Finite Group Databasebeta

← #41012493

Subgroups — #41012493

Order 27008.

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

normal subgroups only

This group has 210 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#27213875C211yesmaximal, characteristic, complemented
2#27213492C211yesmaximal, characteristic
3#27213343C211yesmaximal, characteristic
4#27212891C211yesmaximal, characteristic, complemented
5Q8⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6#27212876C211yesmaximal, characteristic, complemented
7#460×C211C211yesmaximal, characteristic
8C2×#16809950—21no—
9#20269977—21no—
10#20269977C2211yescharacteristic
11#20269968—22no—
12#20269968—21no—
13Q8×Dic211—21no—
14C4×Dic422—21no—
15C4×Dic422C2211yescharacteristic
16C4⋊(C211⋊C8)C2211yescharacteristic
17#20269898—21no—
18#20269898—22no—
19C4×C211⋊C8C2211yescharacteristic
20C2×SD16×C211—21no—
21#127×C211C2211yescharacteristic
22C4×Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24#102×C211—21no—
25C2×Dic422—21no—
26C2×Dic422D411yescharacteristic
27#16809950—42no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C4⋊Dic211—22no—
31C4⋊Dic211—22no—
32C4⋊Dic211—21no—
33C4⋊Dic211D411yescharacteristic
34C4⋊Dic211D412yes—
35Dic211⋊C4—22no—
36Dic211⋊C4—21no—
37C4×Dic211—22no—
38C4×Dic211—21no—
39C2×Q8×C211D411yescharacteristic
40C2×D4×C211—21no—
41C2×D4×C211—21no—
42C2×D4×C211—21no—
43SD16×C211—42no—
44C2×C8×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211D411yescharacteristic
47C42×C211—21no—
48C42×C211C2311yescharacteristic
49Dic422—21no—
50Dic422—21no—
51Dic422—41no—
52Dic422—41no—
53C2×Dic211—21no—
54C2×Dic211—21no—
55C2×Dic211—22no—
56C211⋊C8—41no—
57C211⋊C8—41no—
58C23×C211—41no—
59Q8×C211—21no—
60Q8×C211SD1612yescomplemented
61D4×C211—42no—
62D4×C211—41no—
63D4×C211—41no—
64D4×C211—41no—
65D4×C211—41no—
66C2×C4×C211—21no—
67C2×C4×C211—21no—
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211C2×D411yescharacteristic
71C8×C211—41no—
72C22×C211—81no—
73C22×C211—81no—
74C22×C211—81no—
75C22×C211#12011yescharacteristic
76C4×C211—22no—
77C4×C211—21no—
78C4×C211—41no—
79C4×C211C2×SD1611yescharacteristic
80C4×C211#13511yescharacteristic
81C4×C211AGL(1,ℤ/8)11yescharacteristic
82C4×C211#13711yescharacteristic
83Dic211—41no—
84Dic211—42no—
85Dic211—41no—
86C2×C211—161no—
87C2×C211#44811yescharacteristic
88C2×C211#44911yescharacteristic
89C2×C211#45411yescharacteristic
90C211#162611yescharacteristic, complemented, Sylow
91#1626—2111nomaximal, Sylow
92#557—2111no—
93#486—2111no—
94#460—2111no—
95#460D21111yescharacteristic
96#332—2111no—
97Q8⋊C8—2111no—
98Q8⋊C8—2111no—
99C2×SD16—21no—
100C2×SD16—4221no—
101C4⋊Q8—4221no—
102C4⋊Q8—2111no—
103#127C2×D21111yescharacteristic
104#125—4222no—
105#125—4221no—
106C4×Q8—4221no—
107C4×Q8—4221no—
108C4×Q8—2111no—
109C4×Q8C2×D21111yescharacteristic
110C4⋊C8—2111no—
111C4⋊C8C2×D21111yescharacteristic
112#102—21no—
113#102—4221no—
114#102—4222no—
115C4×C8—2111no—
116C2×Q8—4221no—
117C2×Q8—2111no—
118C2×Q8C22⋊D21111yescharacteristic
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122SD16—42no—
123SD16—8442no—
124C2×C8—21no—
125C2×C8—4221no—
126C2×C8—4221no—
127C4⋊C4—21no—
128C4⋊C4—4222no—
129C4⋊C4—4222no—
130C4⋊C4—4222no—
131C4⋊C4—4221no—
132C4⋊C4—4221no—
133C4⋊C4—2111no—
134C4⋊C4—2112no—
135C4⋊C4C22⋊D21111yescharacteristic
136C42—21no—
137C42—4222no—
138C42—4221no—
139C42C22×D21111yescharacteristic
140C23—41no—
141Q8—21no—
142Q8—8441no—
143Q8—8441no—
144Q8—4221no—
145Q8—4221no—
146Q8#1680995012yescomplemented
147D4—42no—
148D4—41no—
149D4—41no—
150D4—41no—
151D4—41no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—4221no—
155C2×C4—4221no—
156C2×C4—4222no—
157C2×C4D4×D21111yescharacteristic
158C2×C4D4×D21111yescharacteristic
159C2×C4C2×C22⋊D21111yescharacteristic
160C8—41no—
161C8—8441no—
162C8—8441no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22#2026994511yescharacteristic
167C4—21no—
168C4—22no—
169C4—41no—
170C4—8441no—
171C4—8442no—
172C4—8441no—
173C4C2×#1680995011yescharacteristic
174C4#2026999811yescharacteristic
175C4#2027000411yescharacteristic
176C4#2027000711yescharacteristic
177C2—161no—
178C2#2721328411yescharacteristic
179C2#2721329011yescharacteristic
180C2#2721331711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#460×C211C2
socleC22×C211#120
prime radical#127×C211C22
hypercenter#460D211
hypocenterC211#1626
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #27213875C2semidirect
4. #27212891C2semidirect
5. Q8⋊(C211⋊C8)C2semidirect
6. #27212876C2semidirect
60. Q8×C211SD16semidirect
90. C211#1626semidirect
146. Q8#16809950semidirect

Maximal subgroups

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

subgroupindex
#272138752
#272134922
#272133432
#272128912
Q8⋊(C211⋊C8)2
#272128762
#460×C2112
#1626211