Finite Group Databasebeta

← #44398658

Subgroups — #44398658

Order 29056.

316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 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#28409657C211yesmaximal, characteristic, complemented
2#28409185C211yesmaximal, characteristic, complemented
3#28409182C211yesmaximal, characteristic, complemented
4#28408942C211yesmaximal, characteristic
5#28408768C211yesmaximal, characteristic, complemented
6#28408755C211yesmaximal, characteristic, complemented
7#328×C227C211yesmaximal, characteristic, complemented
8C22×D908—21no—
9C22×D908C2211yescharacteristic
10C2×#16813873—21no—
11C2×#16813871—21no—
12#20300268—21no—
13#20300267C2211yescharacteristic
14#20300266—21no—
15#20300266—21no—
16#20300265—21no—
17#20300264—21no—
18C2×(C2×D227)⋊C4—21no—
19C2×(C2×D227)⋊C4C2211yescharacteristic
20C22⋊C4×D227—21no—
21#20300210—21no—
22#20300208—21no—
23C22⋊(C227⋊C8)C2211yescharacteristic
24C22⋊Q8×C227C2211yescharacteristic
25C22⋊C8×C227C2211yescharacteristic
26#95×C227C2211yescharacteristic
27C23×D227—21no—
28C23×D227D411yescharacteristic
29C2×Dic454—21no—
30C2×Dic454D411yescharacteristic, complemented
31C2×D908—21no—
32C2×D908—21no—
33C2×D908—42no—
34C2×D908—42no—
35C2×D908—42no—
36C2×D908—42no—
37C22×Dic227D411yescharacteristic
38C2×C4×D227—21no—
39C2×C4×D227—21no—
40#16813873—42no—
41#16813871—42no—
42C2×C227⋊C8—21no—
43C4⋊Dic227—21no—
44C4⋊Dic227D411yescharacteristic, complemented
45Dic227⋊C4—21no—
46Dic227⋊C4—21no—
47Dic227⋊C4—21no—
48C22⋊Dic227—21no—
49(C2×D227)⋊C4—21no—
50(C2×D227)⋊C4—21no—
51(C2×D227)⋊C4—21no—
52(C2×D227)⋊C4—21no—
53(C2×D227)⋊C4—42no—
54C2×Q8×C227D411yescharacteristic, complemented
55C22×C4×C227—21no—
56C22×C4×C227C2311yescharacteristic
57C2×C8×C227—21no—
58C4⋊C4×C227—21no—
59C4⋊C4×C227D411yescharacteristic, complemented
60C22⋊C4×C227—21no—
61C22×D227—21no—
62C22×D227—21no—
63C22×D227—22no—
64C22×D227—41no—
65C22×D227—41no—
66C22×D227—41no—
67C22×D227—41no—
68C22×D227—41no—
69C22×D227—41no—
70C22×D227—41no—
71C22×D227—41no—
72C22×D227SD1612yescomplemented
73Dic454—41no—
74Dic454—81no—
75D908—22no—
76D908—22no—
77D908—41no—
78D908—41no—
79D908—48no—
80C2×Dic227—21no—
81C2×Dic227—21no—
82C2×Dic227—21no—
83C2×Dic227—41no—
84C4×D227—81no—
85C4×D227—81no—
86C227⋊C8—41no—
87C23×C227C2×D411yescharacteristic
88Q8×C227—41no—
89C2×C4×C227—22no—
90C2×C4×C227—21no—
91C2×C4×C227—21no—
92C2×C4×C227—41no—
93C2×C4×C227—41no—
94C2×C4×C227—41no—
95C2×C4×C227C2×D411yescharacteristic
96C2×C4×C227C2×D411yescharacteristic
97C8×C227—41no—
98C22×C227—21no—
99C22×C227—21no—
100C22×C227#12011yescharacteristic
101C22×C227C2×SD1611yescharacteristic
102C22×C227#13711yescharacteristic
103C2×D227—22no—
104C2×D227—41no—
105C2×D227—41no—
106C2×D227—42no—
107C2×D227—42no—
108C2×D227—42no—
109C2×D227—42no—
110C2×D227—81no—
111C2×D227—81no—
112C2×D227—81no—
113C2×D227—81no—
114C2×D227—81no—
115C4×C227—21no—
116C4×C227—21no—
117C4×C227—42no—
118C4×C227—41no—
119C4×C227—41no—
120Dic227—41no—
121Dic227—41no—
122Dic227—41no—
123C2×C227—21no—
124C2×C227—21no—
125C2×C227#44811yescharacteristic
126C2×C227#45311yescharacteristic
127C2×C227B(4,𝔽2)11yescharacteristic
128D227—42no—
129D227—82no—
130C227#159611yescharacteristic, complemented, Sylow
