Finite Group Databasebeta

← #44409097

Subgroups — #44409097

Order 29056.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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×#20300336C211yesmaximal, characteristic, complemented
2#28409830C212yesmaximal, complemented
3#28409214C214yesmaximal, complemented
4#28409204C214yesmaximal, complemented
5#28409050C212yesmaximal, complemented
6C2×#20300207C211yesmaximal, characteristic, complemented
7#536×C227C211yesmaximal, characteristic, complemented
8#20300336—24no—
9#20300336—22no—
10#20300336C2214yes—
11C2×#16813885—22no—
12C2×#16813884—22no—
13C2×#16813884C2212yes—
14C2×Q8×D227—22no—
15#20300319C2214yescomplemented
16#20300318—22no—
17#20300314C2214yescomplemented
18#20300313—22no—
19#20300308—24no—
20C2×#16813868C2212yes—
21#20300216C2218yescomplemented
22#20300207C2214yescomplemented
23C2×#41×C227C2211yescharacteristic
24C22×Q8×C227C2211yescharacteristic
25C4⋊Q8×C227C2212yescomplemented
26#124×C227C2212yescomplemented
27#123×C227—22no—
28C22⋊Q8×C227—22no—
29#117×C227C2211yescharacteristic
30#16813885—28no—
31#16813885—28no—
32#16813884—28no—
33#16813884—28no—
34#16813884—22no—
35#16813884—22no—
36#16813884D414yes—
37#16813884C2314yes—
38C2×Dic454—22no—
39C2×Dic454D412yes—
40Q8×D227—28no—
41Q8×D227—28no—
42C2×D908—22no—
43C2×D908D412yes—
44C2×C22⋊D227—22no—
45C2×C4×D227—24no—
46C2×C4×D227—22no—
47#16813874—24no—
48#16813872—24no—
49#16813871—24no—
50#16813869—24no—
51#16813868—22no—
52#16813868C2314yes—
53C2×C227⋊C8—22no—
54#41×C227—22no—
55#41×C227—22no—
56C2×Q8×C227—22no—
57C2×Q8×C227—22no—
58C2×Q8×C227D412yes—
59C2×Q8×C227C2311yescharacteristic
60C2×Q8×C227C2312yes—
61C2×D4×C227—21no—
62C2×D4×C227C2311yescharacteristic
63C22×C4×C227—21no—
64C22×C4×C227D412yes—
65C22×C4×C227C2311yescharacteristic
66C4⋊C4×C227—21no—
67C4⋊C4×C227—24no—
68C22⋊C4×C227—21no—
69C22⋊C4×C227—24no—
70C42×C227C2312yes—
71C22×D227—22no—
72Dic454—28no—
73Dic454—22no—
74Dic454C2×D414yes—
75D908—28no—
76D908—22no—
77D908C2×D414yes—
78C22⋊D227—24no—
79C22⋊D227—24no—
80C2×Dic227—22no—
81C4×D227—216no—
82C4×D227—24no—
83C4×D227—24no—
84C227⋊C8—24no—
85C23×C227—21no—
86C23×C227C2×D411yescharacteristic
87Q8×C227—24no—
88Q8×C227—24no—
89Q8×C227—22no—
90D4×C227—22no—
91D4×C227—41no—
92D4×C227—41no—
93C2×C4×C227—22no—
94C2×C4×C227—22no—
95C2×C4×C227—22no—
96C2×C4×C227—21no—
97C2×C4×C227—42no—
98C2×C4×C227C2×D414yes—
99C2×C4×C227C2×D411yescharacteristic
100C2×C4×C227C2×D412yes—
101C2×C4×C227C2×D412yes—
102C2×C4×C227C2411yescharacteristic
103C22×C227—41no—
104C22×C227—42no—
105C22×C227—41no—
106C22×C227#12011yescharacteristic
107C22×C227#12011yescharacteristic
108C22×C227C22×D411yescharacteristic
109C2×D227—24no—
110C2×D227—44no—
111C4×C227—24no—
112C4×C227—42no—
113C4×C227—41no—
114C4×C227#12012yes—
115C4×C227C22×D412yes—
116Dic227—24no—
117C2×C227—21no—
118C2×C227—21no—
119C2×C227—21no—
120C2×C227—81no—
121C2×C227C2×#12011yescharacteristic
122D227—44no—
123C227#302011yescharacteristic, complemented, Sylow
