Finite Group Databasebeta

← C2×#469×C211 #41010672

Subgroups — C2×#469×C211 #41010672

Order 27008.

596 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 216 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×D8×C211C211yesmaximal, characteristic, complemented
2C22×M16×C211C211yesmaximal, characteristic, complemented
3C2×#121×C211C212yesmaximal, complemented
4#469×C211C218yesmaximal, direct product factor
5C2×#106×C211C211yesmaximal, characteristic, complemented
6C2×#102×C211C212yesmaximal, complemented
7C22×D4×C211—22no—
8C22×D4×C211C2212yes—
9C23×C4×C211C2211yescharacteristic
10C2×D8×C211—24no—
11C2×D8×C211C2214yescomplemented
12C2×M16×C211—24no—
13C2×M16×C211C2214yescomplemented
14C22×C8×C211C2211yescharacteristic
15C22×C8×C211C2211yescharacteristic
16#121×C211—24no—
17#121×C211C2218yescomplemented
18C2×C4⋊C4×C211C2212yes—
19C2×C22⋊C4×C211—22no—
20#106×C211C2214yescomplemented
21#102×C211C2218yescomplemented
22C24×C211—22no—
23C24×C211D411yescharacteristic
24C2×D4×C211—28no—
25C2×D4×C211—24no—
26C2×D4×C211—22no—
27C2×D4×C211—24no—
28C2×D4×C211—48no—
29C2×D4×C211C2314yes—
30C22×C4×C211—22no—
31C22×C4×C211—22no—
32C22×C4×C211—22no—
33C22×C4×C211D412yes—
34C22×C4×C211D411yescharacteristic
35C22×C4×C211C2312yes—
36C22×C4×C211C2311yescharacteristic
37D8×C211—28no—
38M16×C211—28no—
39C2×C8×C211—22no—
40C2×C8×C211D414yescomplemented
41C2×C8×C211C2312yes—
42C2×C8×C211C2312yes—
43C4⋊C4×C211—22no—
44C4⋊C4×C211C2314yes—
45C22⋊C4×C211—28no—
46C23×C211—22no—
47C23×C211—24no—
48C23×C211—24no—
49C23×C211—44no—
50C23×C211—44no—
51C23×C211—44no—
52C23×C211C2×D412yes—
53C23×C211C2×D411yescharacteristic
54D4×C211—24no—
55D4×C211—24no—
56D4×C211—416no—
57D4×C211—44no—
58C2×C4×C211—22no—
59C2×C4×C211—24no—
60C2×C4×C211—22no—
61C2×C4×C211—22no—
62C2×C4×C211—24no—
63C2×C4×C211—44no—
64C2×C4×C211C2×D412yes—
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D412yes—
67C2×C4×C211#4112yes—
68C2×C4×C211C2411yescharacteristic
69C8×C211—22no—
70C8×C211C2×D414yescomplemented
71C22×C211—22no—
72C22×C211—24no—
73C22×C211—44no—
74C22×C211—44no—
75C22×C211—44no—
76C22×C211—44no—
77C22×C211—88no—
78C22×C211#12112yes—
79C22×C211AGL(1,ℤ/8)14yes—
80C22×C211C22×D411yescharacteristic
81C4×C211—22no—
82C4×C211—44no—
83C4×C211#12112yes—
84C4×C211C22×D411yescharacteristic
85C4×C211C2×#4111yescharacteristic
86C2×C211—42no—
87C2×C211—84no—
88C2×C211#46914yesdirect product factor
89C2×C211C2×#12111yescharacteristic
90C2×C211C2×AGL(1,ℤ/8)12yes—
91C211C2×#46911yescharacteristic, direct product factor, Sylow
92C2×#469C21111yesmaximal, characteristic, direct product factor, Sylow
93C22×D8C2×C21111yescharacteristic, complemented
94C22×M16C2×C21111yescharacteristic, complemented
95C2×#121C2×C21112yescomplemented
96#469C2×C21118yesdirect product factor
97C2×#106C2×C21111yescharacteristic, complemented
98C2×#102C2×C21112yescomplemented
99C22×D4—22no—
