Finite Group Databasebeta

← #41014224

Subgroups — #41014224

Order 27008.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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#27213692C211yesmaximal, characteristic
2#27213169C211yesmaximal, characteristic, complemented
3C2×#20269905C211yesmaximal, characteristic, complemented
4C2×#20269904C211yesmaximal, characteristic, complemented
5C2×#20269903C211yesmaximal, characteristic, complemented
6C42×Dic211C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269975C411yescharacteristic, complemented
11#20269973C411yescharacteristic, complemented
12#20269966C411yescharacteristic, complemented
13#20269964C411yescharacteristic, complemented
14#20269949—22no—
15C4×C22⋊D211—22no—
16C2×C4×Dic211—22no—
17C2×C4×Dic211C2211yescharacteristic
18C2×C4×Dic211C2211yescharacteristic
19Dic211⋊C8—22no—
20#20269905—22no—
21#20269904—22no—
22#20269903—22no—
23C2×#41×C211C2211yescharacteristic
24C2×M16×C211C2211yescharacteristic
25C2×C42×C211C2211yescharacteristic
26#104×C211—22no—
27#16809962—21no—
28#16809962—21no—
29#16809962—21no—
30#16809962—21no—
31C2×Dic422C2×C411yescharacteristic
32C2×D844C2×C411yescharacteristic
33C2×C22⋊D211—22no—
34C2×C22⋊D211—21no—
35C22×Dic211—21no—
36C22×Dic211D412yes—
37C2×C4×D211—21no—
38#16809946—21no—
39#16809946—21no—
40C2×C211⋊C8—21no—
41Dic211⋊C4—22no—
42C22⋊Dic211—22no—
43(C2×D211)⋊C4—22no—
44C4×Dic211—28no—
45C4×Dic211—21no—
46C4×Dic211C2×C411yescharacteristic
47C4×Dic211C2×C411yescharacteristic
48C4×Dic211D412yes—
49C4×Dic211D412yescomplemented
50#41×C211—21no—
51#41×C211—21no—
52#41×C211—21no—
53#41×C211—21no—
54C2×Q8×C211C2×C411yescharacteristic
55C2×D4×C211—21no—
56C2×D4×C211C2×C411yescharacteristic
57C22×C4×C211—21no—
58C22×C4×C211—21no—
59C22×C4×C211C2311yescharacteristic
60M16×C211—21no—
61M16×C211—21no—
62C2×C8×C211—21no—
63C42×C211—21no—
64C42×C211D412yescomplemented
65C22×D211—21no—
66Dic422—21no—
67Dic422—21no—
68D844—21no—
69D844—21no—
70C22⋊D211—41no—
71C22⋊D211—41no—
72C22⋊D211—44no—
73C2×Dic211—24no—
74C2×Dic211—22no—
75C2×Dic211—24no—
76C2×Dic211—21no—
77C2×Dic211C22⋊C412yes—
78C2×Dic211C22⋊C412yes—
79C4×D211—41no—
80C4×D211—41no—
81C211⋊C8—41no—
82C23×C211—21no—
83C23×C211C2×D411yescharacteristic
84Q8×C211—21no—
85Q8×C211—21no—
86D4×C211—21no—
87D4×C211—21no—
88D4×C211—41no—
89D4×C211—41no—
90C2×C4×C211—24no—
91C2×C4×C211—22no—
92C2×C4×C211—21no—
93C2×C4×C211—41no—
94C2×C4×C211—41no—
95C2×C4×C211C22×C411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211C2×D411yescharacteristic
99C2×C4×C211#4111yescharacteristic
100C2×C4×C211#4111yescharacteristic
101C8×C211—41no—
102C22×C211—22no—
103C22×C211—41no—
104C22×C211—42no—
105C22×C211C2×C22⋊C411yescharacteristic
106C22×C211#12111yescharacteristic
107C22×C211#12111yescharacteristic
108C2×D211—41no—
109C2×D211—42no—
110C4×C211—24no—
111C4×C211—41no—
112C4×C211C4×D411yescharacteristic
113C4×C211C4×D411yescharacteristic
114C4×C211#12411yescharacteristic
115C4×C211#12711yescharacteristic
116Dic211—24no—
117Dic211—22no—
118Dic211—41no—
119Dic211#10414yescomplemented
120C2×C211—21no—
121C2×C211—21no—
122C2×C211—81no—
123C2×C211#39011yescharacteristic
124C2×C211C2×#10412yes—
125D211—81no—
