Finite Group Databasebeta

← #41013859

Subgroups — #41013859

Order 27008.

390 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 93 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×#16809946C211yesmaximal, characteristic, complemented
2C2×#20269951C211yesmaximal, characteristic
3C2×#20269893C212yesmaximal, complemented
4#27212914C212yesmaximal, complemented
5#407×C211C211yesmaximal, characteristic, complemented
6C22×D844C2211yescharacteristic
7C22×C22⋊D211C412yes—
8C2×#16809946—24no—
9C2×#16809946—21no—
10C2×#16809946C2212yes—
11C22×C211⋊C8—21no—
12#20269951—21no—
13#20269951—21no—
14#20269951—21no—
15#20269951—21no—
16C2×C4⋊Dic211C2211yescharacteristic
17C2×(C2×D211)⋊C4C412yes—
18#20269893—24no—
19C23×C4×C211C2211yescharacteristic
20C2×M16×C211C2212yes—
21C22⋊C8×C211—21no—
22C22⋊C8×C211—21no—
23C23×D211C2×C412yes—
24C2×D844—22no—
25C2×D844—21no—
26C2×D844—21no—
27C2×D844—22no—
28C2×C22⋊D211—22no—
29C2×C22⋊D211—22no—
30C2×C22⋊D211—44no—
31C22×Dic211C2×C412yes—
32#16809946—24no—
33#16809946—22no—
34#16809946D414yescomplemented
35C2×C211⋊C8—24no—
36C2×C211⋊C8—22no—
37C4⋊Dic211—21no—
38C4⋊Dic211—21no—
39(C2×D211)⋊C4—22no—
40(C2×D211)⋊C4—22no—
41C24×C211C2×C411yescharacteristic
42C22×C4×C211—22no—
43C22×C4×C211—22no—
44C22×C4×C211C2×C411yescharacteristic
45C22×C4×C211D411yescharacteristic
46C22×C4×C211D411yescharacteristic
47C22×C4×C211D411yescharacteristic
48C22×C4×C211D411yescharacteristic
49C22×C4×C211C2311yescharacteristic
50M16×C211—22no—
51M16×C211—22no—
52C2×C8×C211—22no—
53C22×D211—22no—
54C22×D211—24no—
55C22×D211—42no—
56C22×D211—42no—
57D844—41no—
58D844—42no—
59D844—41no—
60C22⋊D211—44no—
61C22⋊D211—44no—
62C2×Dic211—22no—
63C2×Dic211—42no—
64C211⋊C8—24no—
65C23×C211—22no—
66C23×C211—22no—
67C23×C211—41no—
68C23×C211C22⋊C411yescharacteristic
69C23×C211C22⋊C411yescharacteristic
70C23×C211C22×C411yescharacteristic
71C2×C4×C211—22no—
72C2×C4×C211—21no—
73C2×C4×C211—21no—
74C2×C4×C211—22no—
75C2×C4×C211—22no—
76C2×C4×C211—21no—
77C2×C4×C211—21no—
78C2×C4×C211C22⋊C411yescharacteristic
79C2×C4×C211C22⋊C411yescharacteristic
80C2×C4×C211C2×D412yes—
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211#4112yes—
84C8×C211—42no—
85C22×C211—21no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211—21no—
89C22×C211—21no—
90C22×C211—21no—
91C22×C211—21no—
92C22×C211—44no—
93C22×C211#10012yescomplemented
94C22×C211C2×C22⋊C411yescharacteristic
95C22×C211C4×D412yes—
96C2×D211—42no—
97C2×D211—42no—
98C2×D211—42no—
99C2×D211—84no—
100C4×C211—21no—
101C4×C211—21no—
102C4×C211#12011yescharacteristic
103C4×C211#12112yes—
104C4×C211#12311yescharacteristic
105Dic211—42no—
106C2×C211—22no—
107C2×C211—22no—
108C2×C211—41no—
109C2×C211#38611yescharacteristic
110C2×C211C2×#10011yescharacteristic
111C2×C211#41311yescharacteristic
112D211—82no—
113C211#190011yescharacteristic, complemented, Sylow
