Finite Group Databasebeta

← #27142125

Subgroups — #27142125

Order 12416.

360 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 97 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#20261210C211yesmaximal, characteristic
2#20260744C212yesmaximal, complemented
3C2×#15690727C211yesmaximal, characteristic
4#20260495C212yesmaximal, complemented
5C2×#124×C97C211yesmaximal, characteristic
6C2×#13095495C2212yes—
7D4×Dic97—24no—
8#15690769C2212yes—
9C2×C22⋊Dic97—22no—
10#15690734—24no—
11#15690727—22no—
12C22×Q8×C97C2211yescharacteristic
13C22×D4×C97C2211yescharacteristic
14#124×C97—22no—
15#124×C97C414yescomplemented
16C2×C22⋊C4×C97—22no—
17C2×C42×C97C2211yescharacteristic
18C22×Dic97—24no—
19#13095495—22no—
20#13095495—22no—
21C2×#11715895—22no—
22C4⋊Dic97D414yes—
23C22⋊Dic97—24no—
24C22⋊Dic97—22no—
25C4×Dic97—22no—
26C24×C97D412yes—
27C2×Q8×C97—21no—
28C2×Q8×C97—22no—
29C2×Q8×C97—22no—
30C2×Q8×C97C2×C412yescomplemented
31C2×D4×C97—22no—
32C2×D4×C97—22no—
33C2×D4×C97C2×C412yes—
34C2×D4×C97D412yes—
35C22×C4×C97—21no—
36C22×C4×C97—21no—
37C22×C4×C97C2311yescharacteristic
38C22⋊C4×C97—28no—
39C42×C97—21no—
40C42×C97C2×C412yescomplemented
41C23×C97—22no—
42C23×C97—22no—
43C23×C97—44no—
44C23×C97C22⋊C414yes—
45C23×C97C2×D411yescharacteristic
46Q8×C97—24no—
47Q8×C97—42no—
48D4×C97—24no—
49D4×C97—24no—
50C2×C4×C97—22no—
51C2×C4×C97—22no—
52C2×C4×C97—42no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×C4×C97C22×C411yescharacteristic
56C2×C4×C97C2×D412yes—
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97#4112yes—
59C2×Dic97—24no—
60C2×Dic97—84no—
61#11715895—42no—
62C22×C97—22no—
63C22×C97—24no—
64C22×C97—44no—
65C22×C97—44no—
66C22×C97—82no—
67C22×C97C2×C22⋊C411yescharacteristic
68C22×C97#12011yescharacteristic
69C22×C97#12311yescharacteristic
70C4×C97—42no—
71C4×C97—42no—
72C4×C97C4×D412yes—
73C4×C97#12112yes—
74Dic97—44no—
75C2×C97—21no—
76C2×C97—21no—
77C2×C97—44no—
78C2×C97#38611yescharacteristic
79C2×C97#45512yes—
80#1885—971nomaximal, Sylow
81C97#188511yescharacteristic, complemented, Sylow
82C2×#124D9711yescharacteristic
83#518—971no—
84#419—972no—
85C2×#100—971no—
86#339—972no—
87C22×Q8C2×D9711yescharacteristic
88C22×D4C2×D9711yescharacteristic
89C2×M16—972no—
90#124—22no—
91#124Dic9714yescomplemented
92C4×D4—1944no—
93#117—972no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—1942no—
96C2×C42C2×D9711yescharacteristic
97#103—1944no—
98#100—1942no—
99C24C22⋊D9712yes—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8C2×Dic9712yescomplemented
104C2×D4—22no—
105C2×D4—22no—
106C2×D4C2×Dic9712yes—
107C2×D4C22⋊D9712yes—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—1944no—
111C22×C4C22×D9711yescharacteristic
112M16—1942no—
113M16—1942no—
114C2×C8—1942no—
115C4⋊C4—974no—
116C22⋊C4—28no—
117C22⋊C4—1944no—
118C22⋊C4—1942no—
119C42—21no—
120C42—1942no—
121C42C2×Dic9712yescomplemented
122C23—22no—
123C23—22no—
124C23—44no—
125C23C22⋊Dic9714yes—
126C23C2×C22⋊D9711yescharacteristic
127Q8—24no—
128Q8—42no—
129D4—24no—
130D4—24no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—42no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—7764no—
137C2×C4—1944no—
138C2×C4C22×Dic9711yescharacteristic
139C2×C4D4×D9712yes—
140C2×C4C2×C22⋊D9711yescharacteristic
141C2×C4#1309551012yes—
142C8—3882no—
143C22—22no—
144C22—24no—
145C22—44no—
146C22—44no—
147C22—82no—
148C22C2×C22⋊Dic9711yescharacteristic
149C22#1569077811yescharacteristic
150C22#1569079111yescharacteristic
151C4—42no—
152C4—42no—
153C4—3884no—
154C4D4×Dic9712yes—
155C4#1569078212yes—
156C2—21no—
157C2—21no—
158C2—44no—
159C2#2026061511yescharacteristic
160C2#2026091212yes—
subgroupG/H
centerC22#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#124×C97C2
socleC22×C97#120
prime radicalC22×D4×C97C22
cosocleC22×C4×C97C23
hypercenterC2×#124D97
hypocenterC97#1885
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. #20260744C2semidirect
4. #20260495C2semidirect
15. #124×C97C4semidirect
30. C2×Q8×C97C2×C4semidirect
40. C42×C97C2×C4semidirect
81. C97#1885semidirect
91. #124Dic97semidirect
103. C2×Q8C2×Dic97semidirect
121. C42C2×Dic97semidirect

Maximal subgroups

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

subgroupindex
#202612102
#202607442
C2×#156907272
#202604952
C2×#124×C972
#188597