Finite Group Databasebeta

← #41011421

Subgroups — #41011421

Order 27008.

154 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 35 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#27213082C211yesmaximal, characteristic
2#27212870C211yesmaximal, characteristic
3#342×C211C211yesmaximal, characteristic
4C2×C4⋊Dic211C411yescharacteristic, complemented
5C2×Dic211⋊C4—21no—
6C2×C22⋊Dic211—21no—
7C22⋊(C211⋊C8)—21no—
8C211⋊#95C411yescharacteristic, complemented
9C2×C22⋊C4×C211—21no—
10C2×C22⋊C4×C211C2211yescharacteristic
11C22×Dic211—21no—
12C22×Dic211—21no—
13C2×C211⋊C8—41no—
14C4⋊Dic211—41no—
15Dic211⋊C4—42no—
16C22⋊Dic211—42no—
17C24×C211D411yescharacteristic
18C22×C4×C211—41no—
19C22×C4×C211C2×C411yescharacteristic
20C22×C4×C211D411yescharacteristic
21C22⋊C4×C211—22no—
22C22⋊C4×C211—41no—
23C22⋊C4×C211—41no—
24C2×Dic211—22no—
25C2×Dic211—41no—
26C2×Dic211—41no—
27C2×Dic211—41no—
28C2×Dic211—41no—
29C2×Dic211—41no—
30C211⋊C8—81no—
31C23×C211—21no—
32C23×C211—21no—
33C23×C211—41no—
34C23×C211—41no—
35C23×C211C22⋊C411yescharacteristic
36C23×C211SD1611yescharacteristic
37C23×C211Q1611yescharacteristic
38C2×C4×C211—21no—
39C2×C4×C211—21no—
40C2×C4×C211—41no—
41C2×C4×C211—41no—
42C2×C4×C211—42no—
43C2×C4×C211—81no—
44C2×C4×C211—81no—
45C22×C211—21no—
46C22×C211—21no—
47C22×C211—21no—
48C22×C211—21no—
49C22×C211—41no—
50C22×C211—41no—
51C22×C211—41no—
52C22×C211—41no—
53C22×C211—81no—
54C22×C211#9911yescharacteristic
55C22×C211#10311yescharacteristic
56C22×C211#10411yescharacteristic
57C4×C211—41no—
58C4×C211—41no—
59C4×C211—82no—
60Dic211—42no—
61Dic211—81no—
62C2×C211—21no—
63C2×C211—21no—
64C2×C211—41no—
65C2×C211—41no—
66C2×C211#32811yescharacteristic
67C2×C211#35211yescharacteristic
68C2×C211#35311yescharacteristic
69C211#135211yescharacteristic, complemented, Sylow
70#1352—2111nomaximal, Sylow
71#399—2111no—
72#342D21111yescharacteristic
73#323—2111no—
74C2×C4⋊C4—4221no—
75C2×C4⋊C4—2111no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—4221no—
78C2×C22⋊C4C2×D21111yescharacteristic
79C22⋊C8—4221no—
80#95—2111no—
81C24C22⋊D21111yescharacteristic
82C22×C4—41no—
83C22×C4—4221no—
84C22×C4—4221no—
85C22×C4C4×D21111yescharacteristic
86C22×C4D84411yescharacteristic
87C2×C8—8441no—
88C4⋊C4—8441no—
89C4⋊C4—8442no—
90C22⋊C4—22no—
91C22⋊C4—41no—
92C22⋊C4—41no—
93C22⋊C4—8442no—
94C23—21no—
95C23—21no—
96C23—41no—
97C23—41no—
98C23(C2×D211)⋊C411yescharacteristic
99C23#1680995011yescharacteristic
100C23#1680995211yescharacteristic
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—81no—
107C2×C4—81no—
108C2×C4—8441no—
109C2×C4—8441no—
110C2×C4—8441no—
111C2×C4—8441no—
112C2×C4—8441no—
113C2×C4—4222no—
114C8—16881no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—41no—
123C22—81no—
124C22#2026989111yescharacteristic
125C22#2026990011yescharacteristic
126C22#2026990511yescharacteristic
127C4—41no—
128C4—41no—
129C4—82no—
130C4—8442no—
131C4—16881no—
132C2—21no—
133C2—21no—
134C2—41no—
135C2—41no—
136C2#2721288211yescharacteristic
137C2#2721293511yescharacteristic
138C2#2721293811yescharacteristic
subgroupG/H
centerC22#20269891
commutatorC22×C4×C211C2×C4
FrattiniC2×C22⋊C4C2×D211
Fitting#342×C211C2
socleC22×C211#99
prime radicalC24×C211D4
cosocleC2×C22⋊C4×C211C22
hypercenter#342D211
hypocenterC211#1352
LCSG ▷ C22×C4×C211 ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #342

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊Dic211C4semidirect
8. C211⋊#95C4semidirect
69. C211#1352semidirect

Maximal subgroups

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

subgroupindex
#272130822
#272128702
#342×C2112
#1352211