Finite Group Databasebeta

← #41011419

Subgroups — #41011419

Order 27008.

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

normal subgroups only

This group has 162 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#393×C211C211yesmaximal, characteristic
4C2×C22⋊Dic211—21no—
5C22⋊(C211⋊C8)—21no—
6C22×Q8×C211C411yescharacteristic, complemented
7C2×C22⋊C4×C211—21no—
8C2×C22⋊C4×C211C2211yescharacteristic
9#95×C211—21no—
10#95×C211C411yescharacteristic, complemented
11C22×Dic211—41no—
12C2×C211⋊C8—41no—
13C22⋊Dic211—22no—
14C22⋊Dic211—41no—
15C24×C211D411yescharacteristic
16C2×Q8×C211—42no—
17C2×Q8×C211—41no—
18C22×C4×C211—21no—
19C22×C4×C211—21no—
20C22×C4×C211C2×C411yescharacteristic
21C22×C4×C211D411yescharacteristic
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
35Q8×C211—42no—
36Q8×C211—42no—
37C2×C4×C211—22no—
38C2×C4×C211—21no—
39C2×C4×C211—21no—
40C2×C4×C211—41no—
41C2×C4×C211—41no—
42C2×C4×C211—41no—
43C2×C4×C211—41no—
44C2×C4×C211—41no—
45C2×C4×C211—41no—
46C2×C4×C211—41no—
47C22×C211—21no—
48C22×C211—21no—
49C22×C211—21no—
50C22×C211—21no—
51C22×C211—41no—
52C22×C211—41no—
53C22×C211—41no—
54C22×C211—41no—
55C22×C211—81no—
56C22×C211#9911yescharacteristic
57C22×C211#10311yescharacteristic
58C22×C211#10411yescharacteristic
59C4×C211—42no—
60C4×C211—41no—
61C4×C211—41no—
62C4×C211—81no—
63Dic211—82no—
64C2×C211—21no—
65C2×C211—21no—
66C2×C211—41no—
67C2×C211—41no—
68C2×C211#32811yescharacteristic
69C2×C211AΣL(1,𝔽16)11yescharacteristic
70C2×C211#35411yescharacteristic
71C211#135111yescharacteristic, complemented, Sylow
72#1351—2111nomaximal, Sylow
73#393D21111yescharacteristic
74#342—2111no—
75#323—2111no—
76C22×Q8Dic21111yescharacteristic, complemented
77C2×C22⋊C4—21no—
78C2×C22⋊C4—4221no—
79C2×C22⋊C4C2×D21111yescharacteristic
80C22⋊C8—4221no—
81#95—21no—
82#95Dic21111yescharacteristic, complemented
83C24C22⋊D21111yescharacteristic
84C2×Q8—42no—
85C2×Q8—41no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—8441no—
89C22×C4C2×Dic21111yescharacteristic
90C22×C4C22⋊D21111yescharacteristic
91C2×C8—8441no—
92C22⋊C4—41no—
93C22⋊C4—42no—
94C22⋊C4—8441no—
95C22⋊C4—4222no—
96C23—21no—
97C23—21no—
98C23—41no—
99C23—41no—
100C23C22⋊Dic21111yescharacteristic
101C23#1680994911yescharacteristic
102C23#1680995211yescharacteristic
103Q8—42no—
104Q8—42no—
105C2×C4—22no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—8442no—
116C2×C4—16881no—
117C2×C4—16881no—
118C8—16881no—
119C22—21no—
120C22—21no—
121C22—21no—
122C22—21no—
123C22—41no—
124C22—41no—
125C22—41no—
126C22—41no—
127C22—81no—
128C22#2026989211yescharacteristic
129C22#2026990111yescharacteristic
130C22#2026990411yescharacteristic
131C4—42no—
132C4—41no—
133C4—41no—
134C4—81no—
135C4—16882no—
136C2—21no—
137C2—21no—
138C2—41no—
139C2—41no—
140C2#2721288311yescharacteristic
141C2#2721293311yescharacteristic
142C2#2721294111yescharacteristic
subgroupG/H
centerC22#20269892
commutatorC22×C4×C211C2×C4
FrattiniC2×C22⋊C4C2×D211
Fitting#393×C211C2
socleC22×C211#99
prime radicalC24×C211D4
cosocleC2×C22⋊C4×C211C22
hypercenter#393D211
hypocenterC211#1351
LCSG ▷ C22×C4×C211 ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #393

Complements

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

NH ≅ G/Ntype
6. C22×Q8×C211C4semidirect
10. #95×C211C4semidirect
71. C211#1351semidirect
76. C22×Q8Dic211semidirect
82. #95Dic211semidirect

Maximal subgroups

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

subgroupindex
#272129132
#272128702
#393×C2112
#1351211