Finite Group Databasebeta

← #41012681

Subgroups — #41012681

Order 27008.

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

normal subgroups only

This group has 240 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#27213800C211yesmaximal, characteristic
2#27213542C211yesmaximal, characteristic, complemented
3#27213348C211yesmaximal, characteristic, complemented
4#27212893C211yesmaximal, characteristic, complemented
5D4⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6#27212873C211yesmaximal, characteristic, complemented
7#474×C211C211yesmaximal, characteristic, complemented
8C2×#16809961—21no—
9C2×#16809949—21no—
10#20269977C2211yescharacteristic
11#20269961—22no—
12#20269957—21no—
13#20269951—21no—
14C4×D844C2211yescharacteristic
15C2×C4⋊Dic211—21no—
16#20269908—22no—
17C4⋊(C211⋊C8)C2211yescharacteristic
18#20269901—21no—
19C4×C211⋊C8C2211yescharacteristic
20C4⋊Q8×C211C2211yescharacteristic
21C4×D4×C211C2211yescharacteristic
22#107×C211—21no—
23C4⋊C8×C211C2211yescharacteristic
24#102×C211—21no—
25#16809961—42no—
26#16809961—42no—
27C2×Dic422—21no—
28C2×D844D411yescharacteristic
29C2×C22⋊D211—22no—
30C22×Dic211—22no—
31C2×C4×D211—21no—
32#16809949—42no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C4⋊Dic211—22no—
36C4⋊Dic211—21no—
37C4⋊Dic211D411yescharacteristic
38C4⋊Dic211D412yes—
39(C2×D211)⋊C4—22no—
40(C2×D211)⋊C4—21no—
41C2×Q8×C211—21no—
42C2×D4×C211D411yescharacteristic
43C22×C4×C211—21no—
44C2×C8×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211D411yescharacteristic
48C22⋊C4×C211—21no—
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51Dic422—41no—
52Dic422—41no—
53D844—21no—
54D844—21no—
55C22⋊D211—44no—
56C2×Dic211—21no—
57C2×Dic211—22no—
58C2×Dic211—44no—
59C4×D211—41no—
60C4×D211—41no—
61C211⋊C8—22no—
62C211⋊C8—41no—
63C23×C211—21no—
64Q8×C211—41no—
65Q8×C211—41no—
66D4×C211—21no—
67D4×C211SD1612yescomplemented
68C2×C4×C211—21no—
69C2×C4×C211—41no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C2×C4×C211C2×D411yescharacteristic
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C8×C211—41no—
76C22×C211—22no—
77C22×C211—41no—
78C22×C211—41no—
79C22×C211#12011yescharacteristic
80C2×D211—41no—
81C2×D211—41no—
82C2×D211—41no—
83C4×C211—21no—
84C4×C211—41no—
85C4×C211—81no—
86C4×C211C2×SD1611yescharacteristic
87C4×C211#13511yescharacteristic
88C4×C211AGL(1,ℤ/8)11yescharacteristic
89C4×C211AGL(1,ℤ/8)11yescharacteristic
90Dic211—41no—
91Dic211—42no—
92C2×C211—42no—
93C2×C211#44911yescharacteristic
94C2×C211#45011yescharacteristic
95C2×C211#45611yescharacteristic
96D211—81no—
97C211#165411yescharacteristic, complemented, Sylow
98#1654—2111nomaximal, Sylow
99#547—2111no—
100#498—2111no—
101#474D21111yescharacteristic, complemented
102#461—2111no—
103#332—2111no—
104D4⋊C8—2111no—
105D4⋊C8—2111no—
106C2×#41—4221no—
107C2×SD16—4221no—
108C4⋊Q8—2111no—
109C4⋊Q8C2×D21111yescharacteristic
110#123—4222no—
111C22⋊Q8—4221no—
112#121—4221no—
113C4×D4—2111no—
114C4×D4C2×D21111yescharacteristic
115C2×C4⋊C4—4221no—
116#107—21no—
117#106—4222no—
118C4⋊C8—2111no—
119C4⋊C8C2×D21111yescharacteristic
120#103—4221no—
121#102—21no—
122C4×C8—2111no—
123#41—8442no—
124#41—8442no—
125C2×Q8—21no—
126C2×Q8—4221no—
127C2×D4—4222no—
128C2×D4—2111no—
129C2×D4C22⋊D21111yescharacteristic
130C22×C4—21no—
131C22×C4—4222no—
132C22×C4—4221no—
133SD16—8442no—
134C2×C8—21no—
135C2×C8—4221no—
136C2×C8—4221no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—4222no—
140C4⋊C4—4221no—
141C4⋊C4—2111no—
142C4⋊C4—2112no—
143C4⋊C4C22⋊D21111yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—4222no—
146C22⋊C4—4221no—
147C42C22×D21111yescharacteristic
148C23—21no—
149C23—4221no—
150Q8—41no—
151Q8—41no—
152Q8—8441no—
153Q8—8441no—
154D4—21no—
155D4—8444no—
156D4—4221no—
157D4—4221no—
158D4#1680994912yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—8444no—
164C2×C4—8441no—
165C2×C4—8441no—
166C2×C4—4221no—
167C2×C4—4222no—
168C2×C4D4×D21111yescharacteristic
169C2×C4D4×D21111yescharacteristic
170C2×C4C2×C22⋊D21111yescharacteristic
171C8—41no—
172C8—8441no—
173C8—4222no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—8441no—
178C22—8441no—
179C22—8441no—
180C22#2026994511yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—8441no—
185C4—8442no—
186C4C2×#1680994911yescharacteristic
187C4#2026999611yescharacteristic
188C4#2027000111yescharacteristic
189C4#2027000211yescharacteristic
190C2—42no—
191C2—16881no—
192C2#2721328811yescharacteristic
193C2#2721329611yescharacteristic
194C2#2721332711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#474×C211C2
socleC22×C211#120
prime radical#27213800C2
hypercenter#474D211
hypocenterC211#1654
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #27213542C2semidirect
3. #27213348C2semidirect
4. #27212893C2semidirect
5. D4⋊(C211⋊C8)C2semidirect
6. #27212873C2semidirect
7. #474×C211C2semidirect
67. D4×C211SD16semidirect
97. C211#1654semidirect
101. #474D211semidirect
158. D4#16809949semidirect

Maximal subgroups

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

subgroupindex
#272138002
#272135422
#272133482
#272128932
D4⋊(C211⋊C8)2
#272128732
#474×C2112
#1654211