Finite Group Databasebeta

← #44400766

Subgroups — #44400766

Order 29056.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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×#20300321C211yesmaximal, characteristic, complemented
2#28409601C211yesmaximal, characteristic
3C2×#20300276C211yesmaximal, characteristic, complemented
4C2×#20300216C211yesmaximal, characteristic, complemented
5#28409040C211yesmaximal, characteristic, complemented
6C2×#20300204C211yesmaximal, characteristic, complemented
7#339×C227C211yesmaximal, characteristic, complemented
8C2×#16813884C2211yescharacteristic
9C22×Dic454C2211yescharacteristic
10C22×D908C2211yescharacteristic
11#20300321—22no—
12#20300321—22no—
13C2×Dic908—21no—
14C2×#16813873—21no—
15C2×#16813868C2211yescharacteristic
16#20300285—21no—
17#20300276—21no—
18#20300276—22no—
19#20300276—22no—
20#20300263—22no—
21#20300258—22no—
22C2×(C2×D227)⋊C4—22no—
23#20300216—22no—
24#20300211—22no—
25#20300204—22no—
26C2×M16×C227C2211yescharacteristic
27#117×C227C2211yescharacteristic
28C2×C42×C227C2211yescharacteristic
29C23×D227D412yes—
30#16813884—21no—
31#16813884—21no—
32#16813884—21no—
33#16813884—21no—
34C2×Dic454—21no—
35C2×Dic454—21no—
36C2×Dic454—21no—
37C2×Dic454—21no—
38C2×Dic454—21no—
39C2×Dic454D411yescharacteristic
40C2×Dic454D412yescomplemented
41C2×D908—24no—
42C2×D908—21no—
43C2×D908—21no—
44C2×D908—21no—
45C2×D908—21no—
46C2×D908—21no—
47C2×D908—21no—
48C2×D908D411yescharacteristic
49C2×C22⋊D227—21no—
50C22×Dic227—21no—
51C2×C4×D227—21no—
52Dic908—24no—
53#16813873—24no—
54#16813868—21no—
55#16813868—21no—
56C2×C227⋊C8—21no—
57(C2×D227)⋊C4—28no—
58(C2×D227)⋊C4—24no—
59C22×C4×C227—21no—
60C22×C4×C227C2311yescharacteristic
61M16×C227—21no—
62M16×C227D412yescomplemented
63C2×C8×C227—21no—
64C4⋊C4×C227D412yes—
65C22⋊C4×C227—21no—
66C42×C227—21no—
67C42×C227—21no—
68C42×C227D412yescomplemented
69C22×D227—22no—
70C22×D227—22no—
71C22×D227—21no—
72C22×D227—22no—
73C22×D227—44no—
74Dic454—22no—
75Dic454—21no—
76Dic454—22no—
77Dic454—21no—
78Dic454—41no—
79Dic454—41no—
80D908—21no—
81D908—21no—
82D908—41no—
83D908—41no—
84D908—41no—
85D908—41no—
86D908—84no—
87C22⋊D227—41no—
88C22⋊D227—41no—
89C2×Dic227—21no—
90C2×Dic227—22no—
91C2×Dic227—41no—
92C2×Dic227—41no—
93C4×D227—41no—
94C4×D227—41no—
95C227⋊C8—41no—
96C23×C227C2×D411yescharacteristic
97C2×C4×C227—22no—
98C2×C4×C227—22no—
99C2×C4×C227—42no—
100C2×C4×C227C2×D411yescharacteristic
101C2×C4×C227C2×D411yescharacteristic
102C2×C4×C227C2×D411yescharacteristic
103C2×C4×C227C2×D411yescharacteristic
104C2×C4×C227C2×D411yescharacteristic
105C2×C4×C227#4111yescharacteristic
106C8×C227—22no—
107C22×C227—22no—
108C22×C227#12011yescharacteristic
109C22×C227#12011yescharacteristic
110C22×C227#12111yescharacteristic
111C2×D227—44no—
112C2×D227—41no—
113C2×D227—42no—
114C2×D227—42no—
115C2×D227—44no—
116C2×D227—82no—
117C4×C227—42no—
118C4×C227—42no—
119C4×C227#12011yescharacteristic
120C4×C227#12111yescharacteristic
121C4×C227#12111yescharacteristic
122C4×C227#12711yescharacteristic
123Dic227—41no—
124Dic227—42no—
125C2×C227—21no—
126C2×C227—21no—
127C2×C227#39211yescharacteristic
128C2×C227#45512yes—
129D227—81no—
130D227—82no—
131C227#200411yescharacteristic, complemented, Sylow
132#2004—2271nomaximal, Sylow
