Finite Group Databasebeta

← #3192×C211 #41010812

Subgroups — #3192×C211 #41010812

Order 27008.

514 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 176 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
1C2×#121×C211C212yesmaximal, complemented
2#481×C211C214yesmaximal, complemented
3#469×C211C214yesmaximal, complemented
4#447×C211C214yesmaximal, complemented
5#407×C211C211yesmaximal, characteristic, complemented
6C22×D4×C211—22no—
7C22×D4×C211C2212yescomplemented
8C23×C4×C211C2211yescharacteristic
9C2×D8×C211C2214yescomplemented
10C2×M16×C211C2212yescomplemented
11#121×C211—22no—
12#121×C211—22no—
13#121×C211C2214yes—
14#121×C211C2214yescomplemented
15C2×C4⋊C4×C211C2212yescomplemented
16C2×C22⋊C4×C211—22no—
17#106×C211C2214yescomplemented
18#102×C211C2218yescomplemented
19C22⋊C8×C211C2214yescomplemented
20C24×C211—22no—
21C24×C211D411yescharacteristic
22C2×D4×C211—24no—
23C2×D4×C211—24no—
24C2×D4×C211—22no—
25C2×D4×C211—44no—
26C2×D4×C211—48no—
27C2×D4×C211C2314yes—
28C22×C4×C211—22no—
29C22×C4×C211—42no—
30C22×C4×C211D412yes—
31C22×C4×C211D411yescharacteristic
32C22×C4×C211C2312yes—
33C22×C4×C211C2311yescharacteristic
34D8×C211—44no—
35M16×C211—42no—
36C2×C8×C211C2314yes—
37C4⋊C4×C211—22no—
38C4⋊C4×C211C2314yes—
39C22⋊C4×C211—24no—
40C22⋊C4×C211—24no—
41C23×C211—24no—
42C23×C211—24no—
43C23×C211—41no—
44C23×C211—44no—
45C23×C211—44no—
46C23×C211—44no—
47C23×C211C2×D411yescharacteristic
48C23×C211C2×D412yes—
49D4×C211—28no—
50D4×C211—44no—
51D4×C211—48no—
52D4×C211—48no—
53C2×C4×C211—22no—
54C2×C4×C211—24no—
55C2×C4×C211—24no—
56C2×C4×C211—42no—
57C2×C4×C211—41no—
58C2×C4×C211—44no—
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211C2×D412yes—
61C2×C4×C211C2411yescharacteristic
62C8×C211—24no—
63C22×C211—22no—
64C22×C211—22no—
65C22×C211—21no—
66C22×C211—22no—
67C22×C211—44no—
68C22×C211—44no—
69C22×C211—44no—
70C22×C211—44no—
71C22×C211—88no—
72C22×C211AGL(1,ℤ/8)14yes—
73C22×C211C22×D411yescharacteristic
74C4×C211—21no—
75C4×C211—22no—
76C4×C211—44no—
77C4×C211H(5,𝔽2)12yes—
78C2×C211—24no—
79C2×C211—41no—
80C2×C211—84no—
81C2×C211#53411yescharacteristic
82C2×C211C2×AGL(1,ℤ/8)12yes—
83C211#319211yescharacteristic, direct product factor, Sylow
84#3192C21111yesmaximal, characteristic, direct product factor, Sylow
85C2×#121C2×C21112yescomplemented
86#481C2×C21114yescomplemented
87#469C2×C21114yescomplemented
88#447C2×C21114yescomplemented
89#407C2×C21111yescharacteristic, complemented
90C22×D4—22no—
91C22×D4C22×C21112yescomplemented
92C23×C4C22×C21111yescharacteristic
93C2×D8C22×C21114yescomplemented
94C2×M16C22×C21112yescomplemented
95#121—22no—
96#121—22no—
97#121C22×C21114yes—
98#121C22×C21114yescomplemented
99C2×C4⋊C4C22×C21112yescomplemented
100C2×C22⋊C4—22no—
101#106C22×C21114yescomplemented
102#102C22×C21118yescomplemented
103C22⋊C8C22×C21114yescomplemented
104C24—22no—
105C24D4×C21111yescharacteristic
106C2×D4—24no—
107C2×D4—24no—
108C2×D4—22no—
109C2×D4—44no—
110C2×D4—48no—
111C2×D4C23×C21114yes—
112C22×C4—22no—
113C22×C4—42no—
114C22×C4D4×C21112yes—
115C22×C4D4×C21111yescharacteristic
116C22×C4C23×C21112yes—
117C22×C4C23×C21111yescharacteristic
118D8—44no—
119M16—42no—
120C2×C8C23×C21114yes—
121C4⋊C4—22no—
122C4⋊C4C23×C21114yes—
123C22⋊C4—24no—
124C22⋊C4—24no—
125C23—24no—
126C23—24no—
127C23—41no—
128C23—44no—
129C23—44no—
130C23—44no—
131C23C2×D4×C21111yescharacteristic
132C23C2×D4×C21112yes—
133D4—28no—
134D4—44no—
135D4—48no—
136D4—48no—
137C2×C4—22no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—44no—
143C2×C4C2×D4×C21111yescharacteristic
144C2×C4C2×D4×C21112yes—
145C2×C4C24×C21111yescharacteristic
146C8—24no—
147C22—22no—
148C22—22no—
149C22—21no—
150C22—22no—
151C22—44no—
152C22—44no—
153C22—44no—
154C22—44no—
155C22—88no—
156C22AGL(1,ℤ/8)×C21114yes—
157C22C22×D4×C21111yescharacteristic
158C4—22no—
159C4—21no—
160C4—44no—
161C4H(5,𝔽2)×C21112yes—
162C2—24no—
163C2—41no—
164C2—84no—
165C2#534×C21111yescharacteristic
166C2C2×AGL(1,ℤ/8)×C21112yes—
subgroupG/H
center, socleC22×C211C22×D4
commutator, Frattini, cosocleC2×C4C24×C211
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#121×C211C2semidirect
2. #481×C211C2semidirect
3. #469×C211C2semidirect
4. #447×C211C2semidirect
5. #407×C211C2semidirect
7. C22×D4×C211C22semidirect
9. C2×D8×C211C22semidirect
10. C2×M16×C211C22semidirect
14. #121×C211C22semidirect
15. C2×C4⋊C4×C211C22semidirect
17. #106×C211C22semidirect
18. #102×C211C22semidirect
19. C22⋊C8×C211C22semidirect
83. C211#3192direct (N × H)
84. #3192C211direct (N × H)
85. C2×#121C2×C211semidirect
86. #481C2×C211semidirect
87. #469C2×C211semidirect
88. #447C2×C211semidirect
89. #407C2×C211semidirect
91. C22×D4C22×C211semidirect
93. C2×D8C22×C211semidirect
94. C2×M16C22×C211semidirect
98. #121C22×C211semidirect
99. C2×C4⋊C4C22×C211semidirect
101. #106C22×C211semidirect
102. #102C22×C211semidirect
103. C22⋊C8C22×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#121×C2112
#481×C2112
#469×C2112
#447×C2112
#407×C2112
#3192211