Finite Group Databasebeta

← #41011928

Subgroups — #41011928

Order 27008.

326 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 73 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×#20269951C211yesmaximal, characteristic
2#27213009C211yesmaximal, characteristic, complemented
3#27212881C214yesmaximal, complemented
4#407×C211C211yesmaximal, characteristic, complemented
5C22×D844C411yescharacteristic, complemented
6C22×C22⋊D211—21no—
7#20269951—21no—
8#20269951—21no—
9#20269951C412yescomplemented
10#20269951C2212yes—
11C2×C4⋊Dic211C411yescharacteristic, complemented
12C2×C22⋊Dic211—21no—
13C2×(C2×D211)⋊C4—21no—
14C211⋊#95—21no—
15C211⋊#95C2212yescomplemented
16C23×C4×C211C2211yescharacteristic
17C2×M16×C211—21no—
18C22⋊C8×C211—21no—
19C22⋊C8×C211C2212yescomplemented
20C23×D211—21no—
21C2×D844—21no—
22C2×D844—42no—
23C2×D844C2×C412yes—
24C2×C22⋊D211—22no—
25C2×C22⋊D211—22no—
26C2×C22⋊D211—44no—
27C22×Dic211—21no—
28C22×Dic211—22no—
29C4⋊Dic211—21no—
30C4⋊Dic211C2×C412yes—
31C22⋊Dic211—42no—
32(C2×D211)⋊C4—22no—
33(C2×D211)⋊C4—22no—
34C24×C211D411yescharacteristic
35C22×C4×C211—42no—
36C22×C4×C211C2×C411yescharacteristic
37C22×C4×C211C2×C411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211D411yescharacteristic
40C22×C4×C211D411yescharacteristic
41C22×C4×C211C2311yescharacteristic
42M16×C211—42no—
43C2×C8×C211—22no—
44C22×D211—22no—
45C22×D211—42no—
46C22×D211—42no—
47C22×D211—42no—
48D844—42no—
49D844—42no—
50C22⋊D211—44no—
51C22⋊D211—44no—
52C2×Dic211—22no—
53C2×Dic211—42no—
54C2×Dic211—42no—
55C2×Dic211—42no—
56C2×Dic211—42no—
57C23×C211—22no—
58C23×C211—22no—
59C23×C211—41no—
60C23×C211C22⋊C411yescharacteristic
61C23×C211C22⋊C411yescharacteristic
62C23×C211C2×D411yescharacteristic
63C2×C4×C211—22no—
64C2×C4×C211—22no—
65C2×C4×C211—41no—
66C2×C4×C211—41no—
67C2×C4×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211C22⋊C411yescharacteristic
70C2×C4×C211C22⋊C411yescharacteristic
71C2×C4×C211C22×C411yescharacteristic
72C2×C4×C211C2×D411yescharacteristic
73C8×C211—42no—
74C22×C211—21no—
75C22×C211—21no—
76C22×C211—21no—
77C22×C211—21no—
78C22×C211—21no—
79C22×C211—21no—
80C22×C211—21no—
81C22×C211—44no—
82C22×C211#9912yes—
83C22×C211C2×C22⋊C411yescharacteristic
84C22×C211AGL(1,ℤ/8)12yes—
85C2×D211—42no—
86C2×D211—42no—
87C2×D211—42no—
88C2×D211—84no—
89C4×C211—21no—
90C4×C211—21no—
91C4×C211—21no—
92C4×C211—21no—
93Dic211—42no—
94Dic211—82no—
95C2×C211—22no—
96C2×C211—22no—
97C2×C211—41no—
98C2×C211C2×#9911yescharacteristic
99C2×C211#41811yescharacteristic
100C2×C211#42111yescharacteristic
101D211—82no—
102C211#151011yescharacteristic, complemented, Sylow
103#1510—2111nomaximal, Sylow
104C2×#121—2111no—
105#407D21111yescharacteristic, complemented
106#381—2111no—
107#327—2114no—
108C22×D4—4221no—
109C22×D4—2111no—
110C23×C4C2×D21111yescharacteristic
111C2×M16—21no—
112#121—4221no—
113#121—4221no—
114#121—2112no—
115#121—2112no—
116C2×C4⋊C4—2111no—
117C2×C22⋊C4—4221no—
118C2×C22⋊C4—4221no—
119C22⋊C8—21no—
120C22⋊C8C2×D21112yescomplemented
121#95—4221no—
122#95—2112no—
123C24—4221no—
124C24D84411yescharacteristic
125C2×D4—8442no—
126C2×D4—8444no—
127C2×D4—4222no—
128C2×D4—4222no—
129C2×D4—4221no—
130C2×D4—2112no—
131C22×C4—42no—
132C22×C4—4222no—
133C22×C4—4221no—
134C22×C4C4×D21111yescharacteristic
135C22×C4C4×D21111yescharacteristic
136C22×C4C22⋊D21111yescharacteristic
137C22×C4C22⋊D21111yescharacteristic
138C22×C4D84411yescharacteristic
139C22×C4C22×D21111yescharacteristic
140M16—42no—
141C2×C8—22no—
142C4⋊C4—4221no—
143C4⋊C4—2112no—
144C22⋊C4—8442no—
145C22⋊C4—4222no—
146C22⋊C4—4222no—
147C23—22no—
148C23—22no—
149C23—41no—
150C23—8442no—
151C23—8442no—
152C23—8442no—
153C23—4222no—
154C23(C2×D211)⋊C411yescharacteristic
155C23(C2×D211)⋊C411yescharacteristic
156C23C2×D84411yescharacteristic
157D4—8442no—
158D4—8442no—
159D4—8444no—
160D4—8444no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—8442no—
168C2×C4—8442no—
169C2×C4—8442no—
170C2×C4—8442no—
171C2×C4—4222no—
172C2×C4(C2×D211)⋊C411yescharacteristic
173C2×C4(C2×D211)⋊C411yescharacteristic
174C2×C4C2×C4×D21111yescharacteristic
175C2×C4C2×C22⋊D21111yescharacteristic
176C8—42no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—44no—
185C22—8442no—
186C22—8442no—
187C22—8442no—
188C22—16884no—
189C22#2026989112yes—
190C22C2×(C2×D211)⋊C411yescharacteristic
191C22#2027000512yes—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—21no—
196C4—8442no—
197C4—16882no—
198C2—22no—
199C2—22no—
200C2—41no—
201C2—16882no—
202C2C2×#2026989111yescharacteristic
203C2#2721315211yescharacteristic
204C2#2721316211yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#407×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC2×#20269951C2
cosocleC22×C4×C211C23
hypercenter#407D211
hypocenterC211#1510
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #27213009C2semidirect
3. #27212881C2semidirect
4. #407×C211C2semidirect
5. C22×D844C4semidirect
9. #20269951C4semidirect
11. C2×C4⋊Dic211C4semidirect
15. C211⋊#95C22semidirect
19. C22⋊C8×C211C22semidirect
102. C211#1510semidirect
105. #407D211semidirect
120. C22⋊C8C2×D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699512
#272130092
#272128812
#407×C2112
#1510211