Finite Group Databasebeta

← #27158512

Subgroups — #27158512

Order 12416.

452 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 158 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#20261813C212yesmaximal, complemented
2#20261782C214yesmaximal, complemented
3#20261777C214yesmaximal, complemented
4D97⋊#108C211yesmaximal, characteristic, complemented
5#20261733C212yesmaximal, complemented
6#20261698C211yesmaximal, characteristic, complemented
7C2×#15690829C211yesmaximal, characteristic
8#15690897—24no—
9#15690897C2218yescomplemented
10D97⋊M16C2212yes—
11D97⋊M16C2211yescharacteristic
12C2×#13095520—24no—
13#15690889C2211yescharacteristic
14#15690881C2212yescomplemented
15#15690880C2212yescomplemented
16#15690874C2218yescomplemented
17#15690864C2212yescomplemented
18C8×#87604C2212yescomplemented
19C2×#13095510C2212yes—
20#15690829C412yescomplemented
21#15690829C412yescomplemented
22#15690829C2212yes—
23#15690829C2212yes—
24C2×Dic388C411yescharacteristic, complemented
25C2×#13095499C412yescomplemented
26C2×C8×D97C2211yescharacteristic
27C2×D8×C97C411yescharacteristic, complemented
28#13095521D412yes—
29#13095521C2314yes—
30#13095520—22no—
31#13095520—28no—
32#13095520D412yes—
33#13095519—22no—
34#13095519C2312yes—
35C2×#11715897—28no—
36C4⋊#87604—21no—
37C22⋊#87604—21no—
38C4×#87604C2312yes—
39#13095510—22no—
40#13095510—22no—
41#13095510C2×C414yes—
42#13095510C2314yes—
43C2×Dic194C2×C412yes—
44C2×C22⋊D97—22no—
45C22×Dic97—22no—
46C2×C4×D97C2311yescharacteristic
47Dic388C2×C412yes—
48Dic388C2×C412yes—
49#13095499C2×C414yes—
50#13095499C2×C414yes—
51C2×#11715895C2×C411yescharacteristic
52C8×D97C2×C411yescharacteristic
53C8×D97C2×C411yescharacteristic
54C8×D97C2311yescharacteristic
55C8×D97C2311yescharacteristic
56C2×D4×C97C2×C412yes—
57D8×C97C2×C412yes—
58D8×C97C2×C412yes—
59C2×C8×C97C2×C411yescharacteristic
60C23×C97—22no—
61C22×D97#4111yescharacteristic
62C2×#87604—22no—
63D4×C97—22no—
64D4×C97C22×C414yes—
65C2×C4×C97C22×C411yescharacteristic
66C22⋊D97—24no—
67C22⋊D97—24no—
68C2×Dic97—22no—
69C2×Dic97—24no—
70C2×Dic97—24no—
71C2×Dic97C2×D411yescharacteristic
72C4×D97C22×C411yescharacteristic
73C4×D97C2×D411yescharacteristic
74C4×D97#4111yescharacteristic
75C4×D97C2411yescharacteristic
76Dic194—22no—
77Dic194C22×C414yes—
78#11715897—24no—
79#11715897C2×D414yes—
80C8×C97C22×C411yescharacteristic
81C8×C97C22×C411yescharacteristic
82#11715895C22×C411yescharacteristic
83#11715895C22×C411yescharacteristic
84C22×C97—24no—
85C22×C97—82no—
86C22×C97C4×D411yescharacteristic
87C2×D97—41no—
88C2×D97C4×D411yescharacteristic
89C2×D97C2×#4111yescharacteristic
90#87604—42no—
91C4×C97C4×D411yescharacteristic
92C4×C97C23×C411yescharacteristic
93Dic97—24no—
94Dic97C4×D411yescharacteristic
95Dic97C22×D411yescharacteristic
96C2×C97—21no—
97C2×C97—44no—
