Finite Group Databasebeta

← #44559592

Subgroups — #44559592

Order 32128.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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×#22548741C212yesmaximal, complemented
2#30632198C211yesmaximal, characteristic, complemented
3#30631838C214yesmaximal, complemented
4#30631821C214yesmaximal, complemented
5#30631692C212yesmaximal, complemented
6#30631651C211yesmaximal, characteristic, complemented
7C23×D4×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic, complemented
9#22548741—28no—
10C2×#16860798C2214yescomplemented
11C2×#16860795C2214yescomplemented
12C2×#16860794C2212yescomplemented
13#22548698—21no—
14#22548694C2212yescomplemented
15#22548689C2212yescomplemented
16#22548684—21no—
17C4×C22⋊D251—22no—
18#22548674C2211yescharacteristic, complemented
19#22548636C2218yescomplemented
20C22⋊(C251⋊C8)C2214yescomplemented
21C25×C251—21no—
22C22×D4×C251—28no—
23C22×D4×C251—22no—
24C22×D4×C251—21no—
25C22×D4×C251C2214yes—
26C22×D4×C251C2212yes—
27C23×C4×C251C2211yescharacteristic
28#16860810—24no—
29C2×Dic502C2311yescharacteristic, complemented
30C2×D1004C2311yescharacteristic, complemented
31C2×C22⋊D251—21no—
32C2×C22⋊D251—22no—
33C2×C4×D251—21no—
34#16860798—28no—
35#16860795—28no—
36#16860794—24no—
37C2×C251⋊C8C2314yescomplemented
38C4⋊Dic251C2312yescomplemented
39Dic251⋊C4—22no—
40C22⋊Dic251—21no—
41C22⋊Dic251—22no—
42(C2×D251)⋊C4—22no—
43C4×Dic251—21no—
44C24×C251—28no—
45C24×C251—24no—
46C24×C251—22no—
47C24×C251—48no—
48C24×C251D411yescharacteristic
49C2×D4×C251—28no—
50C2×D4×C251—216no—
51C2×D4×C251—216no—
52C2×D4×C251—24no—
53C2×D4×C251—22no—
54C2×D4×C251—24no—
55C2×D4×C251D418yescomplemented
56C2×D4×C251C2314yes—
57C22×C4×C251—24no—
58C22×C4×C251D412yes—
59C22×C4×C251D411yescharacteristic
60C22×C4×C251C2312yes—
61C22×C4×C251C2311yescharacteristic
62C22×D251—21no—
63Dic502—22no—
64D1004—22no—
65C22⋊D251—42no—
66C22⋊D251—44no—
67C2×Dic251—22no—
68C2×Dic251—21no—
69C4×D251—42no—
70C251⋊C8—24no—
71C23×C251—22no—
72C23×C251—24no—
73C23×C251—216no—
74C23×C251—28no—
75C23×C251—24no—
76C23×C251—432no—
77C23×C251—48no—
78C23×C251—48no—
79C23×C251—48no—
80C23×C251C2×D411yescharacteristic
81C23×C251C2×D412yes—
82D4×C251—216no—
83D4×C251—28no—
84D4×C251—24no—
85D4×C251C2×D418yes—
86C2×C4×C251—22no—
87C2×C4×C251—24no—
88C2×C4×C251—24no—
89C2×C4×C251C2×D412yes—
90C2×C4×C251C2×D411yescharacteristic
91C2×C4×C251C2×D414yes—
92C2×C4×C251C2411yescharacteristic
93C22×C251—28no—
94C22×C251—22no—
95C22×C251—22no—
96C22×C251—22no—
97C22×C251—28no—
98C22×C251—416no—
99C22×C251—432no—
100C22×C251—48no—
101C22×C251#12012yes—
102C22×C251AGL(1,ℤ/8)14yescomplemented
103C22×C251C22×D411yescharacteristic
104C2×D251—41no—
105C2×D251—42no—
106C4×C251—22no—
107C4×C251#12012yes—
108C4×C251C22×D412yes—
109Dic251—41no—
110Dic251—42no—
111C2×C251—22no—
112C2×C251—24no—
113C2×C251—48no—
114C2×C251C2×#12011yescharacteristic
115C2×C251C2×AGL(1,ℤ/8)12yes—
116D251—81no—
117C251#300311yescharacteristic, complemented, Sylow
118#3003—2511nomaximal, Sylow
119C23×D4D25111yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—2512no—
