Finite Group Databasebeta

← #44398248

Subgroups — #44398248

Order 29056.

346 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 65 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#121×D227C211yesmaximal, characteristic
2#28408994C211yesmaximal, characteristic, complemented
3#28408879C211yesmaximal, characteristic, complemented
4#28408767C211yesmaximal, characteristic, complemented
5#28408766C211yesmaximal, characteristic, complemented
6#28408765C211yesmaximal, characteristic, complemented
7#327×C227C211yesmaximal, characteristic, complemented
8C2×D4×D227—21no—
9C2×D4×D227C411yescharacteristic, complemented
10C22×C4×D227C2211yescharacteristic
11C2×#16813867—21no—
12#20300263C411yescharacteristic, complemented
13#20300262C2211yescharacteristic
14#20300261—21no—
15#20300260—21no—
16#20300259C411yescharacteristic, complemented
17C4⋊C4×D227C411yescharacteristic, complemented
18C2×(C2×D227)⋊C4—21no—
19C22⋊C4×D227—21no—
20C2×C4×Dic227—21no—
21C22⋊(C227⋊C8)C2211yescharacteristic, complemented
22(C2×D227)⋊C8—21no—
23C227⋊#95C2211yescharacteristic, complemented
24#121×C227C2211yescharacteristic
25C22⋊C8×C227C2211yescharacteristic, complemented
26#95×C227C2211yescharacteristic, complemented
27C23×D227—21no—
28C23×D227D411yescharacteristic
29C2×D908—21no—
30C2×D908C2×C411yescharacteristic
31C2×C22⋊D227—21no—
32C2×C22⋊D227—21no—
33C2×C22⋊D227—21no—
34D4×D227—42no—
35D4×D227—42no—
36D4×D227—42no—
37C22×Dic227—21no—
38C22×Dic227D411yescharacteristic
39C2×C4×D227—21no—
40C2×C4×D227—42no—
41C2×C4×D227C2×C411yescharacteristic
42C2×C4×D227C2×C411yescharacteristic
43C2×C4×D227D411yescharacteristic
44C2×C4×D227D411yescharacteristic
45#16813867—42no—
46C2×C227⋊C8—21no—
47C4⋊Dic227C2×C411yescharacteristic
48Dic227⋊C4—21no—
49C22⋊Dic227—21no—
50(C2×D227)⋊C4—24no—
51(C2×D227)⋊C4—21no—
52(C2×D227)⋊C4—21no—
53C4×Dic227—42no—
54C2×D4×C227—21no—
55C2×D4×C227C2×C411yescharacteristic
56C22×C4×C227—21no—
57C22×C4×C227C2311yescharacteristic
58C2×C8×C227—21no—
59C4⋊C4×C227C2×C411yescharacteristic
60C22⋊C4×C227—21no—
61C22×D227—21no—
62C22×D227—21no—
63C22×D227—22no—
64C22×D227—21no—
65C22×D227—41no—
66C22×D227—42no—
67C22×D227—42no—
68C22×D227—42no—
69C22×D227C22⋊C411yescharacteristic
70C22×D227C22⋊C411yescharacteristic
71D908—41no—
72D908—42no—
73C22⋊D227—42no—
74C22⋊D227—41no—
75C22⋊D227—41no—
76C22⋊D227—42no—
77C2×Dic227—22no—
78C2×Dic227—22no—
79C2×Dic227—21no—
80C2×Dic227—41no—
81C2×Dic227—41no—
82C2×Dic227C22⋊C411yescharacteristic
83C2×Dic227C22⋊C411yescharacteristic
84C4×D227—22no—
85C4×D227—22no—
86C4×D227—41no—
87C4×D227—41no—
88C4×D227—42no—
89C227⋊C8—41no—
90C23×C227—21no—
91C23×C227C2×D411yescharacteristic
92D4×C227—41no—
93D4×C227—41no—
94D4×C227—41no—
95C2×C4×C227—22no—
96C2×C4×C227—21no—
97C2×C4×C227—41no—
98C2×C4×C227—41no—
99C2×C4×C227—41no—
100C2×C4×C227C22×C411yescharacteristic
101C2×C4×C227C2×D411yescharacteristic
102C8×C227—41no—
103C22×C227—21no—
104C22×C227—21no—
105C22×C227—41no—
106C22×C227—41no—
107C22×C227—41no—
108C22×C227C2×C22⋊C411yescharacteristic
109C22×C227AGL(1,ℤ/8)11yescharacteristic
110C22×C227AGL(1,ℤ/8)11yescharacteristic
111C2×D227—21no—
112C2×D227—21no—
113C2×D227—21no—
114C2×D227—21no—
115C2×D227—21no—
116C2×D227—42no—
117C2×D227—42no—
118C2×D227—41no—
119C2×D227—41no—
120C2×D227—41no—
121C2×D227—84no—
122C2×D227#10412yes—
123C4×C227—21no—
124C4×C227—21no—
125C4×C227—42no—
126C4×C227—41no—
127Dic227—21no—
128Dic227—42no—
129Dic227—41no—
130Dic227#9912yes—
131C2×C227—21no—
132C2×C227—21no—
133C2×C227—81no—
134C2×C227C2×#9911yescharacteristic
135C2×C227#41811yescharacteristic
136C2×C227C2×#10411yescharacteristic
