Finite Group Databasebeta

← #44409213

Subgroups — #44409213

Order 29056.

584 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 137 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×#20300316C211yesmaximal, characteristic, complemented
2C2×#20300314C211yesmaximal, characteristic, complemented
3C2×#20300284C211yesmaximal, characteristic, complemented
4#28409500C211yesmaximal, characteristic, complemented
5#28409231C214yesmaximal, complemented
6#28409226C214yesmaximal, complemented
7#28409055C211yesmaximal, characteristic, complemented
8#28409036C211yesmaximal, characteristic, complemented
9#418×C227C211yesmaximal, characteristic, complemented
10C2×#16813884C2211yescharacteristic
11C22×D908C2211yescharacteristic
12C22×C22⋊D227—22no—
13#20300316—24no—
14#20300314—24no—
15C2×#16813873C2212yescomplemented
16C2×#16813872C2212yescomplemented
17C2×D1816C2212yescomplemented
18C2×#16813869C2212yescomplemented
19C2×#16813868C2211yescharacteristic
20#20300284—21no—
21#20300284—21no—
22#20300284C2214yes—
23#20300253C2212yescomplemented
24#20300250C2212yescomplemented
25(C22⋊D227)⋊C4—22no—
26#20300245—21no—
27C2×C4×Dic227C2211yescharacteristic
28C42×D227—21no—
29Dic227⋊C8C2214yescomplemented
30#20300208C2214yescomplemented
31C22×D4×C227C2211yescharacteristic
32C2×M16×C227C2211yescharacteristic
33#117×C227C2211yescharacteristic
34#102×C227C2214yescomplemented
35C23×D227—21no—
36#16813884—22no—
37#16813884D414yescomplemented
38C2×Dic454C2311yescharacteristic
39C2×D908—22no—
40C2×D908—21no—
41C2×D908—22no—
42C2×D908C2312yes—
43C2×D908C2311yescharacteristic
44C2×C22⋊D227—24no—
45C2×C22⋊D227—24no—
46C2×C22⋊D227—21no—
47C2×C22⋊D227—48no—
48C22×Dic227D412yes—
49C2×C4×D227—22no—
50C2×C4×D227—21no—
51#16813873—24no—
52#16813872—24no—
53D1816—24no—
54#16813869—24no—
55#16813868—22no—
56C2×C227⋊C8C2312yes—
57Dic227⋊C4—22no—
58(C2×D227)⋊C4—22no—
59C4×Dic227—22no—
60C4×Dic227D411yescharacteristic
61C4×Dic227D411yescharacteristic
62C4×Dic227C2312yes—
63C24×C227—21no—
64C2×D4×C227—22no—
65C2×D4×C227—21no—
66C2×D4×C227—22no—
67C2×D4×C227C2312yes—
68C22×C4×C227C2311yescharacteristic
69M16×C227—22no—
70C2×C8×C227C2312yes—
71C4⋊C4×C227C2312yes—
72C22⋊C4×C227—21no—
73C42×C227—21no—
74C22×D227—21no—
75C22×D227—22no—
76C22×D227—44no—
77C22×D227—42no—
78Dic454—21no—
79Dic454C2×D412yescomplemented
80D908—22no—
81D908—21no—
82D908—22no—
83D908—42no—
84D908C2×D412yescomplemented
85C22⋊D227—24no—
86C22⋊D227—48no—
87C22⋊D227—48no—
88C2×Dic227—21no—
89C2×Dic227—24no—
90C2×Dic227C2×D412yes—
91C2×Dic227C2×D412yes—
92C4×D227—24no—
93C4×D227—44no—
94C227⋊C8—22no—
95C23×C227—22no—
96C23×C227—44no—
97C23×C227—42no—
98C23×C227C2×D411yescharacteristic
99D4×C227—22no—
100D4×C227—22no—
101D4×C227—42no—
102C2×C4×C227—22no—
103C2×C4×C227C2×D411yescharacteristic
104C2×C4×C227C2×D412yes—
105C2×C4×C227#4112yes—
106C2×C4×C227C2411yescharacteristic
107C8×C227—22no—
108C22×C227—22no—
109C22×C227—42no—
110C22×C227—44no—
111C22×C227—84no—
112C22×C227#12112yes—
113C22×C227C22×D411yescharacteristic
114C2×D227—22no—
115C2×D227—42no—
116C2×D227—42no—
117C2×D227—44no—
118C2×D227—84no—
119C4×C227—42no—
120C4×C227#12112yes—
121C4×C227C22×D411yescharacteristic
122C4×C227C2×#4111yescharacteristic
123Dic227—22no—
124Dic227—22no—
125Dic227AGL(1,ℤ/8)14yescomplemented
126C2×C227—22no—
127C2×C227—82no—
128C2×C227C2×#12111yescharacteristic
129C2×C227C2×AGL(1,ℤ/8)12yes—
130D227—42no—
131D227—82no—
132C227#303811yescharacteristic, complemented, Sylow
