Finite Group Databasebeta

← #44400752

Subgroups — #44400752

Order 29056.

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

normal subgroups only

This group has 478 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×#20300316C211yesmaximal, characteristic, complemented
2#28409601C211yesmaximal, characteristic, complemented
3C2×#20300285C211yesmaximal, characteristic, complemented
4C2×#20300216C211yesmaximal, characteristic, complemented
5#28409036C211yesmaximal, characteristic, complemented
6C2×#20300204C211yesmaximal, characteristic, complemented
7#339×C227C211yesmaximal, characteristic, complemented
8C2×#16813884C2211yescharacteristic
9C22×D908—22no—
10C22×D908C2211yescharacteristic
11C22×D908C2211yescharacteristic
12#20300316—22no—
13#20300316—22no—
14C2×#16813873—21no—
15C2×D1816—21no—
16C2×#16813868C2211yescharacteristic
17#20300285—21no—
18#20300285—22no—
19#20300285—22no—
20#20300276—21no—
21#20300263—22no—
22#20300258—22no—
23#20300216—22no—
24#20300208—22no—
25#20300204—22no—
26C2×M16×C227C2211yescharacteristic
27#117×C227C2211yescharacteristic
28C2×C42×C227C2211yescharacteristic
29C23×D227—21no—
30C23×D227D412yes—
31#16813884—21no—
32#16813884—21no—
33#16813884—21no—
34#16813884—21no—
35C2×Dic454D411yescharacteristic, complemented
36C2×D908—28no—
37C2×D908—24no—
38C2×D908—21no—
39C2×D908—21no—
40C2×D908—21no—
41C2×D908—21no—
42C2×D908—21no—
43C2×D908—21no—
44C2×D908—21no—
45C2×D908—21no—
46C2×D908—21no—
47C2×D908—21no—
48C2×D908—21no—
49C2×D908—48no—
50C2×D908D411yescharacteristic, complemented
51C2×D908D412yescomplemented
52C2×C22⋊D227—21no—
53C2×C4×D227—21no—
54#16813873—24no—
55D1816—24no—
56#16813868—21no—
57#16813868—21no—
58C2×C227⋊C8—21no—
59(C2×D227)⋊C4—24no—
60C22×C4×C227—21no—
61C22×C4×C227C2311yescharacteristic
62M16×C227—21no—
63M16×C227D412yescomplemented
64C2×C8×C227—21no—
65C4⋊C4×C227D412yescomplemented
66C22⋊C4×C227—21no—
67C42×C227—21no—
68C42×C227—21no—
69C42×C227D412yescomplemented
70C22×D227—22no—
71C22×D227—21no—
72C22×D227—22no—
73C22×D227—22no—
74C22×D227—22no—
75C22×D227—44no—
76C22×D227—44no—
77C22×D227—41no—
78C22×D227—41no—
79Dic454—21no—
80Dic454—21no—
81D908—21no—
82D908—22no—
83D908—22no—
84D908—21no—
85D908—416no—
86D908—41no—
87D908—41no—
88D908—41no—
89D908—41no—
90D908—41no—
91D908—41no—
92D908—84no—
93C22⋊D227—41no—
94C22⋊D227—41no—
95C2×Dic227—21no—
96C4×D227—41no—
97C4×D227—41no—
98C227⋊C8—41no—
99C23×C227C2×D411yescharacteristic
100C2×C4×C227—24no—
101C2×C4×C227—22no—
102C2×C4×C227—22no—
103C2×C4×C227C2×D411yescharacteristic
104C2×C4×C227C2×D411yescharacteristic
105C2×C4×C227C2×D411yescharacteristic
106C2×C4×C227C2×D411yescharacteristic
107C2×C4×C227C2×D411yescharacteristic
108C2×C4×C227#4111yescharacteristic
109C8×C227—22no—
110C22×C227—22no—
111C22×C227#12011yescharacteristic
112C22×C227#12011yescharacteristic
113C22×C227#12111yescharacteristic
114C2×D227—44no—
115C2×D227—44no—
116C2×D227—41no—
117C2×D227—42no—
118C2×D227—42no—
119C2×D227—42no—
120C2×D227—44no—
121C2×D227—82no—
122C2×D227—82no—
123C2×D227—82no—
124C4×C227—24no—
125C4×C227—42no—
126C4×C227#12011yescharacteristic
127C4×C227#12111yescharacteristic
128C4×C227#12111yescharacteristic
129C4×C227#12711yescharacteristic
130Dic227—41no—
131C2×C227—21no—
132C2×C227—21no—
133C2×C227#39211yescharacteristic
134C2×C227#45312yes—
135D227—81no—
136D227—82no—
137D227—82no—
138C227#200211yescharacteristic, complemented, Sylow
139#2002—2271nomaximal, Sylow
