Finite Group Databasebeta

← #30677755

Subgroups — #30677755

Order 16768.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 125 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#22554124C212yesmaximal, complemented
2#22554119C212yesmaximal, complemented
3#22553685C211yesmaximal, characteristic, complemented
4#22553576C212yesmaximal, complemented
5#22553575C212yesmaximal, complemented
6#22553496C212yesmaximal, complemented
7#22553481C212yesmaximal, complemented
8#22553430C211yesmaximal, characteristic, complemented
9#443×C131C211yesmaximal, characteristic, complemented
10#16862293C2212yes—
11#16862292C2212yescomplemented
12#16862279—22no—
13#16862279C2212yescomplemented
14#16862278—22no—
15#16862276C2212yescomplemented
16#16862274—22no—
17#16862274C2212yes—
18#16862271C2212yes—
19#16862270—22no—
20#16862268—22no—
21#16862268C2211yescharacteristic
22#16862267—22no—
23C2×#14523502—22no—
24C2×#14523502C2212yescomplemented
25C2×#14523502C2212yescomplemented
26SD16×D131—22no—
27#16862252C2212yescomplemented
28#16862246C2211yescharacteristic, complemented
29#16862243C2211yescharacteristic
30#16862182C2211yescharacteristic, complemented
31#16862174C2212yescomplemented
32#16862164C2212yescomplemented
33#16862162C2212yescomplemented
34#135×C131C2211yescharacteristic
35#131×C131C2212yescomplemented
36#108×C131C2211yescharacteristic, complemented
37#104×C131C2212yescomplemented
38C4×C8×C131C2211yescharacteristic, complemented
39#14523514—22no—
40#14523513—22no—
41#14523513—22no—
42#14523513C2312yes—
43#14523512—22no—
44#14523512—42no—
45C2×Dic262—22no—
46C2×Dic262C2312yes—
47Q8×D131—22no—
48C2×D524—22no—
49C2×D524—41no—
50C2×D524C2312yes—
51D4×D131—22no—
52D4×D131—42no—
53Dic524—22no—
54Dic524D411yescharacteristic, complemented
55#14523503—22no—
56#14523502—22no—
57#14523502—22no—
58#14523502—22no—
59#14523502—22no—
60#14523502—22no—
61#14523502—22no—
62#14523502D412yescomplemented
63#14523501—22no—
64#14523500—22no—
65D1048—22no—
66D1048D411yescharacteristic, complemented
67#14523498—22no—
68#14523497C2312yes—
69#14523496—22no—
70C8×D131—22no—
71C4⋊Dic131—41no—
72#41×C131C2312yes—
73Q16×C131D411yescharacteristic, complemented
74SD16×C131D412yescomplemented
75D8×C131D411yescharacteristic, complemented
76M16×C131—22no—
77M16×C131C2312yes—
78C2×C8×C131—22no—
79C2×C8×C131C2311yescharacteristic
80C2×C8×C131C2311yescharacteristic
81C42×C131C2311yescharacteristic
82C22×D131—22no—
83C22×D131—42no—
84Dic262—22no—
85Dic262—22no—
86Dic262—42no—
87Dic262C2×D412yes—
88D524—22no—
89D524—22no—
90D524—42no—
91D524—82no—
92D524C2×D412yes—
93C22⋊D131—22no—
94C22⋊D131—42no—
95C22⋊D131—42no—
96C2×Dic131—22no—
97C2×Dic131—42no—
98C4×D131—22no—
99C4×D131—42no—
100C4×D131—42no—
101C131⋊C8—22no—
102Q8×C131C2×D412yes—
103D4×C131—22no—
104D4×C131C2×D412yes—
105C2×C4×C131—21no—
106C2×C4×C131—22no—
107C2×C4×C131C2411yescharacteristic
108C8×C131—22no—
109C8×C131C2×D411yescharacteristic
110C8×C131C2×D411yescharacteristic
111C8×C131C2×D411yescharacteristic
112C8×C131C2×D411yescharacteristic
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×C131C22×D411yescharacteristic
122C4×C131C22×D411yescharacteristic
123Dic131—22no—
124Dic131—42no—
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—1312no—
138#453—1312no—
139#443—1311no—
140#443D13111yescharacteristic, complemented
1412−1+4—1312no—
142H(5,𝔽2)—1312no—
143#137—2622no—
144#137—2622no—
145#137—1312no—
146#137—1312no—
147AGL(1,ℤ/8)—2622no—
148AGL(1,ℤ/8)—2622no—
149AGL(1,ℤ/8)—1312no—
150AGL(1,ℤ/8)—1312no—
151#135—2622no—
