Finite Group Databasebeta

← #44409098

Subgroups — #44409098

Order 29056.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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#28409830C212yesmaximal, complemented
2#28409606C211yesmaximal, characteristic, complemented
3#28409215C214yesmaximal, complemented
4#28409205C214yesmaximal, complemented
5#28409049C212yesmaximal, complemented
6C2×#20300207C211yesmaximal, characteristic, complemented
7C2×2−1+4×C227C211yesmaximal, characteristic, complemented
8C2×#16813884C2211yescharacteristic
9#20300319—22no—
10#20300318C2214yescomplemented
11#20300314—22no—
12#20300313C2214yescomplemented
13#20300308—24no—
14C2×#16813868C2212yes—
15#20300286C2212yescomplemented
16#20300277C2212yescomplemented
17#20300271—22no—
18#20300264—22no—
19#20300247C2211yescharacteristic
20#20300215C2218yescomplemented
21#20300207C2214yescomplemented
222−1+4×C227—24no—
232−1+4×C227—22no—
242−1+4×C227C2214yes—
25C2×#41×C227—22no—
26C2×#41×C227—22no—
27C2×#41×C227C2212yes—
28C22×Q8×C227—22no—
29C22×Q8×C227C2211yescharacteristic
30#16813884—22no—
31#16813884—22no—
32C2×Dic454C2311yescharacteristic
33C2×D908C2311yescharacteristic
34C2×C22⋊D227—21no—
35C2×C4×D227—21no—
36#16813874—24no—
37#16813872—24no—
38#16813871—24no—
39#16813869—24no—
40#16813868—22no—
41#16813868C2314yes—
42C2×C227⋊C8—22no—
43C4⋊Dic227—21no—
44Dic227⋊C4—24no—
45C22⋊Dic227—21no—
46(C2×D227)⋊C4—24no—
47C4×Dic227C2312yes—
48#41×C227—28no—
49#41×C227—28no—
50#41×C227—28no—
51#41×C227—28no—
52#41×C227—22no—
53#41×C227—22no—
54#41×C227D414yes—
55#41×C227C2314yes—
56C2×Q8×C227—22no—
57C2×Q8×C227—28no—
58C2×Q8×C227—28no—
59C2×Q8×C227—22no—
60C2×Q8×C227—22no—
61C2×Q8×C227D412yes—
62C2×Q8×C227D412yes—
63C2×Q8×C227C2312yes—
64C2×D4×C227—22no—
65C2×D4×C227—22no—
66C2×D4×C227D412yes—
67C22×C4×C227—24no—
68C22×C4×C227—22no—
69C22×C4×C227D412yes—
70C22×C4×C227C2311yescharacteristic
71C22×D227—21no—
72Dic454—22no—
73D908—22no—
74C22⋊D227—41no—
75C22⋊D227—41no—
76C2×Dic227—22no—
77C2×Dic227—21no—
78C4×D227—42no—
79C227⋊C8—24no—
80C23×C227—22no—
81C23×C227C2×D411yescharacteristic
82Q8×C227—28no—
83Q8×C227—22no—
84Q8×C227—24no—
85Q8×C227—24no—
86Q8×C227C2×D414yes—
87D4×C227—28no—
88D4×C227—24no—
89D4×C227—24no—
90D4×C227—22no—
91D4×C227C2×D414yes—
92C2×C4×C227—216no—
93C2×C4×C227—22no—
94C2×C4×C227—22no—
95C2×C4×C227—22no—
96C2×C4×C227—24no—
97C2×C4×C227—24no—
98C2×C4×C227C2×D414yes—
99C2×C4×C227C2×D411yescharacteristic
100C2×C4×C227C2×D412yes—
101C2×C4×C227C2×D412yes—
102C2×C4×C227C2411yescharacteristic
103C22×C227—24no—
104C22×C227—41no—
105C22×C227—44no—
106C22×C227#12011yescharacteristic
107C22×C227#12011yescharacteristic
108C22×C227C22×D411yescharacteristic
109C2×D227—42no—
110C2×D227—41no—
111C4×C227—24no—
112C4×C227—24no—
113C4×C227#12012yes—
114C4×C227C22×D412yes—
115Dic227—42no—
116Dic227—41no—
117C2×C227—21no—
118C2×C227—21no—
119C2×C227—21no—
120C2×C227—44no—
121C2×C227C2×#12011yescharacteristic
122D227—81no—
123C227#302011yescharacteristic, complemented, Sylow
