Finite Group Databasebeta

← #44400142

Subgroups — #44400142

Order 29056.

578 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 93 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×#20300256C211yesmaximal, characteristic
2C2×#20300203C211yesmaximal, characteristic, complemented
3#28408887C212yesmaximal, complemented
4#28408799C212yesmaximal, complemented
5#518×C227C211yesmaximal, characteristic, complemented
6C24×D227C2211yescharacteristic
7C22×C22⋊D227—21no—
8#20300256—22no—
9#20300256C414yescomplemented
10C2×C22⋊Dic227C2212yes—
11C2×C22⋊Dic227C2211yescharacteristic
12C2×(C2×D227)⋊C4—21no—
13C2×(C2×D227)⋊C4—24no—
14#20300203—22no—
15C22×D4×C227C2211yescharacteristic
16C4×D4×C227—24no—
17#117×C227—21no—
18C2×C22⋊C4×C227—21no—
19C2×C22⋊C4×C227C2212yes—
20C23×D227—21no—
21C23×D227—22no—
22C23×D227—21no—
23C23×D227—22no—
24C23×D227—44no—
25C23×D227C2×C412yescomplemented
26C2×C22⋊D227—24no—
27C2×C22⋊D227—44no—
28C22×Dic227—21no—
29C22×Dic227—22no—
30C22⋊Dic227—21no—
31C22⋊Dic227—22no—
32C22⋊Dic227—22no—
33C22⋊Dic227C2×C412yescomplemented
34(C2×D227)⋊C4—24no—
35(C2×D227)⋊C4—48no—
36C24×C227D411yescharacteristic
37C24×C227C2311yescharacteristic
38C2×D4×C227—22no—
39C2×D4×C227—22no—
40C2×D4×C227C2×C412yes—
41C2×D4×C227D412yes—
42C22×C4×C227—22no—
43C22×C4×C227D411yescharacteristic
44C22×C4×C227D414yescomplemented
45C4⋊C4×C227—22no—
46C22⋊C4×C227—22no—
47C22⋊C4×C227—22no—
48C22⋊C4×C227—22no—
49C42×C227—22no—
50C22×D227—28no—
51C22×D227—24no—
52C22×D227—22no—
53C22×D227—24no—
54C22×D227—44no—
55C22×D227—44no—
56C22×D227—48no—
57C22×D227—44no—
58C22×D227—42no—
59C22×D227—41no—
60C22×D227—42no—
61C22×D227—41no—
62C22⋊D227—48no—
63C2×Dic227—22no—
64C2×Dic227—42no—
65C2×Dic227—44no—
66C2×Dic227—42no—
67C23×C227—21no—
68C23×C227—21no—
69C23×C227—42no—
70C23×C227—42no—
71C23×C227C22⋊C412yescomplemented
72C23×C227C22×C411yescharacteristic
73C23×C227C2×D411yescharacteristic
74C23×C227C2×D412yes—
75C23×C227C2×D411yescharacteristic
76C23×C227#4112yes—
77D4×C227—24no—
78D4×C227—24no—
79C2×C4×C227—21no—
80C2×C4×C227—21no—
81C2×C4×C227—24no—
82C2×C4×C227—44no—
83C2×C4×C227—82no—
84C2×C4×C227C22⋊C412yescomplemented
85C22×C227—24no—
86C22×C227—22no—
87C22×C227—22no—
88C22×C227—41no—
89C22×C227—41no—
90C22×C227—42no—
91C22×C227—42no—
92C22×C227—42no—
93C22×C227—81no—
94C22×C227C2×C22⋊C411yescharacteristic
95C22×C227C4×D412yes—
96C22×C227#12011yescharacteristic
97C22×C227#12112yes—
98C22×C227#12311yescharacteristic
99C2×D227—24no—
100C2×D227—28no—
101C2×D227—42no—
102C2×D227—48no—
103C2×D227—44no—
104C2×D227—44no—
105C2×D227—42no—
106C2×D227—44no—
107C4×C227—22no—
108C4×C227—44no—
109Dic227—42no—
110Dic227—82no—
111C2×C227—21no—
112C2×C227—22no—
113C2×C227—22no—
114C2×C227—21no—
115C2×C227—42no—
116C2×C227#38611yescharacteristic
117C2×C227B(4,𝔽2)12yes—
118D227—44no—
119C227#188911yescharacteristic, complemented, Sylow
120#1889—2271nomaximal, Sylow
121C2×#120—2271no—
122#518D22711yescharacteristic, complemented
