Finite Group Databasebeta

← #41012452

Subgroups — #41012452

Order 27008.

336 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 65 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#27213522C211yesmaximal, characteristic, complemented
2#27213341C212yesmaximal, complemented
3#27212890C211yesmaximal, characteristic, complemented
4D4⋊(C211⋊C8)C212yesmaximal, complemented
5D42×C211C211yesmaximal, characteristic, complemented
6C2×#16809947—22no—
7C2×#16809947—22no—
8#20269974C2211yescharacteristic, complemented
9C4⋊(C211⋊C8)C2212yescomplemented
10#20269897—22no—
11C4×C211⋊C8C2211yescharacteristic, complemented
12C22×D4×C211—22no—
13#127×C211C2211yescharacteristic
14#121×C211—22no—
15#120×C211—22no—
16C4×D4×C211C2212yes—
17C2×D844—22no—
18C2×D844—21no—
19#16809947—44no—
20#16809947—44no—
21C2×C211⋊C8—21no—
22C2×C211⋊C8—22no—
23C24×C211—22no—
24C2×D4×C211—24no—
25C2×D4×C211—22no—
26C2×D4×C211—44no—
27C2×D4×C211—44no—
28C2×D4×C211D412yes—
29C2×D4×C211D412yes—
30C22×C4×C211—22no—
31C4⋊C4×C211D412yes—
32C22⋊C4×C211—22no—
33C22⋊C4×C211—22no—
34C42×C211C2311yescharacteristic
35C22×D211—41no—
36D844—42no—
37D844—42no—
38D844—42no—
39C211⋊C8—22no—
40C211⋊C8—42no—
41C23×C211—22no—
42C23×C211—22no—
43C23×C211—48no—
44C23×C211—82no—
45D4×C211—22no—
46D4×C211—42no—
47D4×C211—48no—
48D4×C211—42no—
49D4×C211—42no—
50D4×C211D814yescomplemented
51C2×C4×C211—22no—
52C2×C4×C211—42no—
53C2×C4×C211—42no—
54C2×C4×C211C2×D411yescharacteristic
55C2×C4×C211C2×D412yes—
56C22×C211—24no—
57C22×C211—42no—
58C22×C211—42no—
59C22×C211—44no—
60C22×C211—42no—
61C22×C211—88no—
62C22×C211#12011yescharacteristic
63C2×D211—82no—
64C2×D211—81no—
65C4×C211—21no—
66C4×C211—42no—
67C4×C211C2×D812yes—
68C4×C211AGL(1,ℤ/8)12yes—
69C2×C211—44no—
70C2×C211—82no—
71C2×C211#44712yes—
72C2×C211#45311yescharacteristic
73D211—161no—
74C211#161911yescharacteristic, complemented, Sylow
75#1619—2111nomaximal, Sylow
76D42D21111yescharacteristic, complemented
77#493—2111no—
78#459—2112no—
79#331—2111no—
80D4⋊C8—2112no—
81C22×D4—22no—
82C2×D8—4222no—
83C2×D8—4222no—
84#127—2111no—
85#127C2×D21111yescharacteristic
86#121—22no—
87#120—22no—
88C4×D4C2×D21112yes—
89C4⋊C8—2112no—
90#102—4222no—
91C4×C8—2111no—
92C24—22no—
93C2×D4—24no—
94C2×D4—22no—
95C2×D4—44no—
96C2×D4—44no—
97C2×D4—4222no—
98C2×D4—4221no—
99C2×D4C22⋊D21112yes—
100C2×D4C22⋊D21112yes—
101C22×C4—22no—
102D8—8444no—
103D8—8444no—
104C2×C8—4222no—
105C2×C8—4221no—
106C4⋊C4C22⋊D21112yes—
107C22⋊C4—22no—
108C22⋊C4—22no—
109C42C22×D21111yescharacteristic
110C23—22no—
111C23—22no—
112C23—48no—
113C23—82no—
114C23—8441no—
115D4—22no—
116D4—42no—
117D4—48no—
118D4—42no—
119D4—42no—
120D4—8442no—
121D4—8442no—
122D4—8442no—
123D4#1680994714yescomplemented
124C2×C4—22no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4C2×C22⋊D21111yescharacteristic
128C2×C4C2×C22⋊D21112yes—
129C8—8442no—
130C8—4222no—
131C22—24no—
132C22—42no—
133C22—42no—
134C22—42no—
135C22—44no—
136C22—88no—
137C22—16881no—
138C22—16882no—
139C22#2026994611yescharacteristic
140C4—21no—
141C4—42no—
142C4C2×#1680994712yes—
143C4#2027000312yes—
144C2—44no—
145C2—82no—
146C2—33761no—
147C2#2721328012yes—
148C2#2721331411yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingD42×C211C2
socleC22×C211#120
hypercenterD42D211
hypocenterC211#1619
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #27213522C2semidirect
2. #27213341C2semidirect
3. #27212890C2semidirect
4. D4⋊(C211⋊C8)C2semidirect
5. D42×C211C2semidirect
8. #20269974C22semidirect
9. C4⋊(C211⋊C8)C22semidirect
11. C4×C211⋊C8C22semidirect
50. D4×C211D8semidirect
74. C211#1619semidirect
76. D42D211semidirect
123. D4#16809947semidirect

Maximal subgroups

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

subgroupindex
#272135222
#272133412
#272128902
D4⋊(C211⋊C8)2
D42×C2112
#1619211