124#3020—2271nomaximal, Sylow
125C2×2−1+4D22711yescharacteristic, complemented
126#575—2272no—
127#536—2271no—
128#456—2274no—
129#454—2274no—
130#421—2272no—
131C2×#101—2271no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D22714yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—2271no—
138C2×#41C2×D22712yes—
139C22×Q8—22no—
140C22×Q8C2×D22711yescharacteristic
141#137—4542no—
142#137—2274no—
143AGL(1,ℤ/8)—4542no—
144AGL(1,ℤ/8)—2274no—
145#135—4544no—
146C2×M16—2272no—
147C4⋊Q8—2272no—
148#124—2272no—
149#123—4542no—
150C22⋊Q8—4542no—
151#117—2271no—
152#104—2278no—
153#101—2274no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—4542no—
161#41—4542no—
162#41C22⋊D22714yes—
163#41C22×D22714yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—2271no—
170C2×Q8C22⋊D22712yes—
171C2×Q8C22⋊D22712yes—
172C2×Q8C22×D22712yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—4541no—
176C2×D4—2271no—
177C2×D4C22⋊D22712yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—4541no—
181C22×C4C22⋊D22712yes—
182C22×C4C22×D22711yescharacteristic
183Q16—4544no—
184SD16—4544no—
185SD16—4544no—
186D8—4544no—
187M16—4542no—
188M16—2274no—
189C2×C8—4542no—
190C4⋊C4—4544no—
191C4⋊C4—4541no—
192C22⋊C4—4541no—
193C22⋊C4—4544no—
194C42—2272no—
195C23—22no—
196C23—4541no—
197C23C2×C22⋊D22711yescharacteristic
198Q8—28no—
199Q8—24no—
200Q8—22no—
201Q8—24no—
202Q8—4542no—
203Q8C2×C22⋊D22714yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—9081no—
209D4—9081no—
210D4—4542no—
211D4C2×C22⋊D22714yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—9082no—
219C2×C4—4542no—
220C2×C4—4541no—
221C2×C4C2×C22⋊D22714yes—
222C2×C4C2×C22⋊D22712yes—
223C2×C4C2×C22⋊D22712yes—
224C2×C4C2×C22⋊D22711yescharacteristic
225C2×C4C23×D22711yescharacteristic
226C8—4544no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—9082no—
231C22—9081no—
232C22#2030025711yescharacteristic
233C22#2030025711yescharacteristic
234C22C22×C22⋊D22711yescharacteristic
235C4—24no—
236C4—24no—
237C4—9082no—
238C4—9081no—
239C4#2030025712yes—
240C4C22×C22⋊D22712yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—18161no—
246C2C2×#2030025711yescharacteristic
subgroupG/H
centerC2C2×#20300257
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
FittingC2×2−1+4×C227C2
socleC2×C227C2×#120
hypercenterC2×2−1+4D227
hypocenterC227#3020
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #28409830C2semidirect
2. #28409606C2semidirect
3. #28409215C2semidirect
4. #28409205C2semidirect
5. #28409049C2semidirect
6. C2×#20300207C2semidirect
7. C2×2−1+4×C227C2semidirect
10. #20300318C22semidirect
12. #20300313C22semidirect
15. #20300286C22semidirect
16. #20300277C22semidirect
20. #20300215C22semidirect
21. #20300207C22semidirect
123. C227#3020semidirect
125. C2×2−1+4D227semidirect

Maximal subgroups

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

subgroupindex
#284098302
#284096062
#284092152
#284092052
#284090492
C2×#203002072
C2×2−1+4×C2272
#3020227