121#525—2511no—
122#450—2514no—
123#447—2514no—
124#418—2512no—
125#407—2511no—
126C25—21no—
127C2×#41—2511no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D25114yes—
132C22×D4C2×D25112yes—
133C23×C4C2×D25111yescharacteristic
134AGL(1,ℤ/8)—5028no—
135C2×SD16—2514no—
136C2×D8—2514no—
137C2×M16—2512no—
138#123—5021no—
139C22⋊Q8—2512no—
140#121—2512no—
141#120—5021no—
142C4×D4—5022no—
143#117—2511no—
144#102—2518no—
145C22⋊C8—2514no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D25111yescharacteristic
151#41—5024no—
152C2×Q8—2511no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—22no—
159C2×D4—5022no—
160C2×D4—5021no—
161C2×D4—2511no—
162C2×D4C22⋊D25118yescomplemented
163C2×D4C22×D25114yes—
164C22×C4—24no—
165C22×C4—5021no—
166C22×C4C22⋊D25112yes—
167C22×C4C22⋊D25111yescharacteristic
168C22×C4C22×D25112yes—
169C22×C4C22×D25111yescharacteristic
170SD16—5028no—
171D8—5028no—
172M16—5024no—
173C2×C8—2514no—
174C4⋊C4—5022no—
175C4⋊C4—2512no—
176C22⋊C4—5021no—
177C22⋊C4—5022no—
178C22⋊C4—5022no—
179C42—5021no—
180C23—24no—
181C23—22no—
182C23—216no—
183C23—24no—
184C23—28no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—5021no—
190C23C2×C22⋊D25112yes—
191C23C2×C22⋊D25111yescharacteristic
192Q8—5022no—
193D4—216no—
194D4—24no—
195D4—28no—
196D4—10044no—
197D4—10042no—
198D4—5022no—
199D4C2×C22⋊D25118yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—10042no—
204C2×C4—5022no—
205C2×C4—5021no—
206C2×C4C2×C22⋊D25112yes—
207C2×C4C2×C22⋊D25114yes—
208C2×C4C2×C22⋊D25111yescharacteristic
209C2×C4C23×D25111yescharacteristic
210C8—5024no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—432no—
217C22—416no—
218C22—48no—
219C22—10042no—
220C22—10041no—
221C22#2254868412yes—
222C22#2254874114yescomplemented
223C22C22×C22⋊D25111yescharacteristic
224C4—22no—
225C4—10042no—
226C4—10041no—
227C4#2254868412yes—
228C4C22×C22⋊D25112yes—
229C2—24no—
230C2—22no—
231C2—48no—
232C2—20081no—
233C2C2×#2254868411yescharacteristic
234C2C2×#2254874112yes—
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC23×D4×C251C2
socleC22×C251C22×D4
hypercenterC23×D4D251
hypocenterC251#3003
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#22548741C2semidirect
2. #30632198C2semidirect
3. #30631838C2semidirect
4. #30631821C2semidirect
5. #30631692C2semidirect
6. #30631651C2semidirect
7. C23×D4×C251C2semidirect
8. C2×#16860810C22semidirect
10. C2×#16860798C22semidirect
11. C2×#16860795C22semidirect
12. C2×#16860794C22semidirect
14. #22548694C22semidirect
15. #22548689C22semidirect
18. #22548674C22semidirect
19. #22548636C22semidirect
20. C22⋊(C251⋊C8)C22semidirect
29. C2×Dic502C23semidirect
30. C2×D1004C23semidirect
37. C2×C251⋊C8C23semidirect
38. C4⋊Dic251C23semidirect
55. C2×D4×C251D4semidirect
102. C22×C251AGL(1,ℤ/8)semidirect
117. C251#3003semidirect
119. C23×D4D251semidirect
162. C2×D4C22⋊D251semidirect
222. C22#22548741semidirect

Maximal subgroups

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

subgroupindex
C2×#225487412
#306321982
#306318382
#306318212
#306316922
#306316512
C23×D4×C2512
#3003251