Finite Group Databasebeta

← #41012785

Subgroups — #41012785

Order 27008.

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

normal subgroups only

This group has 172 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#27213549C211yesmaximal, characteristic, complemented
2#27213351C211yesmaximal, characteristic
3D4×C211⋊C8C211yesmaximal, characteristic
4#27212898C211yesmaximal, characteristic, complemented
5#27212893C211yesmaximal, characteristic, complemented
6#27212892C211yesmaximal, characteristic, complemented
7#461×C211C211yesmaximal, characteristic
8C2×#16809950—21no—
9C22×C211⋊C8—21no—
10#20269977C2211yescharacteristic
11#20269910—21no—
12#20269910—21no—
13C4⋊(C211⋊C8)C2211yescharacteristic
14C4⋊(C211⋊C8)C2211yescharacteristic
15#20269900—21no—
16C22⋊(C211⋊C8)—21no—
17C4×C211⋊C8C2211yescharacteristic
18C2×SD16×C211—21no—
19C4⋊Q8×C211C2211yescharacteristic
20C4×D4×C211C2211yescharacteristic
21C4⋊C8×C211C2211yescharacteristic
22#103×C211—21no—
23C2×Dic422—21no—
24#16809950—42no—
25C2×C211⋊C8—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—42no—
28C2×C211⋊C8D411yescharacteristic
29C2×C211⋊C8D411yescharacteristic
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C2×Q8×C211—21no—
33C2×D4×C211D411yescharacteristic
34C22×C4×C211—21no—
35SD16×C211—42no—
36C2×C8×C211—21no—
37C4⋊C4×C211—21no—
38C4⋊C4×C211—21no—
39C4⋊C4×C211D411yescharacteristic
40C22⋊C4×C211—21no—
41C42×C211C2311yescharacteristic
42Dic422—41no—
43Dic422—41no—
44C2×Dic211—41no—
45C211⋊C8—21no—
46C211⋊C8—41no—
47C211⋊C8—41no—
48C211⋊C8SD1612yescomplemented
49C23×C211—21no—
50Q8×C211—41no—
51Q8×C211—41no—
52D4×C211—21no—
53D4×C211D812yes—
54C2×C4×C211—21no—
55C2×C4×C211—41no—
56C2×C4×C211—41no—
57C2×C4×C211—41no—
58C2×C4×C211C2×D411yescharacteristic
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211#4111yescharacteristic
61C8×C211—41no—
62C22×C211—22no—
63C22×C211—41no—
64C22×C211—41no—
65C22×C211#12111yescharacteristic
66C4×C211—21no—
67C4×C211—41no—
68C4×C211—81no—
69C4×C211C2×D811yescharacteristic
70C4×C211C2×SD1611yescharacteristic
71C4×C211#13511yescharacteristic
72C4×C211#13711yescharacteristic
73Dic211—81no—
74C2×C211—42no—
75C2×C211#46111yescharacteristic
76C2×C211#46611yescharacteristic
77C2×C211#47311yescharacteristic
78C211#167311yescharacteristic, complemented, Sylow
79#1673—2111nomaximal, Sylow
80#500—2111no—
81#461—2111no—
82#461D21111yescharacteristic
83C8×D4—2111no—
84#335—2111no—
85#332—2111no—
86#332—2111no—
87C2×SD16—21no—
88C2×SD16—4221no—
89C22×C8—4221no—
90C4⋊Q8—2111no—
91C4⋊Q8C2×D21111yescharacteristic
92C4×D4C2×D21111yescharacteristic
93#107—4221no—
94#107—4221no—
95C4⋊C8—2111no—
96C4⋊C8—2111no—
97C4⋊C8C2×D21111yescharacteristic
98#103—21no—
99#103—4221no—
100C22⋊C8—4221no—
101C4×C8—2111no—
102C2×Q8—21no—
103C2×Q8—4221no—
104C2×D4C22⋊D21111yescharacteristic
105C22×C4—21no—
106SD16—42no—
107SD16—8442no—
108C2×C8—21no—
109C2×C8—8442no—
110C2×C8—4221no—
111C2×C8—4221no—
112C2×C8—2111no—
113C2×C8—2111no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—4221no—
117C4⋊C4—4221no—
118C4⋊C4C22⋊D21111yescharacteristic
119C22⋊C4—21no—
120C42C22×D21111yescharacteristic
121C23—21no—
122Q8—41no—
123Q8—41no—
124Q8—8441no—
125Q8—8441no—
126D4—21no—
127D4#1680994712yes—
128C2×C4—21no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—8441no—
133C2×C4D4×D21111yescharacteristic
134C2×C4C2×C22⋊D21111yescharacteristic
135C2×C4#1680996111yescharacteristic
136C8—41no—
137C8—8441no—
138C8—8441no—
139C8—4221no—
140C8—2112no—
141C22—22no—
142C22—41no—
143C22—41no—
144C22#2026994911yescharacteristic
145C4—21no—
146C4—41no—
147C4—81no—
148C4—16881no—
149C4C2×#1680994711yescharacteristic
150C4SD16×D21111yescharacteristic
151C4#2026999711yescharacteristic
152C4#2027000611yescharacteristic
153C2—42no—
154C2#2721334911yescharacteristic
155C2#2721338011yescharacteristic
156C2#2721341911yescharacteristic
subgroupG/H
centerC22#20269949
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#461×C211C2
socleC22×C211#121
prime radicalC2×D4×C211D4
hypercenter#461D211
hypocenterC211#1673
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #27213549C2semidirect
4. #27212898C2semidirect
5. #27212893C2semidirect
6. #27212892C2semidirect
48. C211⋊C8SD16semidirect
78. C211#1673semidirect

Maximal subgroups

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

subgroupindex
#272135492
#272133512
D4×C211⋊C82
#272128982
#272128932
#272128922
#461×C2112
#1673211