Finite Group Databasebeta

← #44398239

Subgroups — #44398239

Order 29056.

334 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 77 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#28409543C211yesmaximal, characteristic
2C2×C227⋊#95C211yesmaximal, characteristic, complemented
3#28408770C212yesmaximal, complemented
4#28408767C212yesmaximal, complemented
5#407×C227C211yesmaximal, characteristic, complemented
6C2×#16813884C411yescharacteristic, complemented
7C23×Dic227—21no—
8#20300271—21no—
9#20300267C411yescharacteristic, complemented
10#20300267C2211yescharacteristic
11#20300262C411yescharacteristic, complemented
12#20300262C2211yescharacteristic
13#20300257—21no—
14C4×C22⋊D227—21no—
15C4×C22⋊D227—21no—
16#20300247C411yescharacteristic, complemented
17C227⋊#95—21no—
18C227⋊#95C2212yes—
19C23×C4×C227C2211yescharacteristic
20C2×M16×C227—21no—
21C22⋊C8×C227—21no—
22C22⋊C8×C227C2212yes—
23#16813884—42no—
24C2×Dic454C2×C411yescharacteristic, complemented
25C2×D908C2×C411yescharacteristic, complemented
26C2×C22⋊D227—21no—
27C2×C22⋊D227—21no—
28C2×C22⋊D227—21no—
29C22×Dic227—24no—
30C22×Dic227—22no—
31C22×Dic227—42no—
32C22×Dic227—42no—
33C2×C4×D227—21no—
34C4⋊Dic227C2×C411yescharacteristic, complemented
35C4⋊Dic227C2×C411yescharacteristic, complemented
36Dic227⋊C4—21no—
37Dic227⋊C4—21no—
38C22⋊Dic227—21no—
39C22⋊Dic227—21no—
40C22⋊Dic227—21no—
41(C2×D227)⋊C4—21no—
42(C2×D227)⋊C4—21no—
43C4×Dic227—21no—
44C24×C227D411yescharacteristic
45C22×C4×C227—42no—
46C22×C4×C227C2×C411yescharacteristic
47C22×C4×C227C2×C411yescharacteristic
48C22×C4×C227D411yescharacteristic
49C22×C4×C227D411yescharacteristic
50C22×C4×C227D411yescharacteristic
51C22×C4×C227C2311yescharacteristic
52M16×C227—42no—
53C2×C8×C227—22no—
54C22×D227—21no—
55Dic454—41no—
56D908—41no—
57C22⋊D227—42no—
58C22⋊D227—42no—
59C22⋊D227—41no—
60C22⋊D227—41no—
61C2×Dic227—24no—
62C2×Dic227—21no—
63C2×Dic227—21no—
64C2×Dic227—21no—
65C2×Dic227—48no—
66C2×Dic227—42no—
67C2×Dic227—42no—
68C4×D227—42no—
69C23×C227—22no—
70C23×C227—22no—
71C23×C227—21no—
72C23×C227—21no—
73C23×C227C22⋊C411yescharacteristic
74C23×C227C22⋊C411yescharacteristic
75C23×C227C2×D411yescharacteristic
76C2×C4×C227—22no—
77C2×C4×C227—22no—
78C2×C4×C227—41no—
79C2×C4×C227—41no—
80C2×C4×C227—41no—
81C2×C4×C227—41no—
82C2×C4×C227C22⋊C411yescharacteristic
83C2×C4×C227C22⋊C411yescharacteristic
84C2×C4×C227C22×C411yescharacteristic
85C2×C4×C227C2×D411yescharacteristic
86C8×C227—42no—
87C22×C227—24no—
88C22×C227—24no—
89C22×C227—21no—
90C22×C227—21no—
91C22×C227—21no—
92C22×C227—21no—
93C22×C227—21no—
94C22×C227—21no—
95C22×C227#9912yes—
96C22×C227#10412yes—
97C22×C227C2×C22⋊C411yescharacteristic
98C22×C227AGL(1,ℤ/8)11yescharacteristic
99C22×C227#13711yescharacteristic
100C2×D227—41no—
101C2×D227—41no—
102C2×D227—41no—
103C4×C227—21no—
104C4×C227—21no—
105C4×C227—21no—
106C4×C227—21no—
107Dic227—44no—
108Dic227—41no—
109Dic227—41no—
110Dic227—41no—
111C2×C227—21no—
112C2×C227—22no—
113C2×C227—22no—
114C2×C227—21no—
115C2×C227C2×#9911yescharacteristic
116C2×C227#41711yescharacteristic
117C2×C227C2×#10411yescharacteristic
