Finite Group Databasebeta

← #30677751

Subgroups — #30677751

Order 16768.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#22554121C212yesmaximal, complemented
2#22554120C212yesmaximal, complemented
3#22553686C211yesmaximal, characteristic, complemented
4#22553578C212yesmaximal, complemented
5#22553577C212yesmaximal, complemented
6#22553494C211yesmaximal, characteristic, complemented
7#22553481C212yesmaximal, complemented
8#22553426C212yesmaximal, complemented
9#456×C131C211yesmaximal, characteristic, complemented
10#16862294C2211yescharacteristic
11#16862292C2212yescomplemented
12#16862278—22no—
13#16862277—22no—
14#16862277C2212yescomplemented
15#16862274C2212yes—
16#16862273—22no—
17#16862271—22no—
18#16862271C2212yes—
19#16862269—22no—
20#16862266—22no—
21#16862266C2212yes—
22C2×#14523500—22no—
23C2×#14523500C2212yescomplemented
24C2×#14523500C2212yescomplemented
25SD16×D131—22no—
26#16862254C2212yescomplemented
27#16862253C2212yescomplemented
28#16862243C2211yescharacteristic
29#16862181C2212yescomplemented
30#16862174C2212yescomplemented
31#16862165C2211yescharacteristic, complemented
32#16862162C2212yescomplemented
33C4×C131⋊C8C2211yescharacteristic, complemented
342−1+4×C131C2211yescharacteristic
35#137×C131C2212yescomplemented
36C4⋊Q8×C131C2211yescharacteristic, complemented
37#104×C131C2212yescomplemented
38#101×C131C2211yescharacteristic, complemented
39#14523514—22no—
40#14523514—22no—
41#14523513—22no—
42#14523513—41no—
43#14523513C2312yes—
44Q8×D131—22no—
45C2×D524—22no—
46C2×D524—41no—
47C2×D524C2312yes—
48D4×D131—22no—
49D4×D131—42no—
50#14523503—22no—
51#14523503D412yescomplemented
52#14523502—22no—
53#14523501D412yescomplemented
54#14523500—22no—
55#14523500—22no—
56#14523500—22no—
57#14523500—22no—
58#14523500—22no—
59#14523500—22no—
60#14523500D412yescomplemented
61D1048—22no—
62#14523498—22no—
63#14523498D412yescomplemented
64#14523497—22no—
65#14523497C2312yes—
66#14523496—22no—
67C2×C131⋊C8—22no—
68C2×C131⋊C8C2312yes—
69C8×D131—22no—
70#41×C131—22no—
71#41×C131—41no—
72#41×C131C2312yes—
73C2×Q8×C131—22no—
74C2×Q8×C131C2311yescharacteristic
75C2×Q8×C131C2311yescharacteristic
76Q16×C131—22no—
77SD16×C131—22no—
78M16×C131C2312yes—
79C4⋊C4×C131—41no—
80C42×C131C2311yescharacteristic
81C22×D131—22no—
82C22×D131—42no—
83Dic262—41no—
84Dic262C2×D412yes—
85D524—22no—
86D524—22no—
87D524—42no—
88D524—41no—
89D524—82no—
90D524C2×D412yes—
91C22⋊D131—22no—
92C22⋊D131—42no—
93C4×D131—22no—
94C4×D131—42no—
95C4×D131—42no—
96C131⋊C8—22no—
97C131⋊C8C2×D412yes—
98C131⋊C8C2×D412yes—
99Q8×C131—22no—
100Q8×C131—22no—
101Q8×C131—41no—
102Q8×C131C2×D412yes—
103D4×C131—22no—
104D4×C131—41no—
105D4×C131C2×D412yes—
106C2×C4×C131—21no—
107C2×C4×C131—22no—
108C2×C4×C131—21no—
109C2×C4×C131—21no—
110C2×C4×C131—42no—
111C2×C4×C131C2411yescharacteristic
112C8×C131—22no—
113C22×C131—22no—
114C22×C131H(5,𝔽2)11yescharacteristic
115C2×D131—22no—
116C2×D131—42no—
117C2×D131—82no—
118C2×D131—162no—
119C4×C131—22no—
120C4×C131—22no—
121C4×C131—41no—
122C4×C131—41no—
123C4×C131C22×D412yes—
124Dic131—22no—
125C2×C131—21no—
126C2×C131—42no—
127C2×C131D4211yescharacteristic
128D131—42no—
129D131—82no—
130C131#329111yescharacteristic, complemented, Sylow
131#3291—1311nomaximal, Sylow
132#577—1312no—
133#577—1312no—
134#492—1311no—
135#471—1312no—
136#471—1312no—
137#456—1311no—
138#456D13111yescharacteristic, complemented
