Finite Group Databasebeta

← #1789×C211 #41009409

Subgroups — #1789×C211 #41009409

Order 27008.

618 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 140 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×#41×C211C211yesmaximal, characteristic
2C2×#104×C211C214yesmaximal, complemented
3#407×C211C211yesmaximal, characteristic, complemented
4C4×C22⋊C4×C211C211yesmaximal, characteristic, complemented
5C2×#41×C211—28no—
6C2×#41×C211—21no—
7C2×#41×C211—21no—
8C2×#41×C211C412yescomplemented
9C2×#41×C211C2212yes—
10C22×Q8×C211C411yescharacteristic, complemented
11C22×D4×C211—21no—
12C22×D4×C211C411yescharacteristic, complemented
13C23×C4×C211—21no—
14C23×C4×C211C2211yescharacteristic
15C2×M16×C211C2212yescomplemented
16C2×C22⋊C4×C211—21no—
17C2×C42×C211C2212yescomplemented
18#104×C211—28no—
19C22⋊C8×C211—21no—
20C22⋊C8×C211—21no—
21#95×C211—21no—
22C24×C211—21no—
23C24×C211D411yescharacteristic
24#41×C211—28no—
25#41×C211—28no—
26#41×C211—24no—
27#41×C211—24no—
28#41×C211—24no—
29#41×C211D418yes—
30C2×Q8×C211—24no—
31C2×Q8×C211—21no—
32C2×Q8×C211C2×C412yes—
33C2×D4×C211—28no—
34C2×D4×C211—22no—
35C2×D4×C211—22no—
36C2×D4×C211—24no—
37C2×D4×C211—21no—
38C2×D4×C211C2×C412yes—
39C22×C4×C211—28no—
40C22×C4×C211—21no—
41C22×C4×C211—22no—
42C22×C4×C211—24no—
43C22×C4×C211—22no—
44C22×C4×C211—22no—
45C22×C4×C211C2×C411yescharacteristic
46C22×C4×C211C2×C411yescharacteristic
47C22×C4×C211D411yescharacteristic
48C22×C4×C211D411yescharacteristic
49C22×C4×C211D411yescharacteristic
50C22×C4×C211C2311yescharacteristic
51M16×C211—24no—
52C2×C8×C211—22no—
53C22⋊C4×C211—24no—
54C42×C211—24no—
55C23×C211—24no—
56C23×C211—22no—
57C23×C211—22no—
58C23×C211—22no—
59C23×C211—41no—
60C23×C211—42no—
61C23×C211—42no—
62C23×C211C22⋊C411yescharacteristic
63C23×C211C22⋊C411yescharacteristic
64C23×C211C2×D411yescharacteristic
65Q8×C211—24no—
66Q8×C211—22no—
67Q8×C211C22⋊C414yescomplemented
68D4×C211—28no—
69D4×C211—24no—
70D4×C211—28no—
71D4×C211—22no—
72D4×C211C22⋊C414yescomplemented
73C2×C4×C211—24no—
74C2×C4×C211—28no—
75C2×C4×C211—22no—
76C2×C4×C211—22no—
77C2×C4×C211—22no—
78C2×C4×C211—22no—
79C2×C4×C211—22no—
80C2×C4×C211—28no—
81C2×C4×C211—22no—
82C2×C4×C211—24no—
83C2×C4×C211—42no—
84C2×C4×C211—42no—
85C2×C4×C211C22⋊C411yescharacteristic
86C2×C4×C211C22⋊C411yescharacteristic
87C2×C4×C211C22×C411yescharacteristic
88C2×C4×C211C2×D414yes—
89C2×C4×C211C2×D411yescharacteristic
90C8×C211—42no—
91C22×C211—21no—
92C22×C211—21no—
93C22×C211—21no—
94C22×C211—21no—
95C22×C211—21no—
96C22×C211—24no—
97C22×C211—21no—
98C22×C211—21no—
99C22×C211—44no—
100C22×C211—48no—
101C22×C211—42no—
102C22×C211—42no—
103C22×C211#10412yescomplemented
104C22×C211C2×C22⋊C411yescharacteristic
105C22×C211#12012yes—
106C4×C211—21no—
107C4×C211—24no—
108C4×C211—21no—
109C4×C211—44no—
110C4×C211C2×C22⋊C412yes—
111C4×C211#12012yes—
112C2×C211—22no—
113C2×C211—22no—
114C2×C211—41no—
115C2×C211—44no—
116C2×C211#37911yescharacteristic
117C2×C211C2×#10411yescharacteristic
118C2×C211#42111yescharacteristic
119C211#178911yescharacteristic, direct product factor, Sylow
120#1789C21111yesmaximal, characteristic, direct product factor, Sylow
121C22×#41C2×C21111yescharacteristic
122C2×#104C2×C21114yescomplemented
123#407C2×C21111yescharacteristic, complemented
