Finite Group Databasebeta

← #41011763

Subgroups — #41011763

Order 27008.

270 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 77 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×#20269956C211yesmaximal, characteristic
2C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
3#27212871C214yesmaximal, complemented
4#407×C211C211yesmaximal, characteristic
5C22×Dic422C411yescharacteristic
6C22×C211⋊C8—21no—
7#20269956—21no—
8#20269956—21no—
9#20269956—21no—
10#20269956—21no—
11C2×C4⋊Dic211C411yescharacteristic
12C2×Dic211⋊C4C2212yes—
13C2×C22⋊Dic211C412yes—
14C22⋊(C211⋊C8)—21no—
15C22⋊(C211⋊C8)C2212yes—
16C23×C4×C211C2211yescharacteristic
17C2×M16×C211—21no—
18C22⋊C8×C211—21no—
19C22⋊C8×C211C2212yes—
20C2×Dic422—21no—
21C2×Dic422—21no—
22C2×Dic422—42no—
23C22×Dic211C2×C412yes—
24C22×Dic211C2×C412yes—
25C2×C211⋊C8—22no—
26C2×C211⋊C8—42no—
27C4⋊Dic211—21no—
28C4⋊Dic211—21no—
29Dic211⋊C4—22no—
30Dic211⋊C4—22no—
31C22⋊Dic211—22no—
32C22⋊Dic211—22no—
33C24×C211C2×C411yescharacteristic
34C22×C4×C211—42no—
35C22×C4×C211C2×C411yescharacteristic
36C22×C4×C211D411yescharacteristic
37C22×C4×C211D411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211D411yescharacteristic
40C22×C4×C211C2311yescharacteristic
41M16×C211—42no—
42C2×C8×C211—22no—
43Dic422—42no—
44Dic422—42no—
45C2×Dic211—22no—
46C2×Dic211—22no—
47C2×Dic211—24no—
48C2×Dic211—42no—
49C211⋊C8—42no—
50C23×C211—22no—
51C23×C211—22no—
52C23×C211—41no—
53C23×C211C22⋊C411yescharacteristic
54C23×C211C22⋊C411yescharacteristic
55C23×C211C22×C411yescharacteristic
56C2×C4×C211—22no—
57C2×C4×C211—22no—
58C2×C4×C211—22no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211C22⋊C411yescharacteristic
63C2×C4×C211C22⋊C411yescharacteristic
64C2×C4×C211C2×D411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C8×C211—42no—
67C22×C211—21no—
68C22×C211—21no—
69C22×C211—21no—
70C22×C211—21no—
71C22×C211—21no—
72C22×C211—21no—
73C22×C211—21no—
74C22×C211—44no—
75C22×C211#10112yes—
76C22×C211C2×C22⋊C411yescharacteristic
77C22×C211#13112yes—
78C4×C211—21no—
79C4×C211—21no—
80C4×C211—21no—
81C4×C211#9912yes—
82Dic211—42no—
83Dic211—42no—
84C2×C211—22no—
85C2×C211—22no—
86C2×C211—41no—
87C2×C211#40811yescharacteristic
88C2×C211C2×#9911yescharacteristic
89C2×C211C2×#10111yescharacteristic
90C211#147211yescharacteristic, complemented, Sylow
91#1472—2111nomaximal, Sylow
92C2×C22⋊Q8—2111no—
93#407D21111yescharacteristic
94C2×C22⋊C8—2111no—
95#324—2114no—
96C22×Q8—2111no—
97C23×C4C2×D21111yescharacteristic
98C2×M16—21no—
99C22×C8—4221no—
100C22⋊Q8—4221no—
101C22⋊Q8—4221no—
102C22⋊Q8—4221no—
103C22⋊Q8—4221no—
104C2×C4⋊C4—2111no—
105C2×C4⋊C4—2112no—
106C2×C22⋊C4—2112no—
107C22⋊C8—21no—
108C22⋊C8—4221no—
109C22⋊C8—2112no—
110C22⋊C8C2×D21112yes—
111C24C4×D21111yescharacteristic
112C2×Q8—8442no—
113C2×Q8—4221no—
114C2×Q8—4221no—
115C22×C4—42no—
116C22×C4—2112no—
117C22×C4—2112no—
118C22×C4C4×D21111yescharacteristic
119C22×C4C22⋊D21111yescharacteristic
120C22×C4C22⋊D21111yescharacteristic
121C22×C4D84411yescharacteristic
122C22×C4D84411yescharacteristic
123C22×C4C22×D21111yescharacteristic
124M16—42no—
125C2×C8—22no—
126C2×C8—8442no—
127C2×C8—4222no—
128C4⋊C4—4221no—
129C4⋊C4—4221no—
130C4⋊C4—4222no—
131C4⋊C4—4222no—
132C22⋊C4—4222no—
133C22⋊C4—4222no—
134C23—22no—
135C23—22no—
136C23—41no—
137C23(C2×D211)⋊C411yescharacteristic
138C23(C2×D211)⋊C411yescharacteristic
139C23C2×C4×D21111yescharacteristic
140Q8—8442no—
141Q8—8442no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—8442no—
149C2×C4—4222no—
150C2×C4—4222no—
151C2×C4—4224no—
152C2×C4(C2×D211)⋊C411yescharacteristic
153C2×C4(C2×D211)⋊C411yescharacteristic
154C2×C4C2×C22⋊D21111yescharacteristic
155C2×C4C2×D84411yescharacteristic
156C8—42no—
157C8—8442no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—44no—
166C22#2026989512yes—
167C22C2×(C2×D211)⋊C411yescharacteristic
168C22#2026998512yes—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—8442no—
173C4—8442no—
174C4#2026989112yes—
175C2—22no—
176C2—22no—
177C2—41no—
178C2#2721311411yescharacteristic
179C2C2×#2026989111yescharacteristic
180C2C2×#2026989511yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC23×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#407×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC24×C211C2×C4
cosocleC22×C4×C211C23
hypercenter#407D211
hypocenterC211#1472
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
3. #27212871C2semidirect
90. C211#1472semidirect

Maximal subgroups

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

subgroupindex
C2×#202699562
C2×C22⋊(C211⋊C8)2
#272128712
#407×C2112
#1472211