Finite Group Databasebeta

← #41012833

Subgroups — #41012833

Order 27008.

182 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 55 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#27213492C211yesmaximal, characteristic, complemented
2#27213351C211yesmaximal, characteristic, complemented
3D4×C211⋊C8C211yesmaximal, characteristic, complemented
4#27212898C211yesmaximal, characteristic, complemented
5#27212893C211yesmaximal, characteristic, complemented
6#27212890C211yesmaximal, characteristic, complemented
7#459×C211C211yesmaximal, characteristic
8C2×#16809950—21no—
9C22×C211⋊C8—21no—
10#20269977C2211yescharacteristic
11C4⋊(C211⋊C8)C2211yescharacteristic
12C4⋊(C211⋊C8)C2211yescharacteristic
13#20269900—21no—
14#20269898—21no—
15#20269898—21no—
16C22⋊(C211⋊C8)—21no—
17C4×C211⋊C8C2211yescharacteristic
18C2×D8×C211—21no—
19#127×C211C2211yescharacteristic
20C4×D4×C211C2211yescharacteristic
21C4⋊C8×C211C2211yescharacteristic
22#102×C211—21no—
23C2×Dic422—21no—
24#16809950—42no—
25C2×C211⋊C8—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—42no—
28C2×C211⋊C8D411yescharacteristic
29C2×C211⋊C8D411yescharacteristic
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C2×D4×C211—21no—
33C2×D4×C211—21no—
34C2×D4×C211—21no—
35C2×D4×C211D411yescharacteristic
36C22×C4×C211—21no—
37D8×C211—42no—
38C2×C8×C211—21no—
39C4⋊C4×C211D411yescharacteristic
40C22⋊C4×C211—21no—
41C42×C211C2311yescharacteristic
42Dic422—41no—
43Dic422—41no—
44C2×Dic211—41no—
45C211⋊C8—21no—
46C211⋊C8—21no—
47C211⋊C8—41no—
48C211⋊C8—41no—
49C23×C211—21no—
50C23×C211—41no—
51D4×C211—21no—
52D4×C211—41no—
53D4×C211—41no—
54D4×C211—41no—
55D4×C211—41no—
56D4×C211—81no—
57D4×C211D812yes—
58C2×C4×C211—21no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211C2×D411yescharacteristic
63C2×C4×C211#4111yescharacteristic
64C8×C211—41no—
65C22×C211—22no—
66C22×C211—41no—
67C22×C211—41no—
68C22×C211—81no—
69C22×C211—81no—
70C22×C211—81no—
71C22×C211#12111yescharacteristic
72C4×C211—21no—
73C4×C211—41no—
74C4×C211C2×D811yescharacteristic
75C4×C211#13511yescharacteristic
76C4×C211#13511yescharacteristic
77C4×C211AGL(1,ℤ/8)11yescharacteristic
78Dic211—81no—
79C2×C211—42no—
80C2×C211—161no—
81C2×C211#46311yescharacteristic
82C2×C211#46611yescharacteristic
83C2×C211#47111yescharacteristic
84C211#168111yescharacteristic, complemented, Sylow
85#1681—2111nomaximal, Sylow
86#486—2111no—
87#461—2111no—
88#459D21111yescharacteristic
89C8×D4—2111no—
90#335—2111no—
91#332—2111no—
92#331—2111no—
93C2×SD16—4221no—
94C2×D8—21no—
95C22×C8—4221no—
96C4⋊Q8—2111no—
97#127C2×D21111yescharacteristic
98C4×D4C2×D21111yescharacteristic
99C4⋊C8—2111no—
100C4⋊C8—2111no—
101C4⋊C8C2×D21111yescharacteristic
102#103—4221no—
103#102—21no—
104#102—4221no—
105#102—4221no—
106C22⋊C8—4221no—
107C4×C8—2111no—
108C2×Q8—4221no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4C22⋊D21111yescharacteristic
113C22×C4—21no—
114SD16—8442no—
115D8—42no—
116C2×C8—21no—
117C2×C8—8442no—
118C2×C8—4221no—
119C2×C8—4221no—
120C2×C8—2111no—
121C2×C8—2111no—
122C4⋊C4—4221no—
123C4⋊C4—4221no—
124C4⋊C4C22⋊D21111yescharacteristic
125C22⋊C4—21no—
126C42C22×D21111yescharacteristic
127C23—21no—
128C23—41no—
129Q8—8441no—
130Q8—8441no—
131D4—21no—
132D4—41no—
133D4—41no—
134D4—41no—
135D4—41no—
136D4—81no—
137D4#1680994712yes—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—8441no—
142C2×C4D4×D21111yescharacteristic
143C2×C4C2×C22⋊D21111yescharacteristic
144C2×C4#1680996111yescharacteristic
145C8—41no—
146C8—8441no—
147C8—8441no—
148C8—4221no—
149C8—4221no—
150C22—22no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22—81no—
155C22—81no—
156C22#2026994911yescharacteristic
157C4—21no—
158C4—41no—
159C4—16881no—
160C4C2×#1680994711yescharacteristic
161C4#2026999611yescharacteristic
162C4#2026999711yescharacteristic
163C4#2027000111yescharacteristic
164C2—42no—
165C2—161no—
166C2#2721335911yescharacteristic
167C2#2721338011yescharacteristic
168C2#2721340711yescharacteristic
subgroupG/H
centerC22#20269949
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting, prime radical#459×C211C2
socleC22×C211#121
hypercenter#459D211
hypocenterC211#1681
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #27213492C2semidirect
2. #27213351C2semidirect
3. D4×C211⋊C8C2semidirect
4. #27212898C2semidirect
5. #27212893C2semidirect
6. #27212890C2semidirect
84. C211#1681semidirect

Maximal subgroups

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

subgroupindex
#272134922
#272133512
D4×C211⋊C82
#272128982
#272128932
#272128902
#459×C2112
#1681211