Finite Group Databasebeta

← #27142261

Subgroups — #27142261

Order 12416.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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#20261502C211yesmaximal, characteristic, complemented
2#20261299C211yesmaximal, characteristic
3#20260739C212yesmaximal, complemented
4#20260491C212yesmaximal, complemented
5#410×C97C211yesmaximal, characteristic, complemented
6C22×D388C2211yescharacteristic
7#15690817—24no—
8C2×#13095495—22no—
9C2×#13095495C2212yes—
10#15690806C412yescomplemented
11#15690799C412yescomplemented
12#15690784—22no—
13#15690778—22no—
14#15690769C2211yescharacteristic
15#15690730—24no—
16C2×#41×C97C2211yescharacteristic
17#117×C97C2212yes—
18#99×C97—22no—
19C23×D97—21no—
20C2×D388—22no—
21C2×D388—22no—
22C2×D388—21no—
23C2×D388C2×C412yes—
24C2×C22⋊D97—24no—
25#13095495—24no—
26#13095495—22no—
27#13095495—22no—
28C2×#11715895—24no—
29C2×#11715895D414yescomplemented
30C4⋊Dic97—21no—
31C22⋊Dic97—21no—
32#13095488—24no—
33C4×Dic97C2×C412yes—
34#41×C97—22no—
35#41×C97—22no—
36C2×Q8×C97C2×C411yescharacteristic
37C2×D4×C97C2×C411yescharacteristic
38C2×D4×C97D412yes—
39C22×C4×C97D412yes—
40C22×C4×C97C2311yescharacteristic
41C4⋊C4×C97—22no—
42C22⋊C4×C97—22no—
43C42×C97—22no—
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×C97—42no—
55C2×C4×C97C22⋊C412yes—
56C2×C4×C97C22×C411yescharacteristic
57C2×C4×C97C2×D412yes—
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97#4112yes—
60C22⋊D97—84no—
61C2×Dic97—22no—
62D388—24no—
63D388—41no—
64D388—41no—
65#11715895—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×C97—82no—
76C4×C97C4×D412yes—
77C4×C97#12011yescharacteristic
78C4×C97#12311yescharacteristic
79Dic97—42no—
80C2×C97—21no—
81C2×C97—22no—
82C2×C97—42no—
83C2×C97#38611yescharacteristic
84D97—82no—
85#1910—971nomaximal, Sylow
86C97#191011yescharacteristic, complemented, Sylow
87#568—971no—
88#535—971no—
89#418—972no—
90#410D9711yescharacteristic, complemented
91H(3,ℤ/4)—972no—
92C2×#41C2×D9711yescharacteristic
93C22×D4—971no—
94#131—1944no—
95C2×M16—972no—
96C2×M16—1942no—
97#127—972no—
98#124—972no—
99#121—1942no—
100#120—1942no—
101#117—971no—
102#117C2×D9712yes—
103#102—1944no—
104#99—22no—
105C24—1941no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D9711yescharacteristic
109C2×D4—972no—
110C2×D4—1944no—
111C2×D4—1942no—
112C2×D4—1941no—
113C2×D4—1942no—
114C2×D4C4×D9711yescharacteristic
115C2×D4C22⋊D9712yes—
116C22×C4D38812yes—
117C22×C4C22×D9711yescharacteristic
118M16—1944no—
119M16—1942no—
120M16—1942no—
121C2×C8—974no—
122C2×C8—1944no—
123C4⋊C4—22no—
124C4⋊C4—1941no—
125C22⋊C4—22no—
126C22⋊C4—1941no—
127C22⋊C4—1944no—
128C42—22no—
129C42—972no—
130C23—7762no—
131C23—3882no—
132C23—1942no—
133C23#1309548812yes—
134C23C2×C22⋊D9711yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—7764no—
139D4—3881no—
140D4—3881no—
141D4—1944no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—1942no—
146C2×C4#1309548812yes—
147C2×C4C2×C4×D9711yescharacteristic
148C2×C4D4×D9712yes—
149C2×C4C2×D38811yescharacteristic
150C2×C4#1309551012yes—
151C8—1944no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—7764no—
156C22—3884no—
157C22—3882no—
158C22C2×#1309548811yescharacteristic
159C22#1569078212yes—
160C4—22no—
161C4—82no—
162C4—3882no—
163C4#1569077012yes—
164C4#1569078011yescharacteristic
165C4#1569079411yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—7762no—
170C2#2026061211yescharacteristic
subgroupG/H
centerC2#20260612
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#410×C97C2
socleC2×C97#386
prime radical#20261299C2
cosocleC22×C4×C97C23
hypercenter#410D97
hypocenterC97#1910
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #20261502C2semidirect
3. #20260739C2semidirect
4. #20260491C2semidirect
5. #410×C97C2semidirect
10. #15690806C4semidirect
11. #15690799C4semidirect
29. C2×#11715895D4semidirect
86. C97#1910semidirect
90. #410D97semidirect

Maximal subgroups

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

subgroupindex
#202615022
#202612992
#202607392
#202604912
#410×C972
#191097