Finite Group Databasebeta

← #27140326

Subgroups — #27140326

Order 12416.

270 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 63 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#20261284C211yesmaximal, characteristic
2#20260687C212yesmaximal, complemented
3#20260475C211yesmaximal, characteristic, complemented
4#20260474C212yesmaximal, complemented
5#329×C97C211yesmaximal, characteristic, complemented
6C2×#13095512C411yescharacteristic
7C2×#13095510C411yescharacteristic
8C2×#13095494—22no—
9#15690808C411yescharacteristic
10#15690806C411yescharacteristic
11#15690798C2211yescharacteristic
12#15690783—22no—
13#15690770—22no—
14C42×D97C2211yescharacteristic
15#15690721C2212yes—
16C8×Dic97—22no—
17#124×C97C2211yescharacteristic
18#97×C97C2212yes—
19#13095512—42no—
20#13095510—42no—
21C2×Dic194C2×C411yescharacteristic
22C2×D388—21no—
23C2×D388C2×C411yescharacteristic
24C2×C22⋊D97—21no—
25C2×C22⋊D97—22no—
26C22×Dic97—21no—
27C2×C4×D97—21no—
28C2×C4×D97C2×C411yescharacteristic
29C2×C4×D97D412yes—
30#13095494—28no—
31C2×#11715895—22no—
32#13095490—22no—
33#13095488—22no—
34C4×Dic97C2×C411yescharacteristic
35C4×Dic97D412yes—
36C2×Q8×C97C2×C411yescharacteristic
37C2×D4×C97C2×C411yescharacteristic
38C2×C8×C97—22no—
39C22⋊C4×C97—22no—
40C42×C97C2311yescharacteristic
41C23×C97—21no—
42C22×D97—21no—
43C22×D97C22⋊C411yescharacteristic
44Q8×C97—41no—
45D4×C97—41no—
46C2×C4×C97—21no—
47C2×C4×C97C22×C411yescharacteristic
48C2×C4×C97C2×D412yes—
49C22⋊D97—42no—
50C22⋊D97—44no—
51C2×Dic97—21no—
52C2×Dic97—42no—
53C2×Dic97C22⋊C411yescharacteristic
54C2×Dic97C22⋊C412yes—
55D388—41no—
56D388—42no—
57C4×D97—22no—
58C4×D97—24no—
59C4×D97—42no—
60Dic194—41no—
61C8×C97—24no—
62#11715895—24no—
63C22×C97—41no—
64C22×C97—42no—
65C22×C97C2×C22⋊C411yescharacteristic
66C2×D97—21no—
67C2×D97—22no—
68C2×D97—41no—
69C2×D97—42no—
70C4×C97—21no—
71C4×C97—22no—
72C4×C97—41no—
73Dic97—41no—
74Dic97#10012yes—
75Dic97#10414yes—
76C2×C97—81no—
77C2×C97C2×#10011yescharacteristic
78C2×C97C2×#10412yes—
79D97—41no—
80D97—81no—
81#1525—971nomaximal, Sylow
82C97#152511yescharacteristic, complemented, Sylow
83#532—971no—
84C4×M16—972no—
85#329—971no—
86#329—972no—
87#329D9711yescharacteristic, complemented
88C2×#41—971no—
89C2×#41—971no—
90C2×M16—1942no—
91C4⋊Q8—971no—
92#127—971no—
93#124—971no—
94#124C2×D9711yescharacteristic
95#121—1942no—
96C4×D4—1942no—
97C2×C42—971no—
98#97—972no—
99#97C2×D9712yes—
100C4×C8—1942no—
101#41—3882no—
102#41—3882no—
103C2×Q8—971no—
104C2×Q8C4×D9711yescharacteristic
105C2×D4—971no—
106C2×D4—1941no—
107C2×D4—1942no—
108C2×D4—1941no—
109C2×D4C4×D9711yescharacteristic
110C22×C4—971no—
111C22×C4—972no—
112C22×C4—1941no—
113C22×C4—1941no—
114M16—1948no—
115C2×C8—22no—
116C2×C8—1942no—
117C4⋊C4—1942no—
118C22⋊C4—22no—
119C22⋊C4—1942no—
120C42—971no—
121C42—972no—
122C42C22×D9711yescharacteristic
123C23—21no—
124C23—971no—
125C23—1941no—
126Q8—41no—
127Q8—3881no—
128D4—41no—
129D4—3881no—
130D4—3882no—
131D4—3884no—
132D4—3882no—
133C2×C4—21no—
134C2×C4—972no—
135C2×C4—971no—
136C2×C4—3882no—
137C2×C4—3882no—
138C2×C4—1944no—
139C2×C4—1942no—
140C2×C4—1941no—
141C2×C4C2×C4×D9711yescharacteristic
142C2×C4D4×D9712yes—
143C8—24no—
144C8—1944no—
145C22—41no—
146C22—42no—
147C22—3881no—
148C22—3882no—
149C22—1941no—
150C22—1942no—
151C22C22⋊C4×D9711yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—972no—
156C4—974no—
157C4—3881no—
158C2—81no—
159C2—7761no—
160C2—3881no—
161C2#100×D9711yescharacteristic
162C2#104×D9712yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#329×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261284C2
cosocleC42×C97C23
hypercenter#329D97
hypocenterC97#1525
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20260687C2semidirect
3. #20260475C2semidirect
4. #20260474C2semidirect
5. #329×C97C2semidirect
82. C97#1525semidirect
87. #329D97semidirect

Maximal subgroups

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

subgroupindex
#202612842
#202606872
#202604752
#202604742
#329×C972
#152597