Finite Group Databasebeta

← #15686543

Subgroups — #15686543

Order 2944.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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#13093231C211yesmaximal, characteristic, complemented
2#13093218C211yesmaximal, characteristic, complemented
3#13092860C211yesmaximal, characteristic, complemented
4#13092856C212yesmaximal, complemented
5#13092851C212yesmaximal, complemented
6#13092847C211yesmaximal, characteristic, complemented
7#13092846C212yesmaximal, complemented
8C2×#10625009C211yesmaximal, characteristic, complemented
9SD32×D23C212yesmaximal, complemented
10#13092835C211yesmaximal, characteristic, complemented
11C23×#509C211yesmaximal, characteristic, complemented
12#10625161—21no—
13#10625155—21no—
14C23×#135C2211yescharacteristic, complemented
15C2×D8×C23C2211yescharacteristic
16#10625122—21no—
17#10625103—21no—
18Q16×D23C2212yescomplemented
19#10625086—22no—
20#10625083C2212yes—
21#10625083C2212yescomplemented
22#10625082—22no—
23D8×D23C2212yes—
24C2×Dic92C2211yescharacteristic, complemented
25#10625075C2211yescharacteristic
26#10625072C2211yescharacteristic
27C23×SD32C2212yescomplemented
28C23×D16C2212yescomplemented
29C23×M32C2211yescharacteristic, complemented
30#10625011C2211yescharacteristic, complemented
31#10625010C2211yescharacteristic, complemented
32#10625009C2212yescomplemented
33#10625009C2212yescomplemented
34#10625009C2211yescharacteristic, complemented
35#10625008C2211yescharacteristic, complemented
36C2×C23⋊C16C2211yescharacteristic, complemented
37Dic184C2212yescomplemented
38#10624982C2212yescomplemented
39#10624980C2212yescomplemented
40C16×D23C2212yescomplemented
41#41×C23—21no—
42C2×D4×C23—21no—
43C2×C22⋊D23—42no—
44Q8×D23—22no—
45#67290—22no—
46#67290—22no—
47#67290—42no—
48#67290—41no—
49D4×D23—22no—
50D4×D23—41no—
51#67288—21no—
52#67288—41no—
53#67288D411yescharacteristic
54C2×Dic46—21no—
55C2×Dic46—41no—
56Q16×C23C2311yescharacteristic
57SD16×C23—21no—
58D8×C23—21no—
59D8×C23C2311yescharacteristic
60D8×C23C2312yes—
61C2×C8×C23C2311yescharacteristic
62#67275—22no—
63#67273—22no—
64#67273—22no—
65#67272—22no—
66#67267D411yescharacteristic
67Dic92—21no—
68Dic92C2311yescharacteristic
69Dic92C2312yes—
70D184C2311yescharacteristic
71#67263—21no—
72#67262D412yes—
73C8×D23C2312yes—
74C16×C23C2312yes—
75C23⋊C16C2311yescharacteristic
76C23⋊C16C2311yescharacteristic
77C23×C23—41no—
78C22×D23—42no—
79Q8×C23—21no—
80D4×C23—22no—
81D4×C23—21no—
82D4×C23—41no—
83D4×C23—41no—
84C2×C4×C23—41no—
85C2×C4×C23C2×D411yescharacteristic
86C22⋊D23—41no—
87C22⋊D23—42no—
88C22⋊D23—42no—
89C22⋊D23—42no—
90C22⋊D23D812yes—
91C2×Dic23—41no—
92C2×Dic23—42no—
93C2×Dic23—42no—
94D92—21no—
95D92D811yescharacteristic
96C4×D23—42no—
97C4×D23—41no—
98C4×D23C2×D412yes—
99Dic46—21no—
100Dic46—22no—
101Dic46—41no—
102Dic46—42no—
103Dic46D811yescharacteristic
104C8×C23C2×D411yescharacteristic
105C8×C23C2411yescharacteristic
106C23⋊C8C2×D412yes—
107#3416—231nomaximal, Sylow
108C22×C23—41no—
109C22×C23—42no—
110C22×C23—161no—
111C22×C23C2×D811yescharacteristic
112C2×D23—41no—
113C2×D23—162no—
114C2×D23C2×D812yes—
115C4×C23—41no—
116C4×C23C2×D811yescharacteristic
117C4×C23C22×D411yescharacteristic
118Dic23—42no—
119Dic23—41no—
120Dic23C2×D812yes—
121#578—231no—
122#576—231no—
123#510—232no—
124#510—231no—
125#509—232no—
126#509D2311yescharacteristic, complemented
127#508—231no—
128#508—232no—
129C2×SD32—231no—
130C2×SD32—232no—
131#504—231no—
