Finite Group Databasebeta

← #27157783

Subgroups — #27157783

Order 12416.

314 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 66 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#20261803C211yesmaximal, characteristic, complemented
2#20261734C212yesmaximal
3D97⋊#101C211yesmaximal, characteristic, complemented
4#20261602C212yesmaximal, complemented
5#20261310C211yesmaximal, characteristic
6D97⋊M16C2212yes—
7Q8×#87604—24no—
8#15690889—22no—
9#15690889C2212yes—
10#15690875—24no—
11#15690871—22no—
12C2×#13095512C2211yescharacteristic
13C2×Q8×D97C2211yescharacteristic
14#15690808C411yescharacteristic, complemented
15#15690799C411yescharacteristic, complemented
16#15690798C411yescharacteristic, complemented
17#15690795—22no—
18#15690788—22no—
19#15690767C2211yescharacteristic
20C4⋊Q8×C97C411yescharacteristic, complemented
21#13095521—22no—
22#13095520—22no—
23#13095519—22no—
24C4⋊#87604—24no—
25C4⋊#87604—22no—
26C22⋊#87604—22no—
27C4×#87604—24no—
28C4×#87604D414yes—
29#13095512—22no—
30#13095512—22no—
31C2×Dic194C2×C411yescharacteristic
32C2×Dic194D412yes—
33Q8×D97—22no—
34Q8×D97—22no—
35C2×D388—21no—
36C2×D388C2×C411yescharacteristic
37C2×C4×D97—21no—
38C2×C4×D97D412yes—
39C2×C4×D97C2311yescharacteristic
40#13095490—22no—
41#13095490—21no—
42#13095488—21no—
43#13095488—22no—
44#13095488—22no—
45C4×Dic97C2×C411yescharacteristic
46C2×Q8×C97C2×C411yescharacteristic
47C2×Q8×C97C2×C411yescharacteristic
48C4⋊C4×C97—22no—
49C42×C97C2×C411yescharacteristic
50C22×D97—21no—
51C22×D97C2×D411yescharacteristic
52C2×#87604—24no—
53Q8×C97—22no—
54Q8×C97—22no—
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97C22⋊C412yes—
58C2×C4×C97C22×C411yescharacteristic
59C2×Dic97—21no—
60C2×Dic97C22⋊C412yes—
61C2×Dic97C2×D411yescharacteristic
62D388—22no—
63D388—41no—
64D388—41no—
65C4×D97—22no—
66C4×D97—22no—
67C4×D97—42no—
68C4×D97C2×D412yes—
69C4×D97#4112yes—
70Dic194—22no—
71Dic194—24no—
72#11715897—42no—
73C22×C97C2×C22⋊C411yescharacteristic
74C2×D97—41no—
75C2×D97—42no—
76C2×D97—41no—
77C2×D97#12011yescharacteristic
78C2×D97#12311yescharacteristic
79#87604—44no—
80C4×C97—22no—
81C4×C97—41no—
82C4×C97—41no—
83C4×C97C4×D412yes—
84Dic97—22no—
85Dic97—41no—
86Dic97#12112yes—
87C2×C97—21no—
88C2×C97#38611yescharacteristic
89D97—21no—
90D97—21no—
91D97—81no—
92#1912—971nomaximal, Sylow
93C97#191211yescharacteristic, complemented, Sylow
94#536—971no—
95#519—971no—
96#421—972no—
97C2×#101—971no—
98H(3,ℤ/4)—972no—
99C2×#41—971no—
100C22×Q8—971no—
101C2×M16—972no—
102C4⋊Q8—971no—
103C4⋊Q8#8760411yescharacteristic, complemented
104#124—971no—
105#124—971no—
106#123—1942no—
107C22⋊Q8—1942no—
108C4×Q8—1944no—
109#117—972no—
110#117—971no—
111#117—1942no—
112#104—1944no—
113#101—1942no—
114#41—1942no—
115#41—1942no—
116C2×Q8—972no—
117C2×Q8—971no—
118C2×Q8—1942no—
119C2×Q8—1942no—
120C2×Q8C2×#8760411yescharacteristic
121C2×Q8C2×#8760411yescharacteristic
122C2×D4—971no—
123C2×D4—1941no—
124C22×C4—971no—
125C22×C4—972no—
126C22×C4—1941no—
127M16—1942no—
128M16—1942no—
129C2×C8—1942no—
130C4⋊C4—22no—
131C4⋊C4—1941no—
132C4⋊C4—1942no—
133C4⋊C4—1942no—
134C4⋊C4—1944no—
135C22⋊C4—1942no—
136C22⋊C4—1941no—
137C22⋊C4—1942no—
138C22⋊C4—1942no—
139C42—974no—
140C42—971no—
141C42—1944no—
142C42C2×#8760411yescharacteristic
143C23—971no—
144C23—1941no—
145Q8—22no—
146Q8—22no—
147Q8—1944no—
148Q8—1942no—
149D4—3881no—
150D4—3881no—
151D4—1942no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—971no—
155C2×C4—972no—
156C2×C4—972no—
157C2×C4—972no—
158C2×C4—3882no—
159C2×C4—1942no—
160C2×C4—1942no—
161C2×C4—1944no—
162C2×C4—1941no—
163C2×C4C22⋊#8760412yes—
164C2×C4C22×#8760411yescharacteristic
165C8—3882no—
166C22—971no—
167C22—971no—
168C22—3882no—
169C22—3881no—
170C22—3881no—
171C22C2×C22⋊#8760411yescharacteristic
172C4—22no—
173C4—41no—
174C4—41no—
175C4—972no—
176C4—3881no—
177C4—3884no—
178C4—1942no—
179C4D4×#8760412yes—
180C2—21no—
181C2—7761no—
182C2—1941no—
183C2—1941no—
184C2#2026168311yescharacteristic
subgroupG/H
centerC2#20261683
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC2×C97#386
prime radicalC2×#13095512C22
cosocleC2×C4×D97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1912
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20261803C2semidirect
3. D97⋊#101C2semidirect
4. #20261602C2semidirect
14. #15690808C4semidirect
15. #15690799C4semidirect
16. #15690798C4semidirect
20. C4⋊Q8×C97C4semidirect
93. C97#1912semidirect
103. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202618032
#202617342
D97⋊#1012
#202616022
#202613102
#191297