Finite Group Databasebeta

← #27142263

Subgroups — #27142263

Order 12416.

344 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 89 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#20261299C211yesmaximal, characteristic
2#20260741C212yesmaximal, complemented
3#20260709C211yesmaximal, characteristic, complemented
4#20260492C212yesmaximal, complemented
5#568×C97C211yesmaximal, characteristic, complemented
6C22×D388C2211yescharacteristic
7#15690806C412yescomplemented
8#15690799C412yescomplemented
9#15690784—22no—
10#15690778—22no—
11#15690769C2212yes—
12#15690769C2211yescharacteristic
13#15690732—24no—
14#15690725—22no—
15C2×#41×C97C2211yescharacteristic
16#131×C97—24no—
17C2×M16×C97—22no—
18C2×M16×C97C2212yes—
19C23×D97—21no—
20C2×D388—22no—
21C2×D388—22no—
22C2×D388—21no—
23C2×D388C2×C412yes—
24C2×C22⋊D97—24no—
25C4⋊Dic97—21no—
26C4⋊Dic97—22no—
27C22⋊Dic97—21no—
28C22⋊Dic97—22no—
29#13095488—24no—
30C4×Dic97—22no—
31C4×Dic97C2×C412yes—
32#41×C97—22no—
33#41×C97—22no—
34C2×Q8×C97C2×C411yescharacteristic
35C2×D4×C97C2×C411yescharacteristic
36C2×D4×C97D412yes—
37C22×C4×C97D412yes—
38C22×C4×C97C2311yescharacteristic
39M16×C97—24no—
40M16×C97—22no—
41M16×C97—22no—
42C2×C8×C97—24no—
43C2×C8×C97D414yescomplemented
44C23×C97C22⋊C412yes—
45C23×C97C2×D411yescharacteristic
46C22×D97—22no—
47C22×D97—42no—
48C22×D97—82no—
49Q8×C97—22no—
50D4×C97—24no—
51D4×C97—22no—
52C2×C4×C97—22no—
53C2×C4×C97—22no—
54C2×C4×C97C22⋊C412yes—
55C2×C4×C97C22×C411yescharacteristic
56C2×C4×C97C2×D412yes—
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97#4112yes—
59C22⋊D97—84no—
60C2×Dic97—22no—
61C2×Dic97—42no—
62D388—24no—
63D388—41no—
64D388—41no—
65C8×C97—24no—
66C22×C97—22no—
67C22×C97—41no—
68C22×C97—42no—
69C22×C97C2×C22⋊C411yescharacteristic
70C22×C97#12112yes—
71C2×D97—44no—
72C2×D97—42no—
73C2×D97—84no—
74C4×C97—22no—
75C4×C97C4×D412yes—
76C4×C97#12011yescharacteristic
77C4×C97#12311yescharacteristic
78Dic97—42no—
79Dic97—82no—
80C2×C97—21no—
81C2×C97—22no—
82C2×C97—42no—
83C2×C97#38611yescharacteristic
84D97—82no—
85#1910—971nomaximal, Sylow
86C97#191011yescharacteristic, complemented, Sylow
87#568D9711yescharacteristic, complemented
88#535—971no—
89#418—972no—
90#410—971no—
91H(3,ℤ/4)—972no—
92C2×#41C2×D9711yescharacteristic
93C22×D4—971no—
94#131—24no—
95C2×M16—22no—
96C2×M16C2×D9712yes—
97#127—972no—
98#124—972no—
99#121—1942no—
100#120—1942no—
101#117—971no—
102#117—972no—
103#102—1944no—
104#99—1942no—
105C24—1941no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D9711yescharacteristic
109C2×D4—972no—
110C2×D4—1944no—
111C2×D4—1941no—
112C2×D4—1942no—
113C2×D4—1942no—
114C2×D4C4×D9711yescharacteristic
115C2×D4D38812yes—
116C22×C4C22⋊D9712yes—
117C22×C4C22×D9711yescharacteristic
118M16—24no—
119M16—22no—
120M16—22no—
121C2×C8—24no—
122C2×C8C22⋊D9714yescomplemented
123C4⋊C4—1942no—
124C4⋊C4—1941no—
125C22⋊C4—1942no—
126C22⋊C4—1941no—
127C22⋊C4—1944no—
128C42—972no—
129C42—1942no—
130C23—7762no—
131C23—3882no—
132C23—1942no—
133C23#1309548812yes—
134C23C2×D38811yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—7764no—
139D4—3881no—
140D4—3881no—
141D4—1944no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—3882no—
145C2×C4—1942no—
146C2×C4#1309548812yes—
147C2×C4C2×C4×D9711yescharacteristic
148C2×C4C2×C22⋊D9712yes—
149C2×C4C2×C22⋊D9711yescharacteristic
150C2×C4#1309551112yes—
151C8—24no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—7764no—
156C22—3884no—
157C22—3882no—
158C22C2×#1309548811yescharacteristic
159C22#1569078412yes—
160C4—22no—
161C4—7762no—
162C4—3882no—
163C4C4×C22⋊D9712yes—
164C4#1569077911yescharacteristic
165C4#1569079311yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—7762no—
170C2#2026061411yescharacteristic
subgroupG/H
centerC2#20260614
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#568×C97C2
socleC2×C97#386
prime radical#20261299C2
cosocleC22×C4×C97C23
hypercenter#568D97
hypocenterC97#1910
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20260741C2semidirect
3. #20260709C2semidirect
4. #20260492C2semidirect
5. #568×C97C2semidirect
7. #15690806C4semidirect
8. #15690799C4semidirect
43. C2×C8×C97D4semidirect
86. C97#1910semidirect
87. #568D97semidirect
122. C2×C8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202612992
#202607412
#202607092
#202604922
#568×C972
#191097