Finite Group Databasebeta

← C2×#327×C211 #41009118

Subgroups — C2×#327×C211 #41009118

Order 27008.

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

normal subgroups only

This group has 388 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
1C2×#121×C211C211yesmaximal, characteristic
2C2×C22⋊C8×C211C211yesmaximal, characteristic, complemented
3C2×#95×C211C211yesmaximal, characteristic, complemented
4#327×C211C214yesmaximal, direct product factor
5C22×D4×C211—21no—
6C22×D4×C211C411yescharacteristic, complemented
7C23×C4×C211—21no—
8C23×C4×C211C2211yescharacteristic
9C22×C8×C211—21no—
10#121×C211—21no—
11#121×C211—21no—
12#121×C211C412yescomplemented
13#121×C211C2212yes—
14C2×C4⋊C4×C211C411yescharacteristic, complemented
15C2×C22⋊C4×C211—21no—
16C22⋊C8×C211—21no—
17C22⋊C8×C211C2212yescomplemented
18#95×C211—21no—
19#95×C211C2212yescomplemented
20C24×C211—21no—
21C24×C211D411yescharacteristic
22C2×D4×C211—22no—
23C2×D4×C211—22no—
24C2×D4×C211—21no—
25C2×D4×C211—42no—
26C2×D4×C211—42no—
27C2×D4×C211—42no—
28C2×D4×C211C2×C412yescomplemented
29C22×C4×C211—24no—
30C22×C4×C211—21no—
31C22×C4×C211—22no—
32C22×C4×C211—42no—
33C22×C4×C211—42no—
34C22×C4×C211—42no—
35C22×C4×C211C2×C411yescharacteristic
36C22×C4×C211C2×C411yescharacteristic
37C22×C4×C211D411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211D411yescharacteristic
40C22×C4×C211C2311yescharacteristic
41C2×C8×C211—22no—
42C2×C8×C211—42no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211C2×C412yescomplemented
45C22⋊C4×C211—22no—
46C22⋊C4×C211—22no—
47C23×C211—22no—
48C23×C211—22no—
49C23×C211—22no—
50C23×C211—42no—
51C23×C211—42no—
52C23×C211—42no—
53C23×C211C22⋊C411yescharacteristic
54C23×C211C22⋊C411yescharacteristic
55C23×C211D812yes—
56C23×C211SD1612yes—
57C23×C211C2×D411yescharacteristic
58D4×C211—42no—
59D4×C211—42no—
60D4×C211—44no—
61D4×C211—42no—
62D4×C211—42no—
63C2×C4×C211—22no—
64C2×C4×C211—22no—
65C2×C4×C211—24no—
66C2×C4×C211—22no—
67C2×C4×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211—41no—
70C2×C4×C211—41no—
71C2×C4×C211—48no—
72C2×C4×C211—42no—
73C2×C4×C211—42no—
74C2×C4×C211—42no—
75C2×C4×C211C22⋊C411yescharacteristic
76C2×C4×C211C22⋊C411yescharacteristic
77C2×C4×C211C22×C411yescharacteristic
78C2×C4×C211C2×D411yescharacteristic
79C8×C211—42no—
80C22×C211—24no—
81C22×C211—24no—
82C22×C211—21no—
83C22×C211—21no—
84C22×C211—21no—
85C22×C211—21no—
86C22×C211—42no—
87C22×C211—42no—
88C22×C211—42no—
89C22×C211—84no—
90C22×C211#9912yes—
91C22×C211#10212yes—
92C22×C211#10212yes—
93C22×C211#10412yes—
94C22×C211C2×C22⋊C411yescharacteristic
95C22×C211C2×D811yescharacteristic
96C22×C211C2×SD1611yescharacteristic
97C4×C211—21no—
98C4×C211—21no—
99C4×C211—21no—
100C4×C211—21no—
101C4×C211—44no—
102C4×C211—42no—
103C2×C211—21no—
104C2×C211—22no—
105C2×C211—21no—
106C2×C211—82no—
107C2×C211#32714yesdirect product factor
108C2×C211C2×#9911yescharacteristic
109C2×C211C2×#10211yescharacteristic
110C2×C211C2×#10411yescharacteristic
111C211C2×#32711yescharacteristic, direct product factor, Sylow
112C2×#327C21111yesmaximal, characteristic, direct product factor, Sylow
