Finite Group Databasebeta

← #41013844

Subgroups — #41013844

Order 27008.

358 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20269936C211yesmaximal, characteristic, complemented
3C2×#20269903C212yesmaximal, complemented
4#27212907C212yesmaximal, complemented
5#407×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C2211yescharacteristic
7#20269960—21no—
8#20269956C412yescomplemented
9#20269951C412yescomplemented
10#20269946—21no—
11C4×C22⋊D211—22no—
12#20269936—24no—
13#20269936C2211yescharacteristic
14C2×C4⋊Dic211—21no—
15C2×C22⋊Dic211—21no—
16C2×C4×Dic211C2212yes—
17#20269903—24no—
18C23×C4×C211C2211yescharacteristic
19C2×M16×C211C2212yes—
20C22⋊C8×C211—22no—
21#16809962—22no—
22#16809962—22no—
23C2×Dic422C2×C411yescharacteristic
24C2×D844C2×C411yescharacteristic
25C2×C22⋊D211—22no—
26C2×C22⋊D211—21no—
27C22×Dic211—22no—
28C2×C4×D211—21no—
29C4⋊Dic211—24no—
30C4⋊Dic211C2×C412yes—
31Dic211⋊C4—22no—
32C22⋊Dic211—24no—
33C22⋊Dic211—22no—
34C22⋊Dic211—21no—
35(C2×D211)⋊C4—22no—
36C4×Dic211—22no—
37C4×Dic211—21no—
38C4×Dic211D414yescomplemented
39C24×C211D411yescharacteristic
40C22×C4×C211—22no—
41C22×C4×C211—22no—
42C22×C4×C211C2×C412yes—
43C22×C4×C211D412yes—
44C22×C4×C211D411yescharacteristic
45C22×C4×C211C2311yescharacteristic
46M16×C211—22no—
47M16×C211—22no—
48C2×C8×C211—22no—
49C22×D211—21no—
50Dic422—22no—
51D844—22no—
52C22⋊D211—44no—
53C22⋊D211—41no—
54C22⋊D211—41no—
55C2×Dic211—22no—
56C2×Dic211—21no—
57C2×Dic211—24no—
58C2×Dic211—44no—
59C4×D211—42no—
60C23×C211—21no—
61C23×C211—21no—
62C23×C211—42no—
63C23×C211C22⋊C412yes—
64C23×C211C2×D411yescharacteristic
65C2×C4×C211—21no—
66C2×C4×C211—21no—
67C2×C4×C211—24no—
68C2×C4×C211—24no—
69C2×C4×C211C22⋊C412yes—
70C2×C4×C211C22×C411yescharacteristic
71C2×C4×C211C2×D412yes—
72C2×C4×C211C2×D411yescharacteristic
73C2×C4×C211#4112yes—
74C8×C211—42no—
75C22×C211—22no—
76C22×C211—22no—
77C22×C211—22no—
78C22×C211—22no—
79C22×C211—42no—
80C22×C211—42no—
81C22×C211C2×C22⋊C411yescharacteristic
82C22×C211#12011yescharacteristic
83C22×C211#12311yescharacteristic
84C2×D211—41no—
85C2×D211—42no—
86C4×C211—22no—
87C4×C211C4×D412yes—
88C4×C211#12112yes—
89Dic211—44no—
90Dic211—41no—
91Dic211—42no—
92C2×C211—21no—
93C2×C211—21no—
94C2×C211—42no—
95C2×C211#38611yescharacteristic
96C2×C211#42112yes—
97D211—81no—
98C211#189711yescharacteristic, complemented, Sylow
99#1897—2111nomaximal, Sylow
100#525—2111no—
101C2×#117—2111no—
102C2×#104—2112no—
103#407D21111yescharacteristic, complemented
104#339—2112no—
105C2×#41—2111no—
106C23×C4C2×D21111yescharacteristic
107C2×M16C2×D21112yes—
108#123—4221no—
109C22⋊Q8—2112no—
110#121—2112no—
111#120—4221no—
112C4×D4—4222no—
113#117—4224no—
114#117—2111no—
115C2×C4⋊C4—4221no—
116C2×C22⋊C4—4221no—
117C2×C42—2112no—
118#104—4224no—
119C22⋊C8—22no—
120C24D84411yescharacteristic
121#41—4222no—
122#41—4222no—
123C2×Q8—2111no—
124C2×D4—4222no—
125C2×D4—4221no—
126C2×D4—2111no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—4222no—
130C22×C4—4221no—
131C22×C4C4×D21112yes—
132C22×C4C22⋊D21112yes—
133C22×C4D84411yescharacteristic
134C22×C4C22×D21111yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—4224no—
139C4⋊C4—4222no—
140C4⋊C4—2112no—
141C22⋊C4—4224no—
142C22⋊C4—4222no—
143C22⋊C4—4221no—
144C22⋊C4—4222no—
145C42—4222no—
146C42—4221no—
147C42—2114no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—4221no—
152C23(C2×D211)⋊C412yes—
153C23C2×D84411yescharacteristic
154Q8—4222no—
155D4—8444no—
156D4—8441no—
157D4—8441no—
158D4—4222no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—8444no—
164C2×C4—8442no—
165C2×C4—4222no—
166C2×C4—4221no—
167C2×C4—4224no—
168C2×C4(C2×D211)⋊C412yes—
169C2×C4C2×C4×D21111yescharacteristic
170C2×C4D4×D21112yes—
171C2×C4C2×C22⋊D21111yescharacteristic
172C2×C4#1680996112yes—
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—8442no—
181C22—8441no—
182C22C2×(C2×D211)⋊C411yescharacteristic
183C22#2026994711yescharacteristic
184C22#2026996111yescharacteristic
185C4—22no—
186C4—8444no—
187C4—8442no—
188C4—8441no—
189C4(C22⋊D211)⋊C412yes—
190C4#2026994912yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—16881no—
195C2#2721302311yescharacteristic
196C2#2721316212yes—
subgroupG/H
centerC2×C4C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#407×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213657C2
cosocleC22×C4×C211C23
hypercenter#407D211
hypocenterC211#1897
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#20269936C2semidirect
3. C2×#20269903C2semidirect
4. #27212907C2semidirect
5. #407×C211C2semidirect
8. #20269956C4semidirect
9. #20269951C4semidirect
38. C4×Dic211D4semidirect
98. C211#1897semidirect
103. #407D211semidirect

Maximal subgroups

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

subgroupindex
#272136572
C2×#202699362
C2×#202699032
#272129072
#407×C2112
#1897211