Finite Group Databasebeta

← #27157148

Subgroups — #27157148

Order 12416.

222 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 56 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#20261757C211yesmaximal, characteristic, complemented
2#20261679C211yesmaximal, characteristic, complemented
3#20261589C212yesmaximal, complemented
4#20261586C212yesmaximal
5#20261264C211yesmaximal, characteristic
6C2×C4⋊#87604—21no—
7C4⋊#11715897—21no—
8#15690867C2212yes—
9#15690865—21no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4C2212yes—
12#15690803—21no—
13#15690802C411yescharacteristic, complemented
14#15690792—21no—
15#15690786C2211yescharacteristic
16#15690781C2211yescharacteristic
17Q8×Dic97C411yescharacteristic, complemented
18D4×Dic97C411yescharacteristic, complemented
19#15690770—21no—
20#15690766—21no—
21C42×D97C2211yescharacteristic
22#124×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24C2×#11715897—22no—
25C4⋊#87604—42no—
26C2×C22⋊D97—21no—
27C22×Dic97—21no—
28C2×C4×D97D411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97C2311yescharacteristic
31C4⋊Dic97—21no—
32C4⋊Dic97C2×C411yescharacteristic
33C4⋊Dic97C2×C411yescharacteristic
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39#13095488—21no—
40C4×Dic97—21no—
41C4×Dic97C2×C411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C2×Q8×C97C2×C411yescharacteristic
45C2×D4×C97C2×C411yescharacteristic
46C22⋊C4×C97—21no—
47C22⋊C4×C97—21no—
48C42×C97C2×C411yescharacteristic
49C23×C97—21no—
50C22×D97C2×D411yescharacteristic
51C2×#87604—42no—
52C2×#87604—42no—
53C2×#87604—42no—
54Q8×C97—41no—
55D4×C97—41no—
56C2×C4×C97—21no—
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22×C411yescharacteristic
60C22⋊D97—41no—
61C22⋊D97—41no—
62C2×Dic97—21no—
63C2×Dic97—21no—
64C2×Dic97—42no—
65C2×Dic97C22⋊C411yescharacteristic
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C2×D411yescharacteristic
68C4×D97—21no—
69C4×D97—21no—
70C4×D97—41no—
71C4×D97—41no—
72#11715897—42no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97—41no—
76C22×C97C2×C22⋊C411yescharacteristic
77C2×D97—21no—
78C2×D97—21no—
79C2×D97AGL(1,ℤ/8)11yescharacteristic
80C2×D97#13711yescharacteristic
81#87604—82no—
82C4×C97—21no—
83C4×C97—21no—
84C4×C97—21no—
85C4×C97—41no—
86Dic97—21no—
87Dic97—21no—
88Dic97—21no—
89Dic97—41no—
90Dic97—41no—
91C2×C97—81no—
92C2×C97#41011yescharacteristic
93C2×C97#41711yescharacteristic
94C2×C97#42111yescharacteristic
95D97—21no—
96D97—21no—
97#1512—971nomaximal, Sylow
98C97#151211yescharacteristic, complemented, Sylow
99#529—971no—
100#432—971no—
101#382—971no—
102#328—972no—
103#327—972no—
104#126—1941no—
105#125—971no—
106#124#8760411yescharacteristic, complemented
107#123—1941no—
108C22⋊Q8—971no—
109#121—971no—
110C4×Q8—971no—
111C4×D4—971no—
112C4×D4—1941no—
113#117—1941no—
114C2×C4⋊C4—1941no—
115C2×C42—971no—
116C4⋊C8—1941no—
117C22⋊C8—972no—
118#97—1941no—
119#95—972no—
120#95—1941no—
121C2×Q8C2×#8760411yescharacteristic
122C2×D4—1941no—
123C2×D4C2×#8760411yescharacteristic
124C22×C4—971no—
125C22×C4—971no—
126C22×C4—971no—
127C22×C4—1941no—
128C22×C4—1942no—
129C2×C8—1942no—
130C4⋊C4—971no—
131C4⋊C4—971no—
132C4⋊C4—3882no—
133C4⋊C4—1941no—
134C4⋊C4—1941no—
135C4⋊C4—1941no—
136C4⋊C4—1941no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—1941no—
140C22⋊C4—1941no—
141C22⋊C4—1941no—
142C42—971no—
143C42—971no—
144C42—971no—
145C42—1941no—
146C42C2×#8760411yescharacteristic
147C23—21no—
148C23—971no—
149Q8—41no—
150D4—41no—
151D4—3881no—
152D4—3881no—
153C2×C4—21no—
154C2×C4—971no—
155C2×C4—971no—
156C2×C4—971no—
157C2×C4—3881no—
158C2×C4—3881no—
159C2×C4—3882no—
160C2×C4—3882no—
161C2×C4—3882no—
162C2×C4—3882no—
163C2×C4—1941no—
164C2×C4—1941no—
165C2×C4—1941no—
166C2×C4—1941no—
167C2×C4C22⋊#8760411yescharacteristic
168C2×C4C22⋊#8760411yescharacteristic
169C2×C4C22×#8760411yescharacteristic
170C8—3882no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—971no—
175C22—971no—
176C22—1941no—
177C22—1941no—
178C22C2×C22⋊#8760411yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—41no—
183C4—7762no—
184C4—3881no—
185C4—3881no—
186C4—1941no—
187C4—1941no—
188C4—1941no—
189C2—81no—
190C2—1941no—
191C2—1941no—
192C2#2026171511yescharacteristic
193C2#2026172611yescharacteristic
194C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#124×C97C4
socleC22×C97C2×C22⋊C4
prime radical#15690781C22
cosocleC2×C4×D97C23
hypercenter#124#87604
hypocenterC97#1512
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20261757C2semidirect
2. #20261679C2semidirect
3. #20261589C2semidirect
13. #15690802C4semidirect
17. Q8×Dic97C4semidirect
18. D4×Dic97C4semidirect
22. #124×C97C4semidirect
98. C97#1512semidirect
106. #124#87604semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202616792
#202615892
#202615862
#202612642
#151297