Finite Group Databasebeta

← C4×#99×C211 #41009374

Subgroups — C4×#99×C211 #41009374

Order 27008.

412 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 156 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
1C2×C4×D4×C211C211yesmaximal, characteristic
2C2×#99×C211C212yesmaximal
3C4×C22⋊C4×C211C212yesmaximal, complemented
4#342×C211C212yesmaximal, complemented
5C22×D4×C211C2211yescharacteristic
6C23×C4×C211C411yescharacteristic, complemented
7C23×C4×C211C2211yescharacteristic
8C4×D4×C211—22no—
9C4×D4×C211—22no—
10C2×C4⋊C4×C211C411yescharacteristic, complemented
11C2×C22⋊C4×C211C411yescharacteristic, complemented
12C2×C22⋊C4×C211C2211yescharacteristic
13C2×C22⋊C4×C211C2214yes—
14C2×C42×C211—22no—
15C2×C42×C211C411yescharacteristic, complemented
16#99×C211C418yesdirect product factor
17#95×C211—22no—
18C24×C211C2×C411yescharacteristic
19C24×C211C2311yescharacteristic
20C2×D4×C211—22no—
21C2×D4×C211—22no—
22C2×D4×C211C2×C414yes—
23C22×C4×C211—24no—
24C22×C4×C211—21no—
25C22×C4×C211—21no—
26C22×C4×C211—42no—
27C22×C4×C211—42no—
28C22×C4×C211C2×C411yescharacteristic
29C22×C4×C211C2×C411yescharacteristic
30C22×C4×C211C2×C411yescharacteristic
31C22×C4×C211C2×C411yescharacteristic
32C22×C4×C211C2×C411yescharacteristic
33C22×C4×C211D412yes—
34C22×C4×C211D412yes—
35C4⋊C4×C211—22no—
36C22⋊C4×C211—22no—
37C22⋊C4×C211—24no—
38C22⋊C4×C211—22no—
39C22⋊C4×C211C2×C418yescomplemented
40C42×C211—21no—
41C42×C211—21no—
42C42×C211—44no—
43C23×C211—21no—
44C23×C211—21no—
45C23×C211—41no—
46C23×C211—41no—
47C23×C211—41no—
48C23×C211—41no—
49C23×C211C4212yescomplemented
50C23×C211C22×C411yescharacteristic
51C23×C211C22×C412yes—
52C23×C211C2×D412yes—
53C23×C211#4112yes—
54D4×C211—28no—
55C2×C4×C211—21no—
56C2×C4×C211—21no—
57C2×C4×C211—24no—
58C2×C4×C211—22no—
59C2×C4×C211—21no—
60C2×C4×C211—21no—
61C2×C4×C211—28no—
62C2×C4×C211—41no—
63C2×C4×C211—42no—
64C2×C4×C211—42no—
65C2×C4×C211—44no—
66C2×C4×C211—44no—
67C2×C4×C211—42no—
68C2×C4×C211—41no—
69C2×C4×C211—41no—
70C2×C4×C211C4212yescomplemented
71C2×C4×C211C22⋊C411yescharacteristic
72C2×C4×C211C22⋊C412yes—
73C2×C4×C211C22⋊C411yescharacteristic
74C22×C211—22no—
75C22×C211—21no—
76C22×C211—22no—
77C22×C211—21no—
78C22×C211—22no—
79C22×C211—42no—
80C22×C211—42no—
81C22×C211—42no—
82C22×C211—42no—
83C22×C211—41no—
84C22×C211—41no—
85C22×C211C2×C4211yescharacteristic
86C22×C211C2×C22⋊C411yescharacteristic
87C22×C211#11711yescharacteristic
88C22×C211C4×D414yes—
89C4×C211—22no—
90C4×C211—21no—
91C4×C211—22no—
92C4×C211—48no—
93C4×C211—41no—
94C4×C211—41no—
95C4×C211#9912yesdirect product factor
96C2×C211—24no—
97C2×C211—21no—
98C2×C211—21no—
99C2×C211—42no—
100C2×C211C4×C22⋊C411yescharacteristic
101C2×C211C2×#9911yescharacteristic
102C2×C211#41011yescharacteristic
103C211C4×#9911yescharacteristic, direct product factor, Sylow
104C4×#99C21111yesmaximal, characteristic, direct product factor, Sylow
105C2×C4×D4C2×C21111yescharacteristic
106C2×#99C2×C21112yes—
107C4×C22⋊C4C2×C21112yescomplemented
108#342C2×C21112yescomplemented
109C22×D4C22×C21111yescharacteristic
