Finite Group Databasebeta

← #41012675

Subgroups — #41012675

Order 27008.

194 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 61 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#27213542C211yesmaximal, characteristic
2#27213431C211yesmaximal, characteristic
3#27213356C211yesmaximal, characteristic
4#27212892C211yesmaximal, characteristic
5Q8⋊(C211⋊C8)C211yesmaximal, characteristic
6Q8⋊(C211⋊C8)C211yesmaximal, characteristic
7#557×C211C211yesmaximal, characteristic
8C2×#16809952—21no—
9#20269977C2211yescharacteristic
10#20269908—21no—
11#20269908—22no—
12C4⋊(C211⋊C8)C2211yescharacteristic
13C4⋊(C211⋊C8)C2211yescharacteristic
14#20269901—21no—
15#20269900—21no—
16C4×C211⋊C8C2211yescharacteristic
17C4⋊Q8×C211—21no—
18C4⋊Q8×C211C2211yescharacteristic
19#125×C211—22no—
20#125×C211—21no—
21C4×Q8×C211—21no—
22C4×Q8×C211—21no—
23C4×Q8×C211C2211yescharacteristic
24C4×Q8×C211C2211yescharacteristic
25C2×Dic422—21no—
26#16809952—42no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C2×Q8×C211—21no—
33C2×Q8×C211D411yescharacteristic
34C2×Q8×C211D411yescharacteristic
35C4⋊C4×C211—22no—
36C4⋊C4×C211—22no—
37C4⋊C4×C211—22no—
38C4⋊C4×C211—21no—
39C4⋊C4×C211—21no—
40C4⋊C4×C211—21no—
41C4⋊C4×C211D411yescharacteristic
42C4⋊C4×C211D411yescharacteristic
43C4⋊C4×C211D412yes—
44C42×C211—22no—
45C42×C211—21no—
46C42×C211—21no—
47C42×C211C2311yescharacteristic
48Dic422—41no—
49Dic422—41no—
50C2×Dic211—41no—
51C211⋊C8—22no—
52C211⋊C8—41no—
53C211⋊C8—41no—
54Q8×C211—21no—
55Q8×C211—21no—
56Q8×C211—21no—
57Q8×C211—41no—
58Q8×C211—41no—
59Q8×C211SD1612yes—
60C2×C4×C211—21no—
61C2×C4×C211—21no—
62C2×C4×C211—21no—
63C2×C4×C211—21no—
64C2×C4×C211—22no—
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C22×C211#12011yescharacteristic
69C4×C211—22no—
70C4×C211—21no—
71C4×C211—41no—
72C4×C211—42no—
73C4×C211—41no—
74C4×C211—41no—
75C4×C211C2×SD1611yescharacteristic
76C4×C211#13511yescharacteristic
77C4×C211#13711yescharacteristic
78C4×C211#13711yescharacteristic
79Dic211—81no—
80C2×C211#44811yescharacteristic
81C2×C211#45211yescharacteristic
82C2×C211#45611yescharacteristic
83C211#165311yescharacteristic, complemented, Sylow
84#1653—2111nomaximal, Sylow
85#557D21111yescharacteristic
86#498—2111no—
87#475—2111no—
88#462—2111no—
89#332—2111no—
90Q8⋊C8—2111no—
91Q8⋊C8—2111no—
92C2×Q16—4221no—
93C4⋊Q8—21no—
94C4⋊Q8—2111no—
95C4⋊Q8C2×D21111yescharacteristic
96#125—22no—
97#125—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8C2×D21111yescharacteristic
101C4×Q8C2×D21111yescharacteristic
102#106—4221no—
103#106—4222no—
104C4⋊C8—2111no—
105C4⋊C8—2111no—
106#103—4221no—
107#103—4221no—
108C4×C8—2111no—
109C2×Q8—21no—
110C2×Q8—4221no—
111C2×Q8C22⋊D21111yescharacteristic
112C2×Q8C22⋊D21111yescharacteristic
113Q16—8442no—
114C2×C8—4221no—
115C2×C8—4221no—
116C2×C8—4221no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—4221no—
124C4⋊C4—4221no—
125C4⋊C4C22⋊D21111yescharacteristic
126C4⋊C4C22⋊D21111yescharacteristic
127C4⋊C4C22⋊D21112yes—
128C42—22no—
129C42—21no—
130C42—21no—
131C42C22×D21111yescharacteristic
132Q8—21no—
133Q8—21no—
134Q8—21no—
135Q8—41no—
136Q8—41no—
137Q8—8441no—
138Q8—8441no—
139Q8#1680994912yes—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—22no—
145C2×C4—8441no—
146C2×C4C2×C22⋊D21111yescharacteristic
147C2×C4C2×C22⋊D21111yescharacteristic
148C2×C4C2×C22⋊D21111yescharacteristic
149C8—8441no—
150C8—8441no—
151C8—4222no—
152C22#2026994611yescharacteristic
153C4—21no—
154C4—22no—
155C4—41no—
156C4—41no—
157C4—41no—
158C4—42no—
159C4—16881no—
160C4C2×#1680994911yescharacteristic
161C4#2026999711yescharacteristic
162C4#2027000711yescharacteristic
163C4#2027000811yescharacteristic
164C2#2721328511yescharacteristic
165C2#2721330811yescharacteristic
166C2#2721332911yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#557×C211C2
socle, prime radicalC22×C211#120
hypercenter#557D211
hypocenterC211#1653
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #557

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
#272135422
#272134312
#272133562
#272128922
Q8⋊(C211⋊C8)2
#557×C2112
#1653211