123C2×#99—2271no—
124#379—2272no—
125#342—2272no—
126C25—2271no—
127C22×D4—4541no—
128C22×D4C2×D22711yescharacteristic
129#120—4542no—
130#120—2274no—
131C4×D4—24no—
132#117—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—4544no—
135C2×C22⋊C4—4541no—
136C2×C22⋊C4—2272no—
137C2×C22⋊C4—2271no—
138C2×C22⋊C4C2×D22712yes—
139#99—4542no—
140C24—9084no—
141C24—4541no—
142C24—4542no—
143C24—4542no—
144C24—4541no—
145C24—2272no—
146C24C22⋊D22711yescharacteristic
147C24C22×D22711yescharacteristic
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—9084no—
151C2×D4—4544no—
152C2×D4C4×D22712yes—
153C2×D4D90812yes—
154C22×C4—22no—
155C22×C4—4542no—
156C22×C4—4541no—
157C22×C4C22⋊D22711yescharacteristic
158C22×C4C22⋊D22714yescomplemented
159C4⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—9088no—
164C22⋊C4—4542no—
165C22⋊C4—4542no—
166C22⋊C4—4541no—
167C22⋊C4—4544no—
168C22⋊C4—2272no—
169C42—22no—
170C23—21no—
171C23—21no—
172C23—42no—
173C23—42no—
174C23—9084no—
175C23—9082no—
176C23—9084no—
177C23—9088no—
178C23—9084no—
179C23—9081no—
180C23—9082no—
181C23—9081no—
182C23—4548no—
183C23—4544no—
184C23—4542no—
185C23—4544no—
186C23(C2×D227)⋊C412yescomplemented
187C23C2×C4×D22711yescharacteristic
188C23C2×C22⋊D22711yescharacteristic
189C23C2×C22⋊D22712yes—
190C23C2×D90811yescharacteristic
191C23#1681388412yes—
192D4—24no—
193D4—24no—
194D4—9088no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—44no—
199C2×C4—82no—
200C2×C4—9084no—
201C2×C4—9082no—
202C2×C4—9082no—
203C2×C4—4542no—
204C2×C4(C2×D227)⋊C412yescomplemented
205C22—24no—
206C22—22no—
207C22—22no—
208C22—41no—
209C22—41no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—81no—
214C22—9088no—
215C22—9082no—
216C22—9084no—
217C22—9084no—
218C22—9082no—
219C22—9084no—
220C22—4544no—
221C22—4548no—
222C22C2×(C2×D227)⋊C411yescharacteristic
223C22C4×C22⋊D22712yes—
224C22#2030025711yescharacteristic
225C22#2030026212yes—
226C22#2030027111yescharacteristic
227C4—22no—
228C4—44no—
229C4—18162no—
230C4—9082no—
231C2—22no—
232C2—21no—
233C2—21no—
234C2—22no—
235C2—42no—
236C2—9084no—
237C2#2840891111yescharacteristic
238C2#2840921912yes—
subgroupG/H
centerC22#20300257
commutatorC23×C227C22×C4
FrattiniC24C22×D227
Fitting#518×C227C2
socleC22×C227#120
prime radicalC2×#20300256C2
cosocleC24×C227C23
hypercenter#518D227
hypocenterC227#1889
LCSG ▷ C23×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C2×#20300203C2semidirect
3. #28408887C2semidirect
4. #28408799C2semidirect
5. #518×C227C2semidirect
9. #20300256C4semidirect
25. C23×D227C2×C4semidirect
33. C22⋊Dic227C2×C4semidirect
44. C22×C4×C227D4semidirect
71. C23×C227C22⋊C4semidirect
84. C2×C4×C227C22⋊C4semidirect
119. C227#1889semidirect
122. #518D227semidirect
158. C22×C4C22⋊D227semidirect
186. C23(C2×D227)⋊C4semidirect
204. C2×C4(C2×D227)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#203002562
C2×#203002032
#284088872
#284087992
#518×C2272
#1889227