127—971no—
123#125—971no—
124#123—1942no—
125#121—972no—
126#121—1942no—
127C4×D4—972no—
128C4×D4—972no—
129C4×D4—972no—
130C4×D4—1942no—
131#117—971no—
132#117C2×D9711yescharacteristic
133C2×C22⋊C4—1942no—
134#107—972no—
135#106—972no—
136C4⋊C8C2×D9712yescomplemented
137#102—972no—
138#102—972no—
139#102—972no—
140#102—972no—
141C22⋊C8C2×D9712yes—
142#97C2×D9711yescharacteristic, complemented
143C4×C8C2×D9711yescharacteristic, complemented
144C24—1941no—
145C2×D4—972no—
146C2×D4—972no—
147C2×D4—1944no—
148C2×D4—1942no—
149C2×D4—1942no—
150C2×D4—1941no—
151C22×C4—1942no—
152C22×C4—1942no—
153C22×C4—1942no—
154C22×C4C22×D9711yescharacteristic
155D8—1944no—
156D8—1944no—
157C2×C8C22×D9712yes—
158C2×C8C22×D9712yes—
159C4⋊C4—972no—
160C4⋊C4—972no—
161C4⋊C4—1942no—
162C4⋊C4D38812yes—
163C22⋊C4—1941no—
164C22⋊C4—1942no—
165C22⋊C4—1942no—
166C22⋊C4—1942no—
167C22⋊C4—1942no—
168C22⋊C4D38812yes—
169C42—1941no—
170C42C22×D9711yescharacteristic
171C42C22×D9711yescharacteristic
172C23—7762no—
173C23—1944no—
174C23—1942no—
175C23—1942no—
176C23C2×D38811yescharacteristic
177D4—7762no—
178D4—7762no—
179D4—974no—
180D4—1944no—
181D4—1942no—
182D4—1942no—
183D4—1942no—
184C2×C4—22no—
185C2×C4—7762no—
186C2×C4—7762no—
187C2×C4—7762no—
188C2×C4—1942no—
189C2×C4—1942no—
190C2×C4C2×D38811yescharacteristic
191C2×C4C2×D38812yes—
192C2×C4C2×D38812yes—
193C2×C4C23×D9711yescharacteristic
194C8—22no—
195C8—22no—
196C22—21no—
197C22—22no—
198C22—7764no—
199C22—3882no—
200C22—3882no—
201C22—3882no—
202C22—3882no—
203C22—3882no—
204C22—1944no—
205C22C22×D38811yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—3882no—
210C4—3882no—
211C4#41×D9712yes—
212C2—41no—
213C2—7762no—
214C2—3884no—
215C2#2026124711yescharacteristic
216C2#2026154012yes—
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#433×C97C2
socleC22×C97C22×D4
hypercenter#433D97
hypocenterC97#3122
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. #20261340C2semidirect
2. #20261209C2semidirect
3. #20261060C2semidirect
4. #20261036C2semidirect
5. #20260931C2semidirect
6. #20260871C2semidirect
7. #20260848C2semidirect
8. C4×D776C2semidirect
9. #433×C97C2semidirect
13. #15690807C22semidirect
14. #15690802C22semidirect
20. C4×D388C22semidirect
24. #15690744C22semidirect
25. #15690742C22semidirect
28. #15690732C22semidirect
29. #15690732C22semidirect
31. C4⋊C8×C97C22semidirect
33. #97×C97C22semidirect
34. C4×C8×C97C22semidirect
109. C97#3122semidirect
118. #433D97semidirect
136. C4⋊C8C2×D97semidirect
142. #97C2×D97semidirect
143. C4×C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613402
#202612092
#202610602
#202610362
#202609312
#202608712
#202608482
C4×D7762
#433×C972
#312297