Finite Group Databasebeta

← #41012869

Subgroups — #41012869

Order 27008.

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

normal subgroups only

This group has 168 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#27213518C211yesmaximal, characteristic
2#27213432C211yesmaximal, characteristic, complemented
3#27213343C211yesmaximal, characteristic
4#27213273C211yesmaximal, characteristic, complemented
5#27212898C211yesmaximal, characteristic, complemented
6#27212893C211yesmaximal, characteristic, complemented
7#331×C211C211yesmaximal, characteristic
8C2×#16809950—21no—
9C2×#16809950—21no—
10C2×#16809950—21no—
11#20269977C2211yescharacteristic
12C4×Dic422C2211yescharacteristic
13#20269909—21no—
14C4⋊(C211⋊C8)C2211yescharacteristic
15Dic211⋊C8—21no—
16#20269902—21no—
17#20269898—21no—
18#20269887—21no—
19C4×C211⋊C8C2211yescharacteristic
20#127×C211C2211yescharacteristic
21C4⋊C8×C211C2211yescharacteristic
22C4⋊C8×C211C2211yescharacteristic
23C2×Dic422—21no—
24C2×Dic422D411yescharacteristic
25#16809950—42no—
26#16809950—42no—
27#16809950—42no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8D411yescharacteristic
30C2×C211⋊C8D411yescharacteristic
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211D411yescharacteristic
34Dic211⋊C4—21no—
35C4×Dic211—21no—
36C2×D4×C211—21no—
37C2×D4×C211—21no—
38C2×D4×C211—21no—
39C2×C8×C211—21no—
40C2×C8×C211—21no—
41C42×C211C2311yescharacteristic
42Dic422—21no—
43Dic422—21no—
44Dic422—41no—
45Dic422—41no—
46C2×Dic211—21no—
47C2×Dic211—21no—
48C2×Dic211—41no—
49C211⋊C8—21no—
50C211⋊C8—41no—
51C211⋊C8SD1612yescomplemented
52C23×C211—41no—
53D4×C211—41no—
54D4×C211—41no—
55D4×C211—41no—
56D4×C211—41no—
57D4×C211—81no—
58C2×C4×C211C2×D411yescharacteristic
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211#4111yescharacteristic
61C8×C211—41no—
62C8×C211—41no—
63C22×C211—81no—
64C22×C211—81no—
65C22×C211—81no—
66C22×C211#12111yescharacteristic
67C4×C211—21no—
68C4×C211C2×SD1611yescharacteristic
69C4×C211AGL(1,ℤ/8)11yescharacteristic
70C4×C211AGL(1,ℤ/8)11yescharacteristic
71C4×C211#13711yescharacteristic
72Dic211—41no—
73Dic211—41no—
74Dic211—81no—
75C2×C211—161no—
76C2×C211#46011yescharacteristic
77C2×C211#46811yescharacteristic
78C2×C211#47111yescharacteristic
79C211#168611yescharacteristic, complemented, Sylow
80#1686—2111nomaximal, Sylow
81#492—2111no—
82#475—2111no—
83#460—2111no—
84#446—2111no—
85#335—2111no—
86#332—2111no—
87#331D21111yescharacteristic
88C2×SD16—4221no—
89C2×SD16—4221no—
90C2×SD16—4221no—
91C4⋊Q8—2111no—
92#127C2×D21111yescharacteristic
93C4×Q8—2111no—
94#106—4221no—
95C4⋊C8—4221no—
96C4⋊C8—2111no—
97C4⋊C8C2×D21111yescharacteristic
98C4⋊C8C2×D21111yescharacteristic
99#103—4221no—
100#102—4221no—
101#97—4221no—
102C4×C8—2111no—
103C2×Q8—4221no—
104C2×Q8—2111no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108SD16—8442no—
109SD16—8442no—
110SD16—8442no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—4221no—
114C2×C8—2111no—
115C2×C8—2111no—
116C4⋊C4—4221no—
117C4⋊C4—4221no—
118C4⋊C4—4221no—
119C4⋊C4—2111no—
120C42—4221no—
121C42C22×D21111yescharacteristic
122C23—41no—
123Q8—8441no—
124Q8—8441no—
125Q8—4221no—
126Q8—4221no—
127D4—41no—
128D4—41no—
129D4—41no—
130D4—41no—
131D4—81no—
132C2×C4—8441no—
133C2×C4—4221no—
134C2×C4—4221no—
135C2×C4D4×D21111yescharacteristic
136C2×C4D4×D21111yescharacteristic
137C2×C4#1680996211yescharacteristic
138C8—41no—
139C8—41no—
140C8—8441no—
141C8—4221no—
142C8—2112no—
143C22—81no—
144C22—81no—
145C22—81no—
146C22#2026995011yescharacteristic
147C4—21no—
148C4—8441no—
149C4—8441no—
150C4—16881no—
151C4SD16×D21111yescharacteristic
152C4#2027000111yescharacteristic
153C4#2027000111yescharacteristic
154C4#2027000611yescharacteristic
155C2—161no—
156C2#2721334511yescharacteristic
157C2#2721339111yescharacteristic
158C2#2721340811yescharacteristic
subgroupG/H
centerC22#20269950
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#331×C211C2
socleC22×C211#121
prime radical#127×C211C22
hypercenter#331D211
hypocenterC211#1686
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27213432C2semidirect
4. #27213273C2semidirect
5. #27212898C2semidirect
6. #27212893C2semidirect
51. C211⋊C8SD16semidirect
79. C211#1686semidirect

Maximal subgroups

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

subgroupindex
#272135182
#272134322
#272133432
#272132732
#272128982
#272128932
#331×C2112
#1686211