124C4×C22⋊C4C2×C21111yescharacteristic, complemented
125C2×#41—28no—
126C2×#41—21no—
127C2×#41—21no—
128C2×#41C4×C21112yescomplemented
129C2×#41C22×C21112yes—
130C22×Q8C4×C21111yescharacteristic, complemented
131C22×D4—21no—
132C22×D4C4×C21111yescharacteristic, complemented
133C23×C4—21no—
134C23×C4C22×C21111yescharacteristic
135C2×M16C22×C21112yescomplemented
136C2×C22⋊C4—21no—
137C2×C42C22×C21112yescomplemented
138#104—28no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141#95—21no—
142C24—21no—
143C24D4×C21111yescharacteristic
144#41—28no—
145#41—28no—
146#41—24no—
147#41—24no—
148#41—24no—
149#41D4×C21118yes—
150C2×Q8—24no—
151C2×Q8—21no—
152C2×Q8C2×C4×C21112yes—
153C2×D4—28no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—24no—
157C2×D4—21no—
158C2×D4C2×C4×C21112yes—
159C22×C4—28no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4C2×C4×C21111yescharacteristic
166C22×C4C2×C4×C21111yescharacteristic
167C22×C4D4×C21111yescharacteristic
168C22×C4D4×C21111yescharacteristic
169C22×C4D4×C21111yescharacteristic
170C22×C4C23×C21111yescharacteristic
171M16—24no—
172C2×C8—22no—
173C22⋊C4—24no—
174C42—24no—
175C23—24no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—41no—
180C23—42no—
181C23—42no—
182C23C22⋊C4×C21111yescharacteristic
183C23C22⋊C4×C21111yescharacteristic
184C23C2×D4×C21111yescharacteristic
185Q8—24no—
186Q8—22no—
187Q8C22⋊C4×C21114yescomplemented
188D4—28no—
189D4—24no—
190D4—28no—
191D4—22no—
192D4C22⋊C4×C21114yescomplemented
193C2×C4—24no—
194C2×C4—28no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—28no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4C22⋊C4×C21111yescharacteristic
206C2×C4C22⋊C4×C21111yescharacteristic
207C2×C4C22×C4×C21111yescharacteristic
208C2×C4C2×D4×C21114yes—
209C2×C4C2×D4×C21111yescharacteristic
210C8—42no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—24no—
217C22—21no—
218C22—21no—
219C22—44no—
220C22—48no—
221C22—42no—
222C22—42no—
223C22#104×C21112yescomplemented
224C22C2×C22⋊C4×C21111yescharacteristic
225C22#120×C21112yes—
226C4—21no—
227C4—24no—
228C4—21no—
229C4—44no—
230C4C2×C22⋊C4×C21112yes—
231C4#120×C21112yes—
232C2—22no—
233C2—22no—
234C2—41no—
235C2—44no—
236C2#379×C21111yescharacteristic
237C2C2×#104×C21111yescharacteristic
238C2#421×C21111yescharacteristic
subgroupG/H
centerC2×C4×C211C2×D4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
socleC22×C211C2×C22⋊C4
prime radicalC22×#41×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. C2×#104×C211C2semidirect
3. #407×C211C2semidirect
4. C4×C22⋊C4×C211C2semidirect
8. C2×#41×C211C4semidirect
10. C22×Q8×C211C4semidirect
12. C22×D4×C211C4semidirect
15. C2×M16×C211C22semidirect
17. C2×C42×C211C22semidirect
67. Q8×C211C22⋊C4semidirect
72. D4×C211C22⋊C4semidirect
103. C22×C211#104semidirect
119. C211#1789direct (N × H)
120. #1789C211direct (N × H)
122. C2×#104C2×C211semidirect
123. #407C2×C211semidirect
124. C4×C22⋊C4C2×C211semidirect
128. C2×#41C4×C211semidirect
130. C22×Q8C4×C211semidirect
132. C22×D4C4×C211semidirect
135. C2×M16C22×C211semidirect
137. C2×C42C22×C211semidirect
187. Q8C22⋊C4×C211semidirect
192. D4C22⋊C4×C211semidirect
223. C22#104×C211semidirect

Maximal subgroups

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

subgroupindex
C22×#41×C2112
C2×#104×C2112
#407×C2112
C4×C22⋊C4×C2112
#1789211