Finite Group Databasebeta

← #27157916

Subgroups — #27157916

Order 12416.

260 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 78 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#20261736C211yesmaximal, characteristic
2C4×C4⋊#87604C211yesmaximal, characteristic
3#20261608C212yesmaximal
4#20261608C212yesmaximal
5#125×D97C211yesmaximal, characteristic
6C2×#13095519C2212yes—
7C2×C4⋊#87604C2212yes—
8C2×C4×#87604—21no—
9#15690877—21no—
10C4⋊#11715897—21no—
11(C2×#87604)⋊C4—21no—
12#15690802C411yescharacteristic, complemented
13#15690801C411yescharacteristic, complemented
14#15690801C411yescharacteristic, complemented
15#15690800—21no—
16#15690800—21no—
17C4⋊C4×D97—21no—
18C4⋊C4×D97—21no—
19C4⋊C4×D97C2211yescharacteristic
20C4⋊C4×D97C2211yescharacteristic
21C42×D97C2211yescharacteristic
22#125×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24#13095519—24no—
25C2×#11715897—22no—
26C4⋊#87604—22no—
27C4⋊#87604Q814yes—
28C4×#87604—42no—
29C2×C4×D97—21no—
30C2×C4×D97—21no—
31C2×C4×D97D411yescharacteristic
32C2×C4×D97D411yescharacteristic
33C2×C4×D97C2311yescharacteristic
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97C2×C411yescharacteristic
37C4⋊Dic97C2×C411yescharacteristic
38#13095490—21no—
39#13095490—21no—
40#13095490—21no—
41#13095490—21no—
42#13095490—21no—
43#13095490—21no—
44C4×Dic97C2×C411yescharacteristic
45C4×Dic97D411yescharacteristic
46C4×Dic97D411yescharacteristic
47C4⋊C4×C97—21no—
48C4⋊C4×C97—21no—
49C4⋊C4×C97C2×C411yescharacteristic
50C4⋊C4×C97C2×C411yescharacteristic
51C42×C97C2×C411yescharacteristic
52C22×D97C2×D411yescharacteristic
53C2×#87604—24no—
54C2×#87604—42no—
55C2×#87604—42no—
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22⋊C411yescharacteristic
60C2×C4×C97C22×C411yescharacteristic
61C2×Dic97—21no—
62C2×Dic97—21no—
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C22⋊C411yescharacteristic
65C2×Dic97C2×D411yescharacteristic
66C4×D97—22no—
67C4×D97—21no—
68C4×D97—21no—
69C4×D97—42no—
70C4×D97—41no—
71C4×D97—41no—
72C4×D97C2×Q812yes—
73C4×D97#4112yes—
74#11715897—42no—
75C22×C97C2×C22⋊C411yescharacteristic
76C2×D97C22⋊Q812yes—
77C2×D97#13512yes—
78C2×D97AGL(1,ℤ/8)11yescharacteristic
79C2×D97#13711yescharacteristic
80#87604—44no—
81C4×C97—21no—
82C4×C97—21no—
83C4×C97—41no—
84C4×C97—41no—
85C4×C97C4×Q812yes—
86Dic97—21no—
87Dic97—21no—
88Dic97—41no—
89Dic97—41no—
90Dic97#12411yescharacteristic
91Dic97C4⋊Q811yescharacteristic
92C2×C97#39111yescharacteristic
93C2×C97#41611yescharacteristic
94C2×C97#41711yescharacteristic
95D97#47812yes—
96D97#47912yes—
97#1989—971nomaximal, Sylow
98C97#198911yescharacteristic, complemented, Sylow
99C2×#125—971no—
100C2×C4⋊C8—971no—
101C4×C4⋊C4—971no—
102#340—972no—
103#340—972no—
104C22×C8—972no—
105#125—971no—
106#125—971no—
107#125—971no—
108#125—1941no—
109#125—1941no—
110#125#8760411yescharacteristic, complemented
111C2×C4⋊C4—972no—
112C2×C4⋊C4—971no—
113C2×C4⋊C4—971no—
114C2×C4⋊C4—1941no—
115C2×C4⋊C4—1941no—
116C2×C42—971no—
117C2×C42—1941no—
118C4⋊C8—1941no—
119C4⋊C8—1941no—
120#95—1941no—
121C22×C4—971no—
122C22×C4—971no—
123C22×C4—971no—
124C22×C4—1941no—
125C22×C4—1941no—
126C22×C4—1942no—
127C2×C8—1942no—
128C2×C8—1944no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—974no—
132C4⋊C4—971no—
133C4⋊C4—971no—
134C4⋊C4—1941no—
135C4⋊C4—1941no—
136C4⋊C4—1941no—
137C4⋊C4—1941no—
138C4⋊C4—1941no—
139C4⋊C4—1941no—
140C4⋊C4—1942no—
141C4⋊C4—1941no—
142C4⋊C4—1941no—
143C4⋊C4C2×#8760411yescharacteristic
144C4⋊C4C2×#8760411yescharacteristic
145C42—971no—
146C42—971no—
147C42—971no—
148C42—3882no—
149C42C2×#8760411yescharacteristic
150C23—971no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—971no—
154C2×C4—972no—
155C2×C4—972no—
156C2×C4—971no—
157C2×C4—971no—
158C2×C4—3882no—
159C2×C4—3881no—
160C2×C4—3881no—
161C2×C4—3882no—
162C2×C4—3882no—
163C2×C4—1941no—
164C2×C4—1942no—
165C2×C4—1941no—
166C2×C4—1944no—
167C2×C4—1941no—
168C2×C4—1941no—
169C2×C4C22⋊#8760411yescharacteristic
170C2×C4C22⋊#8760411yescharacteristic
171C2×C4C22×#8760411yescharacteristic
172C8—3882no—
173C22—972no—
174C22—972no—
175C22—971no—
176C22—971no—
177C22C2×C22⋊#8760411yescharacteristic
178C4—21no—
179C4—21no—
180C4—41no—
181C4—41no—
182C4—971no—
183C4—971no—
184C4—3884no—
185C4—3881no—
186C4—3881no—
187C4—1941no—
188C4—1941no—
189C4Q8×#8760412yes—
190C2—972no—
191C2—972no—
192C2(C2×#87604)⋊Q811yescharacteristic
193C2#2026172511yescharacteristic
194C2#2026172611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#125×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenter#125#87604
hypocenterC97#1989
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
12. #15690802C4semidirect
13. #15690801C4semidirect
14. #15690801C4semidirect
22. #125×C97C4semidirect
98. C97#1989semidirect
110. #125#87604semidirect

Maximal subgroups

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

subgroupindex
#202617362
C4×C4⋊#876042
#202616082
#125×D972
#198997