Finite Group Databasebeta

← #41011265

Subgroups — #41011265

Order 27008.

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

normal subgroups only

This group has 228 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×C22⋊(C211⋊C8)C211yesmaximal, characteristic
2#27213006C211yesmaximal, characteristic
3#380×C211C211yesmaximal, characteristic
4C22×C211⋊C8C412yescomplemented
5C2×C4⋊Dic211—21no—
6C2×C22⋊Dic211—21no—
7C22⋊(C211⋊C8)—21no—
8C22⋊(C211⋊C8)—21no—
9C211⋊#95C412yescomplemented
10C23×C4×C211C2211yescharacteristic
11C2×C4⋊C4×C211—21no—
12C2×C22⋊C4×C211—21no—
13#95×C211C412yescomplemented
14C22×Dic211—22no—
15C2×C211⋊C8—22no—
16C2×C211⋊C8—42no—
17C4⋊Dic211—42no—
18C22⋊Dic211—41no—
19C22⋊Dic211—41no—
20C24×C211D411yescharacteristic
21C22×C4×C211—22no—
22C22×C4×C211—42no—
23C22×C4×C211C2×C411yescharacteristic
24C22×C4×C211C2×C411yescharacteristic
25C22×C4×C211C2×C411yescharacteristic
26C22×C4×C211D411yescharacteristic
27C22×C4×C211D411yescharacteristic
28C22×C4×C211Q811yescharacteristic
29C4⋊C4×C211—42no—
30C22⋊C4×C211—41no—
31C22⋊C4×C211—41no—
32C2×Dic211—42no—
33C2×Dic211—42no—
34C2×Dic211—42no—
35C211⋊C8—42no—
36C23×C211—41no—
37C23×C211—41no—
38C23×C211C22⋊C411yescharacteristic
39C23×C211C22⋊C411yescharacteristic
40C23×C211C4⋊C411yescharacteristic
41C23×C211D811yescharacteristic
42C23×C211SD1611yescharacteristic
43C23×C211SD1611yescharacteristic
44C23×C211Q1611yescharacteristic
45C2×C4×C211—22no—
46C2×C4×C211—22no—
47C2×C4×C211—22no—
48C2×C4×C211—41no—
49C2×C4×C211—41no—
50C2×C4×C211—41no—
51C2×C4×C211—42no—
52C2×C4×C211—42no—
53C2×C4×C211—42no—
54C2×C4×C211C4211yescharacteristic
55C2×C4×C211C22⋊C411yescharacteristic
56C2×C4×C211C4⋊C411yescharacteristic
57C2×C4×C211C4⋊C411yescharacteristic
58C22×C211—21no—
59C22×C211—21no—
60C22×C211—21no—
61C22×C211—21no—
62C22×C211—21no—
63C22×C211—21no—
64C22×C211—41no—
65C22×C211—41no—
66C22×C211—41no—
67C22×C211—41no—
68C22×C211#9511yescharacteristic
69C22×C211#10211yescharacteristic
70C22×C211#10211yescharacteristic
71C22×C211#10311yescharacteristic
72C22×C211#10311yescharacteristic
73C22×C211#10611yescharacteristic
74C22×C211#10711yescharacteristic
75C4×C211—21no—
76C4×C211—21no—
77C4×C211—21no—
78C4×C211—82no—
79C4×C211#9912yes—
80Dic211—82no—
81C2×C211—21no—
82C2×C211—21no—
83C2×C211—21no—
84C2×C211—21no—
85C2×C211—21no—
86C2×C211—21no—
87C2×C211H(3,ℤ/4)11yescharacteristic
88C2×C211#34011yescharacteristic
89C2×C211#34211yescharacteristic
90C211#128911yescharacteristic, complemented, Sylow
91#1289—2111nomaximal, Sylow
92C2×C22⋊C8—2111no—
93#380—2111no—
94#380D21111yescharacteristic
95C23×C4C2×D21111yescharacteristic
96C22×C8—2112no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—4221no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—4221no—
101C22⋊C8—4221no—
102C22⋊C8—4221no—
103#95—2112no—
104#95Dic21112yescomplemented
105C24C22⋊D21111yescharacteristic
106C22×C4—22no—
107C22×C4—42no—
108C22×C4—4222no—
109C22×C4C4×D21111yescharacteristic
110C22×C4C4×D21111yescharacteristic
111C22×C4C2×Dic21111yescharacteristic
112C22×C4C22⋊D21111yescharacteristic
113C22×C4D84411yescharacteristic
114C22×C4Dic42211yescharacteristic
115C2×C8—8442no—
116C2×C8—4222no—
117C4⋊C4—42no—
118C4⋊C4—8442no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—8441no—
122C22⋊C4—8441no—
123C23—41no—
124C23—41no—
125C23(C2×D211)⋊C411yescharacteristic
126C23C22⋊Dic21111yescharacteristic
127C23Dic211⋊C411yescharacteristic
128C23#1680994711yescharacteristic
129C23#1680994911yescharacteristic
130C23#1680995011yescharacteristic
131C23#1680995211yescharacteristic
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—8442no—
142C2×C4—8442no—
143C2×C4—8442no—
144C2×C4C4×Dic21111yescharacteristic
145C2×C4(C2×D211)⋊C411yescharacteristic
146C2×C4Dic211⋊C411yescharacteristic
147C2×C4C4⋊Dic21111yescharacteristic
148C8—8442no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22—41no—
159C22C211⋊#9511yescharacteristic
160C22#2026989711yescharacteristic
161C22#2026989811yescharacteristic
162C22#2026990011yescharacteristic
163C22#2026990111yescharacteristic
164C22#2026990811yescharacteristic
165C22#2026991011yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—82no—
170C4—16882no—
171C4#2026989112yes—
172C2—21no—
173C2—21no—
174C2—21no—
175C2—21no—
176C2—21no—
177C2—21no—
178C2#2721290211yescharacteristic
179C2#2721290811yescharacteristic
180C2#2721291211yescharacteristic
subgroupG/H
centerC22C211⋊#95
commutatorC2×C4×C211C42
FrattiniC23×C4C2×D211
Fitting#380×C211C2
socleC22×C211#95
prime radicalC24×C211D4
cosocleC23×C4×C211C22
hypercenter#380D211
hypocenterC211#1289
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
4. C22×C211⋊C8C4semidirect
9. C211⋊#95C4semidirect
13. #95×C211C4semidirect
90. C211#1289semidirect
104. #95Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C211⋊C8)2
#272130062
#380×C2112
#1289211