133C2×#137—2271no—
134#535—2271no—
135C2×#124—2271no—
136C2×#104—2271no—
137#419—2271no—
138C2×#100—2271no—
139#339D22711yescharacteristic, complemented
140C2×#41—2271no—
141C22×Q8—2271no—
142C22×D4—2271no—
143#137—4542no—
144#137—4542no—
145C2×Q16—4541no—
146C2×SD16—4541no—
147C2×M16—2271no—
148C2×M16C2×D22711yescharacteristic
149#127—4541no—
150#124—4542no—
151#124—4541no—
152#124—4542no—
153#121—4542no—
154#120—4542no—
155#117C2×D22711yescharacteristic
156C2×C22⋊C4—4542no—
157C2×C42C2×D22711yescharacteristic
158#104—4542no—
159#103—4542no—
160#100—4542no—
161C24—2272no—
162#41—4541no—
163#41—4541no—
164#41—4541no—
165#41—4541no—
166C2×Q8—4541no—
167C2×Q8—4541no—
168C2×Q8—4541no—
169C2×Q8—4541no—
170C2×Q8—4541no—
171C2×Q8—2271no—
172C2×Q8—2272no—
173C2×D4—4544no—
174C2×D4—4541no—
175C2×D4—4541no—
176C2×D4—4541no—
177C2×D4—4541no—
178C2×D4—4541no—
179C2×D4—4541no—
180C2×D4—4541no—
181C2×D4—2271no—
182C22×C4—21no—
183C22×C4—4541no—
184C22×C4—4541no—
185C22×C4C22×D22711yescharacteristic
186Q16—4544no—
187SD16—4544no—
188M16—21no—
189M16—4541no—
190M16—4541no—
191M16D90812yescomplemented
192C2×C8—21no—
193C2×C8—4541no—
194C4⋊C4D90812yes—
195C22⋊C4—21no—
196C22⋊C4—4548no—
197C22⋊C4—4544no—
198C42—21no—
199C42—21no—
200C42D90812yescomplemented
201C23—9084no—
202C23—4542no—
203C23—4542no—
204C23—4542no—
205C23—4541no—
206C23D4×D22711yescharacteristic
207Q8—9081no—
208Q8—9081no—
209Q8—4542no—
210Q8—4541no—
211Q8—4542no—
212Q8—4541no—
213D4—18164no—
214D4—9081no—
215D4—9081no—
216D4—9081no—
217D4—9081no—
218D4—9081no—
219D4—9081no—
220D4—4541no—
221D4—4541no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—9081no—
226C2×C4—9081no—
227C2×C4—9081no—
228C2×C4—9081no—
229C2×C4—4542no—
230C2×C4—4541no—
231C2×C4D4×D22711yescharacteristic
232C2×C4D4×D22711yescharacteristic
233C2×C4C2×D90811yescharacteristic
234C2×C4C2×D90811yescharacteristic
235C2×C4C2×D90811yescharacteristic
236C2×C4#1681388511yescharacteristic
237C8—22no—
238C8—9081no—
239C22—22no—
240C22—18162no—
241C22—9084no—
242C22—9081no—
243C22—9082no—
244C22—9082no—
245C22—9084no—
246C22#2030025811yescharacteristic
247C22#2030025811yescharacteristic
248C22#2030026311yescharacteristic
249C4—42no—
250C4—42no—
251C4—9081no—
252C4—9082no—
253C4#2030025811yescharacteristic
254C4#2030026311yescharacteristic
255C4#2030026311yescharacteristic
256C4#2030028511yescharacteristic
257C2—21no—
258C2—21no—
259C2—18162no—
260C2—18161no—
261C2#2840894211yescharacteristic
262C2#2840921112yes—
subgroupG/H
centerC22#20300258
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
Fitting#339×C227C2
socleC22×C227#120
prime radical#28409601C2
hypercenter#339D227
hypocenterC227#2004
LCSG ▷ C22×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. C2×#20300321C2semidirect
3. C2×#20300276C2semidirect
4. C2×#20300216C2semidirect
5. #28409040C2semidirect
6. C2×#20300204C2semidirect
7. #339×C227C2semidirect
40. C2×Dic454D4semidirect
62. M16×C227D4semidirect
68. C42×C227D4semidirect
131. C227#2004semidirect
139. #339D227semidirect
191. M16D908semidirect
200. C42D908semidirect

Maximal subgroups

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

subgroupindex
C2×#203003212
#284096012
C2×#203002762
C2×#203002162
#284090402
C2×#203002042
#339×C2272
#2004227