Finite Group Databasebeta

← #27140940

Subgroups — #27140940

Order 12416.

208 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261446C211yesmaximal, characteristic
2#20261017C211yesmaximal, characteristic, complemented
3#20260937C211yesmaximal, characteristic
4#20260484C211yesmaximal, characteristic, complemented
5#20260467C211yesmaximal, characteristic, complemented
6#20260464C211yesmaximal, characteristic
7#487×C97C211yesmaximal, characteristic, complemented
8C2×#13095498—21no—
9#15690801—21no—
10#15690790—21no—
11#15690786—22no—
12C4×Dic194C2211yescharacteristic
13C4×D388C2211yescharacteristic
14#15690772—21no—
15C4⋊C4×D97—21no—
16#15690741—21no—
17C4⋊#11715895C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690734—21no—
20#15690733—21no—
21C4⋊Q8×C97C2211yescharacteristic
22C4⋊Q8×C97C2211yescharacteristic
23#103×C97—22no—
24C4×C8×C97C2211yescharacteristic
25C2×Dic194D411yescharacteristic
26C2×D388D411yescharacteristic
27C2×C4×D97—21no—
28C2×C4×D97—22no—
29#13095498—42no—
30C2×#11715895—21no—
31C2×#11715895—21no—
32C4⋊Dic97—22no—
33C4⋊Dic97D411yescharacteristic
34C4⋊Dic97D411yescharacteristic
35#13095490—22no—
36#13095490—21no—
37#13095488—22no—
38#13095488—21no—
39C4×Dic97—21no—
40C2×Q8×C97—21no—
41C2×Q8×C97D412yes—
42C2×C8×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C42×C97C2311yescharacteristic
48C22×D97—21no—
49Q8×C97—42no—
50Q8×C97—41no—
51Q8×C97—41no—
52C2×C4×C97—22no—
53C2×C4×C97—41no—
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97—21no—
60D388—21no—
61D388SD1612yes—
62C4×D97—41no—
63C4×D97—41no—
64C4×D97—82no—
65Dic194—21no—
66Dic194—21no—
67C8×C97—41no—
68#11715895—41no—
69#11715895—41no—
70C22×C97#12011yescharacteristic
71C2×D97—22no—
72C2×D97—41no—
73C2×D97—41no—
74C4×C97—21no—
75C4×C97—42no—
76C4×C97—81no—
77C4×C97C2×SD1611yescharacteristic
78C4×C97#13511yescharacteristic
79C4×C97#13711yescharacteristic
80C4×C97#13711yescharacteristic
81Dic97—41no—
82Dic97—41no—
83Dic97—41no—
84C2×C97#44811yescharacteristic
85C2×C97#45211yescharacteristic
86C2×C97#45511yescharacteristic
87D97—42no—
88#1643—971nomaximal, Sylow
89C97#164311yescharacteristic, complemented, Sylow
90#554—971no—
91#487D9711yescharacteristic, complemented
92#475—971no—
93#461—971no—
94#333—971no—
95Q8⋊C8—971no—
96D4⋊C8—971no—
97C2×SD16—1941no—
98C4⋊Q8C2×D9711yescharacteristic
99C4⋊Q8C2×D9711yescharacteristic
100#125—1941no—
101C22⋊Q8—1941no—
102C22⋊Q8—1942no—
103C4×Q8—971no—
104C4×D4—971no—
105C4×D4—1941no—
106C2×C4⋊C4—1941no—
107#106—1941no—
108C4⋊C8—971no—
109C4⋊C8—971no—
110#103—22no—
111#103—1941no—
112#103—1941no—
113C4×C8C2×D9711yescharacteristic
114C2×Q8—21no—
115C2×Q8—971no—
116C2×Q8C22⋊D9712yes—
117C2×D4—971no—
118C22×C4—1942no—
119C22×C4—1941no—
120SD16—3882no—
121C2×C8—21no—
122C2×C8—1941no—
123C2×C8—1941no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—971no—
129C4⋊C4—971no—
130C4⋊C4—1942no—
131C4⋊C4—1942no—
132C4⋊C4—1941no—
133C22⋊C4—1942no—
134C22⋊C4—1941no—
135C42—1941no—
136C42C22×D9711yescharacteristic
137C23—1941no—
138Q8—42no—
139Q8—41no—
140Q8—41no—
141Q8—1941no—
142Q8—1941no—
143D4—972no—
144D4—1941no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—7762no—
148C2×C4—3881no—
149C2×C4—3881no—
150C2×C4—1941no—
151C2×C4—1941no—
152C2×C4—1941no—
153C2×C4D4×D9711yescharacteristic
154C2×C4D4×D9711yescharacteristic
155C2×C4C2×C22⋊D9711yescharacteristic
156C8—41no—
157C8—3881no—
158C8—3881no—
159C22—3881no—
160C22—3881no—
161C22—1942no—
162C22#1569077811yescharacteristic
163C4—21no—
164C4—42no—
165C4—81no—
166C4—3881no—
167C4—3881no—
168C4—3881no—
169C4SD16×D9711yescharacteristic
170C4#1569083311yescharacteristic
171C4#1569083911yescharacteristic
172C4#1569084211yescharacteristic
173C2—3882no—
174C2#2026087211yescharacteristic
175C2#2026089611yescharacteristic
176C2#2026091211yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#487×C97C2
socleC22×C97#120
prime radicalC2×D388D4
hypercenter#487D97
hypocenterC97#1643
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #20261017C2semidirect
4. #20260484C2semidirect
5. #20260467C2semidirect
7. #487×C97C2semidirect
89. C97#1643semidirect
91. #487D97semidirect

Maximal subgroups

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

subgroupindex
#202614462
#202610172
#202609372
#202604842
#202604672
#202604642
#487×C972
#164397