Finite Group Databasebeta

← #41012849

Subgroups — #41012849

Order 27008.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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#27213497C211yesmaximal, characteristic, complemented
2#27213436C211yesmaximal, characteristic
3#27213426C211yesmaximal, characteristic
4#27212899C211yesmaximal, characteristic, complemented
5#27212894C211yesmaximal, characteristic, complemented
6#27212893C211yesmaximal, characteristic, complemented
7C8×D4×C211C211yesmaximal, characteristic
8#20269977C2211yescharacteristic
9#20269977C2211yescharacteristic
10#20269910—21no—
11#20269908—21no—
12C4⋊(C211⋊C8)C2211yescharacteristic
13C4⋊(C211⋊C8)C2211yescharacteristic
14#20269902—21no—
15#20269902—21no—
16#20269898—21no—
17#20269898—21no—
18C22×C8×C211—21no—
19C4×D4×C211C2211yescharacteristic
20C4⋊C8×C211C2211yescharacteristic
21C22⋊C8×C211—21no—
22C4×C8×C211C2211yescharacteristic
23C2×Dic422—21no—
24C2×Dic422—21no—
25C2×C211⋊C8—21no—
26C2×C211⋊C8—21no—
27C4⋊Dic211—21no—
28C4⋊Dic211—21no—
29C4⋊Dic211—21no—
30C4⋊Dic211—21no—
31C2×D4×C211D411yescharacteristic
32C22×C4×C211—21no—
33C2×C8×C211—21no—
34C2×C8×C211—42no—
35C2×C8×C211D411yescharacteristic
36C2×C8×C211D411yescharacteristic
37C4⋊C4×C211D411yescharacteristic
38C22⋊C4×C211—21no—
39C42×C211C2311yescharacteristic
40Dic422—41no—
41Dic422—41no—
42Dic422—81no—
43C2×Dic211—41no—
44C2×Dic211—41no—
45C211⋊C8—41no—
46C211⋊C8—41no—
47C23×C211—21no—
48D4×C211—21no—
49D4×C211Q1612yes—
50C2×C4×C211—21no—
51C2×C4×C211—41no—
52C2×C4×C211—41no—
53C2×C4×C211C2×D411yescharacteristic
54C2×C4×C211C2×D411yescharacteristic
55C2×C4×C211#4111yescharacteristic
56C8×C211—21no—
57C8×C211—21no—
58C8×C211—41no—
59C22×C211—22no—
60C22×C211—41no—
61C22×C211—41no—
62C22×C211#12111yescharacteristic
63C4×C211—21no—
64C4×C211—41no—
65C4×C211C2×Q1611yescharacteristic
66C4×C211#13511yescharacteristic
67C4×C211#13511yescharacteristic
68C4×C211AGL(1,ℤ/8)11yescharacteristic
69Dic211—81no—
70Dic211—81no—
71C2×C211—42no—
72C2×C211#46311yescharacteristic
73C2×C211#46711yescharacteristic
74C2×C211#47111yescharacteristic
75C211#168311yescharacteristic, complemented, Sylow
76#1683—2111nomaximal, Sylow
77#487—2111no—
78#476—2111no—
79#474—2111no—
80C8×D4D21111yescharacteristic
81#335—2111no—
82#333—2111no—
83#332—2111no—
84C22×C8—21no—
85C4⋊Q8—2111no—
86C4⋊Q8—2111no—
87C4×D4C2×D21111yescharacteristic
88#107—4221no—
89#106—4221no—
90C4⋊C8—2111no—
91C4⋊C8—2111no—
92C4⋊C8C2×D21111yescharacteristic
93#103—4221no—
94#103—4221no—
95#102—4221no—
96#102—4221no—
97C22⋊C8—21no—
98C4×C8C2×D21111yescharacteristic
99C2×Q8—4221no—
100C2×Q8—4221no—
101C2×D4D84411yescharacteristic
102C22×C4—21no—
103C2×C8—21no—
104C2×C8—42no—
105C2×C8—4221no—
106C2×C8—4221no—
107C2×C8C22⋊D21111yescharacteristic
108C2×C8C22⋊D21111yescharacteristic
109C4⋊C4—4221no—
110C4⋊C4—4221no—
111C4⋊C4—4221no—
112C4⋊C4—4221no—
113C4⋊C4D84411yescharacteristic
114C22⋊C4—21no—
115C42C22×D21111yescharacteristic
116C23—21no—
117Q8—8441no—
118Q8—8441no—
119Q8—16881no—
120D4—21no—
121D4Dic84412yes—
122C2×C4—21no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—8441no—
126C2×C4—8441no—
127C2×C4C2×C22⋊D21111yescharacteristic
128C2×C4C2×D84411yescharacteristic
129C2×C4#1680996211yescharacteristic
130C8—21no—
131C8—21no—
132C8—41no—
133C8—8441no—
134C8—8441no—
135C22—22no—
136C22—41no—
137C22—41no—
138C22#2026995111yescharacteristic
139C4—21no—
140C4—41no—
141C4—16881no—
142C4—16881no—
143C4C2×Dic84411yescharacteristic
144C4#2026999711yescharacteristic
145C4#2026999911yescharacteristic
146C4#2027000311yescharacteristic
147C2—42no—
148C2#2721336111yescharacteristic
149C2#2721338711yescharacteristic
150C2#2721340911yescharacteristic
subgroupG/H
centerC22#20269951
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingC8×D4×C211C2
socleC22×C211#121
prime radicalC2×D4×C211D4
hypercenterC8×D4D211
hypocenterC211#1683
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #27213497C2semidirect
4. #27212899C2semidirect
5. #27212894C2semidirect
6. #27212893C2semidirect
75. C211#1683semidirect

Maximal subgroups

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

subgroupindex
#272134972
#272134362
#272134262
#272128992
#272128942
#272128932
C8×D4×C2112
#1683211