133#3038—2271nomaximal, Sylow
134C2×AGL(1,ℤ/8)—2271no—
135C2×AGL(1,ℤ/8)—2271no—
136C2×#127—2271no—
137C4×#41—2271no—
138#460—2274no—
139#459—2274no—
140#423—2271no—
141#418—2271no—
142#418D22711yescharacteristic, complemented
143C2×#41—2271no—
144C22×D4—4542no—
145C22×D4—2271no—
146C22×D4C2×D22711yescharacteristic
147AGL(1,ℤ/8)—4544no—
148AGL(1,ℤ/8)—4544no—
149C2×SD16—2272no—
150C2×SD16—2272no—
151C2×D8—2272no—
152C2×D8—2272no—
153C2×M16—2271no—
154C2×M16C2×D22711yescharacteristic
155#127—4541no—
156#127—4541no—
157#127—2274no—
158C4×Q8—2272no—
159C4×D4—4542no—
160C4×D4—2272no—
161#117—4541no—
162#117C2×D22711yescharacteristic
163C2×C42—4541no—
164C2×C42—2271no—
165C4⋊C8—2274no—
166#102—2274no—
167#102C2×D22714yescomplemented
168C24—21no—
169C24—4541no—
170#41—4542no—
171#41—2274no—
172C2×Q8—2271no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—9088no—
177C2×D4—4541no—
178C2×D4—4544no—
179C2×D4—4544no—
180C2×D4—4541no—
181C2×D4—4542no—
182C2×D4—4542no—
183C2×D4—2272no—
184C2×D4—2271no—
185C2×D4C22×D22712yes—
186C22×C4—4542no—
187C22×C4—4541no—
188C22×C4—2272no—
189C22×C4C22×D22711yescharacteristic
190SD16—4544no—
191SD16—4544no—
192D8—4544no—
193D8—4544no—
194M16—22no—
195M16—4542no—
196C2×C8—2272no—
197C2×C8C22×D22712yes—
198C4⋊C4—4542no—
199C4⋊C4C22×D22712yes—
200C22⋊C4—21no—
201C22⋊C4—4542no—
202C42—21no—
203C42—4542no—
204C42—2271no—
205C42—2271no—
206C42—2272no—
207C23—22no—
208C23—44no—
209C23—42no—
210C23—9084no—
211C23—9082no—
212C23—4542no—
213C23—4541no—
214C23D4×D22711yescharacteristic
215Q8—4541no—
216Q8—2272no—
217D4—22no—
218D4—22no—
219D4—42no—
220D4—9088no—
221D4—9088no—
222D4—9082no—
223D4—4544no—
224D4—4542no—
225D4—4542no—
226D4—4541no—
227D4—2272no—
228C2×C4—22no—
229C2×C4—9084no—
230C2×C4—4541no—
231C2×C4—4544no—
232C2×C4—4544no—
233C2×C4—2272no—
234C2×C4—2272no—
235C2×C4D4×D22712yes—
236C2×C4D4×D22711yescharacteristic
237C2×C4#1681388412yes—
238C2×C4C23×D22711yescharacteristic
239C8—22no—
240C8—4542no—
241C22—22no—
242C22—42no—
243C22—44no—
244C22—84no—
245C22—18164no—
246C22—9082no—
247C22—9084no—
248C22—9082no—
249C22—4542no—
250C22#2030026112yes—
251C22C2×D4×D22711yescharacteristic
252C4—42no—
253C4—4542no—
254C4—4542no—
255C4—2274no—
256C4#2030026112yes—
257C4C2×D4×D22711yescharacteristic
258C4C2×#1681388411yescharacteristic
259C2—22no—
260C2—82no—
261C2—18162no—
262C2—9082no—
263C2C2×#2030026111yescharacteristic
264C2AGL(1,ℤ/8)×D22712yes—
subgroupG/H
centerC22C2×D4×D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#418×C227C2
socleC22×C227C22×D4
hypercenter#418D227
hypocenterC227#3038
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C2×#20300316C2semidirect
2. C2×#20300314C2semidirect
3. C2×#20300284C2semidirect
4. #28409500C2semidirect
5. #28409231C2semidirect
6. #28409226C2semidirect
7. #28409055C2semidirect
8. #28409036C2semidirect
9. #418×C227C2semidirect
15. C2×#16813873C22semidirect
16. C2×#16813872C22semidirect
17. C2×D1816C22semidirect
18. C2×#16813869C22semidirect
23. #20300253C22semidirect
24. #20300250C22semidirect
29. Dic227⋊C8C22semidirect
30. #20300208C22semidirect
34. #102×C227C22semidirect
37. #16813884D4semidirect
79. Dic454C2×D4semidirect
84. D908C2×D4semidirect
125. Dic227AGL(1,ℤ/8)semidirect
132. C227#3038semidirect
142. #418D227semidirect
167. #102C2×D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003162
C2×#203003142
C2×#203002842
#284095002
#284092312
#284092262
#284090552
#284090362
#418×C2272
#3038227