Finite Group Databasebeta

← #27157742

Subgroups — #27157742

Order 12416.

324 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 68 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
2#20261728C212yesmaximal
3D97⋊#101C211yesmaximal, characteristic
4#20261606C212yesmaximal
5C4⋊Q8×D97C211yesmaximal, characteristic
6D97⋊M16C2212yes—
7Q8×#87604—24no—
8#15690889—22no—
9#15690889C2212yes—
10#15690873—24no—
11#15690871—22no—
12C2×Q8×D97C2211yescharacteristic
13C2×Q8×D97C2211yescharacteristic
14#15690810C411yescharacteristic, complemented
15#15690809—22no—
16#15690808C411yescharacteristic, complemented
17#15690808C411yescharacteristic, complemented
18C4⋊C4×D97—22no—
19C42×D97C2211yescharacteristic
20C4⋊Q8×C97C411yescharacteristic, complemented
21#13095521—22no—
22#13095520—22no—
23#13095519—22no—
24C4⋊#87604—24no—
25C4⋊#87604D414yes—
26C22⋊#87604—22no—
27C4×#87604—24no—
28C4×#87604—22no—
29C2×Dic194—21no—
30C2×Dic194C2×C411yescharacteristic
31C2×Dic194C2×C411yescharacteristic
32C2×Dic194D412yes—
33Q8×D97—22no—
34Q8×D97—22no—
35Q8×D97—22no—
36Q8×D97—22no—
37C2×C4×D97—21no—
38C2×C4×D97—21no—
39C2×C4×D97D412yes—
40C2×C4×D97C2311yescharacteristic
41C4⋊Dic97—22no—
42#13095490—22no—
43#13095490—22no—
44C4×Dic97—21no—
45C4×Dic97C2×C411yescharacteristic
46C2×Q8×C97C2×C411yescharacteristic
47C2×Q8×C97C2×C411yescharacteristic
48C4⋊C4×C97—22no—
49C42×C97C2×C411yescharacteristic
50C22×D97C2×D411yescharacteristic
51C2×#87604—24no—
52Q8×C97—22no—
53Q8×C97—22no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97C22⋊C412yes—
57C2×C4×C97C22×C411yescharacteristic
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97C22⋊C412yes—
61C2×Dic97C2×D411yescharacteristic
62C4×D97—24no—
63C4×D97—22no—
64C4×D97—22no—
65C4×D97—41no—
66C4×D97—41no—
67C4×D97C2×D412yes—
68C4×D97#4112yes—
69Dic194—22no—
70Dic194—24no—
71Dic194—22no—
72Dic194—42no—
73#11715897—42no—
74C22×C97C2×C22⋊C411yescharacteristic
75C2×D97—22no—
76C2×D97#12011yescharacteristic
77C2×D97#12311yescharacteristic
78#87604—44no—
79C4×C97—22no—
80C4×C97—22no—
81C4×C97—41no—
82C4×C97C4×D412yes—
83Dic97—22no—
84Dic97—22no—
85Dic97—41no—
86Dic97#12112yes—
87C2×C97—21no—
88C2×C97#38611yescharacteristic
89D97—21no—
90D97#45612yes—
91#1886—971nomaximal, Sylow
92C97#188611yescharacteristic, complemented, Sylow
93C2×C4⋊Q8—971no—
94#519—971no—
95#419—972no—
96C2×#101—971no—
97#339—972no—
98C22×Q8—971no—
99C22×Q8—971no—
100C2×M16—972no—
101C4⋊Q8—971no—
102C4⋊Q8—971no—
103C4⋊Q8—971no—
104C4⋊Q8—1942no—
105C4⋊Q8#8760411yescharacteristic, complemented
106C4×Q8—1944no—
107#117—972no—
108#117—1942no—
109C2×C4⋊C4—1942no—
110C2×C42—971no—
111#103—1944no—
112#101—1942no—
113C2×Q8—972no—
114C2×Q8—971no—
115C2×Q8—971no—
116C2×Q8—1941no—
117C2×Q8—1942no—
118C2×Q8—1942no—
119C2×Q8—1942no—
120C2×Q8—1942no—
121C2×Q8C2×#8760411yescharacteristic
122C2×Q8C2×#8760411yescharacteristic
123C22×C4—971no—
124C22×C4—972no—
125C22×C4—1941no—
126C22×C4—1941no—
127M16—1942no—
128M16—1942no—
129C2×C8—1942no—
130C4⋊C4—22no—
131C4⋊C4—974no—
132C4⋊C4—1942no—
133C4⋊C4—1942no—
134C4⋊C4—1944no—
135C4⋊C4—1942no—
136C22⋊C4—1942no—
137C42—971no—
138C42—1941no—
139C42—1942no—
140C42—1944no—
141C42C2×#8760411yescharacteristic
142C23—971no—
143Q8—22no—
144Q8—22no—
145Q8—3882no—
146Q8—1944no—
147Q8—1942no—
148Q8—1942no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—971no—
152C2×C4—972no—
153C2×C4—972no—
154C2×C4—972no—
155C2×C4—3881no—
156C2×C4—3881no—
157C2×C4—1942no—
158C2×C4—1942no—
159C2×C4—1944no—
160C2×C4—1944no—
161C2×C4—1941no—
162C2×C4—1941no—
163C2×C4C22⋊#8760412yes—
164C2×C4C22×#8760411yescharacteristic
165C8—3882no—
166C22—971no—
167C22—971no—
168C22—1942no—
169C22C2×C22⋊#8760411yescharacteristic
170C4—22no—
171C4—22no—
172C4—41no—
173C4—972no—
174C4—3884no—
175C4—3881no—
176C4—1942no—
177C4—1942no—
178C4D4×#8760412yes—
179C2—21no—
180C2—972no—
181C2—1941no—
182C2#2026168311yescharacteristic
subgroupG/H
centerC2#20261683
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC2×C97#386
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1886
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
14. #15690810C4semidirect
16. #15690808C4semidirect
17. #15690808C4semidirect
20. C4⋊Q8×C97C4semidirect
92. C97#1886semidirect
105. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202618032
#202617282
D97⋊#1012
#202616062
C4⋊Q8×D972
#188697