Finite Group Databasebeta

← #1896×C211 #41009516

Subgroups — #1896×C211 #41009516

Order 27008.

442 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 120 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#525×C211C211yesmaximal, characteristic
2C22×C42×C211C211yesmaximal, characteristic, complemented
3C2×#104×C211C212yesmaximal, complemented
4#407×C211C211yesmaximal, characteristic, complemented
5#339×C211C212yesmaximal, complemented
6C2×#41×C211C2211yescharacteristic
7C23×C4×C211—21no—
8C23×C4×C211C2211yescharacteristic
9C2×M16×C211C2212yes—
10#123×C211—21no—
11C22⋊Q8×C211C412yescomplemented
12#121×C211C412yescomplemented
13#120×C211—21no—
14C4×D4×C211—22no—
15#117×C211C2211yescharacteristic
16C2×C42×C211—21no—
17C2×C42×C211—24no—
18C2×C42×C211C2212yes—
19#104×C211—24no—
20C22⋊C8×C211—22no—
21C24×C211D411yescharacteristic
22#41×C211—22no—
23#41×C211—22no—
24C2×Q8×C211C2×C411yescharacteristic, complemented
25C2×D4×C211—22no—
26C2×D4×C211—21no—
27C2×D4×C211C2×C411yescharacteristic, complemented
28C22×C4×C211—24no—
29C22×C4×C211—28no—
30C22×C4×C211—22no—
31C22×C4×C211—22no—
32C22×C4×C211—22no—
33C22×C4×C211—21no—
34C22×C4×C211C2×C412yes—
35C22×C4×C211D411yescharacteristic
36C22×C4×C211D412yes—
37C22×C4×C211C2311yescharacteristic
38M16×C211—22no—
39M16×C211—22no—
40C2×C8×C211—22no—
41C4⋊C4×C211—22no—
42C4⋊C4×C211C2×C412yescomplemented
43C22⋊C4×C211—22no—
44C22⋊C4×C211—21no—
45C22⋊C4×C211—22no—
46C42×C211—24no—
47C42×C211—22no—
48C42×C211—21no—
49C42×C211D414yescomplemented
50C23×C211—24no—
51C23×C211—21no—
52C23×C211—21no—
53C23×C211—21no—
54C23×C211C22⋊C412yes—
55C23×C211C2×D411yescharacteristic
56Q8×C211—22no—
57D4×C211—22no—
58D4×C211—44no—
59D4×C211—41no—
60D4×C211—41no—
61C2×C4×C211—216no—
62C2×C4×C211—28no—
63C2×C4×C211—21no—
64C2×C4×C211—24no—
65C2×C4×C211—21no—
66C2×C4×C211—24no—
67C2×C4×C211—24no—
68C2×C4×C211—22no—
69C2×C4×C211—21no—
70C2×C4×C211—42no—
71C2×C4×C211C22⋊C412yes—
72C2×C4×C211C22×C411yescharacteristic
73C2×C4×C211C2×D412yes—
74C2×C4×C211C2×D411yescharacteristic
75C2×C4×C211#4112yes—
76C8×C211—42no—
77C22×C211—24no—
78C22×C211—24no—
79C22×C211—22no—
80C22×C211—22no—
81C22×C211—22no—
82C22×C211—41no—
83C22×C211—42no—
84C22×C211#10414yescomplemented
85C22×C211C2×C22⋊C411yescharacteristic
86C22×C211#12011yescharacteristic
87C22×C211#12311yescharacteristic
88C4×C211—28no—
89C4×C211—22no—
90C4×C211—41no—
91C4×C211—42no—
92C4×C211C4×D412yes—
93C4×C211#12112yes—
94C2×C211—21no—
95C2×C211—24no—
96C2×C211—21no—
97C2×C211—81no—
98C2×C211#38611yescharacteristic
99C2×C211C2×#10412yes—
100C211#189611yescharacteristic, direct product factor, Sylow
101#1896C21111yesmaximal, characteristic, direct product factor, Sylow
102#525C2×C21111yescharacteristic
103C22×C42C2×C21111yescharacteristic, complemented
104C2×#104C2×C21112yescomplemented
105#407C2×C21111yescharacteristic, complemented