132C2×C23—21no—
133C2×C23—82no—
134C2×C23—81no—
135C2×C23C22×D811yescharacteristic
136D23—22no—
137D23—81no—
1382−1+4—461no—
139H(5,𝔽2)—461no—
140#137—462no—
141#137—461no—
142AGL(1,ℤ/8)—462no—
143AGL(1,ℤ/8)—461no—
144#135—232no—
145#135—232no—
146#135—231no—
147#135C2×D2311yescharacteristic, complemented
148C2×Q16—232no—
149C2×Q16—231no—
150C2×D8—232no—
151C2×D8C2×D2311yescharacteristic
152#131—231no—
153Q32—231no—
154Q32—232no—
155SD32—231no—
156SD32—232no—
157SD32—232no—
158SD32—231no—
159SD32—232no—
160SD32C2×D2312yescomplemented
161D16—231no—
162D16C2×D2312yescomplemented
163M32—232no—
164M32C2×D2311yescharacteristic, complemented
165C2×C16—231no—
166C2×C16—232no—
167C23#341611yescharacteristic, complemented, Sylow
168#41—21no—
169#41—231no—
170#41—462no—
171#41—462no—
172#41—461no—
173#41—921no—
174#41—922no—
175#41—921no—
176C2×Q8—462no—
177C2×Q8—461no—
178C2×Q8—921no—
179C2×D4—21no—
180C2×D4—462no—
181C2×D4—921no—
182C2×D4—922no—
183Q16—232no—
184Q16—231no—
185Q16—462no—
186Q16—461no—
187Q16C22×D2311yescharacteristic
188SD16—21no—
189SD16—462no—
190SD16—462no—
191SD16—461no—
192D8—21no—
193D8—231no—
194D8—462no—
195D8C22×D2311yescharacteristic
196D8C22×D2312yes—
197M16—231no—
198M16—232no—
199C2×C8—232no—
200C2×C8C22×D2311yescharacteristic
201C16—231no—
202C16—231no—
203C16C22×D2312yes—
204C23—41no—
205C23—922no—
206Q8—21no—
207Q8—231no—
208Q8—461no—
209Q8—462no—
210Q8—922no—
211Q8—921no—
212D4—22no—
213D4—21no—
214D4—41no—
215D4—41no—
216D4—232no—
217D4—231no—
218D4—461no—
219D4—922no—
220D4—922no—
221D4—922no—
222D4—921no—
223C2×C4—41no—
224C2×C4—232no—
225C2×C4—922no—
226C2×C4—922no—
227C2×C4—922no—
228C2×C4—921no—
229C2×C4—921no—
230C2×C4D4×D2311yescharacteristic
231C8—232no—
232C8D4×D2311yescharacteristic
233C8C23×D2311yescharacteristic
234C22—41no—
235C22—42no—
236C22—161no—
237C22—232no—
238C22—921no—
239C22—3682no—
240C22D8×D2311yescharacteristic
241C4—41no—
242C4—232no—
243C4—921no—
244C4—922no—
245C4D8×D2311yescharacteristic
246C4C2×D4×D2311yescharacteristic
247C2—21no—
248C2—82no—
249C2—81no—
250C2—462no—
251C2—1841no—
252C2C2×D8×D2311yescharacteristic
subgroupG/H
centerC2C2×D8×D23
commutator, cosocleC8×C23C24
FrattiniC8C23×D23
FittingC23×#509C2
socleC2×C23C22×D8
hypercenter#509D23
hypocenterC23#3416
LCSG ▷ C8×C23 ▷ C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #13093231C2semidirect
2. #13093218C2semidirect
3. #13092860C2semidirect
4. #13092856C2semidirect
5. #13092851C2semidirect
6. #13092847C2semidirect
7. #13092846C2semidirect
8. C2×#10625009C2semidirect
9. SD32×D23C2semidirect
10. #13092835C2semidirect
11. C23×#509C2semidirect
14. C23×#135C22semidirect
18. Q16×D23C22semidirect
21. #10625083C22semidirect
24. C2×Dic92C22semidirect
27. C23×SD32C22semidirect
28. C23×D16C22semidirect
29. C23×M32C22semidirect
30. #10625011C22semidirect
31. #10625010C22semidirect
32. #10625009C22semidirect
33. #10625009C22semidirect
34. #10625009C22semidirect
35. #10625008C22semidirect
36. C2×C23⋊C16C22semidirect
37. Dic184C22semidirect
38. #10624982C22semidirect
39. #10624980C22semidirect
40. C16×D23C22semidirect
126. #509D23semidirect
147. #135C2×D23semidirect
160. SD32C2×D23semidirect
162. D16C2×D23semidirect
164. M32C2×D23semidirect
167. C23#3416semidirect

Maximal subgroups

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

subgroupindex
#130932312
#130932182
#130928602
#130928562
#130928512
#130928472
#130928462
C2×#106250092
SD32×D232
#130928352
C23×#5092
#341623