Finite Group Databasebeta

← #41012834

Subgroups — #41012834

Order 27008.

182 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 51 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#27213352C211yesmaximal, characteristic, complemented
3#27213338C211yesmaximal, characteristic
4#27213211C211yesmaximal, characteristic, complemented
5#27212898C211yesmaximal, characteristic, complemented
6#27212891C211yesmaximal, characteristic, complemented
7#331×C211C211yesmaximal, characteristic, complemented
8C2×#16809951—21no—
9C2×#16809947—21no—
10C2×C8×D211—21no—
11#20269977C2211yescharacteristic
12C4×D844C2211yescharacteristic
13C4⋊(C211⋊C8)C2211yescharacteristic
14#20269902—21no—
15#20269898—21no—
16#20269898—21no—
17#20269898—21no—
18(C2×D211)⋊C8—21no—
19C4×C211⋊C8C2211yescharacteristic
20#127×C211C2211yescharacteristic
21C4⋊C8×C211C2211yescharacteristic
22C4⋊C8×C211C2211yescharacteristic
23C2×Dic422—21no—
24C2×D844D411yescharacteristic
25C2×C4×D211—21no—
26#16809951—42no—
27#16809947—42no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8D411yescharacteristic
30C2×C211⋊C8D411yescharacteristic
31C8×D211—42no—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211D411yescharacteristic
35(C2×D211)⋊C4—21no—
36C2×D4×C211—21no—
37C2×D4×C211—21no—
38C2×D4×C211—21no—
39C2×C8×C211—21no—
40C2×C8×C211—21no—
41C42×C211C2311yescharacteristic
42C22×D211—21no—
43Dic422—41no—
44Dic422—41no—
45D844—21no—
46D844D812yes—
47C2×Dic211—21no—
48C2×Dic211—41no—
49C4×D211—41no—
50C4×D211—41no—
51C211⋊C8—21no—
52C211⋊C8—21no—
53C211⋊C8—41no—
54C23×C211—41no—
55D4×C211—41no—
56D4×C211—41no—
57D4×C211—41no—
58D4×C211—41no—
59D4×C211—81no—
60C2×C4×C211C2×D411yescharacteristic
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211#4111yescharacteristic
63C8×C211—41no—
64C8×C211—41no—
65C22×C211—81no—
66C22×C211—81no—
67C22×C211—81no—
68C22×C211#12111yescharacteristic
69C2×D211—22no—
70C2×D211—41no—
71C2×D211—41no—
72C4×C211—21no—
73C4×C211C2×D811yescharacteristic
74C4×C211#13511yescharacteristic
75C4×C211#13511yescharacteristic
76C4×C211AGL(1,ℤ/8)11yescharacteristic
77Dic211—41no—
78Dic211—81no—
79C2×C211—161no—
80C2×C211#46311yescharacteristic
81C2×C211#46611yescharacteristic
82C2×C211#47111yescharacteristic
83D211—42no—
84C211#168111yescharacteristic, complemented, Sylow
85#1681—2111nomaximal, Sylow
86#486—2111no—
87#461—2111no—
88#459—2111no—
89C8×D4—2111no—
90#335—2111no—
91#332—2111no—
92#331D21111yescharacteristic, complemented
93C2×SD16—4221no—
94C2×D8—4221no—
95C22×C8—4221no—
96C4⋊Q8—2111no—
97#127C2×D21111yescharacteristic
98C4×D4—2111no—
99C4⋊C8—2111no—
100C4⋊C8C2×D21111yescharacteristic
101C4⋊C8C2×D21111yescharacteristic
102#103—4221no—
103#102—4221no—
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×D4—2111no—
113C22×C4—4221no—
114SD16—8442no—
115D8—8442no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—8442no—
119C2×C8—4221no—
120C2×C8—2111no—
121C2×C8—2111no—
122C4⋊C4—4221no—
123C4⋊C4—4221no—
124C4⋊C4—2111no—
125C22⋊C4—4221no—
126C42C22×D21111yescharacteristic
127C23—41no—
128C23—4221no—
129Q8—8441no—
130Q8—8441no—
131D4—41no—
132D4—41no—
133D4—41no—
134D4—41no—
135D4—81no—
136D4—4221no—
137D4—2112no—
138C2×C4—8441no—
139C2×C4—8441no—
140C2×C4—8441no—
141C2×C4—4221no—
142C2×C4D4×D21111yescharacteristic
143C2×C4D4×D21111yescharacteristic
144C2×C4#1680996211yescharacteristic
145C8—41no—
146C8—41no—
147C8—8441no—
148C8—4221no—
149C8—4221no—
150C22—81no—
151C22—81no—
152C22—81no—
153C22—8441no—
154C22—8441no—
155C22—4222no—
156C22#2026995011yescharacteristic
157C4—21no—
158C4—8441no—
159C4—16881no—
160C4D8×D21111yescharacteristic
161C4#2026999811yescharacteristic
162C4#2026999811yescharacteristic
163C4#2027000111yescharacteristic
164C2—161no—
165C2—8442no—
166C2#2721336011yescharacteristic
167C2#2721338111yescharacteristic
168C2#2721340811yescharacteristic
subgroupG/H
centerC22#20269950
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#331×C211C2
socleC22×C211#121
prime radical#27213338C2
hypercenter#331D211
hypocenterC211#1681
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. #27213492C2semidirect
2. #27213352C2semidirect
4. #27213211C2semidirect
5. #27212898C2semidirect
6. #27212891C2semidirect
7. #331×C211C2semidirect
84. C211#1681semidirect
92. #331D211semidirect

Maximal subgroups

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

subgroupindex
#272134922
#272133522
#272133382
#272132112
#272128982
#272128912
#331×C2112
#1681211