Finite Group Databasebeta

← #41012821

Subgroups — #41012821

Order 27008.

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

normal subgroups only

This group has 166 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#27213426C211yesmaximal, characteristic
3#27213351C211yesmaximal, characteristic
4#27212897C211yesmaximal, characteristic, complemented
5#27212893C211yesmaximal, characteristic, complemented
6#27212891C211yesmaximal, characteristic, complemented
7C8×D4×C211C211yesmaximal, characteristic
8C2×#16809950—21no—
9#20269977C2211yescharacteristic
10#20269977C2211yescharacteristic
11#20269910—21no—
12C4⋊(C211⋊C8)C2211yescharacteristic
13C4⋊(C211⋊C8)C2211yescharacteristic
14#20269902—21no—
15#20269902—21no—
16#20269900—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—
25#16809950—42no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C4⋊Dic211—21no—
29C4⋊Dic211—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C2×D4×C211D411yescharacteristic
33C22×C4×C211—21no—
34C2×C8×C211—21no—
35C2×C8×C211—42no—
36C2×C8×C211D411yescharacteristic
37C2×C8×C211D411yescharacteristic
38C4⋊C4×C211D411yescharacteristic
39C22⋊C4×C211—21no—
40C42×C211C2311yescharacteristic
41Dic422—41no—
42Dic422—41no—
43Dic422—41no—
44Dic422—41no—
45C2×Dic211—41no—
46C2×Dic211—41no—
47C211⋊C8—41no—
48C211⋊C8—41no—
49C23×C211—21no—
50D4×C211—21no—
51D4×C211SD1612yes—
52C2×C4×C211—21no—
53C2×C4×C211—41no—
54C2×C4×C211—41no—
55C2×C4×C211C2×D411yescharacteristic
56C2×C4×C211C2×D411yescharacteristic
57C2×C4×C211#4111yescharacteristic
58C8×C211—21no—
59C8×C211—21no—
60C8×C211—41no—
61C22×C211—22no—
62C22×C211—41no—
63C22×C211—41no—
64C22×C211#12111yescharacteristic
65C4×C211—21no—
66C4×C211—41no—
67C4×C211C2×SD1611yescharacteristic
68C4×C211#13511yescharacteristic
69C4×C211#13511yescharacteristic
70C4×C211AGL(1,ℤ/8)11yescharacteristic
71Dic211—81no—
72Dic211—81no—
73C2×C211—42no—
74C2×C211#46311yescharacteristic
75C2×C211#46511yescharacteristic
76C2×C211#47311yescharacteristic
77C211#167911yescharacteristic, complemented, Sylow
78#1679—2111nomaximal, Sylow
79#487—2111no—
80#474—2111no—
81#461—2111no—
82C8×D4D21111yescharacteristic
83#334—2111no—
84#332—2111no—
85#332—2111no—
86C2×SD16—4221no—
87C22×C8—21no—
88C4⋊Q8—2111no—
89C4⋊Q8—2111no—
90C4×D4C2×D21111yescharacteristic
91#107—4221no—
92C4⋊C8—2111no—
93C4⋊C8—2111no—
94C4⋊C8C2×D21111yescharacteristic
95#103—4221no—
96#103—4221no—
97#103—4221no—
98#102—4221no—
99C22⋊C8—21no—
100C4×C8C2×D21111yescharacteristic
101C2×Q8—4221no—
102C2×Q8—4221no—
103C2×D4D84411yescharacteristic
104C22×C4—21no—
105SD16—8442no—
106C2×C8—21no—
107C2×C8—42no—
108C2×C8—4221no—
109C2×C8—4221no—
110C2×C8C22⋊D21111yescharacteristic
111C2×C8C22⋊D21111yescharacteristic
112C4⋊C4—4221no—
113C4⋊C4—4221no—
114C4⋊C4—4221no—
115C4⋊C4—4221no—
116C4⋊C4D84411yescharacteristic
117C22⋊C4—21no—
118C42C22×D21111yescharacteristic
119C23—21no—
120Q8—8441no—
121Q8—8441no—
122Q8—8441no—
123Q8—8441no—
124D4—21no—
125D4#1680995112yes—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—8441no—
130C2×C4—8441no—
131C2×C4C2×C22⋊D21111yescharacteristic
132C2×C4C2×D84411yescharacteristic
133C2×C4#1680996211yescharacteristic
134C8—21no—
135C8—21no—
136C8—41no—
137C8—8441no—
138C8—8441no—
139C22—22no—
140C22—41no—
141C22—41no—
142C22#2026995111yescharacteristic
143C4—21no—
144C4—41no—
145C4—16881no—
146C4—16881no—
147C4C2×#1680995111yescharacteristic
148C4#2026999711yescharacteristic
149C4#2026999911yescharacteristic
150C4#2027000311yescharacteristic
151C2—42no—
152C2#2721336111yescharacteristic
153C2#2721337511yescharacteristic
154C2#2721342111yescharacteristic
subgroupG/H
centerC22#20269951
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingC8×D4×C211C2
socleC22×C211#121
prime radicalC2×D4×C211D4
hypercenterC8×D4D211
hypocenterC211#1679
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. #27212897C2semidirect
5. #27212893C2semidirect
6. #27212891C2semidirect
77. C211#1679semidirect

Maximal subgroups

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

subgroupindex
#272134972
#272134262
#272133512
#272128972
#272128932
#272128912
C8×D4×C2112
#1679211