124#3020—2271nomaximal, Sylow
125C2×2−1+4—2271no—
126#575—2272no—
127#536D22711yescharacteristic, complemented
128#456—2274no—
129#454—2274no—
130#421—2272no—
131C2×#101—2271no—
1322−1+4—4544no—
1332−1+4—4542no—
1342−1+4—2274no—
135C2×#41—4542no—
136C2×#41—4542no—
137C2×#41—2272no—
138C2×#41C2×D22711yescharacteristic
139C22×Q8—4542no—
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⋊Q8C2×D22712yescomplemented
148#124C2×D22712yescomplemented
149#123—22no—
150C22⋊Q8—22no—
151#117C2×D22711yescharacteristic
152#104—2278no—
153#101—2274no—
154#41—22no—
155#41—22no—
156#41—4548no—
157#41—4548no—
158#41—4548no—
159#41—4542no—
160#41—4548no—
161#41—4542no—
162#41—2274no—
163#41—2274no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—4542no—
167C2×Q8—4548no—
168C2×Q8—4548no—
169C2×Q8—2272no—
170C2×Q8C22⋊D22712yes—
171C2×Q8C22×D22711yescharacteristic
172C2×Q8C22×D22712yes—
173C2×D4—21no—
174C2×D4—4542no—
175C2×D4—4542no—
176C2×D4—2272no—
177C2×D4C22×D22711yescharacteristic
178C22×C4—21no—
179C22×C4—4544no—
180C22×C4—4542no—
181C22×C4C22⋊D22712yes—
182C22×C4C22×D22711yescharacteristic
183Q16—4544no—
184SD16—4544no—
185SD16—4544no—
186D8—4544no—
187M16—4542no—
188M16—2274no—
189C2×C8—4542no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C22×D22712yes—
195C23—21no—
196C23—4542no—
197C23C2×C22⋊D22711yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—22no—
201Q8—4548no—
202Q8—4542no—
203Q8—2274no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—4548no—
208D4—4544no—
209D4—4544no—
210D4—4542no—
211D4—2274no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—45416no—
218C2×C4—4542no—
219C2×C4—4544no—
220C2×C4—4544no—
221C2×C4D4×D22712yes—
222C2×C4D4×D22712yes—
223C2×C4C2×C22⋊D22714yes—
224C2×C4C2×C22⋊D22711yescharacteristic
225C2×C4C23×D22711yescharacteristic
226C8—4544no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—9084no—
231C22—4544no—
232C22#2030025611yescharacteristic
233C22#2030025611yescharacteristic
234C22C22×C22⋊D22711yescharacteristic
235C4—24no—
236C4—42no—
237C4—41no—
238C4—4544no—
239C4#2030025612yes—
240C4C2×D4×D22712yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2—9084no—
246C2C2×#2030025611yescharacteristic
subgroupG/H
centerC2C2×#20300256
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#536×C227C2
socleC2×C227C2×#120
hypercenter#536D227
hypocenterC227#3020
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#20300336C2semidirect
2. #28409830C2semidirect
3. #28409214C2semidirect
4. #28409204C2semidirect
5. #28409050C2semidirect
6. C2×#20300207C2semidirect
7. #536×C227C2semidirect
15. #20300319C22semidirect
17. #20300314C22semidirect
21. #20300216C22semidirect
22. #20300207C22semidirect
25. C4⋊Q8×C227C22semidirect
26. #124×C227C22semidirect
123. C227#3020semidirect
127. #536D227semidirect
147. C4⋊Q8C2×D227semidirect
148. #124C2×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003362
#284098302
#284092142
#284092042
#284090502
C2×#203002072
#536×C2272
#3020227