Finite Group Databasebeta

← #27140775

Subgroups — #27140775

Order 12416.

246 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261316C211yesmaximal, characteristic
2#20261054C212yesmaximal, complemented
3#20260674C211yesmaximal, characteristic, complemented
4#20260470C212yesmaximal, complemented
5#324×C97C211yesmaximal, characteristic, complemented
6C2×#13095511—21no—
7#15690790—22no—
8#15690785—22no—
9#15690784—21no—
10#15690784C2212yes—
11C2×C4⋊Dic97C2211yescharacteristic
12#15690742—22no—
13#15690732—22no—
14#15690717—21no—
15#15690717—21no—
16#15690717C2211yescharacteristic
17C2×C4⋊C4×C97C2211yescharacteristic
18C22⋊C8×C97C2212yes—
19#13095511—42no—
20#13095511—42no—
21C2×Dic194—21no—
22C2×D388—21no—
23C2×D388—21no—
24C2×D388D412yes—
25C2×C22⋊D97—22no—
26C22×Dic97—21no—
27C22×Dic97—21no—
28C2×C4×D97—22no—
29C4⋊Dic97—21no—
30C4⋊Dic97—22no—
31C4⋊Dic97D412yes—
32#13095488—22no—
33#13095488—22no—
34C22×C4×C97D411yescharacteristic
35C22×C4×C97D411yescharacteristic
36C22×C4×C97C2311yescharacteristic
37C2×C8×C97—22no—
38C4⋊C4×C97—22no—
39C23×C97C2×D411yescharacteristic
40C22×D97—22no—
41C2×C4×C97—21no—
42C2×C4×C97—21no—
43C2×C4×C97—21no—
44C2×C4×C97—41no—
45C2×C4×C97—41no—
46C2×C4×C97SD1612yes—
47C2×C4×C97C2×D412yes—
48C22⋊D97—42no—
49C22⋊D97—42no—
50C2×Dic97—22no—
51C2×Dic97—41no—
52C2×Dic97—41no—
53C2×Dic97—42no—
54C2×Dic97—41no—
55C2×Dic97—41no—
56D388—42no—
57D388—42no—
58D388—81no—
59C4×D97—44no—
60Dic194—42no—
61C8×C97—42no—
62C22×C97—21no—
63C22×C97—21no—
64C22×C97#12011yescharacteristic
65C22×C97C2×SD1611yescharacteristic
66C22×C97AGL(1,ℤ/8)11yescharacteristic
67C2×D97—42no—
68C2×D97—42no—
69C2×D97—42no—
70C4×C97—22no—
71C4×C97—22no—
72C4×C97—41no—
73Dic97—42no—
74Dic97—81no—
75Dic97—81no—
76C2×C97—21no—
77C2×C97—21no—
78C2×C97#45011yescharacteristic
79C2×C97#45411yescharacteristic
80C2×C97#45811yescharacteristic
81D97—82no—
82#1616—971nomaximal, Sylow
83C97#161611yescharacteristic, complemented, Sylow
84#537—971no—
85#481—972no—
86#400—971no—
87#327—972no—
88#324D9711yescharacteristic, complemented
89C2×#41—1941no—
90C22⋊Q8—1942no—
91#121—972no—
92#121—1942no—
93#121—1941no—
94C2×C4⋊C4—971no—
95C2×C4⋊C4C2×D9711yescharacteristic
96#106—1942no—
97#102—1942no—
98C22⋊C8C2×D9712yes—
99#95—971no—
100#95—1941no—
101#95—1941no—
102#41—3882no—
103#41—3882no—
104C2×Q8—1941no—
105C2×D4—972no—
106C2×D4—1941no—
107C2×D4—1941no—
108C2×D4—1942no—
109C22×C4—1942no—
110C22×C4—1941no—
111C22×C4—1941no—
112C22×C4D38811yescharacteristic
113C22×C4D38811yescharacteristic
114C22×C4C22×D9711yescharacteristic
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—972no—
118C4⋊C4—1941no—
119C4⋊C4—1942no—
120C22⋊C4—1942no—
121C22⋊C4—1942no—
122C23—1942no—
123C23C2×D38811yescharacteristic
124Q8—3882no—
125D4—7761no—
126D4—3882no—
127D4—3882no—
128D4—3882no—
129D4—3882no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—3884no—
136C2×C4—3881no—
137C2×C4—3881no—
138C2×C4—3882no—
139C2×C4—3881no—
140C2×C4—3881no—
141C2×C4—1942no—
142C2×C4#1309550012yes—
143C2×C4D4×D9712yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—3882no—
148C22—3882no—
149C22—3882no—
150C22#1569078011yescharacteristic
151C22C2×#1309550011yescharacteristic
152C22#1569083811yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—7761no—
157C4—7761no—
158C4—3882no—
159C2—21no—
160C2—21no—
161C2—7762no—
162C2#2026088811yescharacteristic
163C2#2026091011yescharacteristic
164C2#2026092611yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#324×C97C2
socleC22×C97#120
prime radical#20261316C2
hypercenter#324D97
hypocenterC97#1616
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #20261054C2semidirect
3. #20260674C2semidirect
4. #20260470C2semidirect
5. #324×C97C2semidirect
83. C97#1616semidirect
88. #324D97semidirect

Maximal subgroups

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

subgroupindex
#202613162
#202610542
#202606742
#202604702
#324×C972
#161697