98C2×C97C2×C4×D411yescharacteristic
99D97—21no—
100D97—21no—
101#2957—971nomaximal, Sylow
102C97#295711yescharacteristic, complemented, Sylow
103C2×#135—971no—
104#568—972no—
105#444—974no—
106#443—974no—
107C2×#108—971no—
108#421—972no—
109#405—971no—
110C2×#41—972no—
111#135—972no—
112#135—972no—
113#135—972no—
114#135—972no—
115C2×Q16—971no—
116C2×SD16—972no—
117C2×D8#8760411yescharacteristic, complemented
118#131—978no—
119#131—1944no—
120C2×M16—971no—
121C2×M16—972no—
122C2×M16—1944no—
123C22×C8—971no—
124#117—971no—
125#108—972no—
126#108—972no—
127#104—978no—
128#97—972no—
129C4×C8—972no—
130#41—974no—
131#41—974no—
132#41—1942no—
133#41—1942no—
134C2×Q8—972no—
135C2×D4—1942no—
136C2×D4C2×#8760412yes—
137C22×C4—971no—
138C22×C4—1942no—
139Q16—972no—
140Q16—972no—
141SD16—974no—
142SD16—974no—
143D8C2×#8760412yes—
144D8C2×#8760412yes—
145M16—972no—
146M16—972no—
147M16—974no—
148M16—1948no—
149M16—1942no—
150C2×C8—972no—
151C2×C8—971no—
152C2×C8—971no—
153C2×C8—971no—
154C2×C8—971no—
155C2×C8—971no—
156C2×C8—1948no—
157C2×C8—1942no—
158C2×C8C2×#8760411yescharacteristic
159C4⋊C4—1941no—
160C22⋊C4—1941no—
161C42—972no—
162C23—22no—
163C23—971no—
164Q8—974no—
165Q8—1942no—
166D4—22no—
167D4—1944no—
168D4—1944no—
169D4C22×#8760414yes—
170C2×C4—971no—
171C2×C4—971no—
172C2×C4—971no—
173C2×C4—971no—
174C2×C4—971no—
175C2×C4—1944no—
176C2×C4—1944no—
177C2×C4—1942no—
178C2×C4—1942no—
179C2×C4C22×#8760411yescharacteristic
180C8—974no—
181C8—971no—
182C8—971no—
183C8—1944no—
184C8C22×#8760411yescharacteristic
185C8C22×#8760411yescharacteristic
186C22—24no—
187C22—82no—
188C22—971no—
189C22—971no—
190C22—3881no—
191C22D4×#8760411yescharacteristic
192C4—971no—
193C4—971no—
194C4—3882no—
195C4—1944no—
196C4D4×#8760411yescharacteristic
197C4C23×#8760411yescharacteristic
198C2—21no—
199C2—44no—
200C2—1941no—
201C2—1941no—
202C2C2×D4×#8760411yescharacteristic
subgroupG/H
centerC2C2×D4×#87604
commutatorC4×C97C23×C4
FrattiniC4C23×#87604
FittingC2×D8×C97C4
socleC2×C97C2×C4×D4
prime radicalC2×#15690829C2
cosocleC4×D97C24
hypercenterC2×D8#87604
hypocenterC97#2957
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. #20261813C2semidirect
2. #20261782C2semidirect
3. #20261777C2semidirect
4. D97⋊#108C2semidirect
5. #20261733C2semidirect
6. #20261698C2semidirect
9. #15690897C22semidirect
14. #15690881C22semidirect
15. #15690880C22semidirect
16. #15690874C22semidirect
17. #15690864C22semidirect
18. C8×#87604C22semidirect
20. #15690829C4semidirect
21. #15690829C4semidirect
24. C2×Dic388C4semidirect
25. C2×#13095499C4semidirect
27. C2×D8×C97C4semidirect
102. C97#2957semidirect
117. C2×D8#87604semidirect

Maximal subgroups

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

subgroupindex
#202618132
#202617822
#202617772
D97⋊#1082
#202617332
#202616982
C2×#156908292
#295797