Finite Group Databasebeta

← #41014466

Subgroups — #41014466

Order 27008.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 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#27213946C211yesmaximal, characteristic, complemented
2#27213658C212yesmaximal, complemented
3#27213164C212yesmaximal, complemented
4#27213109C211yesmaximal, characteristic, complemented
5H(3,ℤ/4)×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2212yes—
7C22×C4×D211C2211yescharacteristic
8#20270010—22no—
9#20270005—22no—
10#20269999—22no—
11C2×#16809946C2211yescharacteristic
12#20269961—22no—
13#20269957—22no—
14#20269952—22no—
15#20269947—22no—
16C4×D844—24no—
17#20269905—24no—
18(C2×D211)⋊C8—22no—
19C2×M16×C211C2211yescharacteristic
20#117×C211C2212yes—
21C23×D211D411yescharacteristic
22#16809962—22no—
23#16809962—22no—
24#16809962—22no—
25#16809962—22no—
26C2×Dic422D412yescomplemented
27C2×D844—24no—
28C2×D844D412yescomplemented
29C2×C22⋊D211—22no—
30C22×Dic211D411yescharacteristic
31C2×C4×D211—21no—
32C2×C4×D211—22no—
33C2×C4×D211—22no—
34C2×C4×D211—22no—
35C2×C4×D211—21no—
36Dic844—22no—
37#16809951—24no—
38D1688—22no—
39#16809946—22no—
40C2×C211⋊C8—21no—
41C4⋊Dic211—24no—
42(C2×D211)⋊C4—24no—
43(C2×D211)⋊C4—24no—
44C22×C4×C211C2311yescharacteristic
45M16×C211—22no—
46C2×C8×C211D412yescomplemented
47C4⋊C4×C211—22no—
48C22⋊C4×C211—22no—
49C42×C211D414yescomplemented
50C22×D211—22no—
51C22×D211—21no—
52C22×D211—22no—
53C22×D211—42no—
54Dic422—22no—
55Dic422—22no—
56D844—22no—
57D844—22no—
58D844—48no—
59C22⋊D211—42no—
60C22⋊D211—42no—
61C2×Dic211—22no—
62C2×Dic211—22no—
63C2×Dic211—21no—
64C4×D211—41no—
65C4×D211—42no—
66C4×D211—42no—
67C4×D211—41no—
68C4×D211—42no—
69C211⋊C8—41no—
70C23×C211C2×D411yescharacteristic
71C2×C4×C211—24no—
72C2×C4×C211C2×D412yes—
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D412yes—
75C2×C4×C211#4111yescharacteristic
76C8×C211—22no—
77C22×C211—41no—
78C22×C211#12012yes—
79C22×C211#12111yescharacteristic
80C2×D211—24no—
81C2×D211—22no—
82C2×D211—42no—
83C2×D211—41no—
84C2×D211—41no—
85C2×D211—44no—
86C2×D211—42no—
87C4×C211—44no—
88C4×C211#12011yescharacteristic
89C4×C211#12112yes—
90C4×C211#12711yescharacteristic
91Dic211—41no—
92Dic211—42no—
93C2×C211—22no—
94C2×C211—21no—
95C2×C211#39211yescharacteristic
96D211—42no—
97D211—82no—
98C211#201011yescharacteristic, complemented, Sylow
99#2010—2111nomaximal, Sylow
100#575—2111no—
101#525—2112no—
102#421—2112no—
103#407—2111no—
104H(3,ℤ/4)D21111yescharacteristic, complemented
105C2×#41—2112no—
106C23×C4—2111no—
107#137—4222no—
108AGL(1,ℤ/8)—4222no—
109#135—4222no—
110C2×M16—2111no—
111C2×M16C2×D21111yescharacteristic
112#123—4222no—
113C22⋊Q8—4222no—
114#121—4222no—
115#120—4222no—
116C4×D4—4224no—
117#117C2×D21112yes—
118#104—4224no—
119C22⋊C8—4222no—
120C24—2111no—
121#41—4222no—
122#41—4222no—
123#41—4222no—
124#41—4222no—
125C2×Q8—2112no—
126C2×D4—4224no—
127C2×D4—4222no—
128C2×D4—2112no—
129C22×C4—4222no—
130C22×C4—4221no—
131C22×C4—4222no—
132C22×C4—4222no—
133C22×C4—4221no—
134C22×C4—2111no—
135C22×C4C22×D21111yescharacteristic
136Q16—4222no—
137SD16—4224no—
138D8—4222no—
139M16—22no—
140M16—4222no—
141C2×C8—4221no—
142C2×C8D84412yescomplemented
143C4⋊C4—22no—
144C4⋊C4—4224no—
145C22⋊C4—22no—
146C22⋊C4—4224no—
147C22⋊C4—4224no—
148C42D84414yescomplemented
149C23—8442no—
150C23—4222no—
151C23—4222no—
152C23—4221no—
153C23D4×D21111yescharacteristic
154Q8—4222no—
155Q8—4222no—
156D4—8448no—
157D4—8442no—
158D4—8442no—
159D4—4222no—
160D4—4222no—
161C2×C4—24no—
162C2×C4—8441no—
163C2×C4—8442no—
164C2×C4—8442no—
165C2×C4—8441no—
166C2×C4—8442no—
167C2×C4—4222no—
168C2×C4—4222no—
169C2×C4—4221no—
170C2×C4D4×D21112yes—
171C2×C4C2×D84411yescharacteristic
172C2×C4C2×D84412yes—
173C2×C4#1680996311yescharacteristic
174C8—22no—
175C8—8441no—
176C22—41no—
177C22—8442no—
178C22—8441no—
179C22—8444no—
180C22—8441no—
181C22—8442no—
182C22—4224no—
183C22—4222no—
184C22#2026994712yes—
185C22#2026995211yescharacteristic
186C4—44no—
187C4—8441no—
188C4—8442no—
189C4#2026994711yescharacteristic
190C4#2026995212yes—
191C4#2026997411yescharacteristic
192C2—22no—
193C2—21no—
194C2—8442no—
195C2—16882no—
196C2#2721305611yescharacteristic
subgroupG/H
centerC4#20269974
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingH(3,ℤ/4)×C211C2
socleC2×C211#392
hypercenterH(3,ℤ/4)D211
hypocenterC211#2010
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. #27213946C2semidirect
2. #27213658C2semidirect
3. #27213164C2semidirect
4. #27213109C2semidirect
5. H(3,ℤ/4)×C211C2semidirect
26. C2×Dic422D4semidirect
28. C2×D844D4semidirect
46. C2×C8×C211D4semidirect
49. C42×C211D4semidirect
98. C211#2010semidirect
104. H(3,ℤ/4)D211semidirect
142. C2×C8D844semidirect
148. C42D844semidirect

Maximal subgroups

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

subgroupindex
#272139462
#272136582
#272131642
#272131092
H(3,ℤ/4)×C2112
#2010211