100C22×D4C22×C21112yes—
101C23×C4C22×C21111yescharacteristic
102C2×D8—24no—
103C2×D8C22×C21114yescomplemented
104C2×M16—24no—
105C2×M16C22×C21114yescomplemented
106C22×C8C22×C21111yescharacteristic
107C22×C8C22×C21111yescharacteristic
108#121—24no—
109#121C22×C21118yescomplemented
110C2×C4⋊C4C22×C21112yes—
111C2×C22⋊C4—22no—
112#106C22×C21114yescomplemented
113#102C22×C21118yescomplemented
114C24—22no—
115C24D4×C21111yescharacteristic
116C2×D4—28no—
117C2×D4—24no—
118C2×D4—22no—
119C2×D4—24no—
120C2×D4—48no—
121C2×D4C23×C21114yes—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4D4×C21111yescharacteristic
126C22×C4D4×C21112yes—
127C22×C4C23×C21112yes—
128C22×C4C23×C21111yescharacteristic
129D8—28no—
130M16—28no—
131C2×C8—22no—
132C2×C8D4×C21114yescomplemented
133C2×C8C23×C21112yes—
134C2×C8C23×C21112yes—
135C4⋊C4—22no—
136C4⋊C4C23×C21114yes—
137C22⋊C4—28no—
138C23—22no—
139C23—24no—
140C23—24no—
141C23—44no—
142C23—44no—
143C23—44no—
144C23C2×D4×C21112yes—
145C23C2×D4×C21111yescharacteristic
146D4—24no—
147D4—24no—
148D4—416no—
149D4—44no—
150C2×C4—22no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—24no—
155C2×C4—44no—
156C2×C4C2×D4×C21112yes—
157C2×C4C2×D4×C21111yescharacteristic
158C2×C4C2×D4×C21112yes—
159C2×C4#41×C21112yes—
160C2×C4C24×C21111yescharacteristic
161C8—22no—
162C8C2×D4×C21114yescomplemented
163C22—22no—
164C22—24no—
165C22—44no—
166C22—44no—
167C22—44no—
168C22—44no—
169C22—88no—
170C22#121×C21112yes—
171C22AGL(1,ℤ/8)×C21114yes—
172C22C22×D4×C21111yescharacteristic
173C4—22no—
174C4—44no—
175C4#121×C21112yes—
176C4C22×D4×C21111yescharacteristic
177C4C2×#41×C21111yescharacteristic
178C2—42no—
179C2—84no—
180C2#469×C21114yesdirect product factor
181C2C2×#121×C21111yescharacteristic
182C2C2×AGL(1,ℤ/8)×C21112yes—
subgroupG/H
center, socleC23×C211C2×D4
commutator, Frattini, cosocleC2×C4C24×C211
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×D8×C211C2semidirect
2. C22×M16×C211C2semidirect
3. C2×#121×C211C2semidirect
4. #469×C211C2direct (N × H)
5. C2×#106×C211C2semidirect
6. C2×#102×C211C2semidirect
11. C2×D8×C211C22semidirect
13. C2×M16×C211C22semidirect
17. #121×C211C22semidirect
20. #106×C211C22semidirect
21. #102×C211C22semidirect
40. C2×C8×C211D4semidirect
70. C8×C211C2×D4semidirect
88. C2×C211#469direct (N × H)
91. C211C2×#469direct (N × H)
92. C2×#469C211direct (N × H)
93. C22×D8C2×C211semidirect
94. C22×M16C2×C211semidirect
95. C2×#121C2×C211semidirect
96. #469C2×C211direct (N × H)
97. C2×#106C2×C211semidirect
98. C2×#102C2×C211semidirect
103. C2×D8C22×C211semidirect
105. C2×M16C22×C211semidirect
109. #121C22×C211semidirect
112. #106C22×C211semidirect
113. #102C22×C211semidirect
132. C2×C8D4×C211semidirect
162. C8C2×D4×C211semidirect
180. C2#469×C211direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D8×C2112
C22×M16×C2112
C2×#121×C2112
#469×C2112
C2×#106×C2112
C2×#102×C2112
C2×#469211