114#1900—2111nomaximal, Sylow
115C22×M16—2111no—
116C2×#121—2111no—
117C2×#100—2112no—
118#407D21111yescharacteristic, complemented
119#343—2112no—
120C22×D4—2112no—
121C22×D4—2111no—
122C23×C4C2×D21111yescharacteristic
123C2×M16—4221no—
124C2×M16—4224no—
125C2×M16—2112no—
126C2×M16C2×D21112yes—
127C22×C8—4221no—
128#121—4221no—
129#121—4221no—
130#121—4221no—
131#121—4221no—
132C2×C4⋊C4—2111no—
133C2×C22⋊C4—2112no—
134#100—4224no—
135C22⋊C8—21no—
136C22⋊C8—21no—
137C24—2112no—
138C24C4×D21111yescharacteristic
139C2×D4—8444no—
140C2×D4—4222no—
141C2×D4—4222no—
142C2×D4—4222no—
143C2×D4—4221no—
144C2×D4—4221no—
145C2×D4—4222no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—2112no—
149C22×C4C4×D21111yescharacteristic
150C22×C4C22⋊D21111yescharacteristic
151C22×C4C22⋊D21111yescharacteristic
152C22×C4D84411yescharacteristic
153C22×C4D84411yescharacteristic
154C22×C4C22×D21111yescharacteristic
155M16—22no—
156M16—22no—
157M16—4224no—
158M16—4222no—
159M16—2114no—
160C2×C8—22no—
161C2×C8—4224no—
162C2×C8—4222no—
163C4⋊C4—4221no—
164C4⋊C4—4221no—
165C22⋊C4—4222no—
166C22⋊C4—4222no—
167C23—22no—
168C23—22no—
169C23—41no—
170C23—8442no—
171C23—8442no—
172C23—4222no—
173C23—4224no—
174C23(C2×D211)⋊C411yescharacteristic
175C23(C2×D211)⋊C411yescharacteristic
176C23C2×C4×D21111yescharacteristic
177D4—8444no—
178D4—8444no—
179D4—8441no—
180D4—8442no—
181D4—8441no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—8442no—
190C2×C4—4222no—
191C2×C4(C2×D211)⋊C411yescharacteristic
192C2×C4(C2×D211)⋊C411yescharacteristic
193C2×C4D4×D21112yes—
194C2×C4C2×C22⋊D21111yescharacteristic
195C2×C4C2×D84411yescharacteristic
196C2×C4#1680996112yes—
197C8—42no—
198C8—4224no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—44no—
207C22—8442no—
208C22—8442no—
209C22—8442no—
210C22—16884no—
211C22#2026989312yescomplemented
212C22C2×(C2×D211)⋊C411yescharacteristic
213C22(C22⋊D211)⋊C412yes—
214C4—21no—
215C4—21no—
216C4—8442no—
217C4#2026994711yescharacteristic
218C4#2026994912yes—
219C4#2026996111yescharacteristic
220C2—22no—
221C2—22no—
222C2—41no—
223C2—16882no—
224C2#2721302311yescharacteristic
225C2C2×#2026989311yescharacteristic
226C2#2721312811yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#407×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC2×#20269951C2
cosocleC22×C4×C211C23
hypercenter#407D211
hypocenterC211#1900
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#16809946C2semidirect
3. C2×#20269893C2semidirect
4. #27212914C2semidirect
5. #407×C211C2semidirect
34. #16809946D4semidirect
93. C22×C211#100semidirect
113. C211#1900semidirect
118. #407D211semidirect
211. C22#20269893semidirect

Maximal subgroups

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

subgroupindex
C22×#168099462
C2×#202699512
C2×#202698932
#272129142
#407×C2112
#1900211