126C211#196211yescharacteristic, complemented, Sylow
127#1962—2111nomaximal, Sylow
128#532—2111no—
129#423—2111no—
130C2×#104—2111no—
131C2×#104—2111no—
132C2×#104—2111no—
133C2×#104D21111yescharacteristic, complemented
134C43—2111no—
135C2×#41—2111no—
136C2×#41C2×D21111yescharacteristic
137C2×M16—2111no—
138C2×M16C2×D21111yescharacteristic
139C4⋊Q8—2111no—
140#127—2111no—
141#124—2111no—
142#124—2111no—
143#121—4222no—
144C4×D4—4222no—
145C2×C42—4222no—
146C2×C42—2111no—
147C2×C42—2111no—
148C2×C42C2×D21111yescharacteristic
149C4⋊C8—4222no—
150#104—22no—
151#104—4222no—
152#104—4222no—
153#104—4222no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—21no—
158#41—4221no—
159#41—4221no—
160#41—4221no—
161#41—4221no—
162C2×Q8—2111no—
163C2×Q8C4×D21111yescharacteristic
164C2×D4—21no—
165C2×D4—4221no—
166C2×D4—4222no—
167C2×D4—2111no—
168C2×D4C4×D21111yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—4221no—
172C22×C4—4221no—
173C22×C4—2112no—
174C22×C4C22×D21111yescharacteristic
175M16—21no—
176M16—21no—
177M16—4221no—
178M16—4221no—
179C2×C8—21no—
180C2×C8—4221no—
181C4⋊C4—4222no—
182C22⋊C4—4222no—
183C22⋊C4—4222no—
184C42—21no—
185C42—4228no—
186C42—4221no—
187C42—2112no—
188C42—2111no—
189C42—2111no—
190C42—2112no—
191C42C22⋊D21112yescomplemented
192C23—21no—
193C23—4221no—
194C23D4×D21111yescharacteristic
195Q8—21no—
196Q8—21no—
197Q8—4221no—
198Q8—4221no—
199D4—21no—
200D4—21no—
201D4—41no—
202D4—41no—
203D4—8441no—
204D4—8441no—
205D4—8444no—
206D4—4221no—
207D4—4221no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—8441no—
214C2×C4—8441no—
215C2×C4—4221no—
216C2×C4—4224no—
217C2×C4—4222no—
218C2×C4—4224no—
219C2×C4—2112no—
220C2×C4—2112no—
221C2×C4C2×C4×D21111yescharacteristic
222C2×C4D4×D21111yescharacteristic
223C2×C4D4×D21111yescharacteristic
224C2×C4C2×C22⋊D21111yescharacteristic
225C2×C4#1680996111yescharacteristic
226C2×C4#1680996211yescharacteristic
227C8—41no—
228C8—8441no—
229C22—22no—
230C22—42no—
231C22—41no—
232C22—8442no—
233C22—8441no—
234C22C22⋊C4×D21111yescharacteristic
235C22#2026994911yescharacteristic
236C22#2026995011yescharacteristic
237C4—24no—
238C4—41no—
239C4—8441no—
240C4—4224no—
241C4—4222no—
242C4—2114no—
243C4(C22⋊D211)⋊C411yescharacteristic
244C4C4×C22⋊D21111yescharacteristic
245C4#2026996311yescharacteristic
246C4#2026997311yescharacteristic
247C2—21no—
248C2—21no—
249C2—81no—
250C2—16881no—
251C2#2721304611yescharacteristic
252C2#104×D21112yes—
subgroupG/H
centerC2×C4D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213692C2
cosocleC22×C4×C211C23
hypercenterC2×#104D211
hypocenterC211#1962
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #27213169C2semidirect
3. C2×#20269905C2semidirect
4. C2×#20269904C2semidirect
5. C2×#20269903C2semidirect
6. C42×Dic211C2semidirect
7. C2×#104×C211C2semidirect
10. #20269975C4semidirect
11. #20269973C4semidirect
12. #20269966C4semidirect
13. #20269964C4semidirect
49. C4×Dic211D4semidirect
64. C42×C211D4semidirect
119. Dic211#104semidirect
126. C211#1962semidirect
133. C2×#104D211semidirect
191. C42C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272136922
#272131692
C2×#202699052
C2×#202699042
C2×#202699032
C42×Dic2112
C2×#104×C2112
#1962211