Finite Group Databasebeta

← #41012674

Subgroups — #41012674

Order 27008.

194 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 55 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#27213874C211yesmaximal, characteristic
2#27213542C211yesmaximal, characteristic
3#27213353C211yesmaximal, characteristic
4#27212893C211yesmaximal, characteristic
5Q8⋊(C211⋊C8)C211yesmaximal, characteristic
6#27212876C211yesmaximal, characteristic
7#475×C211C211yesmaximal, characteristic
8C2×#16809952—21no—
9#20269977C2211yescharacteristic
10#20269975—21no—
11#20269969—21no—
12#20269967—22no—
13C4×Dic422—21no—
14C4×Dic422C2211yescharacteristic
15#20269942—21no—
16#20269908—22no—
17C4⋊(C211⋊C8)C2211yescharacteristic
18#20269901—21no—
19C4×C211⋊C8C2211yescharacteristic
20C4⋊Q8×C211C2211yescharacteristic
21C4×Q8×C211C2211yescharacteristic
22#106×C211—21no—
23C4⋊C8×C211C2211yescharacteristic
24#103×C211—21no—
25C2×Dic422—21no—
26C2×Dic422D411yescharacteristic
27#16809952—42no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211D411yescharacteristic
32C4⋊Dic211D412yes—
33Dic211⋊C4—22no—
34Dic211⋊C4—22no—
35Dic211⋊C4—22no—
36Dic211⋊C4—21no—
37C4×Dic211—22no—
38C4×Dic211—21no—
39C2×Q8×C211—21no—
40C2×Q8×C211D411yescharacteristic
41C2×C8×C211—21no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211D411yescharacteristic
46C42×C211—21no—
47C42×C211C2311yescharacteristic
48Dic422—21no—
49Dic422—41no—
50Dic422—41no—
51Dic422SD1612yes—
52C2×Dic211—21no—
53C2×Dic211—21no—
54C2×Dic211—22no—
55C211⋊C8—22no—
56C211⋊C8—41no—
57Q8×C211—21no—
58Q8×C211—21no—
59Q8×C211—41no—
60Q8×C211—41no—
61C2×C4×C211—21no—
62C2×C4×C211—21no—
63C2×C4×C211—41no—
64C2×C4×C211C2×D411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C8×C211—41no—
68C22×C211#12011yescharacteristic
69C4×C211—21no—
70C4×C211—41no—
71C4×C211—41no—
72C4×C211—81no—
73C4×C211C2×SD1611yescharacteristic
74C4×C211#13511yescharacteristic
75C4×C211#13711yescharacteristic
76C4×C211#13711yescharacteristic
77Dic211—22no—
78Dic211—42no—
79Dic211—41no—
80C2×C211#44811yescharacteristic
81C2×C211#45211yescharacteristic
82C2×C211#45611yescharacteristic
83C211#165311yescharacteristic, complemented, Sylow
84#1653—2111nomaximal, Sylow
85#557—2111no—
86#498—2111no—
87#475D21111yescharacteristic
88#462—2111no—
89#332—2111no—
90Q8⋊C8—2111no—
91Q8⋊C8—2111no—
92C2×Q16—4221no—
93C4⋊Q8—4221no—
94C4⋊Q8—2111no—
95C4⋊Q8C2×D21111yescharacteristic
96#125—4221no—
97#125—4222no—
98C4×Q8—4221no—
99C4×Q8—4221no—
100C4×Q8—2111no—
101C4×Q8C2×D21111yescharacteristic
102#106—21no—
103#106—4222no—
104C4⋊C8—2111no—
105C4⋊C8C2×D21111yescharacteristic
106#103—21no—
107#103—4221no—
108C4×C8—2111no—
109C2×Q8—21no—
110C2×Q8—4221no—
111C2×Q8—2111no—
112C2×Q8C22⋊D21111yescharacteristic
113Q16—8442no—
114C2×C8—21no—
115C2×C8—4221no—
116C2×C8—4221no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—4222no—
121C4⋊C4—4222no—
122C4⋊C4—4222no—
123C4⋊C4—4221no—
124C4⋊C4—4221no—
125C4⋊C4—2111no—
126C4⋊C4—2112no—
127C4⋊C4C22⋊D21111yescharacteristic
128C42—21no—
129C42—4222no—
130C42—4221no—
131C42C22×D21111yescharacteristic
132Q8—21no—
133Q8—21no—
134Q8—41no—
135Q8—41no—
136Q8—8441no—
137Q8—8441no—
138Q8—4221no—
139Q8—2112no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—4221no—
144C2×C4—4222no—
145C2×C4—4221no—
146C2×C4D4×D21111yescharacteristic
147C2×C4D4×D21111yescharacteristic
148C2×C4C2×C22⋊D21111yescharacteristic
149C8—41no—
150C8—8441no—
151C8—4222no—
152C22#2026994511yescharacteristic
153C4—21no—
154C4—41no—
155C4—41no—
156C4—81no—
157C4—8441no—
158C4—8442no—
159C4—4222no—
160C4SD16×D21111yescharacteristic
161C4#2026999711yescharacteristic
162C4#2027000611yescharacteristic
163C4#2027000811yescharacteristic
164C2#2721328311yescharacteristic
165C2#2721330611yescharacteristic
166C2#2721332711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#475×C211C2
socle, prime radicalC22×C211#120
hypercenter#475D211
hypocenterC211#1653
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
83. C211#1653semidirect

Maximal subgroups

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

subgroupindex
#272138742
#272135422
#272133532
#272128932
Q8⋊(C211⋊C8)2
#272128762
#475×C2112
#1653211