Finite Group Databasebeta

← #41014233

Subgroups — #41014233

Order 27008.

408 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20269891C214yesmaximal, complemented
3C4×C22⋊Dic211C211yesmaximal, characteristic, complemented
4#380×C211C211yesmaximal, characteristic, complemented
5C22×D844C411yescharacteristic, complemented
6C22×C22⋊D211C2212yes—
7#20269951—21no—
8#20269951—21no—
9#20269951—21no—
10#20269951—21no—
11C2×C4⋊Dic211C411yescharacteristic, complemented
12C2×C22⋊Dic211C2212yes—
13C2×(C2×D211)⋊C4C412yescomplemented
14C2×C4×Dic211—21no—
15#20269891—28no—
16C211⋊#95—21no—
17C23×C4×C211C2211yescharacteristic
18C2×C4⋊C4×C211—21no—
19C2×C22⋊C4×C211C2212yes—
20#95×C211—21no—
21C23×D211C2×C412yes—
22C2×D844—21no—
23C2×D844—21no—
24C2×D844—42no—
25C2×C22⋊D211—24no—
26C2×C22⋊D211—22no—
27C2×C22⋊D211—22no—
28C2×C22⋊D211—24no—
29C22×Dic211—22no—
30C22×Dic211C2×C412yes—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C22⋊Dic211—22no—
34C22⋊Dic211D414yescomplemented
35(C2×D211)⋊C4—22no—
36(C2×D211)⋊C4—22no—
37C4×Dic211—42no—
38C24×C211C2311yescharacteristic
39C22×C4×C211—22no—
40C22×C4×C211—42no—
41C22×C4×C211C2×C411yescharacteristic
42C22×C4×C211C2×C411yescharacteristic
43C22×C4×C211D411yescharacteristic
44C22×C4×C211D411yescharacteristic
45C22×C4×C211D411yescharacteristic
46C22×C4×C211D411yescharacteristic
47C4⋊C4×C211—42no—
48C22⋊C4×C211—24no—
49C22×D211—22no—
50C22×D211—24no—
51C22×D211—42no—
52C22×D211—42no—
53D844—42no—
54D844—42no—
55C22⋊D211—44no—
56C22⋊D211—42no—
57C22⋊D211—42no—
58C2×Dic211—22no—
59C2×Dic211—24no—
60C2×Dic211—24no—
61C2×Dic211—42no—
62C2×Dic211—42no—
63C23×C211—41no—
64C23×C211—41no—
65C23×C211C22×C411yescharacteristic
66C23×C211C2×D412yes—
67C23×C211C2×D411yescharacteristic
68C23×C211C2×D411yescharacteristic
69C23×C211#4112yes—
70C2×C4×C211—22no—
71C2×C4×C211—22no—
72C2×C4×C211—22no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211—42no—
77C2×C4×C211—42no—
78C2×C4×C211—42no—
79C2×C4×C211C22⋊C411yescharacteristic
80C2×C4×C211C22⋊C411yescharacteristic
81C2×C4×C211C22⋊C411yescharacteristic
82C2×C4×C211C22⋊C411yescharacteristic
83C22×C211—21no—
84C22×C211—21no—
85C22×C211—21no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211—21no—
89C22×C211—42no—
90C22×C211—42no—
91C22×C211C2×C22⋊C411yescharacteristic
92C22×C211C4×D412yes—
93C22×C211#12112yes—
94C22×C211#12411yescharacteristic
95C22×C211#12711yescharacteristic
96C2×D211—42no—
97C2×D211—42no—
98C2×D211—42no—
99C2×D211—84no—
100C4×C211—21no—
101C4×C211—21no—
102C4×C211—21no—
103C4×C211—82no—
104C4×C211#9912yescomplemented
105Dic211—44no—
106Dic211—42no—
107C2×C211—22no—
108C2×C211—21no—
109C2×C211—22no—
110C2×C211—21no—
111C2×C211#39011yescharacteristic
112C2×C211C2×#9911yescharacteristic
113C2×C211#41011yescharacteristic
114D211—82no—
115C211#196411yescharacteristic, complemented, Sylow
116#1964—2111nomaximal, Sylow
117C2×#121—2111no—
