Finite Group Databasebeta

← #27158010

Subgroups — #27158010

Order 12416.

258 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 52 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#20261716C212yesmaximal, complemented
2#20261641C212yesmaximal, complemented
3#20261640C212yesmaximal, complemented
4#20261286C211yesmaximal, characteristic
5#15690889—22no—
6#15690869—22no—
7#15690869C2214yes—
8C2×#13095512C411yescharacteristic, complemented
9C2×#13095512C2211yescharacteristic
10#15690807C2211yescharacteristic
11#15690799C411yescharacteristic, complemented
12#15690799C2211yescharacteristic
13#15690785—22no—
14#15690772—22no—
15C42×D97C411yescharacteristic, complemented
16C4⋊Q8×C97C411yescharacteristic, complemented
17C4⋊#87604—42no—
18C22⋊#87604—24no—
19C4×#87604—42no—
20#13095512—41no—
21#13095512—41no—
22#13095512—41no—
23#13095512—41no—
24C2×D388—21no—
25C2×D388—22no—
26C2×D388C2×C411yescharacteristic
27C2×D388D412yes—
28C2×D388C2311yescharacteristic
29C2×C4×D97—21no—
30C2×C4×D97—21no—
31C2×C4×D97C2×C411yescharacteristic
32C2×C4×D97D412yes—
33#13095488—22no—
34#13095488—22no—
35C4×Dic97—21no—
36C4×Dic97C2×C411yescharacteristic
37C2×Q8×C97C2×C411yescharacteristic
38C2×Q8×C97C2×C411yescharacteristic
39C4⋊C4×C97—22no—
40C42×C97C2×C411yescharacteristic
41C22×D97—21no—
42C22×D97C22⋊C411yescharacteristic
43C22×D97C2×D412yes—
44C2×#87604—44no—
45Q8×C97—41no—
46Q8×C97—41no—
47C2×C4×C97—21no—
48C2×C4×C97—21no—
49C2×C4×C97C22⋊C412yes—
50C2×C4×C97C22×C411yescharacteristic
51C2×Dic97—21no—
52C2×Dic97C22⋊C411yescharacteristic
53D388—42no—
54D388—41no—
55D388—41no—
56D388—44no—
57D388—42no—
58C4×D97—21no—
59C4×D97—21no—
60C4×D97—22no—
61C4×D97—22no—
62C4×D97—42no—
63C4×D97—41no—
64C4×D97—41no—
65C22×C97C2×C22⋊C411yescharacteristic
66C2×D97—21no—
67C2×D97—22no—
68C2×D97—41no—
69C2×D97—42no—
70C2×D97—42no—
71C2×D97—41no—
72#87604—84no—
73C4×C97—22no—
74C4×C97—21no—
75C4×C97—22no—
76C4×C97—41no—
77Dic97—41no—
78Dic97#9911yescharacteristic
79Dic97#9911yescharacteristic
80C2×C97—21no—
81C2×C97C2×#9911yescharacteristic
82D97—42no—
83D97—41no—
84D97—81no—
85#2126—971nomaximal, Sylow
86C97#212611yescharacteristic, complemented, Sylow
87#532—971no—
88#410—972no—
89#354—972no—
90#353—972no—
91C2×#41—971no—
92C2×#41—971no—
93C4⋊Q8#8760411yescharacteristic, complemented
94#127—971no—
95#124—971no—
96#124—971no—
97#121—1942no—
98C4×D4—1942no—
99#117—1942no—
100C2×C42—971no—
101#99—974no—
102#99—1942no—
103#41—3881no—
104#41—3881no—
105#41—3881no—
106#41—3881no—
107C2×Q8C2×#8760411yescharacteristic
108C2×Q8C2×#8760411yescharacteristic
109C2×D4—972no—
110C2×D4—971no—
111C2×D4—971no—
112C2×D4—1942no—
113C2×D4—1941no—
114C22×C4—972no—
115C22×C4—971no—
116C22×C4—1941no—
117C22×C4—1941no—
118C4⋊C4—22no—
119C4⋊C4—3882no—
120C22⋊C4—1942no—
121C22⋊C4—1944no—
122C22⋊C4—1942no—
123C42—971no—
124C42—3882no—
125C42—1941no—
126C42C2×#8760411yescharacteristic
127C23—972no—
128C23—971no—
129C23—1941no—
130Q8—41no—
131Q8—41no—
132D4—3881no—
133D4—3884no—
134D4—3882no—
135D4—3881no—
136D4—3882no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—971no—
140C2×C4—3882no—
141C2×C4—3881no—
142C2×C4—3881no—
143C2×C4—3884no—
144C2×C4—1942no—
145C2×C4—1942no—
146C2×C4—1941no—
147C2×C4—1941no—
148C2×C4—1941no—
149C2×C4C22⋊#8760412yes—
150C2×C4C22×#8760411yescharacteristic
151C22—3882no—
152C22—3881no—
153C22—3882no—
154C22—3881no—
155C22—1942no—
156C22—1941no—
157C22C2×C22⋊#8760411yescharacteristic
158C4—22no—
159C4—22no—
160C4—21no—
161C4—41no—
162C4—7764no—
163C4—971no—
164C4—971no—
165C4—3881no—
166C2—21no—
167C2—7761no—
168C2—3882no—
169C2—3881no—
170C2D97⋊#9911yescharacteristic
subgroupG/H
centerC2D97⋊#99
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC2×C97C2×#99
prime radical#20261286C2
cosocleC2×D388C23
hypercenterC4⋊Q8#87604
hypocenterC97#2126
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20261716C2semidirect
2. #20261641C2semidirect
3. #20261640C2semidirect
8. C2×#13095512C4semidirect
11. #15690799C4semidirect
15. C42×D97C4semidirect
16. C4⋊Q8×C97C4semidirect
86. C97#2126semidirect
93. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617162
#202616412
#202616402
#202612862
#212697