Finite Group Databasebeta

← #41011422

Subgroups — #41011422

Order 27008.

154 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 37 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#27212913C211yesmaximal, characteristic
2#27212870C211yesmaximal, characteristic
3#399×C211C211yesmaximal, characteristic
4C2×C22⋊Dic211—21no—
5C22⋊(C211⋊C8)—21no—
6C2×C4⋊C4×C211—21no—
7C2×C4⋊C4×C211C411yescharacteristic, complemented
8C2×C22⋊C4×C211—21no—
9C2×C22⋊C4×C211C2211yescharacteristic
10#95×C211C411yescharacteristic, complemented
11C22×Dic211—41no—
12C2×C211⋊C8—41no—
13C22⋊Dic211—22no—
14C22⋊Dic211—41no—
15C24×C211D411yescharacteristic
16C22×C4×C211—21no—
17C22×C4×C211—21no—
18C22×C4×C211C2×C411yescharacteristic
19C22×C4×C211D411yescharacteristic
20C4⋊C4×C211—41no—
21C4⋊C4×C211—42no—
22C22⋊C4×C211—41no—
23C22⋊C4×C211—42no—
24C2×Dic211—42no—
25C2×Dic211—81no—
26C2×Dic211—81no—
27C211⋊C8—81no—
28C23×C211—21no—
29C23×C211—21no—
30C23×C211—41no—
31C23×C211—41no—
32C23×C211C22⋊C411yescharacteristic
33C23×C211SD1611yescharacteristic
34C23×C211Q1611yescharacteristic
35C2×C4×C211—22no—
36C2×C4×C211—21no—
37C2×C4×C211—21no—
38C2×C4×C211—41no—
39C2×C4×C211—41no—
40C2×C4×C211—41no—
41C2×C4×C211—41no—
42C2×C4×C211—41no—
43C2×C4×C211—41no—
44C2×C4×C211—41no—
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—42no—
58C4×C211—41no—
59C4×C211—41no—
60C4×C211—81no—
61Dic211—82no—
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#399D21111yescharacteristic
72#342—2111no—
73#323—2111no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4Dic21111yescharacteristic, complemented
76C2×C22⋊C4—21no—
77C2×C22⋊C4—4221no—
78C2×C22⋊C4C2×D21111yescharacteristic
79C22⋊C8—4221no—
80#95Dic21111yescharacteristic, complemented
81C24C22⋊D21111yescharacteristic
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—8441no—
85C22×C4C2×Dic21111yescharacteristic
86C22×C4C22⋊D21111yescharacteristic
87C2×C8—8441no—
88C4⋊C4—41no—
89C4⋊C4—42no—
90C22⋊C4—41no—
91C22⋊C4—42no—
92C22⋊C4—8441no—
93C22⋊C4—4222no—
94C23—21no—
95C23—21no—
96C23—41no—
97C23—41no—
98C23C22⋊Dic21111yescharacteristic
99C23#1680994911yescharacteristic
100C23#1680995211yescharacteristic
101C2×C4—22no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—8442no—
112C2×C4—16881no—
113C2×C4—16881no—
114C8—16881no—
115C22—21no—
116C22—21no—
117C22—21no—
118C22—21no—
119C22—41no—
120C22—41no—
121C22—41no—
122C22—41no—
123C22—81no—
124C22#2026989211yescharacteristic
125C22#2026990111yescharacteristic
126C22#2026990411yescharacteristic
127C4—42no—
128C4—41no—
129C4—41no—
130C4—81no—
131C4—16882no—
132C2—21no—
133C2—21no—
134C2—41no—
135C2—41no—
136C2#2721288311yescharacteristic
137C2#2721293611yescharacteristic
138C2#2721293911yescharacteristic
subgroupG/H
centerC22#20269892
commutatorC22×C4×C211C2×C4
FrattiniC2×C22⋊C4C2×D211
Fitting#399×C211C2
socleC22×C211#99
prime radicalC24×C211D4
cosocleC2×C22⋊C4×C211C22
hypercenter#399D211
hypocenterC211#1352
LCSG ▷ C22×C4×C211 ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #399

Complements

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

NH ≅ G/Ntype
7. C2×C4⋊C4×C211C4semidirect
10. #95×C211C4semidirect
69. C211#1352semidirect
75. C2×C4⋊C4Dic211semidirect
80. #95Dic211semidirect

Maximal subgroups

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

subgroupindex
#272129132
#272128702
#399×C2112
#1352211