Finite Group Databasebeta

← #1914×C211 #41009534

Subgroups — #1914×C211 #41009534

Order 27008.

340 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 108 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#568×C211C211yesmaximal, characteristic, complemented
2#535×C211C211yesmaximal, characteristic
3#413×C211C211yesmaximal, characteristic, complemented
4C2×#100×C211C212yesmaximal, complemented
5#344×C211C212yesmaximal, complemented
6C2×#41×C211C2211yescharacteristic
7C22×D4×C211C2211yescharacteristic
8#131×C211—24no—
9C2×M16×C211—22no—
10C2×M16×C211C2212yes—
11C2×M16×C211C2212yes—
12#127×C211—21no—
13#124×C211—21no—
14#121×C211—22no—
15#120×C211C414yes—
16#117×C211C2211yescharacteristic
17#101×C211—21no—
18#100×C211—21no—
19#100×C211—24no—
20C24×C211C2×C412yes—
21#41×C211—22no—
22#41×C211—22no—
23C2×Q8×C211D411yescharacteristic
24C2×D4×C211—24no—
25C2×D4×C211—22no—
26C2×D4×C211—21no—
27C2×D4×C211—21no—
28C2×D4×C211—22no—
29C2×D4×C211C2×C412yes—
30C2×D4×C211D411yescharacteristic
31C22×C4×C211D412yes—
32C22×C4×C211C2311yescharacteristic
33M16×C211—24no—
34M16×C211—22no—
35M16×C211—22no—
36M16×C211—22no—
37M16×C211D414yescomplemented
38C2×C8×C211—24no—
39C2×C8×C211—22no—
40C2×C8×C211—22no—
41C4⋊C4×C211—21no—
42C22⋊C4×C211—24no—
43C22⋊C4×C211C2×C412yes—
44C42×C211—21no—
45C23×C211—22no—
46C23×C211—24no—
47C23×C211—82no—
48C23×C211C22⋊C412yes—
49C23×C211C22×C411yescharacteristic
50Q8×C211—22no—
51D4×C211—24no—
52D4×C211—22no—
53D4×C211—41no—
54D4×C211—42no—
55D4×C211—41no—
56D4×C211—84no—
57C2×C4×C211—22no—
58C2×C4×C211—22no—
59C2×C4×C211—22no—
60C2×C4×C211C22⋊C412yes—
61C2×C4×C211C2×D412yes—
62C2×C4×C211C2×D411yescharacteristic
63C2×C4×C211C2×D411yescharacteristic
64C2×C4×C211#4112yes—
65C8×C211—24no—
66C8×C211—42no—
67C22×C211—22no—
68C22×C211—41no—
69C22×C211—42no—
70C22×C211—44no—
71C22×C211—42no—
72C22×C211—84no—
73C22×C211C2×C22⋊C411yescharacteristic
74C22×C211C4×D412yes—
75C4×C211—22no—
76C4×C211—42no—
77C4×C211#12011yescharacteristic
78C4×C211#12112yes—
79C4×C211#12311yescharacteristic
80C2×C211—21no—
81C2×C211—22no—
82C2×C211—42no—
83C2×C211—82no—
84C2×C211#38611yescharacteristic
85C211#191411yescharacteristic, direct product factor, Sylow
86#1914C21111yesmaximal, characteristic, direct product factor, Sylow
87#568C2×C21111yescharacteristic, complemented
88#535C2×C21111yescharacteristic
89#413C2×C21111yescharacteristic, complemented
90C2×#100C2×C21112yescomplemented
91#344C2×C21112yescomplemented
92C2×#41C22×C21111yescharacteristic
93C22×D4C22×C21111yescharacteristic
94#131—24no—
95C2×M16—22no—
96C2×M16C22×C21112yes—
97C2×M16C22×C21112yes—
98#127—21no—
99#124—21no—
100#121—22no—
101#120C4×C21114yes—
102#117C22×C21111yescharacteristic
103#101—21no—
104#100—21no—
105#100—24no—
106C24C2×C4×C21112yes—
107#41—22no—
108#41—22no—
109C2×Q8D4×C21111yescharacteristic
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4C2×C4×C21112yes—
116C2×D4D4×C21111yescharacteristic
117C22×C4D4×C21112yes—
118C22×C4C23×C21111yescharacteristic
119M16—24no—
120M16—22no—
121M16—22no—
122M16—22no—
123M16D4×C21114yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—22no—
127C4⋊C4—21no—
128C22⋊C4—24no—
129C22⋊C4C2×C4×C21112yes—
130C42—21no—
131C23—22no—
132C23—24no—
133C23—82no—
134C23C22⋊C4×C21112yes—
135C23C22×C4×C21111yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—41no—
140D4—42no—
141D4—41no—
142D4—84no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4C22⋊C4×C21112yes—
147C2×C4C2×D4×C21112yes—
148C2×C4C2×D4×C21111yescharacteristic
149C2×C4C2×D4×C21111yescharacteristic
150C2×C4#41×C21112yes—
151C8—24no—
152C8—42no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—44no—
157C22—42no—
158C22—84no—
159C22C2×C22⋊C4×C21111yescharacteristic
160C22C4×D4×C21112yes—
161C4—22no—
162C4—42no—
163C4#120×C21111yescharacteristic
164C4#121×C21112yes—
165C4#123×C21111yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#386×C21111yescharacteristic
subgroupG/H
center, socleC2×C211#386
commutatorC23C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radical#535×C211C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. #568×C211C2semidirect
3. #413×C211C2semidirect
4. C2×#100×C211C2semidirect
5. #344×C211C2semidirect
37. M16×C211D4semidirect
85. C211#1914direct (N × H)
86. #1914C211direct (N × H)
87. #568C2×C211semidirect
89. #413C2×C211semidirect
90. C2×#100C2×C211semidirect
91. #344C2×C211semidirect
123. M16D4×C211semidirect

Maximal subgroups

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

subgroupindex
#568×C2112
#535×C2112
#413×C2112
C2×#100×C2112
#344×C2112
#1914211