152#135—2622no—
153#135—1311no—
154#135C2×D13111yescharacteristic
155C2×SD16—2622no—
156C2×SD16—2622no—
157C2×SD16—1312no—
158C2×SD16—1312no—
159#131—1312no—
160#131C2×D13112yescomplemented
161C4⋊Q8—1311no—
162#127—1311no—
163#108—1311no—
164#108C2×D13111yescharacteristic, complemented
165#104—1312no—
166#104C2×D13112yescomplemented
167#101—1312no—
168#100—1312no—
169C4×C8C2×D13111yescharacteristic, complemented
170#41—2622no—
171#41—2622no—
172#41—2622no—
173#41—2622no—
174#41—5242no—
175#41—1312no—
176#41C22×D13112yes—
177C2×Q8—2622no—
178C2×Q8—2622no—
179C2×Q8—1312no—
180C2×D4—2622no—
181C2×D4—2622no—
182C2×D4—5241no—
183C2×D4—5242no—
184C2×D4—1312no—
185Q16—2622no—
186Q16—2622no—
187Q16—1311no—
188Q16D52411yescharacteristic, complemented
189SD16—2622no—
190SD16—2622no—
191SD16—2622no—
192SD16—2622no—
193SD16—2622no—
194SD16—2622no—
195SD16—2622no—
196SD16—2622no—
197SD16—1312no—
198SD16D52412yescomplemented
199D8—2622no—
200D8—2622no—
201D8—1311no—
202D8D52411yescharacteristic, complemented
203M16—22no—
204M16—2622no—
205M16—1312no—
206M16C22×D13112yes—
207C2×C8—22no—
208C2×C8—2622no—
209C2×C8C22×D13111yescharacteristic
210C2×C8C22×D13111yescharacteristic
211C4⋊C4—5241no—
212C42C22×D13111yescharacteristic
213C23—2622no—
214C23—5242no—
215Q8—2622no—
216Q8—2622no—
217Q8—5242no—
218Q8—1312no—
219Q8C2×D52412yes—
220D4—22no—
221D4—2622no—
222D4—2622no—
223D4—2622no—
224D4—5242no—
225D4—5242no—
226D4—5242no—
227D4—10482no—
228D4—1312no—
229D4C2×D52412yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—2622no—
233C2×C4—2622no—
234C2×C4—5242no—
235C2×C4—5242no—
236C2×C4—5242no—
237C2×C4C23×D13111yescharacteristic
238C8—22no—
239C8—2622no—
240C8D4×D13111yescharacteristic
241C8D4×D13111yescharacteristic
242C8C2×D52411yescharacteristic
243C8C2×D52411yescharacteristic
244C22—22no—
245C22—2622no—
246C22—5242no—
247C22—10482no—
248C22—20962no—
249C22#1686229211yescharacteristic
250C4—22no—
251C4—22no—
252C4—2622no—
253C4—5242no—
254C4C2×D4×D13111yescharacteristic
255C4C22×D52411yescharacteristic
256C2—21no—
257C2—42no—
258C2—5242no—
259C2—10482no—
260C2D4×D52411yescharacteristic
subgroupG/H
centerC2D4×D524
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#443×C131C2
socleC2×C131D42
hypercenter#443D131
hypocenterC131#3291
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #22554124C2semidirect
2. #22554119C2semidirect
3. #22553685C2semidirect
4. #22553576C2semidirect
5. #22553575C2semidirect
6. #22553496C2semidirect
7. #22553481C2semidirect
8. #22553430C2semidirect
9. #443×C131C2semidirect
11. #16862292C22semidirect
13. #16862279C22semidirect
15. #16862276C22semidirect
24. C2×#14523502C22semidirect
25. C2×#14523502C22semidirect
27. #16862252C22semidirect
28. #16862246C22semidirect
30. #16862182C22semidirect
31. #16862174C22semidirect
32. #16862164C22semidirect
33. #16862162C22semidirect
35. #131×C131C22semidirect
36. #108×C131C22semidirect
37. #104×C131C22semidirect
38. C4×C8×C131C22semidirect
54. Dic524D4semidirect
62. #14523502D4semidirect
66. D1048D4semidirect
73. Q16×C131D4semidirect
74. SD16×C131D4semidirect
75. D8×C131D4semidirect
130. C131#3291semidirect
140. #443D131semidirect
160. #131C2×D131semidirect
164. #108C2×D131semidirect
166. #104C2×D131semidirect
169. C4×C8C2×D131semidirect
188. Q16D524semidirect
198. SD16D524semidirect
202. D8D524semidirect

Maximal subgroups

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

subgroupindex
#225541242
#225541192
#225536852
#225535762
#225535752
#225534962
#225534812
#225534302
#443×C1312
#3291131