118D227—81no—
119C227#150411yescharacteristic, complemented, Sylow
120#1504—2271nomaximal, Sylow
121#525—2271no—
122#407D22711yescharacteristic, complemented
123C2×#95—2271no—
124#328—2272no—
125#327—2272no—
126C2×#41—2271no—
127C23×C4—4541no—
128C23×C4C2×D22711yescharacteristic
129C2×M16—21no—
130#123—4541no—
131C22⋊Q8—2271no—
132C22⋊Q8—2271no—
133#121—2271no—
134#121—2271no—
135#120—4541no—
136C4×D4—4541no—
137C4×D4—4541no—
138#117—2271no—
139C22⋊C8—21no—
140C22⋊C8C2×D22712yes—
141#95—4541no—
142#95—2272no—
143C24D90811yescharacteristic
144#41—9082no—
145C2×Q8—2271no—
146C2×D4—4541no—
147C2×D4—4541no—
148C2×D4—4541no—
149C2×D4—2271no—
150C22×C4—42no—
151C22×C4—9082no—
152C22×C4—9082no—
153C22×C4—4544no—
154C22×C4—4542no—
155C22×C4—4541no—
156C22×C4C4×D22711yescharacteristic
157C22×C4C4×D22711yescharacteristic
158C22×C4C22⋊D22711yescharacteristic
159C22×C4C22⋊D22711yescharacteristic
160C22×C4D90811yescharacteristic
161C22×C4C22×D22711yescharacteristic
162M16—42no—
163C2×C8—22no—
164C4⋊C4—4541no—
165C4⋊C4—4541no—
166C4⋊C4—2271no—
167C4⋊C4—2271no—
168C22⋊C4—4541no—
169C22⋊C4—4541no—
170C22⋊C4—4541no—
171C22⋊C4—4541no—
172C22⋊C4—4541no—
173C42—4541no—
174C23—21no—
175C23—21no—
176C23—22no—
177C23—22no—
178C23—4541no—
179C23(C2×D227)⋊C411yescharacteristic
180C23(C2×D227)⋊C411yescharacteristic
181C23C2×D90811yescharacteristic
182Q8—9081no—
183D4—9081no—
184D4—9081no—
185D4—9081no—
186D4—9082no—
187D4—9082no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—9082no—
195C2×C4—9088no—
196C2×C4—9082no—
197C2×C4—9082no—
198C2×C4—4544no—
199C2×C4—4541no—
200C2×C4—4541no—
201C2×C4—4541no—
202C2×C4(C2×D227)⋊C411yescharacteristic
203C2×C4(C2×D227)⋊C411yescharacteristic
204C2×C4C2×C4×D22711yescharacteristic
205C2×C4C2×C22⋊D22711yescharacteristic
206C8—42no—
207C22—24no—
208C22—24no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—9081no—
216C22—9081no—
217C22—9081no—
218C22#2030020212yes—
219C22#2030021412yes—
220C22C2×(C2×D227)⋊C411yescharacteristic
221C22#2030031611yescharacteristic
222C22#2030032111yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—9084no—
228C4—9081no—
229C4—9081no—
230C4—9081no—
231C2—21no—
232C2—21no—
233C2—22no—
234C2—22no—
235C2—18161no—
236C2C2×#2030020211yescharacteristic
237C2#2840903411yescharacteristic
238C2C2×#2030021411yescharacteristic
subgroupG/H
centerC22C2×(C2×D227)⋊C4
commutatorC2×C4×C227C22×C4
FrattiniC22×C4C22×D227
Fitting#407×C227C2
socleC22×C227C2×C22⋊C4
prime radical#28409543C2
cosocleC22×C4×C227C23
hypercenter#407D227
hypocenterC227#1504
LCSG ▷ C2×C4×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×C227⋊#95C2semidirect
3. #28408770C2semidirect
4. #28408767C2semidirect
5. #407×C227C2semidirect
6. C2×#16813884C4semidirect
9. #20300267C4semidirect
11. #20300262C4semidirect
16. #20300247C4semidirect
24. C2×Dic454C2×C4semidirect
25. C2×D908C2×C4semidirect
34. C4⋊Dic227C2×C4semidirect
35. C4⋊Dic227C2×C4semidirect
119. C227#1504semidirect
122. #407D227semidirect

Maximal subgroups

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

subgroupindex
#284095432
C2×C227⋊#952
#284087702
#284087672
#407×C2272
#1504227