140C2×AGL(1,ℤ/8)—2271no—
141#535—2271no—
142C2×#127—2271no—
143C2×#104—2271no—
144#418—2271no—
145C2×#100—2271no—
146#339D22711yescharacteristic, complemented
147C2×#41—2271no—
148C22×D4—4542no—
149C22×D4—2271no—
150C22×D4—2271no—
151AGL(1,ℤ/8)—4542no—
152AGL(1,ℤ/8)—4542no—
153C2×SD16—4541no—
154C2×D8—4541no—
155C2×M16—2271no—
156C2×M16C2×D22711yescharacteristic
157#127—4542no—
158#127—4541no—
159#127—4542no—
160#124—4541no—
161#121—4542no—
162#120—4542no—
163#117C2×D22711yescharacteristic
164C2×C42C2×D22711yescharacteristic
165#104—4542no—
166#102—4542no—
167#100—4542no—
168C24—4541no—
169C24—2272no—
170#41—4541no—
171#41—4541no—
172#41—4541no—
173#41—4541no—
174C2×Q8—2271no—
175C2×D4—9088no—
176C2×D4—4544no—
177C2×D4—4548no—
178C2×D4—4541no—
179C2×D4—4541no—
180C2×D4—4541no—
181C2×D4—4541no—
182C2×D4—4541no—
183C2×D4—4541no—
184C2×D4—4541no—
185C2×D4—4541no—
186C2×D4—4541no—
187C2×D4—4541no—
188C2×D4—4541no—
189C2×D4—4541no—
190C2×D4—2271no—
191C2×D4—2272no—
192C22×C4—21no—
193C22×C4—4541no—
194C22×C4C22×D22711yescharacteristic
195SD16—4544no—
196D8—4544no—
197M16—21no—
198M16—4541no—
199M16—4541no—
200M16D90812yescomplemented
201C2×C8—21no—
202C2×C8—4541no—
203C4⋊C4D90812yescomplemented
204C22⋊C4—21no—
205C22⋊C4—4544no—
206C42—21no—
207C42—21no—
208C42D90812yescomplemented
209C23—9084no—
210C23—9084no—
211C23—9081no—
212C23—9081no—
213C23—4542no—
214C23—4542no—
215C23—4542no—
216C23—4542no—
217C23—4541no—
218C23D4×D22711yescharacteristic
219Q8—4541no—
220Q8—4541no—
221D4—18164no—
222D4—90816no—
223D4—9081no—
224D4—9081no—
225D4—9081no—
226D4—9081no—
227D4—9081no—
228D4—9081no—
229D4—9081no—
230D4—9081no—
231D4—4541no—
232D4—4542no—
233D4—4542no—
234D4—4541no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—9081no—
239C2×C4—9081no—
240C2×C4—4541no—
241C2×C4D4×D22711yescharacteristic
242C2×C4D4×D22711yescharacteristic
243C2×C4C2×D90811yescharacteristic
244C2×C4C2×D90811yescharacteristic
245C2×C4C2×D90811yescharacteristic
246C2×C4#1681388511yescharacteristic
247C8—22no—
248C8—9081no—
249C22—22no—
250C22—18162no—
251C22—18162no—
252C22—18162no—
253C22—9084no—
254C22—9084no—
255C22—9081no—
256C22—9082no—
257C22—9082no—
258C22—9082no—
259C22—9084no—
260C22#2030025811yescharacteristic
261C22#2030025811yescharacteristic
262C22#2030026311yescharacteristic
263C4—24no—
264C4—42no—
265C4—9081no—
266C4#2030025811yescharacteristic
267C4#2030026311yescharacteristic
268C4#2030026311yescharacteristic
269C4#2030028511yescharacteristic
270C2—21no—
271C2—21no—
272C2—18162no—
273C2—18162no—
274C2—18161no—
275C2#2840894211yescharacteristic
276C2#2840920112yes—
subgroupG/H
centerC22#20300258
commutator, cosocleC22×C4×C227C23
FrattiniC22×C4C22×D227
Fitting#339×C227C2
socleC22×C227#120
hypercenter#339D227
hypocenterC227#2002
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×#20300316C2semidirect
2. #28409601C2semidirect
3. C2×#20300285C2semidirect
4. C2×#20300216C2semidirect
5. #28409036C2semidirect
6. C2×#20300204C2semidirect
7. #339×C227C2semidirect
35. C2×Dic454D4semidirect
50. C2×D908D4semidirect
51. C2×D908D4semidirect
63. M16×C227D4semidirect
65. C4⋊C4×C227D4semidirect
69. C42×C227D4semidirect
138. C227#2002semidirect
146. #339D227semidirect
200. M16D908semidirect
203. C4⋊C4D908semidirect
208. C42D908semidirect

Maximal subgroups

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

subgroupindex
C2×#203003162
#284096012
C2×#203002852
C2×#203002162
#284090362
C2×#203002042
#339×C2272
#2002227