Finite Group Databasebeta

← #41012479

Subgroups — #41012479

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#27213862C211yesmaximal, characteristic, complemented
2#27213338C212yesmaximal, complemented
3#27212895C211yesmaximal, characteristic, complemented
4#27212875C212yesmaximal, complemented
5#492×C211C211yesmaximal, characteristic, complemented
6C2×#16809947—22no—
7#20269966—22no—
8#20269952—22no—
9C4×D844C2212yes—
10#20269935—22no—
11C4⋊(C211⋊C8)C2212yes—
12#20269898—22no—
13C2×SD16×C211—22no—
14C4⋊Q8×C211C2211yescharacteristic
15#127×C211C2211yescharacteristic
16C4×C8×C211C2211yescharacteristic
17C2×D844—22no—
18C2×D844D412yes—
19C2×C4×D211—22no—
20#16809947—44no—
21C2×C211⋊C8—22no—
22C4⋊Dic211D412yes—
23(C2×D211)⋊C4—24no—
24(C2×D211)⋊C4—22no—
25C4×Dic211—22no—
26C2×Q8×C211D412yes—
27C2×D4×C211—21no—
28C2×D4×C211—22no—
29SD16×C211—44no—
30C2×C8×C211—21no—
31C4⋊C4×C211—22no—
32C42×C211C2311yescharacteristic
33C22×D211—22no—
34D844—22no—
35D844—22no—
36D844—82no—
37C2×Dic211—22no—
38C4×D211—42no—
39C4×D211—42no—
40C211⋊C8—42no—
41C23×C211—41no—
42Q8×C211—42no—
43D4×C211—42no—
44D4×C211—42no—
45D4×C211—42no—
46C2×C4×C211—22no—
47C2×C4×C211C2×D412yes—
48C2×C4×C211C2×D411yescharacteristic
49C8×C211—22no—
50C22×C211—82no—
51C22×C211—81no—
52C22×C211#12011yescharacteristic
53C2×D211—42no—
54C2×D211—42no—
55C2×D211—42no—
56C4×C211—21no—
57C4×C211—42no—
58C4×C211#13512yes—
59C4×C211AGL(1,ℤ/8)12yes—
60Dic211—42no—
61C2×C211—161no—
62C2×C211#44912yes—
63C2×C211#45311yescharacteristic
64D211—82no—
65C211#162411yescharacteristic, complemented, Sylow
66#1624—2111nomaximal, Sylow
67#555—2111no—
68#492D21111yescharacteristic, complemented
69#459—2112no—
70#333—2111no—
71D4⋊C8—2112no—
72C2×SD16—22no—
73C2×D8—4222no—
74C4⋊Q8C2×D21111yescharacteristic
75#127C2×D21111yescharacteristic
76#124—4222no—
77#121—4222no—
78C4×D4—2112no—
79#117—4222no—
80C4⋊C8—2112no—
81#102—4222no—
82C4×C8C2×D21111yescharacteristic
83C2×Q8C22⋊D21112yes—
84C2×D4—22no—
85C2×D4—21no—
86C2×D4—4222no—
87C2×D4—2112no—
88C22×C4—4222no—
89SD16—44no—
90D8—8444no—
91C2×C8—21no—
92C2×C8—4222no—
93C4⋊C4—22no—
94C4⋊C4—2112no—
95C22⋊C4—4224no—
96C22⋊C4—4222no—
97C42—4222no—
98C42C22×D21111yescharacteristic
99C23—41no—
100C23—4222no—
101Q8—42no—
102D4—42no—
103D4—42no—
104D4—42no—
105D4—16882no—
106D4—4222no—
107D4—4222no—
108C2×C4—22no—
109C2×C4—8442no—
110C2×C4—8442no—
111C2×C4—4222no—
112C2×C4D4×D21112yes—
113C2×C4C2×C22⋊D21111yescharacteristic
114C8—22no—
115C8—8442no—
116C22—81no—
117C22—82no—
118C22—8442no—
119C22—8442no—
120C22—8442no—
121C22#2026994511yescharacteristic
122C4—21no—
123C4—42no—
124C4—8442no—
125C4#2026999812yes—
126C4#2027000412yes—
127C2—161no—
128C2—16882no—
129C2#2721329012yes—
130C2#2721331311yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#492×C211C2
socleC22×C211#120
hypercenter#492D211
hypocenterC211#1624
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #27213862C2semidirect
2. #27213338C2semidirect
3. #27212895C2semidirect
4. #27212875C2semidirect
5. #492×C211C2semidirect
65. C211#1624semidirect
68. #492D211semidirect

Maximal subgroups

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

subgroupindex
#272138622
#272133382
#272128952
#272128752
#492×C2112
#1624211