137D227—22no—
138D227—41no—
139D227—81no—
140C227#150711yescharacteristic, complemented, Sylow
141#1507—2271nomaximal, Sylow
142C2×#121—2271no—
143#407—2271no—
144C4×C22⋊C4—2271no—
145#327—2271no—
146#327—2271no—
147#327—2271no—
148#327D22711yescharacteristic, complemented
149C22×D4—4541no—
150C22×D4—2271no—
151C23×C4—2271no—
152C2×M16—4541no—
153#121—4541no—
154#121—4541no—
155#121—2271no—
156#121—2271no—
157#121—2271no—
158#121C2×D22711yescharacteristic
159C2×C4⋊C4—2271no—
160C2×C22⋊C4—4541no—
161C2×C22⋊C4—4541no—
162C2×C42—4541no—
163C22⋊C8—4541no—
164C22⋊C8—2271no—
165C22⋊C8C2×D22711yescharacteristic, complemented
166#95—2271no—
167#95C2×D22711yescharacteristic, complemented
168C24—4541no—
169C24—2271no—
170C2×D4—21no—
171C2×D4—9082no—
172C2×D4—9082no—
173C2×D4—9082no—
174C2×D4—4541no—
175C2×D4—4541no—
176C2×D4—4541no—
177C2×D4—4541no—
178C2×D4—2271no—
179C2×D4C4×D22711yescharacteristic
180C22×C4—21no—
181C22×C4—9082no—
182C22×C4—4541no—
183C22×C4—4541no—
184C22×C4—2271no—
185C22×C4—2271no—
186C22×C4—2271no—
187C22×C4—2271no—
188C22×C4—2271no—
189C22×C4C22×D22711yescharacteristic
190M16—9082no—
191C2×C8—21no—
192C2×C8—4541no—
193C4⋊C4—4541no—
194C4⋊C4—2271no—
195C4⋊C4C4×D22711yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—4544no—
198C22⋊C4—4541no—
199C22⋊C4—4541no—
200C22⋊C4—4541no—
201C42—9082no—
202C23—21no—
203C23—9081no—
204C23—9082no—
205C23—9082no—
206C23—9082no—
207C23—4542no—
208C23—4541no—
209C23—4541no—
210C23—4541no—
211C23—2271no—
212C23—2271no—
213C23D4×D22711yescharacteristic
214D4—41no—
215D4—41no—
216D4—41no—
217D4—9081no—
218D4—9082no—
219D4—9082no—
220D4—9081no—
221D4—9081no—
222D4—9082no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—9081no—
229C2×C4—9081no—
230C2×C4—9082no—
231C2×C4—9081no—
232C2×C4—9081no—
233C2×C4—4542no—
234C2×C4—4542no—
235C2×C4—4542no—
236C2×C4—4541no—
237C2×C4—4542no—
238C2×C4—2271no—
239C2×C4—2271no—
240C2×C4C2×C4×D22711yescharacteristic
241C2×C4D4×D22711yescharacteristic
242C8—41no—
243C8—9081no—
244C22—21no—
245C22—21no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—18164no—
250C22—9082no—
251C22—9082no—
252C22—9081no—
253C22—9081no—
254C22—9081no—
255C22—4541no—
256C22—4541no—
257C22—4541no—
258C22—4541no—
259C22—4541no—
260C22—2272no—
261C22C22⋊C4×D22711yescharacteristic
262C22#2030031211yescharacteristic
263C22#2030031511yescharacteristic
264C4—21no—
265C4—21no—
266C4—42no—
267C4—41no—
268C4—9082no—
269C4—9081no—
270C4—4541no—
271C4—2272no—
272C2—21no—
273C2—21no—
274C2—81no—
275C2—18161no—
276C2—9081no—
277C2—4542no—
278C2#99×D22711yescharacteristic
279C2#2840903511yescharacteristic
280C2#104×D22711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D227
commutatorC2×C4×C227C22×C4
FrattiniC22×C4C22×D227
Fitting#327×C227C2
socleC22×C227C2×C22⋊C4
prime radical#121×D227C2
cosocleC22×C4×C227C23
hypercenter#327D227
hypocenterC227#1507
LCSG ▷ C2×C4×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #28408994C2semidirect
3. #28408879C2semidirect
4. #28408767C2semidirect
5. #28408766C2semidirect
6. #28408765C2semidirect
7. #327×C227C2semidirect
9. C2×D4×D227C4semidirect
12. #20300263C4semidirect
16. #20300259C4semidirect
17. C4⋊C4×D227C4semidirect
21. C22⋊(C227⋊C8)C22semidirect
23. C227⋊#95C22semidirect
25. C22⋊C8×C227C22semidirect
26. #95×C227C22semidirect
140. C227#1507semidirect
148. #327D227semidirect
165. C22⋊C8C2×D227semidirect
167. #95C2×D227semidirect

Maximal subgroups

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

subgroupindex
#121×D2272
#284089942
#284088792
#284087672
#284087662
#284087652
#327×C2272
#1507227