106#339C2×C21112yescomplemented
107C2×#41C22×C21111yescharacteristic
108C23×C4—21no—
109C23×C4C22×C21111yescharacteristic
110C2×M16C22×C21112yes—
111#123—21no—
112C22⋊Q8C4×C21112yescomplemented
113#121C4×C21112yescomplemented
114#120—21no—
115C4×D4—22no—
116#117C22×C21111yescharacteristic
117C2×C42—21no—
118C2×C42—24no—
119C2×C42C22×C21112yes—
120#104—24no—
121C22⋊C8—22no—
122C24D4×C21111yescharacteristic
123#41—22no—
124#41—22no—
125C2×Q8C2×C4×C21111yescharacteristic, complemented
126C2×D4—22no—
127C2×D4—21no—
128C2×D4C2×C4×C21111yescharacteristic, complemented
129C22×C4—24no—
130C22×C4—28no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4C2×C4×C21112yes—
136C22×C4D4×C21111yescharacteristic
137C22×C4D4×C21112yes—
138C22×C4C23×C21111yescharacteristic
139M16—22no—
140M16—22no—
141C2×C8—22no—
142C4⋊C4—22no—
143C4⋊C4C2×C4×C21112yescomplemented
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—22no—
147C42—24no—
148C42—22no—
149C42—21no—
150C42D4×C21114yescomplemented
151C23—24no—
152C23—21no—
153C23—21no—
154C23—21no—
155C23C22⋊C4×C21112yes—
156C23C2×D4×C21111yescharacteristic
157Q8—22no—
158D4—22no—
159D4—44no—
160D4—41no—
161D4—41no—
162C2×C4—216no—
163C2×C4—28no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4C22⋊C4×C21112yes—
173C2×C4C22×C4×C21111yescharacteristic
174C2×C4C2×D4×C21112yes—
175C2×C4C2×D4×C21111yescharacteristic
176C2×C4#41×C21112yes—
177C8—42no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—41no—
184C22—42no—
185C22#104×C21114yescomplemented
186C22C2×C22⋊C4×C21111yescharacteristic
187C22#120×C21111yescharacteristic
188C22#123×C21111yescharacteristic
189C4—28no—
190C4—22no—
191C4—41no—
192C4—42no—
193C4C4×D4×C21112yes—
194C4#121×C21112yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—81no—
199C2#386×C21111yescharacteristic
200C2C2×#104×C21112yes—
subgroupG/H
centerC2×C4×C211C2×D4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
socleC22×C211C2×C22⋊C4
prime radical#525×C211C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C42×C211C2semidirect
3. C2×#104×C211C2semidirect
4. #407×C211C2semidirect
5. #339×C211C2semidirect
11. C22⋊Q8×C211C4semidirect
12. #121×C211C4semidirect
24. C2×Q8×C211C2×C4semidirect
27. C2×D4×C211C2×C4semidirect
42. C4⋊C4×C211C2×C4semidirect
49. C42×C211D4semidirect
84. C22×C211#104semidirect
100. C211#1896direct (N × H)
101. #1896C211direct (N × H)
103. C22×C42C2×C211semidirect
104. C2×#104C2×C211semidirect
105. #407C2×C211semidirect
106. #339C2×C211semidirect
112. C22⋊Q8C4×C211semidirect
113. #121C4×C211semidirect
125. C2×Q8C2×C4×C211semidirect
128. C2×D4C2×C4×C211semidirect
143. C4⋊C4C2×C4×C211semidirect
150. C42D4×C211semidirect
185. C22#104×C211semidirect

Maximal subgroups

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

subgroupindex
#525×C2112
C22×C42×C2112
C2×#104×C2112
#407×C2112
#339×C2112
#1896211