Finite Group Databasebeta

← #41013841

Subgroups — #41013841

Order 27008.

442 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 101 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#27213657C211yesmaximal, characteristic
2C2×#20269905C212yesmaximal, complemented
3#27213110C211yesmaximal, characteristic, complemented
4#27212906C212yesmaximal, complemented
5C22×C42×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2211yescharacteristic
7C2×#16809946C2212yes—
8#20269960—21no—
9#20269956C412yescomplemented
10#20269951C412yescomplemented
11#20269946—21no—
12C4×C22⋊D211—22no—
13#20269936C2211yescharacteristic
14#20269905—24no—
15C22⋊(C211⋊C8)—22no—
16C23×C4×C211—21no—
17C23×C4×C211C2211yescharacteristic
18C2×C42×C211—21no—
19C2×C42×C211—24no—
20C2×C42×C211C2212yes—
21#16809962—22no—
22#16809962—22no—
23C2×Dic422C2×C411yescharacteristic, complemented
24C2×D844C2×C411yescharacteristic, complemented
25C2×C22⋊D211—22no—
26C2×C22⋊D211—21no—
27C2×C4×D211—21no—
28#16809946—22no—
29#16809946—22no—
30C2×C211⋊C8—22no—
31C4⋊Dic211C2×C412yescomplemented
32Dic211⋊C4—22no—
33C22⋊Dic211—22no—
34C22⋊Dic211—21no—
35(C2×D211)⋊C4—22no—
36C4×Dic211—21no—
37C24×C211D411yescharacteristic
38C22×C4×C211—24no—
39C22×C4×C211—28no—
40C22×C4×C211—22no—
41C22×C4×C211—22no—
42C22×C4×C211—22no—
43C22×C4×C211C2×C412yes—
44C22×C4×C211D411yescharacteristic
45C22×C4×C211D412yes—
46C22×C4×C211C2311yescharacteristic
47C42×C211—24no—
48C42×C211—22no—
49C42×C211D414yescomplemented
50C22×D211—21no—
51Dic422—22no—
52D844—22no—
53C22⋊D211—44no—
54C22⋊D211—41no—
55C22⋊D211—41no—
56C2×Dic211—22no—
57C2×Dic211—21no—
58C4×D211—42no—
59C211⋊C8—42no—
60C23×C211—24no—
61C23×C211—21no—
62C23×C211—21no—
63C23×C211C22⋊C412yes—
64C23×C211C2×D411yescharacteristic
65C2×C4×C211—216no—
66C2×C4×C211—28no—
67C2×C4×C211—21no—
68C2×C4×C211—24no—
69C2×C4×C211—21no—
70C2×C4×C211—24no—
71C2×C4×C211—24no—
72C2×C4×C211C22⋊C412yes—
73C2×C4×C211C22×C411yescharacteristic
74C2×C4×C211C2×D412yes—
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211#4112yes—
77C22×C211—24no—
78C22×C211—24no—
79C22×C211—22no—
80C22×C211—22no—
81C22×C211—22no—
82C22×C211#10414yescomplemented
83C22×C211C2×C22⋊C411yescharacteristic
84C22×C211#12011yescharacteristic
85C22×C211#12311yescharacteristic
86C2×D211—41no—
87C2×D211—42no—
88C4×C211—28no—
89C4×C211—22no—
90C4×C211C4×D412yes—
91C4×C211#12112yes—
92Dic211—41no—
93Dic211—42no—
94C2×C211—21no—
95C2×C211—24no—
96C2×C211—21no—
97C2×C211#38611yescharacteristic
98C2×C211C2×#10412yes—
99D211—81no—
100C211#189611yescharacteristic, complemented, Sylow
101#1896—2111nomaximal, Sylow
102#525—2111no—
103C22×C42D21111yescharacteristic, complemented
104C2×#104—2112no—
105#407—2111no—
106#339—2112no—
107C2×#41—2111no—
108C23×C4—21no—
109C23×C4C2×D21111yescharacteristic
110C2×M16—2112no—
111#123—4221no—
112C22⋊Q8—2112no—
113#121—2112no—
114#120—4221no—
115C4×D4—4222no—
116#117—2111no—
117C2×C42—21no—
118C2×C42—24no—
119C2×C42C2×D21112yes—
120#104—4224no—
121C22⋊C8—4222no—
122C24C22⋊D21111yescharacteristic
123#41—4222no—
124#41—4222no—
125C2×Q8—2111no—
126C2×D4—4222no—
127C2×D4—4221no—
128C2×D4—2111no—
129C22×C4—24no—
130C22×C4—28no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—4221no—
135C22×C4C4×D21112yes—
136C22×C4C22⋊D21111yescharacteristic
137C22×C4D84412yes—
138C22×C4C22×D21111yescharacteristic
139M16—4222no—
140M16—4222no—
141C2×C8—4222no—
142C4⋊C4—4222no—
143C4⋊C4—2112no—
144C22⋊C4—4222no—
145C22⋊C4—4221no—
146C22⋊C4—4222no—
147C42—24no—
148C42—22no—
149C42—4221no—
150C42C22⋊D21114yescomplemented
151C23—24no—
152C23—21no—
153C23—21no—
154C23—4221no—
155C23(C2×D211)⋊C412yes—
156C23C2×C22⋊D21111yescharacteristic
157Q8—4222no—
158D4—8444no—
159D4—8441no—
160D4—8441no—
161D4—4222no—
162C2×C4—216no—
163C2×C4—28no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—8442no—
170C2×C4—4222no—
171C2×C4—4221no—
172C2×C4(C2×D211)⋊C412yes—
173C2×C4C2×C4×D21111yescharacteristic
174C2×C4C2×C22⋊D21112yes—
175C2×C4C2×D84411yescharacteristic
176C2×C4#1680996212yes—
177C8—8442no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—8442no—
184C22—8441no—
185C22#2026990514yescomplemented
186C22C2×(C2×D211)⋊C411yescharacteristic
187C22#2026994611yescharacteristic
188C22#2026996011yescharacteristic
189C4—28no—
190C4—22no—
191C4—8442no—
192C4—8441no—
193C4C4×C22⋊D21112yes—
194C4#2026995112yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—16881no—
199C2#2721302511yescharacteristic
200C2C2×#2026990512yes—
subgroupG/H
centerC2×C4C2×D844
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC22×C42×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213657C2
cosocleC22×C4×C211C23
hypercenterC22×C42D211
hypocenterC211#1896
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#20269905C2semidirect
3. #27213110C2semidirect
4. #27212906C2semidirect
5. C22×C42×C211C2semidirect
9. #20269956C4semidirect
10. #20269951C4semidirect
23. C2×Dic422C2×C4semidirect
24. C2×D844C2×C4semidirect
31. C4⋊Dic211C2×C4semidirect
49. C42×C211D4semidirect
82. C22×C211#104semidirect
100. C211#1896semidirect
103. C22×C42D211semidirect
150. C42C22⋊D211semidirect
185. C22#20269905semidirect

Maximal subgroups

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

subgroupindex
#272136572
C2×#202699052
#272131102
#272129062
C22×C42×C2112
#1896211