Finite Group Databasebeta

← #41014228

Subgroups — #41014228

Order 27008.

320 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 83 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
6#27212992C211yesmaximal, 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×Dic211C2211yescharacteristic
17C2×C4×Dic211C2211yescharacteristic
18Dic211⋊C8—22no—
19#20269905—22no—
20#20269904—22no—
21#20269903—22no—
22C211⋊#95—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—21no—
45C4×Dic211C2×C411yescharacteristic
46C4×Dic211C2×C411yescharacteristic
47C4×Dic211D412yes—
48C4×Dic211D412yescomplemented
49#41×C211—21no—
50#41×C211—21no—
51#41×C211—21no—
52#41×C211—21no—
53C2×Q8×C211C2×C411yescharacteristic
54C2×D4×C211—21no—
55C2×D4×C211C2×C411yescharacteristic
56C22×C4×C211—21no—
57C22×C4×C211—21no—
58C22×C4×C211C2311yescharacteristic
59M16×C211—21no—
60M16×C211—21no—
61C2×C8×C211—21no—
62C42×C211—21no—
63C42×C211D412yescomplemented
64C22×D211—21no—
65Dic422—21no—
66Dic422—21no—
67D844—21no—
68D844—21no—
69C22⋊D211—44no—
70C22⋊D211—41no—
71C22⋊D211—41no—
72C2×Dic211—21no—
73C2×Dic211—22no—
74C2×Dic211—42no—
75C2×Dic211—42no—
76C2×Dic211C22⋊C412yes—
77C2×Dic211C22⋊C412yes—
78C4×D211—41no—
79C4×D211—41no—
80C211⋊C8—41no—
81C23×C211—21no—
82C23×C211C2×D411yescharacteristic
83Q8×C211—21no—
84Q8×C211—21no—
85D4×C211—21no—
86D4×C211—21no—
87D4×C211—41no—
88D4×C211—41no—
89C2×C4×C211—21no—
90C2×C4×C211—22no—
91C2×C4×C211—42no—
92C2×C4×C211—41no—
93C2×C4×C211—41no—
94C2×C4×C211C22×C411yescharacteristic
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211#4111yescharacteristic
99C2×C4×C211#4111yescharacteristic
100C8×C211—41no—
101C22×C211—22no—
102C22×C211—41no—
103C22×C211—42no—
104C22×C211C2×C22⋊C411yescharacteristic
105C22×C211#12111yescharacteristic
106C22×C211#12111yescharacteristic
107C2×D211—41no—
108C2×D211—42no—
109C4×C211—42no—
110C4×C211—41no—
111C4×C211C4×D411yescharacteristic
112C4×C211C4×D411yescharacteristic
113C4×C211#12411yescharacteristic
114C4×C211#12711yescharacteristic
115Dic211—22no—
116Dic211—22no—
117Dic211—42no—
118Dic211—41no—
119C2×C211—21no—
120C2×C211—21no—
121C2×C211—81no—
122C2×C211#39011yescharacteristic
123C2×C211#42112yes—
124D211—81no—
125C211#196311yescharacteristic, complemented, Sylow
126#1963—2111nomaximal, Sylow
127#532—2111no—
128#423—2111no—
129C2×#104—2111no—
130C2×#104—2111no—
131C2×#104—2111no—
132C2×#104D21111yescharacteristic, complemented
133#376—2111no—
134C2×#41—2111no—
135C2×#41C2×D21111yescharacteristic
136C2×M16—2111no—
137C2×M16C2×D21111yescharacteristic
138C4⋊Q8—2111no—
139#127—2111no—
140#124—2111no—
141#124—2111no—
142#121—4222no—
143C4×D4—4222no—
144C2×C42—2111no—
145C2×C42—2111no—
146C2×C42C2×D21111yescharacteristic
147C4⋊C8—4222no—
148#104—22no—
149#104—4222no—
150#104—4222no—
151#104—4222no—
152#95—4222no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—4221no—
158#41—4221no—
159#41—4221no—
160#41—4221no—
161C2×Q8—2111no—
162C2×Q8C4×D21111yescharacteristic
163C2×D4—21no—
164C2×D4—4221no—
165C2×D4—4222no—
166C2×D4—2111no—
167C2×D4C4×D21111yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—4221no—
171C22×C4—4221no—
172C22×C4—2112no—
173C22×C4C22×D21111yescharacteristic
174M16—21no—
175M16—21no—
176M16—4221no—
177M16—4221no—
178C2×C8—21no—
179C2×C8—4221no—
180C4⋊C4—4222no—
181C22⋊C4—4222no—
182C22⋊C4—4222no—
183C42—21no—
184C42—4221no—
185C42—2112no—
186C42—2111no—
187C42—2111no—
188C42—2112no—
189C42C22⋊D21112yescomplemented
190C23—21no—
191C23—4221no—
192C23D4×D21111yescharacteristic
193Q8—21no—
194Q8—21no—
195Q8—4221no—
196Q8—4221no—
197D4—21no—
198D4—21no—
199D4—41no—
200D4—41no—
201D4—8441no—
202D4—8441no—
203D4—8444no—
204D4—4221no—
205D4—4221no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—8442no—
212C2×C4—8441no—
213C2×C4—8441no—
214C2×C4—8442no—
215C2×C4—4221no—
216C2×C4—4222no—
217C2×C4—2112no—
218C2×C4—2112no—
219C2×C4C2×C4×D21111yescharacteristic
220C2×C4D4×D21111yescharacteristic
221C2×C4D4×D21111yescharacteristic
222C2×C4C2×C22⋊D21111yescharacteristic
223C2×C4#1680996111yescharacteristic
224C2×C4#1680996211yescharacteristic
225C8—41no—
226C8—8441no—
227C22—22no—
228C22—42no—
229C22—41no—
230C22—8442no—
231C22—8441no—
232C22C22⋊C4×D21111yescharacteristic
233C22#2026994911yescharacteristic
234C22#2026995011yescharacteristic
235C4—42no—
236C4—41no—
237C4—8442no—
238C4—8441no—
239C4—4222no—
240C4—4222no—
241C4(C22⋊D211)⋊C411yescharacteristic
242C4C4×C22⋊D21111yescharacteristic
243C4#2026996311yescharacteristic
244C4#2026997311yescharacteristic
245C2—21no—
246C2—21no—
247C2—81no—
248C2—16881no—
249C2#2721304611yescharacteristic
250C2#2721316112yes—
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#1963
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. #27212992C2semidirect
7. C2×#104×C211C2semidirect
10. #20269975C4semidirect
11. #20269973C4semidirect
12. #20269966C4semidirect
13. #20269964C4semidirect
48. C4×Dic211D4semidirect
63. C42×C211D4semidirect
125. C211#1963semidirect
132. C2×#104D211semidirect
189. 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
#272129922
C2×#104×C2112
#1963211