Finite Group Databasebeta

← #27142264

Subgroups — #27142264

Order 12416.

344 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 93 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#20260740C212yesmaximal, complemented
3#20260709C211yesmaximal, characteristic, complemented
4#20260491C212yesmaximal, complemented
5#535×C97C211yesmaximal, characteristic
6#15690817—24no—
7C2×#13095495—22no—
8C2×#13095495C2212yes—
9#15690769C2212yes—
10#15690731—24no—
11#15690725—22no—
12C2×#41×C97C2211yescharacteristic
13C22×D4×C97C2211yescharacteristic
14#127×C97C412yescomplemented
15#124×C97C412yescomplemented
16#121×C97—22no—
17#120×C97—22no—
18#117×C97C2211yescharacteristic
19#13095495—24no—
20#13095495—22no—
21#13095495—22no—
22C2×#11715895—24no—
23C2×#11715895D414yescomplemented
24C4⋊Dic97—22no—
25C22⋊Dic97—22no—
26C4×Dic97—22no—
27C24×C97—21no—
28#41×C97—22no—
29#41×C97—22no—
30C2×Q8×C97C2×C411yescharacteristic
31C2×D4×C97—24no—
32C2×D4×C97—22no—
33C2×D4×C97—21no—
34C2×D4×C97—22no—
35C2×D4×C97C2×C412yes—
36C2×D4×C97C2×C411yescharacteristic
37C2×D4×C97D412yes—
38C22×C4×C97D412yes—
39C22×C4×C97C2311yescharacteristic
40C4⋊C4×C97—21no—
41C22⋊C4×C97—21no—
42C22⋊C4×C97—24no—
43C42×C97C2×C412yes—
44C23×C97—22no—
45C23×C97—42no—
46C23×C97—82no—
47C23×C97C22⋊C412yes—
48C23×C97C2×D411yescharacteristic
49Q8×C97—22no—
50D4×C97—24no—
51D4×C97—22no—
52D4×C97—24no—
53D4×C97—41no—
54D4×C97—41no—
55D4×C97—84no—
56C2×C4×C97—22no—
57C2×C4×C97—22no—
58C2×C4×C97—22no—
59C2×C4×C97C22⋊C412yes—
60C2×C4×C97C22×C411yescharacteristic
61C2×C4×C97C2×D412yes—
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97#4112yes—
64C2×Dic97—42no—
65#11715895—24no—
66C22×C97—22no—
67C22×C97—41no—
68C22×C97—42no—
69C22×C97—44no—
70C22×C97—42no—
71C22×C97—84no—
72C22×C97C2×C22⋊C411yescharacteristic
73C22×C97#12112yes—
74C4×C97—22no—
75C4×C97—42no—
76C4×C97C4×D412yes—
77C4×C97#12011yescharacteristic
78C4×C97#12311yescharacteristic
79Dic97—82no—
80C2×C97—22no—
81C2×C97—21no—
82C2×C97—42no—
83C2×C97—82no—
84C2×C97#38611yescharacteristic
85#1910—971nomaximal, Sylow
86C97#191011yescharacteristic, complemented, Sylow
87#568—971no—
88#535D9711yescharacteristic
89#418—972no—
90#410—971no—
91H(3,ℤ/4)—972no—
92C2×#41C2×D9711yescharacteristic
93C22×D4C2×D9711yescharacteristic
94#131—1944no—
95C2×M16—972no—
96C2×M16—1942no—
97#127Dic9712yescomplemented
98#124Dic9712yescomplemented
99#121—22no—
100#120—22no—
101#117—972no—
102#117C2×D9711yescharacteristic
103#102—1944no—
104#99—1942no—
105C24—21no—
106#41—22no—
107#41—22no—
108C2×Q8C2×Dic9711yescharacteristic
109C2×D4—24no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4C2×Dic9712yes—
114C2×D4C2×Dic9711yescharacteristic
115C2×D4C22⋊D9712yes—
116C22×C4C22⋊D9712yes—
117C22×C4C22×D9711yescharacteristic
118M16—1944no—
119M16—1942no—
120M16—1942no—
121C2×C8—974no—
122C2×C8—1944no—
123C4⋊C4—21no—
124C4⋊C4—1942no—
125C22⋊C4—21no—
126C22⋊C4—24no—
127C22⋊C4—1942no—
128C42—1942no—
129C42C2×Dic9712yes—
130C23—22no—
131C23—42no—
132C23—82no—
133C23C22⋊Dic9712yes—
134C23C2×C22⋊D9711yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—24no—
139D4—41no—
140D4—41no—
141D4—84no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—3882no—
146C2×C4C22⋊Dic9712yes—
147C2×C4C22×Dic9711yescharacteristic
148C2×C4D4×D9712yes—
149C2×C4C2×C22⋊D9711yescharacteristic
150C2×C4#1309551012yes—
151C8—1944no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—44no—
156C22—42no—
157C22—84no—
158C22C2×C22⋊Dic9711yescharacteristic
159C22#1569078212yes—
160C4—22no—
161C4—42no—
162C4—7762no—
163C4D4×Dic9712yes—
164C4#1569077811yescharacteristic
165C4#1569079111yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#2026061511yescharacteristic
subgroupG/H
centerC2#20260615
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting, prime radical#535×C97C2
socleC2×C97#386
cosocleC22×C4×C97C23
hypercenter#535D97
hypocenterC97#1910
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #20261502C2semidirect
2. #20260740C2semidirect
3. #20260709C2semidirect
4. #20260491C2semidirect
14. #127×C97C4semidirect
15. #124×C97C4semidirect
23. C2×#11715895D4semidirect
86. C97#1910semidirect
97. #127Dic97semidirect
98. #124Dic97semidirect

Maximal subgroups

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

subgroupindex
#202615022
#202607402
#202607092
#202604912
#535×C972
#191097