Finite Group Databasebeta

← #41012458

Subgroups — #41012458

Order 27008.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#27213882C211yesmaximal, characteristic, complemented
2#27213522C211yesmaximal, characteristic
3#27213347C212yesmaximal, complemented
4#27212876C212yesmaximal, complemented
5#331×C211C211yesmaximal, characteristic, complemented
6C2×#16809951—22no—
7C2×#16809947—22no—
8#20269974C2211yescharacteristic
9#20269964—22no—
10#20269958—22no—
11C4×Dic422C2212yes—
12D4×Dic211—22no—
13#20269899—22no—
14C4×C211⋊C8C2211yescharacteristic
15#127×C211C2211yescharacteristic
16C4⋊C8×C211C2212yes—
17C2×Dic422D412yes—
18C2×D844—22no—
19C2×D844—21no—
20C22×Dic211—22no—
21#16809951—44no—
22#16809947—44no—
23C2×C211⋊C8—21no—
24C4⋊Dic211D412yes—
25Dic211⋊C4—22no—
26C22⋊Dic211—24no—
27C22⋊Dic211—22no—
28C4×Dic211—22no—
29C2×D4×C211—22no—
30C2×D4×C211D412yes—
31C2×C8×C211—22no—
32C42×C211C2311yescharacteristic
33C22×D211—41no—
34Dic422—22no—
35Dic422—22no—
36D844—42no—
37D844—42no—
38D844—42no—
39C2×Dic211—22no—
40C2×Dic211—22no—
41C2×Dic211—82no—
42C211⋊C8—22no—
43C23×C211—22no—
44D4×C211—42no—
45D4×C211—42no—
46D4×C211—42no—
47C2×C4×C211C2×D412yes—
48C2×C4×C211C2×D411yescharacteristic
49C8×C211—42no—
50C22×C211—44no—
51C22×C211—42no—
52C22×C211#12011yescharacteristic
53C2×D211—82no—
54C2×D211—81no—
55C4×C211—21no—
56C4×C211#13512yes—
57C4×C211AGL(1,ℤ/8)12yes—
58Dic211—42no—
59Dic211—42no—
60C2×C211—82no—
61C2×C211#44912yes—
62C2×C211#45311yescharacteristic
63D211—161no—
64C211#162011yescharacteristic, complemented, Sylow
65#1620—2111nomaximal, Sylow
66#559—2111no—
67#493—2111no—
68#460—2112no—
69#331D21111yescharacteristic, complemented
70Q8⋊C8—2112no—
71C2×SD16—4222no—
72C2×D8—4222no—
73#127—2111no—
74#127C2×D21111yescharacteristic
75#124—4222no—
76#123—4222no—
77C4×Q8—2112no—
78C4×D4—4222no—
79C4⋊C8C2×D21112yes—
80#102—4222no—
81C4×C8—2111no—
82C2×Q8—2112no—
83C2×D4—22no—
84C2×D4—4222no—
85C2×D4—4221no—
86C2×D4D84412yes—
87C22×C4—4222no—
88SD16—8444no—
89D8—8444no—
90C2×C8—22no—
91C2×C8—4221no—
92C4⋊C4—4222no—
93C4⋊C4—2112no—
94C22⋊C4—4224no—
95C22⋊C4—4222no—
96C42—4222no—
97C42C22×D21111yescharacteristic
98C23—22no—
99C23—8441no—
100Q8—4222no—
101Q8—4222no—
102D4—42no—
103D4—42no—
104D4—42no—
105D4—8442no—
106D4—8442no—
107D4—8442no—
108C2×C4—16882no—
109C2×C4—4222no—
110C2×C4—4222no—
111C2×C4D4×D21112yes—
112C2×C4C2×D84411yescharacteristic
113C8—42no—
114C8—4222no—
115C22—42no—
116C22—44no—
117C22—16881no—
118C22—16882no—
119C22#2026994711yescharacteristic
120C4—21no—
121C4—8442no—
122C4—8442no—
123C4#2026999812yes—
124C4#2027000112yes—
125C2—82no—
126C2—33761no—
127C2#2721329212yes—
128C2#2721331611yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#331×C211C2
socleC22×C211#120
prime radical#27213522C2
hypercenter#331D211
hypocenterC211#1620
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #27213882C2semidirect
3. #27213347C2semidirect
4. #27212876C2semidirect
5. #331×C211C2semidirect
64. C211#1620semidirect
69. #331D211semidirect

Maximal subgroups

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

subgroupindex
#272138822
#272135222
#272133472
#272128762
#331×C2112
#1620211