131#1596—2271nomaximal, Sylow
132#543—2271no—
133#450—2271no—
134#450—2271no—
135#392—2271no—
136#328—2271no—
137#328D22711yescharacteristic, complemented
138#323—2271no—
139C22×D4—4541no—
140C22×D4—2271no—
141C2×SD16—4541no—
142C2×SD16—4541no—
143C22⋊Q8—4541no—
144C22⋊Q8—4541no—
145C22⋊Q8—4541no—
146C22⋊Q8—4541no—
147C22⋊Q8—4541no—
148C22⋊Q8—2271no—
149C22⋊Q8C2×D22711yescharacteristic
150C2×C22⋊C4—4541no—
151C2×C22⋊C4—4541no—
152C2×C22⋊C4—2271no—
153#102—4541no—
154#102—4541no—
155C22⋊C8—2271no—
156C22⋊C8C2×D22711yescharacteristic
157#95C2×D22711yescharacteristic
158C24—4541no—
159C24—2271no—
160C2×Q8—4541no—
161C2×Q8—2271no—
162C2×Q8D90811yescharacteristic, complemented
163C2×D4—9082no—
164C2×D4—9082no—
165C2×D4—9082no—
166C2×D4—9082no—
167C2×D4—4541no—
168C2×D4—4541no—
169C22×C4—21no—
170C22×C4—4541no—
171C22×C4—4541no—
172C22×C4—2271no—
173C22×C4C22×D22711yescharacteristic
174SD16—9082no—
175SD16—9082no—
176C2×C8—21no—
177C2×C8—4541no—
178C4⋊C4—21no—
179C4⋊C4—4541no—
180C4⋊C4—4541no—
181C4⋊C4—4541no—
182C4⋊C4—4541no—
183C4⋊C4—2271no—
184C4⋊C4D90811yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—9082no—
187C22⋊C4—4541no—
188C22⋊C4—4541no—
189C22⋊C4—4541no—
190C22⋊C4—4541no—
191C22⋊C4—4541no—
192C23—9081no—
193C23—9081no—
194C23—9081no—
195C23—9081no—
196C23—9081no—
197C23—9081no—
198C23—9081no—
199C23—9081no—
200C23—4542no—
201C23—4541no—
202C23—4541no—
203C23—2272no—
204C23D4×D22711yescharacteristic
205Q8—41no—
206Q8—18161no—
207Q8—9081no—
208D4—9081no—
209D4—9081no—
210D4—9088no—
211D4—4542no—
212D4—4542no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—18161no—
220C2×C4—18161no—
221C2×C4—9081no—
222C2×C4—4541no—
223C2×C4—4541no—
224C2×C4—4541no—
225C2×C4D4×D22711yescharacteristic
226C2×C4C2×D90811yescharacteristic
227C8—41no—
228C8—9081no—
229C22—21no—
230C22—21no—
231C22—18161no—
232C22—18161no—
233C22—18161no—
234C22—18161no—
235C22—18161no—
236C22—9082no—
237C22—9082no—
238C22—9082no—
239C22—9082no—
240C22—9081no—
241C22—9081no—
242C22—4542no—
243C22#2030025811yescharacteristic
244C22SD16×D22711yescharacteristic
245C22#2030032011yescharacteristic
246C4—21no—
247C4—21no—
248C4—42no—
249C4—41no—
250C4—41no—
251C4—9081no—
252C4—9081no—
253C4—9081no—
254C2—21no—
255C2—21no—
256C2—18162no—
257C2—9082no—
258C2#2840917211yescharacteristic
259C2#2840920111yescharacteristic
260C2#2840922011yescharacteristic
subgroupG/H
centerC22#20300258
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
Fitting#328×C227C2
socleC22×C227#120
prime radical#28408942C2
hypercenter#328D227
hypocenterC227#1596
LCSG ▷ C22×C4×C227 ▷ C22×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #28409657C2semidirect
2. #28409185C2semidirect
3. #28409182C2semidirect
5. #28408768C2semidirect
6. #28408755C2semidirect
7. #328×C227C2semidirect
30. C2×Dic454D4semidirect
44. C4⋊Dic227D4semidirect
54. C2×Q8×C227D4semidirect
59. C4⋊C4×C227D4semidirect
72. C22×D227SD16semidirect
130. C227#1596semidirect
137. #328D227semidirect
162. C2×Q8D908semidirect
184. C4⋊C4D908semidirect

Maximal subgroups

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

subgroupindex
#284096572
#284091852
#284091822
#284089422
#284087682
#284087552
#328×C2272
#1596227