110C23×C4C4×C21111yescharacteristic, complemented
111C23×C4C22×C21111yescharacteristic
112C4×D4—22no—
113C4×D4—22no—
114C2×C4⋊C4C4×C21111yescharacteristic, complemented
115C2×C22⋊C4C4×C21111yescharacteristic, complemented
116C2×C22⋊C4C22×C21111yescharacteristic
117C2×C22⋊C4C22×C21114yes—
118C2×C42—22no—
119C2×C42C4×C21111yescharacteristic, complemented
120#99C4×C21118yesdirect product factor
121#95—22no—
122C24C2×C4×C21111yescharacteristic
123C24C23×C21111yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4C2×C4×C21114yes—
127C22×C4—24no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—42no—
131C22×C4—42no—
132C22×C4C2×C4×C21111yescharacteristic
133C22×C4C2×C4×C21111yescharacteristic
134C22×C4C2×C4×C21111yescharacteristic
135C22×C4C2×C4×C21111yescharacteristic
136C22×C4C2×C4×C21111yescharacteristic
137C22×C4D4×C21112yes—
138C22×C4D4×C21112yes—
139C4⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—24no—
142C22⋊C4—22no—
143C22⋊C4C2×C4×C21118yescomplemented
144C42—21no—
145C42—21no—
146C42—44no—
147C23—21no—
148C23—21no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23C42×C21112yescomplemented
154C23C22×C4×C21111yescharacteristic
155C23C22×C4×C21112yes—
156C23C2×D4×C21112yes—
157C23#41×C21112yes—
158D4—28no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—28no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4C42×C21112yescomplemented
175C2×C4C22⋊C4×C21111yescharacteristic
176C2×C4C22⋊C4×C21112yes—
177C2×C4C22⋊C4×C21111yescharacteristic
178C22—22no—
179C22—21no—
180C22—22no—
181C22—21no—
182C22—22no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—41no—
188C22—41no—
189C22C2×C42×C21111yescharacteristic
190C22C2×C22⋊C4×C21111yescharacteristic
191C22#117×C21111yescharacteristic
192C22C4×D4×C21114yes—
193C4—22no—
194C4—21no—
195C4—22no—
196C4—48no—
197C4—41no—
198C4—41no—
199C4#99×C21112yesdirect product factor
200C2—24no—
201C2—21no—
202C2—21no—
203C2—42no—
204C2C4×C22⋊C4×C21111yescharacteristic
205C2C2×#99×C21111yescharacteristic
206C2#410×C21111yescharacteristic
subgroupG/H
centerC2×C4×C211C22⋊C4
commutatorC22C2×C42×C211
Frattini, cosocleC24C23×C211
socleC22×C211C2×C22⋊C4
prime radicalC22×D4×C211C22
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
3. C4×C22⋊C4×C211C2semidirect
4. #342×C211C2semidirect
6. C23×C4×C211C4semidirect
10. C2×C4⋊C4×C211C4semidirect
11. C2×C22⋊C4×C211C4semidirect
15. C2×C42×C211C4semidirect
16. #99×C211C4direct (N × H)
39. C22⋊C4×C211C2×C4semidirect
49. C23×C211C42semidirect
70. C2×C4×C211C42semidirect
95. C4×C211#99direct (N × H)
103. C211C4×#99direct (N × H)
104. C4×#99C211direct (N × H)
107. C4×C22⋊C4C2×C211semidirect
108. #342C2×C211semidirect
110. C23×C4C4×C211semidirect
114. C2×C4⋊C4C4×C211semidirect
115. C2×C22⋊C4C4×C211semidirect
119. C2×C42C4×C211semidirect
120. #99C4×C211direct (N × H)
143. C22⋊C4C2×C4×C211semidirect
153. C23C42×C211semidirect
174. C2×C4C42×C211semidirect
199. C4#99×C211direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×D4×C2112
C2×#99×C2112
C4×C22⋊C4×C2112
#342×C2112
C4×#99211