Finite Group Databasebeta

← #27140744

Subgroups — #27140744

Order 12416.

210 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261063C212yesmaximal
2#20260665C211yesmaximal, characteristic
3#20260472C212yesmaximal
4#20260461C211yesmaximal, characteristic
5#544×C97C211yesmaximal, characteristic
6C2×C4⋊Dic97C2211yescharacteristic
7C2×#13095490—21no—
8#15690741—22no—
9#15690734—22no—
10C22⋊#11715895C2212yes—
11#15690717—21no—
12#15690717C2211yescharacteristic
13C4⋊Q8×C97—22no—
14#125×C97—22no—
15C22⋊Q8×C97—21no—
16C22⋊Q8×C97C2212yes—
17#117×C97—21no—
18C2×C4⋊C4×C97C2211yescharacteristic
19C22×Dic97—21no—
20C22×Dic97—21no—
21C2×#11715895—22no—
22C4⋊Dic97—22no—
23#13095490—42no—
24C2×Q8×C97—21no—
25C2×Q8×C97D412yes—
26C22×C4×C97D411yescharacteristic
27C22×C4×C97D411yescharacteristic
28C22×C4×C97C2311yescharacteristic
29C4⋊C4×C97—21no—
30C4⋊C4×C97—21no—
31C4⋊C4×C97—21no—
32C4⋊C4×C97—22no—
33C4⋊C4×C97—22no—
34C4⋊C4×C97—22no—
35C4⋊C4×C97D412yes—
36C22⋊C4×C97—22no—
37C42×C97—22no—
38C23×C97C2×D411yescharacteristic
39Q8×C97—42no—
40Q8×C97—81no—
41C2×C4×C97—21no—
42C2×C4×C97—21no—
43C2×C4×C97—22no—
44C2×C4×C97—22no—
45C2×C4×C97—21no—
46C2×C4×C97—41no—
47C2×C4×C97—41no—
48C2×C4×C97SD1612yes—
49C2×C4×C97C2×D412yes—
50C2×Dic97—22no—
51C2×Dic97—41no—
52C2×Dic97—41no—
53C2×Dic97—41no—
54C2×Dic97—41no—
55C2×Dic97—41no—
56#11715895—42no—
57C22×C97—21no—
58C22×C97—21no—
59C22×C97#12011yescharacteristic
60C22×C97C2×SD1611yescharacteristic
61C22×C97#13711yescharacteristic
62C4×C97—22no—
63C4×C97—22no—
64C4×C97—42no—
65C4×C97—41no—
66C4×C97—42no—
67Dic97—42no—
68Dic97—81no—
69C2×C97—21no—
70C2×C97—21no—
71C2×C97#44811yescharacteristic
72C2×C97#45611yescharacteristic
73C2×C97#45811yescharacteristic
74#1611—971nomaximal, Sylow
75C97#161111yescharacteristic, complemented, Sylow
76#544D9711yescharacteristic
77#483—972no—
78#398—971no—
79#328—972no—
80#324—971no—
81C4⋊Q8—22no—
82#125—22no—
83C22⋊Q8—21no—
84C22⋊Q8C2×D9712yes—
85#117—21no—
86C2×C4⋊C4—971no—
87C2×C4⋊C4—1941no—
88C2×C4⋊C4C2×D9711yescharacteristic
89#106—1942no—
90#103—1942no—
91C22⋊C8—972no—
92#95—971no—
93#95—1941no—
94C2×Q8—21no—
95C2×Q8C22⋊D9712yes—
96C22×C4—1941no—
97C22×C4—1941no—
98C22×C4C22⋊D9711yescharacteristic
99C22×C4C22⋊D9711yescharacteristic
100C22×C4C22×D9711yescharacteristic
101C2×C8—1942no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—3882no—
109C4⋊C4—1942no—
110C4⋊C4C22⋊D9712yes—
111C22⋊C4—22no—
112C42—22no—
113C23C2×C22⋊D9711yescharacteristic
114Q8—42no—
115Q8—81no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—21no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—3881no—
124C2×C4—3881no—
125C2×C4—3881no—
126C2×C4—3881no—
127C2×C4—3881no—
128C2×C4—1942no—
129C2×C4#1309549812yes—
130C2×C4C2×C22⋊D9712yes—
131C8—3882no—
132C22—21no—
133C22—21no—
134C22#1569077911yescharacteristic
135C22C2×#1309549811yescharacteristic
136C22#1569084111yescharacteristic
137C4—22no—
138C4—22no—
139C4—42no—
140C4—41no—
141C4—42no—
142C4—7761no—
143C4—3882no—
144C2—21no—
145C2—21no—
146C2#2026087411yescharacteristic
147C2#2026091811yescharacteristic
148C2#2026092511yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#544×C97C2
socleC22×C97#120
prime radicalC23×C97C2×D4
hypercenter#544D97
hypocenterC97#1611
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
75. C97#1611semidirect

Maximal subgroups

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

subgroupindex
#202610632
#202606652
#202604722
#202604612
#544×C972
#161197