139#453—1312no—
140#443—1312no—
1412−1+4—1311no—
1422−1+4C2×D13111yescharacteristic
143H(5,𝔽2)—1312no—
144#137—2622no—
145#137—2622no—
146#137—1312no—
147#137C2×D13112yescomplemented
148AGL(1,ℤ/8)—2622no—
149AGL(1,ℤ/8)—2622no—
150AGL(1,ℤ/8)—1312no—
151AGL(1,ℤ/8)—1312no—
152#135—2622no—
153#135—2622no—
154#135—1312no—
155C2×SD16—2622no—
156C2×SD16—2622no—
157C2×SD16—1312no—
158C2×SD16—1312no—
159#131—1312no—
160#131—1312no—
161C4⋊Q8C2×D13111yescharacteristic, complemented
162#127—1311no—
163#108—1312no—
164#104—1312no—
165#104C2×D13112yescomplemented
166#101—1311no—
167#101C2×D13111yescharacteristic, complemented
168#100—1312no—
169C4×C8—1311no—
170#41—22no—
171#41—41no—
172#41—2622no—
173#41—2622no—
174#41—2622no—
175#41—5241no—
176#41—1312no—
177#41C22×D13112yes—
178C2×Q8—22no—
179C2×Q8—2622no—
180C2×Q8C22×D13111yescharacteristic
181C2×Q8C22×D13111yescharacteristic
182C2×D4—2622no—
183C2×D4—2622no—
184C2×D4—5241no—
185C2×D4—5242no—
186C2×D4—1312no—
187Q16—22no—
188Q16—2622no—
189Q16—1312no—
190SD16—22no—
191SD16—2622no—
192SD16—2622no—
193SD16—2622no—
194SD16—2622no—
195SD16—2622no—
196SD16—2622no—
197SD16—2622no—
198SD16—1312no—
199SD16—1312no—
200D8—2622no—
201D8—2622no—
202D8—1312no—
203M16—2622no—
204M16—2622no—
205M16—1312no—
206M16C22×D13112yes—
207C2×C8—2622no—
208C2×C8—2622no—
209C2×C8—1312no—
210C4⋊C4—41no—
211C42C22×D13111yescharacteristic
212C23—2622no—
213C23—5242no—
214Q8—22no—
215Q8—22no—
216Q8—41no—
217Q8—5241no—
218Q8—1312no—
219Q8D4×D13112yes—
220D4—22no—
221D4—41no—
222D4—2622no—
223D4—2622no—
224D4—2622no—
225D4—5242no—
226D4—5241no—
227D4—5242no—
228D4—10482no—
229D4—1312no—
230D4D4×D13112yes—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—2622no—
237C2×C4—5242no—
238C2×C4—5242no—
239C2×C4C23×D13111yescharacteristic
240C8—22no—
241C8—2622no—
242C8—1312no—
243C8—1312no—
244C22—22no—
245C22—2622no—
246C22—5242no—
247C22—10482no—
248C22—20962no—
249C22#1686229111yescharacteristic
250C4—22no—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—2622no—
255C4C2×D4×D13112yes—
256C2—21no—
257C2—42no—
258C2—5242no—
259C2—10482no—
260C2#2255394411yescharacteristic
subgroupG/H
centerC2#22553944
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#456×C131C2
socleC2×C131D42
hypercenter#456D131
hypocenterC131#3291
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #22554121C2semidirect
2. #22554120C2semidirect
3. #22553686C2semidirect
4. #22553578C2semidirect
5. #22553577C2semidirect
6. #22553494C2semidirect
7. #22553481C2semidirect
8. #22553426C2semidirect
9. #456×C131C2semidirect
11. #16862292C22semidirect
14. #16862277C22semidirect
23. C2×#14523500C22semidirect
24. C2×#14523500C22semidirect
26. #16862254C22semidirect
27. #16862253C22semidirect
29. #16862181C22semidirect
30. #16862174C22semidirect
31. #16862165C22semidirect
32. #16862162C22semidirect
33. C4×C131⋊C8C22semidirect
35. #137×C131C22semidirect
36. C4⋊Q8×C131C22semidirect
37. #104×C131C22semidirect
38. #101×C131C22semidirect
51. #14523503D4semidirect
53. #14523501D4semidirect
60. #14523500D4semidirect
63. #14523498D4semidirect
130. C131#3291semidirect
138. #456D131semidirect
147. #137C2×D131semidirect
161. C4⋊Q8C2×D131semidirect
165. #104C2×D131semidirect
167. #101C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225541212
#225541202
#225536862
#225535782
#225535772
#225534942
#225534812
#225534262
#456×C1312
#3291131