113C2×#121C2×C21111yescharacteristic
114C2×C22⋊C8C2×C21111yescharacteristic, complemented
115C2×#95C2×C21111yescharacteristic, complemented
116#327C2×C21114yesdirect product factor
117C22×D4—21no—
118C22×D4C4×C21111yescharacteristic, complemented
119C23×C4—21no—
120C23×C4C22×C21111yescharacteristic
121C22×C8—21no—
122#121—21no—
123#121—21no—
124#121C4×C21112yescomplemented
125#121C22×C21112yes—
126C2×C4⋊C4C4×C21111yescharacteristic, complemented
127C2×C22⋊C4—21no—
128C22⋊C8—21no—
129C22⋊C8C22×C21112yescomplemented
130#95—21no—
131#95C22×C21112yescomplemented
132C24—21no—
133C24D4×C21111yescharacteristic
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4C2×C4×C21112yescomplemented
141C22×C4—24no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4C2×C4×C21111yescharacteristic
148C22×C4C2×C4×C21111yescharacteristic
149C22×C4D4×C21111yescharacteristic
150C22×C4D4×C21111yescharacteristic
151C22×C4D4×C21111yescharacteristic
152C22×C4C23×C21111yescharacteristic
153C2×C8—22no—
154C2×C8—42no—
155C4⋊C4—21no—
156C4⋊C4C2×C4×C21112yescomplemented
157C22⋊C4—22no—
158C22⋊C4—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23C22⋊C4×C21111yescharacteristic
166C23C22⋊C4×C21111yescharacteristic
167C23D8×C21112yes—
168C23SD16×C21112yes—
169C23C2×D4×C21111yescharacteristic
170D4—42no—
171D4—42no—
172D4—44no—
173D4—42no—
174D4—42no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—48no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4C22⋊C4×C21111yescharacteristic
188C2×C4C22⋊C4×C21111yescharacteristic
189C2×C4C22×C4×C21111yescharacteristic
190C2×C4C2×D4×C21111yescharacteristic
191C8—42no—
192C22—24no—
193C22—24no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22#99×C21112yes—
203C22#102×C21112yes—
204C22#102×C21112yes—
205C22#104×C21112yes—
206C22C2×C22⋊C4×C21111yescharacteristic
207C22C2×D8×C21111yescharacteristic
208C22C2×SD16×C21111yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—44no—
214C4—42no—
215C2—21no—
216C2—22no—
217C2—21no—
218C2—82no—
219C2#327×C21114yesdirect product factor
220C2C2×#99×C21111yescharacteristic
221C2C2×#102×C21111yescharacteristic
222C2C2×#104×C21111yescharacteristic
subgroupG/H
center, socleC23×C211C22⋊C4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC2×#121×C211C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C8×C211C2semidirect
3. C2×#95×C211C2semidirect
4. #327×C211C2direct (N × H)
6. C22×D4×C211C4semidirect
12. #121×C211C4semidirect
14. C2×C4⋊C4×C211C4semidirect
17. C22⋊C8×C211C22semidirect
19. #95×C211C22semidirect
28. C2×D4×C211C2×C4semidirect
44. C4⋊C4×C211C2×C4semidirect
107. C2×C211#327direct (N × H)
111. C211C2×#327direct (N × H)
112. C2×#327C211direct (N × H)
114. C2×C22⋊C8C2×C211semidirect
115. C2×#95C2×C211semidirect
116. #327C2×C211direct (N × H)
118. C22×D4C4×C211semidirect
124. #121C4×C211semidirect
126. C2×C4⋊C4C4×C211semidirect
129. C22⋊C8C22×C211semidirect
131. #95C22×C211semidirect
140. C2×D4C2×C4×C211semidirect
156. C4⋊C4C2×C4×C211semidirect
219. C2#327×C211direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#121×C2112
C2×C22⋊C8×C2112
C2×#95×C2112
#327×C2112
C2×#327211