Finite Group Databasebeta

← #41011927

Subgroups — #41011927

Order 27008.

326 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 81 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
1#27213110C211yesmaximal, characteristic, complemented
2#27213009C211yesmaximal, characteristic, complemented
3#27212880C214yesmaximal, complemented
4C2×#121×C211C211yesmaximal, characteristic
5C2×#16809946—21no—
6C2×C22⋊Dic211—21no—
7C22⋊(C211⋊C8)—21no—
8C22⋊(C211⋊C8)C2212yescomplemented
9C211⋊#95—21no—
10C211⋊#95C2212yescomplemented
11C22×D4×C211—21no—
12C22×D4×C211C411yescharacteristic, complemented
13C23×C4×C211C2211yescharacteristic
14#121×C211—21no—
15#121×C211—21no—
16#121×C211C412yescomplemented
17#121×C211C2212yes—
18C2×C4⋊C4×C211C411yescharacteristic, complemented
19C2×C22⋊C4×C211—21no—
20C22×Dic211—22no—
21#16809946—42no—
22C2×C211⋊C8—22no—
23C22⋊Dic211—42no—
24C24×C211—21no—
25C24×C211D411yescharacteristic
26C2×D4×C211—22no—
27C2×D4×C211—22no—
28C2×D4×C211—21no—
29C2×D4×C211—42no—
30C2×D4×C211—44no—
31C2×D4×C211C2×C412yes—
32C22×C4×C211—21no—
33C22×C4×C211—42no—
34C22×C4×C211C2×C411yescharacteristic
35C22×C4×C211C2×C411yescharacteristic
36C22×C4×C211D411yescharacteristic
37C22×C4×C211D411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211C2311yescharacteristic
40C4⋊C4×C211—21no—
41C4⋊C4×C211C2×C412yes—
42C22⋊C4×C211—22no—
43C22⋊C4×C211—22no—
44C2×Dic211—42no—
45C2×Dic211—42no—
46C2×Dic211—42no—
47C211⋊C8—42no—
48C23×C211—22no—
49C23×C211—22no—
50C23×C211—22no—
51C23×C211—41no—
52C23×C211—42no—
53C23×C211—42no—
54C23×C211—42no—
55C23×C211C22⋊C411yescharacteristic
56C23×C211C22⋊C411yescharacteristic
57C23×C211C2×D411yescharacteristic
58D4×C211—42no—
59D4×C211—42no—
60D4×C211—44no—
61D4×C211—44no—
62C2×C4×C211—22no—
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×C211—42no—
70C2×C4×C211C22⋊C411yescharacteristic
71C2×C4×C211C22⋊C411yescharacteristic
72C2×C4×C211C22×C411yescharacteristic
73C2×C4×C211C2×D411yescharacteristic
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—42no—
83C22×C211—42no—
84C22×C211—42no—
85C22×C211—84no—
86C22×C211#9912yes—
87C22×C211C2×C22⋊C411yescharacteristic
88C22×C211AGL(1,ℤ/8)12yes—
89C4×C211—21no—
90C4×C211—21no—
91C4×C211—21no—
92C4×C211—21no—
93C4×C211—42no—
94Dic211—82no—
95C2×C211—22no—
96C2×C211—22no—
97C2×C211—41no—
98C2×C211—82no—
99C2×C211C2×#9911yescharacteristic
100C2×C211#41811yescharacteristic
101C2×C211#42111yescharacteristic
102C211#151011yescharacteristic, complemented, Sylow
103#1510—2111nomaximal, Sylow
104C2×#121D21111yescharacteristic
105#407—2111no—
106#381—2111no—
107#327—2114no—
108C22×D4—21no—
109C22×D4Dic21111yescharacteristic, complemented
110C23×C4C2×D21111yescharacteristic
111C2×M16—4221no—
112#121—21no—
113#121—21no—
114#121Dic21112yescomplemented
115#121C2×D21112yes—
116C2×C4⋊C4Dic21111yescharacteristic, complemented
117C2×C22⋊C4—21no—
118C2×C22⋊C4—4221no—
119C22⋊C8—4221no—
120C22⋊C8—2112no—
121#95—4221no—
122#95—2112no—
123C24—21no—
124C24C22⋊D21111yescharacteristic
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—21no—
128C2×D4—42no—
129C2×D4—44no—
130C2×D4C2×Dic21112yes—
131C22×C4—21no—
132C22×C4—42no—
133C22×C4—4222no—
134C22×C4C2×Dic21111yescharacteristic
135C22×C4C2×Dic21111yescharacteristic
136C22×C4C22⋊D21111yescharacteristic
137C22×C4C22⋊D21111yescharacteristic
138C22×C4C22⋊D21111yescharacteristic
139C22×C4C22×D21111yescharacteristic
140M16—8442no—
141C2×C8—4222no—
142C4⋊C4—21no—
143C4⋊C4C2×Dic21112yes—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—8442no—
147C23—22no—
148C23—22no—
149C23—22no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23C22⋊Dic21111yescharacteristic
155C23C22⋊Dic21111yescharacteristic
156C23C2×C22⋊D21111yescharacteristic
157D4—42no—
158D4—42no—
159D4—44no—
160D4—44no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—8442no—
170C2×C4—8442no—
171C2×C4—8442no—
172C2×C4C22⋊Dic21111yescharacteristic
173C2×C4C22⋊Dic21111yescharacteristic
174C2×C4C22×Dic21111yescharacteristic
175C2×C4C2×C22⋊D21111yescharacteristic
176C8—8442no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—44no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—84no—
189C22#2026989212yes—
190C22C2×C22⋊Dic21111yescharacteristic
191C22#2027000312yes—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—21no—
196C4—42no—
197C4—16882no—
198C2—22no—
199C2—22no—
200C2—41no—
201C2—82no—
202C2C2×#2026989211yescharacteristic
203C2#2721315111yescharacteristic
204C2#2721316311yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting, prime radicalC2×#121×C211C2
socleC22×C211C2×C22⋊C4
cosocleC22×C4×C211C23
hypercenterC2×#121D211
hypocenterC211#1510
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #27213110C2semidirect
2. #27213009C2semidirect
3. #27212880C2semidirect
8. C22⋊(C211⋊C8)C22semidirect
10. C211⋊#95C22semidirect
12. C22×D4×C211C4semidirect
16. #121×C211C4semidirect
18. C2×C4⋊C4×C211C4semidirect
102. C211#1510semidirect
109. C22×D4Dic211semidirect
114. #121Dic211semidirect
116. C2×C4⋊C4Dic211semidirect

Maximal subgroups

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

subgroupindex
#272131102
#272130092
#272128802
C2×#121×C2112
#1510211