Finite Group Databasebeta

← #41012868

Subgroups — #41012868

Order 27008.

168 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 53 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#27213431C211yesmaximal, characteristic, complemented
3#27213277C211yesmaximal, characteristic, complemented
4#27212898C211yesmaximal, characteristic, complemented
5#27212891C211yesmaximal, characteristic, complemented
6#27212890C211yesmaximal, characteristic
7#460×C211C211yesmaximal, characteristic
8C2×#16809950—21no—
9C2×#16809950—21no—
10#20269977C2211yescharacteristic
11#20269908—21no—
12C4⋊(C211⋊C8)—21no—
13C4⋊(C211⋊C8)C2211yescharacteristic
14C4⋊(C211⋊C8)C2211yescharacteristic
15#20269901—21no—
16#20269888—21no—
17C4×C211⋊C8C2211yescharacteristic
18C2×SD16×C211—21no—
19#127×C211C2211yescharacteristic
20C4×Q8×C211C2211yescharacteristic
21C4⋊C8×C211C2211yescharacteristic
22#102×C211—21no—
23C2×Dic422—21no—
24#16809950—42no—
25#16809950—42no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8D411yescharacteristic
29C2×C211⋊C8D411yescharacteristic
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C2×Q8×C211D411yescharacteristic
33C2×D4×C211—21no—
34C2×D4×C211—21no—
35C2×D4×C211—21no—
36SD16×C211—42no—
37C2×C8×C211—21no—
38C4⋊C4×C211—21no—
39C4⋊C4×C211D411yescharacteristic
40C42×C211—21no—
41C42×C211C2311yescharacteristic
42Dic422—41no—
43Dic422—41no—
44C2×Dic211—41no—
45C211⋊C8—21no—
46C211⋊C8—41no—
47C211⋊C8—41no—
48C211⋊C8SD1612yescomplemented
49C23×C211—41no—
50Q8×C211—21no—
51Q8×C211—21no—
52D4×C211—41no—
53D4×C211—41no—
54D4×C211—41no—
55D4×C211—41no—
56D4×C211—81no—
57C2×C4×C211—21no—
58C2×C4×C211—21no—
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211C2×D411yescharacteristic
61C2×C4×C211#4111yescharacteristic
62C8×C211—41no—
63C22×C211—81no—
64C22×C211—81no—
65C22×C211—81no—
66C22×C211#12111yescharacteristic
67C4×C211—21no—
68C4×C211—41no—
69C4×C211—41no—
70C4×C211C2×SD1611yescharacteristic
71C4×C211AGL(1,ℤ/8)11yescharacteristic
72C4×C211AGL(1,ℤ/8)11yescharacteristic
73C4×C211#13711yescharacteristic
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#460D21111yescharacteristic
84#446—2111no—
85#335—2111no—
86#332—2111no—
87#331—2111no—
88C2×SD16—21no—
89C2×SD16—4221no—
90C2×SD16—4221no—
91C4⋊Q8—2111no—
92#127C2×D21111yescharacteristic
93C4×Q8C2×D21111yescharacteristic
94#106—4221no—
95C4⋊C8—4221no—
96C4⋊C8—2111no—
97C4⋊C8—2111no—
98C4⋊C8C2×D21111yescharacteristic
99#103—4221no—
100#102—21no—
101#97—4221no—
102C4×C8—2111no—
103C2×Q8—4221no—
104C2×Q8C22⋊D21111yescharacteristic
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108SD16—42no—
109SD16—8442no—
110SD16—8442no—
111C2×C8—21no—
112C2×C8—4221no—
113C2×C8—4221no—
114C2×C8—2111no—
115C2×C8—2111no—
116C4⋊C4—21no—
117C4⋊C4—4221no—
118C4⋊C4—4221no—
119C4⋊C4C22⋊D21111yescharacteristic
120C42—21no—
121C42C22×D21111yescharacteristic
122C23—41no—
123Q8—21no—
124Q8—21no—
125Q8—8441no—
126Q8—8441no—
127D4—41no—
128D4—41no—
129D4—41no—
130D4—41no—
131D4—81no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—8441no—
135C2×C4D4×D21111yescharacteristic
136C2×C4C2×C22⋊D21111yescharacteristic
137C2×C4#1680996111yescharacteristic
138C8—41no—
139C8—8441no—
140C8—8441no—
141C8—4221no—
142C8—2112no—
143C22—81no—
144C22—81no—
145C22—81no—
146C22#2026994911yescharacteristic
147C4—21no—
148C4—41no—
149C4—41no—
150C4—16881no—
151C4SD16×D21111yescharacteristic
152C4#2027000311yescharacteristic
153C4#2027000411yescharacteristic
154C4#2027000711yescharacteristic
155C2—161no—
156C2#2721334411yescharacteristic
157C2#2721339011yescharacteristic
158C2#2721340711yescharacteristic
subgroupG/H
centerC22#20269949
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#460×C211C2
socleC22×C211#121
prime radical#127×C211C22
hypercenter#460D211
hypocenterC211#1686
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #27213431C2semidirect
3. #27213277C2semidirect
4. #27212898C2semidirect
5. #27212891C2semidirect
48. 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
#272134312
#272132772
#272128982
#272128912
#272128902
#460×C2112
#1686211