Finite Group Databasebeta

← #41012896

Subgroups — #41012896

Order 27008.

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

normal subgroups only

This group has 160 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#27213542C211yesmaximal, characteristic, complemented
2#27213437C211yesmaximal, characteristic
3#27213348C211yesmaximal, characteristic
4#27213223C211yesmaximal, characteristic
5#27212898C211yesmaximal, characteristic, complemented
6#27212894C211yesmaximal, characteristic, complemented
7#332×C211C211yesmaximal, characteristic, complemented
8C2×#16809949—21no—
9C2×#16809945—21no—
10#20269977C2211yescharacteristic
11C4×D844C2211yescharacteristic
12#20269909—21no—
13#20269908—21no—
14#20269908—21no—
15C4⋊(C211⋊C8)C2211yescharacteristic
16#20269901—21no—
17#20269899—21no—
18(C2×D211)⋊C8—21no—
19C4×C211⋊C8C2211yescharacteristic
20C4⋊Q8×C211C2211yescharacteristic
21C4⋊C8×C211C2211yescharacteristic
22C4⋊C8×C211C2211yescharacteristic
23C2×Dic422—21no—
24C2×D844D411yescharacteristic
25C2×C4×D211—21no—
26#16809949—42no—
27#16809945—42no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8D411yescharacteristic
30C2×C211⋊C8D411yescharacteristic
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211D411yescharacteristic
34(C2×D211)⋊C4—21no—
35C2×Q8×C211—21no—
36C2×C8×C211—21no—
37C2×C8×C211—21no—
38C4⋊C4×C211—21no—
39C4⋊C4×C211—21no—
40C42×C211C2311yescharacteristic
41C22×D211—21no—
42Dic422—81no—
43D844—21no—
44D844—21no—
45C2×Dic211—21no—
46C2×Dic211—41no—
47C4×D211—41no—
48C4×D211—41no—
49C211⋊C8—21no—
50C211⋊C8—41no—
51C211⋊C8SD1612yescomplemented
52Q8×C211—41no—
53Q8×C211—41no—
54C2×C4×C211—41no—
55C2×C4×C211C2×D411yescharacteristic
56C2×C4×C211C2×D411yescharacteristic
57C2×C4×C211#4111yescharacteristic
58C8×C211—41no—
59C8×C211—41no—
60C22×C211#12111yescharacteristic
61C2×D211—41no—
62C2×D211—41no—
63C2×D211—41no—
64C4×C211—21no—
65C4×C211—81no—
66C4×C211C2×SD1611yescharacteristic
67C4×C211AGL(1,ℤ/8)11yescharacteristic
68C4×C211#13711yescharacteristic
69C4×C211#13711yescharacteristic
70Dic211—41no—
71Dic211—81no—
72C2×C211#46111yescharacteristic
73C2×C211#46811yescharacteristic
74C2×C211#47111yescharacteristic
75D211—81no—
76C211#169111yescharacteristic, complemented, Sylow
77#1691—2111nomaximal, Sylow
78#498—2111no—
79#476—2111no—
80#461—2111no—
81#436—2111no—
82#335—2111no—
83#333—2111no—
84#332D21111yescharacteristic, complemented
85C2×SD16—4221no—
86C2×M16—4221no—
87C4⋊Q8—2111no—
88C4⋊Q8C2×D21111yescharacteristic
89C4×D4—2111no—
90#106—4221no—
91#106—4221no—
92#106—4221no—
93C4⋊C8—2111no—
94C4⋊C8C2×D21111yescharacteristic
95C4⋊C8C2×D21111yescharacteristic
96#103—4221no—
97#102—4221no—
98C22⋊C8—4221no—
99C4×C8—2111no—
100C2×Q8—21no—
101C2×Q8—4221no—
102C2×D4—2111no—
103C22×C4—4221no—
104SD16—8442no—
105M16—8442no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—4221no—
109C2×C8—2111no—
110C2×C8—2111no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—4221no—
114C4⋊C4—4221no—
115C4⋊C4—2111no—
116C22⋊C4—4221no—
117C42C22×D21111yescharacteristic
118C23—4221no—
119Q8—41no—
120Q8—41no—
121Q8—16881no—
122D4—4221no—
123D4—4221no—
124C2×C4—41no—
125C2×C4—8441no—
126C2×C4—8441no—
127C2×C4—8441no—
128C2×C4—4221no—
129C2×C4D4×D21111yescharacteristic
130C2×C4D4×D21111yescharacteristic
131C2×C4#1680996211yescharacteristic
132C8—41no—
133C8—41no—
134C8—8441no—
135C8—4221no—
136C8—2112no—
137C22—8441no—
138C22—8441no—
139C22—8441no—
140C22#2026995011yescharacteristic
141C4—21no—
142C4—81no—
143C4—8441no—
144C4—16881no—
145C4SD16×D21111yescharacteristic
146C4#2027000111yescharacteristic
147C4#2027000611yescharacteristic
148C4#2027000911yescharacteristic
149C2—16881no—
150C2#2721335011yescharacteristic
151C2#2721339111yescharacteristic
152C2#2721341211yescharacteristic
subgroupG/H
centerC22#20269950
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#332×C211C2
socleC22×C211#121
prime radicalC2×D844D4
hypercenter#332D211
hypocenterC211#1691
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #27213542C2semidirect
5. #27212898C2semidirect
6. #27212894C2semidirect
7. #332×C211C2semidirect
51. C211⋊C8SD16semidirect
76. C211#1691semidirect
84. #332D211semidirect

Maximal subgroups

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

subgroupindex
#272135422
#272134372
#272133482
#272132232
#272128982
#272128942
#332×C2112
#1691211