Finite Group Databasebeta

← #1895×C211 #41009515

Subgroups — #1895×C211 #41009515

Order 27008.

412 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 128 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
1C22×M16×C211C211yesmaximal, characteristic, complemented
2C2×#121×C211C211yesmaximal, characteristic
3C2×#102×C211C212yesmaximal, complemented
4#380×C211C211yesmaximal, characteristic, complemented
5#340×C211C212yesmaximal, complemented
6C22×D4×C211—21no—
7C22×D4×C211C2211yescharacteristic
8C23×C4×C211C2211yescharacteristic
9C2×M16×C211—22no—
10C2×M16×C211—24no—
11C22×C8×C211C2212yes—
12#121×C211—22no—
13#121×C211C414yescomplemented
14C2×C4⋊C4×C211C2212yes—
15C2×C4⋊C4×C211C2211yescharacteristic
16C2×C22⋊C4×C211—21no—
17C2×C22⋊C4×C211—21no—
18#102×C211—24no—
19#95×C211—21no—
20C24×C211—21no—
21C24×C211D411yescharacteristic
22C2×D4×C211—24no—
23C2×D4×C211—22no—
24C2×D4×C211—21no—
25C2×D4×C211—22no—
26C2×D4×C211—44no—
27C2×D4×C211C2×C412yes—
28C22×C4×C211—22no—
29C22×C4×C211—22no—
30C22×C4×C211—21no—
31C22×C4×C211—22no—
32C22×C4×C211C2×C412yes—
33C22×C4×C211D411yescharacteristic
34C22×C4×C211D412yes—
35C22×C4×C211C2311yescharacteristic
36M16×C211—28no—
37C2×C8×C211—22no—
38C2×C8×C211—22no—
39C2×C8×C211D414yescomplemented
40C4⋊C4×C211—21no—
41C4⋊C4×C211—22no—
42C4⋊C4×C211—22no—
43C4⋊C4×C211C2×C412yes—
44C22⋊C4×C211—24no—
45C22⋊C4×C211—42no—
46C23×C211—22no—
47C23×C211—21no—
48C23×C211—22no—
49C23×C211—21no—
50C23×C211—22no—
51C23×C211—42no—
52C23×C211—42no—
53C23×C211—41no—
54C23×C211—41no—
55C23×C211C22⋊C412yes—
56C23×C211C2×D411yescharacteristic
57D4×C211—24no—
58D4×C211—48no—
59D4×C211—41no—
60D4×C211—41no—
61C2×C4×C211—22no—
62C2×C4×C211—24no—
63C2×C4×C211—22no—
64C2×C4×C211—22no—
65C2×C4×C211—22no—
66C2×C4×C211—42no—
67C2×C4×C211—42no—
68C2×C4×C211—42no—
69C2×C4×C211—42no—
70C2×C4×C211C22⋊C412yes—
71C2×C4×C211C22×C411yescharacteristic
72C2×C4×C211C2×D412yes—
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211#4112yes—
75C8×C211—24no—
76C22×C211—22no—
77C22×C211—22no—
78C22×C211—22no—
79C22×C211—42no—
80C22×C211—42no—
81C22×C211—42no—
82C22×C211—42no—
83C22×C211—42no—
84C22×C211—82no—
85C22×C211—82no—
86C22×C211C2×C22⋊C411yescharacteristic
87C22×C211#12112yes—
88C22×C211AGL(1,ℤ/8)11yescharacteristic
89C22×C211AGL(1,ℤ/8)12yes—
90C22×C211#13711yescharacteristic
91C4×C211—22no—
92C4×C211—42no—
93C4×C211—82no—
94C4×C211C4×D412yes—
95C4×C211#12011yescharacteristic
96C4×C211#12311yescharacteristic
97C2×C211—42no—
98C2×C211—82no—
99C2×C211#38611yescharacteristic
100C2×C211#41711yescharacteristic
101C2×C211#41811yescharacteristic
102C2×C211#46812yes—
103C2×C211#46912yes—
104C211#189511yescharacteristic, direct product factor, Sylow
105#1895C21111yesmaximal, characteristic, direct product factor, Sylow
106C22×M16C2×C21111yescharacteristic, complemented
107C2×#121C2×C21111yescharacteristic
108C2×#102C2×C21112yescomplemented
109#380C2×C21111yescharacteristic, complemented
110#340C2×C21112yescomplemented
111C22×D4—21no—
112C22×D4C22×C21111yescharacteristic
113C23×C4C22×C21111yescharacteristic
114C2×M16—22no—
115C2×M16—24no—
116C22×C8C22×C21112yes—
117#121—22no—
118#121C4×C21114yescomplemented
119C2×C4⋊C4C22×C21112yes—
120C2×C4⋊C4C22×C21111yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123#102—24no—
124#95—21no—
125C24—21no—
126C24D4×C21111yescharacteristic
127C2×D4—24no—
128C2×D4—22no—
129C2×D4—21no—
130C2×D4—22no—
131C2×D4—44no—
132C2×D4C2×C4×C21112yes—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4C2×C4×C21112yes—
138C22×C4D4×C21111yescharacteristic
139C22×C4D4×C21112yes—
140C22×C4C23×C21111yescharacteristic
141M16—28no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8D4×C21114yescomplemented
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4C2×C4×C21112yes—
149C22⋊C4—24no—
150C22⋊C4—42no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—21no—
155C23—22no—
156C23—42no—
157C23—42no—
158C23—41no—
159C23—41no—
160C23C22⋊C4×C21112yes—
161C23C2×D4×C21111yescharacteristic
162D4—24no—
163D4—48no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4C22⋊C4×C21112yes—
176C2×C4C22×C4×C21111yescharacteristic
177C2×C4C2×D4×C21112yes—
178C2×C4C2×D4×C21111yescharacteristic
179C2×C4#41×C21112yes—
180C8—24no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—42no—
189C22—82no—
190C22—82no—
191C22C2×C22⋊C4×C21111yescharacteristic
192C22#121×C21112yes—
193C22AGL(1,ℤ/8)×C21111yescharacteristic
194C22AGL(1,ℤ/8)×C21112yes—
195C22#137×C21111yescharacteristic
196C4—22no—
197C4—42no—
198C4—82no—
199C4C4×D4×C21112yes—
200C4#120×C21111yescharacteristic
201C4#123×C21111yescharacteristic
202C2—42no—
203C2—82no—
204C2#386×C21111yescharacteristic
205C2#417×C21111yescharacteristic
206C2#418×C21111yescharacteristic
207C2#468×C21112yes—
208C2#469×C21112yes—
subgroupG/H
center, socleC23×C211C2×D4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC2×#121×C211C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×M16×C211C2semidirect
3. C2×#102×C211C2semidirect
4. #380×C211C2semidirect
5. #340×C211C2semidirect
13. #121×C211C4semidirect
39. C2×C8×C211D4semidirect
104. C211#1895direct (N × H)
105. #1895C211direct (N × H)
106. C22×M16C2×C211semidirect
108. C2×#102C2×C211semidirect
109. #380C2×C211semidirect
110. #340C2×C211semidirect
118. #121C4×C211semidirect
144. C2×C8D4×C211semidirect

Maximal subgroups

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

subgroupindex
C22×M16×C2112
C2×#121×C2112
C2×#102×C2112
#380×C2112
#340×C2112
#1895211