118C2×#99—2114no—
119#380D21111yescharacteristic, complemented
120C4×C22⋊C4—2111no—
121C22×D4—2112no—
122C22×D4—2111no—
123C23×C4C2×D21111yescharacteristic
124#121—4221no—
125#121—4221no—
126#121—4221no—
127#121—4221no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—2111no—
130C2×C22⋊C4—2112no—
131C2×C22⋊C4—2112no—
132C2×C22⋊C4C2×D21112yes—
133C2×C42—4221no—
134#99—4228no—
135#95—21no—
136#95—4221no—
137C24—2112no—
138C24C22×D21111yescharacteristic
139C2×D4—8442no—
140C2×D4—4224no—
141C2×D4—4222no—
142C2×D4—4222no—
143C2×D4—4224no—
144C2×D4—4221no—
145C2×D4—4221no—
146C22×C4—22no—
147C22×C4—42no—
148C22×C4—4222no—
149C22×C4—2112no—
150C22×C4C4×D21111yescharacteristic
151C22×C4C4×D21111yescharacteristic
152C22×C4C22⋊D21111yescharacteristic
153C22×C4C22⋊D21111yescharacteristic
154C22×C4D84411yescharacteristic
155C22×C4D84411yescharacteristic
156C4⋊C4—42no—
157C4⋊C4—4221no—
158C4⋊C4—4221no—
159C22⋊C4—24no—
160C22⋊C4—4222no—
161C22⋊C4—4222no—
162C22⋊C4—4222no—
163C22⋊C4—2114no—
164C42—8442no—
165C23—41no—
166C23—41no—
167C23—8442no—
168C23—8442no—
169C23—4222no—
170C23—4224no—
171C23C2×C4×D21111yescharacteristic
172C23D4×D21112yes—
173C23C2×C22⋊D21111yescharacteristic
174C23C2×D84411yescharacteristic
175C23#1680996312yes—
176D4—8442no—
177D4—8444no—
178D4—8442no—
179D4—8442no—
180D4—8442no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—8442no—
191C2×C4—8442no—
192C2×C4—4222no—
193C2×C4—4224no—
194C2×C4—4224no—
195C2×C4(C2×D211)⋊C411yescharacteristic
196C2×C4(C2×D211)⋊C411yescharacteristic
197C2×C4(C2×D211)⋊C411yescharacteristic
198C2×C4(C2×D211)⋊C411yescharacteristic
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—42no—
206C22—42no—
207C22—8442no—
208C22—8442no—
209C22—8442no—
210C22—16884no—
211C22C2×(C2×D211)⋊C411yescharacteristic
212C22#2026993912yes—
213C22#2026995212yes—
214C22#2026996611yescharacteristic
215C22#2026997311yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—82no—
220C4—8444no—
221C4—8442no—
222C4#2026989112yescomplemented
223C2—21no—
224C2—22no—
225C2—22no—
226C2—21no—
227C2—16882no—
228C2#2721304711yescharacteristic
229C2C2×#2026989111yescharacteristic
230C2#2721311911yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC23×C211C22×C4
FrattiniC24C22×D211
Fitting#380×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC2×#20269951C2
cosocleC24×C211C23
hypercenter#380D211
hypocenterC211#1964
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#20269891C2semidirect
3. C4×C22⋊Dic211C2semidirect
4. #380×C211C2semidirect
5. C22×D844C4semidirect
11. C2×C4⋊Dic211C4semidirect
13. C2×(C2×D211)⋊C4C4semidirect
34. C22⋊Dic211D4semidirect
104. C4×C211#99semidirect
115. C211#1964semidirect
119. #380D211semidirect
222. C4#20269891semidirect

Maximal subgroups

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

subgroupindex
C2×#202699512
C2×#202698912
C4×C22⋊Dic2112
#380×C2112
#1964211