Finite Group Databasebeta

← #41014465

Subgroups — #41014465

Order 27008.

394 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213657C212yesmaximal, complemented
3#27213162C212yesmaximal, complemented
4#27212903C211yesmaximal, characteristic, complemented
5#407×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2212yes—
7#20270010—22no—
8#20270005—22no—
9#20269999—22no—
10#20269960—22no—
11#20269956—22no—
12#20269951—22no—
13#20269946—22no—
14C4×C22⋊D211—24no—
15#20269936C2212yes—
16#20269903—24no—
17C23×C4×C211C2211yescharacteristic
18C2×M16×C211C2211yescharacteristic
19C2×M16×C211C2211yescharacteristic
20C22⋊C8×C211—22no—
21#16809962—22no—
22#16809962—22no—
23#16809962—22no—
24#16809962—22no—
25C2×Dic422D412yescomplemented
26C2×D844D412yescomplemented
27C2×C22⋊D211—24no—
28C2×C22⋊D211—22no—
29C2×C4×D211—22no—
30Dic844—22no—
31#16809951—24no—
32D1688—22no—
33C4⋊Dic211—22no—
34Dic211⋊C4—24no—
35C22⋊Dic211—24no—
36C22⋊Dic211—22no—
37(C2×D211)⋊C4—24no—
38C4×Dic211D414yescomplemented
39C24×C211D411yescharacteristic
40C22×C4×C211—21no—
41C22×C4×C211—22no—
42C22×C4×C211—22no—
43C22×C4×C211—21no—
44C22×C4×C211D411yescharacteristic
45C22×C4×C211C2311yescharacteristic
46M16×C211—22no—
47M16×C211—22no—
48C2×C8×C211—21no—
49C2×C8×C211D412yescomplemented
50C22×D211—22no—
51Dic422—22no—
52Dic422—22no—
53D844—22no—
54D844—22no—
55C22⋊D211—48no—
56C22⋊D211—42no—
57C22⋊D211—42no—
58C2×Dic211—24no—
59C2×Dic211—22no—
60C4×D211—42no—
61C4×D211—42no—
62C23×C211—22no—
63C23×C211—21no—
64C23×C211—42no—
65C23×C211C2×D411yescharacteristic
66C2×C4×C211—22no—
67C2×C4×C211—21no—
68C2×C4×C211—41no—
69C2×C4×C211—42no—
70C2×C4×C211—41no—
71C2×C4×C211C2×D412yes—
72C2×C4×C211C2×D412yes—
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211#4111yescharacteristic
75C8×C211—22no—
76C8×C211—41no—
77C22×C211—22no—
78C22×C211—24no—
79C22×C211—41no—
80C22×C211—42no—
81C22×C211—41no—
82C22×C211—41no—
83C22×C211#12012yes—
84C22×C211#12111yescharacteristic
85C2×D211—44no—
86C2×D211—42no—
87C4×C211—41no—
88C4×C211#12011yescharacteristic
89C4×C211#12112yes—
90C4×C211#12711yescharacteristic
91Dic211—44no—
92Dic211—42no—
93C2×C211—22no—
94C2×C211—21no—
95C2×C211—42no—
96C2×C211#39211yescharacteristic
97D211—82no—
98C211#201011yescharacteristic, complemented, Sylow
99#2010—2111nomaximal, Sylow
100#575—2111no—
101#525—2112no—
102#421—2112no—
103#407D21111yescharacteristic, complemented
104H(3,ℤ/4)—2111no—
105C2×#41—2112no—
106C23×C4C2×D21111yescharacteristic
107#137—4222no—
108AGL(1,ℤ/8)—4222no—
109#135—4222no—
110C2×M16C2×D21111yescharacteristic
111C2×M16C2×D21111yescharacteristic
112#123—4222no—
113C22⋊Q8—4222no—
114#121—4222no—
115#120—4222no—
116C4×D4—4224no—
117#117—2112no—
118#104—4224no—
119C22⋊C8—22no—
120C24D84411yescharacteristic
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—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—4222no—
134C22×C4D84411yescharacteristic
135C22×C4C22×D21111yescharacteristic
136Q16—4222no—
137SD16—4224no—
138D8—4222no—
139M16—22no—
140M16—22no—
141C2×C8—21no—
142C2×C8C22⋊D21112yescomplemented
143C4⋊C4—4224no—
144C4⋊C4—4222no—
145C22⋊C4—4224no—
146C22⋊C4—4224no—
147C22⋊C4—4222no—
148C42—2114no—
149C23—22no—
150C23—21no—
151C23—42no—
152C23—4222no—
153C23C2×D84411yescharacteristic
154Q8—4222no—
155Q8—4222no—
156D4—8448no—
157D4—8442no—
158D4—8442no—
159D4—4222no—
160D4—4222no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—8442no—
167C2×C4—8442no—
168C2×C4—4224no—
169C2×C4—4222no—
170C2×C4D4×D21112yes—
171C2×C4D4×D21112yes—
172C2×C4C2×C22⋊D21111yescharacteristic
173C2×C4#1680996211yescharacteristic
174C8—22no—
175C8—41no—
176C22—24no—
177C22—22no—
178C22—41no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—8444no—
183C22—8442no—
184C22#2026994712yes—
185C22#2026995111yescharacteristic
186C4—41no—
187C4—8444no—
188C4—8442no—
189C4#2026994511yescharacteristic
190C4#2026995012yes—
191C4#2026997311yescharacteristic
192C2—22no—
193C2—21no—
194C2—42no—
195C2—16882no—
196C2#2721305511yescharacteristic
subgroupG/H
centerC4#20269973
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#407×C211C2
socleC2×C211#392
hypercenter#407D211
hypocenterC211#2010
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #27213946C2semidirect
2. #27213657C2semidirect
3. #27213162C2semidirect
4. #27212903C2semidirect
5. #407×C211C2semidirect
25. C2×Dic422D4semidirect
26. C2×D844D4semidirect
38. C4×Dic211D4semidirect
49. C2×C8×C211D4semidirect
98. C211#2010semidirect
103. #407D211semidirect
142. C2×C8C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272139462
#272136572
#272131622
#272129032
#407×C2112
#2010211