Finite Group Databasebeta

← #1891×C211 #41009511

Subgroups — #1891×C211 #41009511

Order 27008.

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

normal subgroups only

This group has 342 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#525×C211C211yesmaximal, characteristic
3#413×C211C211yesmaximal, characteristic, complemented
4#407×C211C212yesmaximal, complemented
5#344×C211C212yesmaximal, complemented
6C2×#41×C211C2211yescharacteristic
7C23×C4×C211C2211yescharacteristic
8#131×C211—24no—
9C2×M16×C211—22no—
10C2×M16×C211C2212yes—
11C2×M16×C211C2212yes—
12#123×C211C412yes—
13C22⋊Q8×C211—21no—
14#121×C211—21no—
15#120×C211C412yes—
16C4×D4×C211—22no—
17#117×C211C2211yescharacteristic
18#101×C211—21no—
19#100×C211—21no—
20C22⋊C8×C211—24no—
21C24×C211C2×C411yescharacteristic
22#41×C211—22no—
23#41×C211—22no—
24C2×Q8×C211D411yescharacteristic
25C2×D4×C211—22no—
26C2×D4×C211C2×C412yes—
27C2×D4×C211D411yescharacteristic
28C22×C4×C211—22no—
29C22×C4×C211—21no—
30C22×C4×C211—21no—
31C22×C4×C211—22no—
32C22×C4×C211C2×C411yescharacteristic
33C22×C4×C211D412yes—
34C22×C4×C211C2311yescharacteristic
35M16×C211—24no—
36M16×C211—22no—
37M16×C211—22no—
38M16×C211—22no—
39M16×C211—22no—
40C2×C8×C211—24no—
41C2×C8×C211—22no—
42C2×C8×C211D414yescomplemented
43C4⋊C4×C211—22no—
44C4⋊C4×C211—21no—
45C22⋊C4×C211—22no—
46C22⋊C4×C211—22no—
47C22⋊C4×C211C2×C412yes—
48C42×C211—21no—
49C23×C211—21no—
50C23×C211—22no—
51C23×C211—42no—
52C23×C211C22⋊C412yes—
53C23×C211C22×C411yescharacteristic
54Q8×C211—22no—
55D4×C211—24no—
56D4×C211—22no—
57D4×C211—44no—
58C2×C4×C211—22no—
59C2×C4×C211—22no—
60C2×C4×C211—22no—
61C2×C4×C211—21no—
62C2×C4×C211—22no—
63C2×C4×C211—41no—
64C2×C4×C211—42no—
65C2×C4×C211—41no—
66C2×C4×C211C22⋊C412yes—
67C2×C4×C211C2×D412yes—
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211#4112yes—
71C8×C211—24no—
72C8×C211—42no—
73C22×C211—22no—
74C22×C211—24no—
75C22×C211—22no—
76C22×C211—41no—
77C22×C211—42no—
78C22×C211—41no—
79C22×C211—42no—
80C22×C211—41no—
81C22×C211C2×C22⋊C411yescharacteristic
82C22×C211C4×D412yes—
83C4×C211—22no—
84C4×C211—42no—
85C4×C211—41no—
86C4×C211#12011yescharacteristic
87C4×C211#12112yes—
88C4×C211#12311yescharacteristic
89C2×C211—21no—
90C2×C211—22no—
91C2×C211—42no—
92C2×C211—42no—
93C2×C211#38611yescharacteristic
94C211#189111yescharacteristic, direct product factor, Sylow
95#1891C21111yesmaximal, characteristic, direct product factor, Sylow
96#568C2×C21111yescharacteristic, complemented
97#525C2×C21111yescharacteristic
98#413C2×C21111yescharacteristic, complemented
99#407C2×C21112yescomplemented
100#344C2×C21112yescomplemented
101C2×#41C22×C21111yescharacteristic
102C23×C4C22×C21111yescharacteristic
103#131—24no—
104C2×M16—22no—
105C2×M16C22×C21112yes—
106C2×M16C22×C21112yes—
107#123C4×C21112yes—
108C22⋊Q8—21no—
109#121—21no—
110#120C4×C21112yes—
111C4×D4—22no—
112#117C22×C21111yescharacteristic
113#101—21no—
114#100—21no—
115C22⋊C8—24no—
116C24C2×C4×C21111yescharacteristic
117#41—22no—
118#41—22no—
119C2×Q8D4×C21111yescharacteristic
120C2×D4—22no—
121C2×D4C2×C4×C21112yes—
122C2×D4D4×C21111yescharacteristic
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4C2×C4×C21111yescharacteristic
128C22×C4D4×C21112yes—
129C22×C4C23×C21111yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—22no—
134M16—22no—
135C2×C8—24no—
136C2×C8—22no—
137C2×C8D4×C21114yescomplemented
138C4⋊C4—22no—
139C4⋊C4—21no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4C2×C4×C21112yes—
143C42—21no—
144C23—21no—
145C23—22no—
146C23—42no—
147C23C22⋊C4×C21112yes—
148C23C22×C4×C21111yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—44no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4C22⋊C4×C21112yes—
162C2×C4C2×D4×C21112yes—
163C2×C4C2×D4×C21111yescharacteristic
164C2×C4C2×D4×C21111yescharacteristic
165C2×C4#41×C21112yes—
166C8—24no—
167C8—42no—
168C22—22no—
169C22—24no—
170C22—22no—
171C22—41no—
172C22—42no—
173C22—41no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊C4×C21111yescharacteristic
177C22C4×D4×C21112yes—
178C4—22no—
179C4—42no—
180C4—41no—
181C4#120×C21111yescharacteristic
182C4#121×C21112yes—
183C4#123×C21111yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—42no—
188C2#386×C21111yescharacteristic
subgroupG/H
centerC4×C211#120
commutatorC23C22×C4×C211
Frattini, cosocleC22×C4C23×C211
socleC2×C211#386
prime radical#525×C211C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C4×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. #407×C211C2semidirect
5. #344×C211C2semidirect
42. C2×C8×C211D4semidirect
94. C211#1891direct (N × H)
95. #1891C211direct (N × H)
96. #568C2×C211semidirect
98. #413C2×C211semidirect
99. #407C2×C211semidirect
100. #344C2×C211semidirect
137. C2×C8D4×C211semidirect

Maximal subgroups

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

subgroupindex
#568×C2112
#525×C2112
#413×C2112
#407×C2112
#344×C2112
#1891211