Finite Group Databasebeta

← #41013720

Subgroups — #41013720

Order 27008.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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
1C2×#20269951C211yesmaximal, characteristic
2C2×#20269897C212yesmaximal, complemented
3C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
4#27212908C212yesmaximal, complemented
5C2×#117×C211C211yesmaximal, characteristic, complemented
6C22×D844C2211yescharacteristic
7C22×C22⋊D211—21no—
8C22×C211⋊C8C2212yes—
9#20269951—22no—
10#20269951C414yescomplemented
11C2×C4⋊Dic211C2211yescharacteristic
12C2×(C2×D211)⋊C4—21no—
13#20269897—24no—
14C22⋊(C211⋊C8)—22no—
15C23×C4×C211C2211yescharacteristic
16#117×C211—24no—
17C2×C4⋊C4×C211C2212yes—
18C2×C22⋊C4×C211—21no—
19C2×C42×C211—21no—
20C23×D211—21no—
21C2×D844—22no—
22C2×D844—21no—
23C2×D844—22no—
24C2×D844C2×C412yes—
25C2×C22⋊D211—24no—
26C2×C22⋊D211—44no—
27C22×Dic211—21no—
28C2×C211⋊C8—22no—
29C2×C211⋊C8—22no—
30C2×C211⋊C8—22no—
31C4⋊Dic211—21no—
32C4⋊Dic211C2×C412yes—
33(C2×D211)⋊C4—24no—
34C24×C211D411yescharacteristic
35C22×C4×C211—22no—
36C22×C4×C211—22no—
37C22×C4×C211—22no—
38C22×C4×C211C2×C412yes—
39C22×C4×C211D411yescharacteristic
40C22×C4×C211D412yes—
41C22×C4×C211C2311yescharacteristic
42C4⋊C4×C211—22no—
43C4⋊C4×C211D414yescomplemented
44C22⋊C4×C211—24no—
45C42×C211—24no—
46C22×D211—22no—
47C22×D211—41no—
48C22×D211—41no—
49C22×D211—42no—
50C22×D211—42no—
51D844—24no—
52D844—42no—
53C22⋊D211—48no—
54C2×Dic211—22no—
55C2×Dic211—42no—
56C211⋊C8—42no—
57C23×C211—21no—
58C23×C211—21no—
59C23×C211—22no—
60C23×C211—22no—
61C23×C211C22⋊C412yes—
62C23×C211C2×D411yescharacteristic
63C2×C4×C211—21no—
64C2×C4×C211—21no—
65C2×C4×C211—24no—
66C2×C4×C211—24no—
67C2×C4×C211—24no—
68C2×C4×C211—42no—
69C2×C4×C211—42no—
70C2×C4×C211C22⋊C412yes—
71C2×C4×C211C22×C411yescharacteristic
72C2×C4×C211C2×D412yes—
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211#4112yes—
75C22×C211—22no—
76C22×C211—22no—
77C22×C211—22no—
78C22×C211—42no—
79C22×C211—42no—
80C22×C211C2×C22⋊C411yescharacteristic
81C22×C211#12011yescharacteristic
82C22×C211#12311yescharacteristic
83C22×C211#13512yes—
84C22×C211AGL(1,ℤ/8)12yes—
85C2×D211—42no—
86C2×D211—42no—
87C2×D211—42no—
88C2×D211—82no—
89C2×D211—82no—
90C4×C211—22no—
91C4×C211—44no—
92C4×C211C4×D412yes—
93C4×C211#12112yes—
94Dic211—42no—
95C2×C211—42no—
96C2×C211#38611yescharacteristic
97C2×C211#41611yescharacteristic
98C2×C211#41811yescharacteristic
99C2×C211#44912yes—
100C2×C211#48212yes—
101D211—82no—
102C211#187511yescharacteristic, complemented, Sylow
103#1875—2111nomaximal, Sylow
104C2×#121—2111no—
105C2×#117D21111yescharacteristic, complemented
106C2×#102—2112no—
107C2×C22⋊C8—2111no—
108#340—2112no—
109C22×D4—4221no—
110C22×D4—2111no—
111C23×C4C2×D21111yescharacteristic
112C22×C8—2112no—
113#121—4222no—
114#121—2114no—
115#117—24no—
116C2×C4⋊C4—2111no—
117C2×C4⋊C4C2×D21112yes—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—4221no—
120C2×C42—21no—
121#102—4224no—
122C22⋊C8—4222no—
123C24—4221no—
124C24C22⋊D21111yescharacteristic
125C2×D4—8444no—
126C2×D4—4224no—
127C2×D4—4222no—
128C2×D4—4221no—
129C2×D4—4222no—
130C2×D4—2112no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—4221no—
135C22×C4C4×D21112yes—
136C22×C4C22⋊D21111yescharacteristic
137C22×C4D84412yes—
138C22×C4C22×D21111yescharacteristic
139C2×C8—4222no—
140C2×C8—4222no—
141C2×C8—4222no—
142C4⋊C4—22no—
143C4⋊C4—4221no—
144C4⋊C4—2112no—
145C4⋊C4C22⋊D21114yescomplemented
146C22⋊C4—24no—
147C22⋊C4—4224no—
148C42—24no—
149C23—21no—
150C23—21no—
151C23—22no—
152C23—22no—
153C23—8441no—
154C23—8441no—
155C23—8442no—
156C23—8442no—
157C23—4222no—
158C23(C2×D211)⋊C412yes—
159C23C2×C22⋊D21111yescharacteristic
160D4—8448no—
161D4—8442no—
162D4—4224no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—8442no—
171C2×C4—4222no—
172C2×C4(C2×D211)⋊C412yes—
173C2×C4C2×C4×D21111yescharacteristic
174C2×C4C2×C22⋊D21112yes—
175C2×C4C2×D84411yescharacteristic
176C2×C4#1680996212yes—
177C8—8442no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—8442no—
184C22—8442no—
185C22—8442no—
186C22—16882no—
187C22—16882no—
188C22C2×(C2×D211)⋊C411yescharacteristic
189C22#2026994611yescharacteristic
190C22#2026996011yescharacteristic
191C22#2026999712yes—
192C22#2027000212yes—
193C4—22no—
194C4—44no—
195C4—8442no—
196C4C4×C22⋊D21112yes—
197C4#2026995112yes—
198C2—42no—
199C2—16882no—
200C2#2721302511yescharacteristic
201C2#2721314011yescharacteristic
202C2#2721315011yescharacteristic
203C2#2721329112yes—
204C2#2721347012yes—
subgroupG/H
centerC23C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#117×C211C2
socleC23×C211C2×D4
prime radicalC2×#20269951C2
cosocleC22×C4×C211C23
hypercenterC2×#117D211
hypocenterC211#1875
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#20269897C2semidirect
3. C2×C22⋊(C211⋊C8)C2semidirect
4. #27212908C2semidirect
5. C2×#117×C211C2semidirect
10. #20269951C4semidirect
43. C4⋊C4×C211D4semidirect
102. C211#1875semidirect
105. C2×#117D211semidirect
145. C4⋊C4C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699512
C2×#202698972
C2×C22⋊(C211⋊C8)2
#272129082
C2×#117×C2112
#1875211