Finite Group Databasebeta

← #30677786

Subgroups — #30677786

Order 16768.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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#22554019C211yesmaximal, characteristic, complemented
2D4×D524C211yesmaximal, characteristic
3#22553659C211yesmaximal, characteristic, complemented
4#22553631C212yesmaximal, complemented
5#22553603C211yesmaximal, characteristic, complemented
6#22553541C212yesmaximal, complemented
7#22553513C211yesmaximal, characteristic, complemented
8#22553465C212yesmaximal, complemented
9C4×#14523502C211yesmaximal, characteristic, complemented
10C2×#16862168C212yesmaximal, complemented
11C8×D4×C131C211yesmaximal, characteristic, complemented
12C22×D524C2212yes—
13C2×D4×D131—21no—
14C2×#14523502C2211yescharacteristic
15C2×#14523502C2212yescomplemented
16#16862246C2211yescharacteristic, complemented
17#16862243C2211yescharacteristic
18#16862237—21no—
19#16862225C2212yescomplemented
20#16862223—22no—
21#16862221—21no—
22#16862220C2212yes—
23#16862216—22no—
24C4×Dic262C2211yescharacteristic, complemented
25C4×D524C2211yescharacteristic
26D4×Dic131—21no—
27C2×C4⋊Dic131C2212yescomplemented
28#16862178C2212yescomplemented
29#16862178C2211yescharacteristic, complemented
30#16862171C2211yescharacteristic, complemented
31#16862168C2212yes—
32#16862168C2211yescharacteristic
33#16862168C2212yescomplemented
34#16862168C2212yescomplemented
35#16862168C2212yescomplemented
36C22×C8×C131C2212yescomplemented
37C4×D4×C131C2211yescharacteristic
38C4⋊C8×C131C2211yescharacteristic, complemented
39C22⋊C8×C131C2212yescomplemented
40C4×C8×C131C2211yescharacteristic, complemented
41C23×D131—22no—
42C2×Dic262—21no—
43C2×Dic262C2311yescharacteristic
44C2×D524—24no—
45C2×D524—22no—
46C2×D524—21no—
47C2×D524—21no—
48C2×D524—42no—
49C2×D524C2312yes—
50C2×D524C2311yescharacteristic
51C2×C22⋊D131—22no—
52D4×D131—44no—
53C22×Dic131—22no—
54C2×C4×D131—21no—
55#14523502—22no—
56#14523502—42no—
57C4⋊Dic131—21no—
58C4⋊Dic131—22no—
59C4⋊Dic131C2311yescharacteristic
60C4⋊Dic131C2311yescharacteristic
61C4⋊Dic131C2312yes—
62Dic131⋊C4—21no—
63Dic131⋊C4—22no—
64C22⋊Dic131—22no—
65(C2×D131)⋊C4—21no—
66(C2×D131)⋊C4—22no—
67C4×Dic131—21no—
68C2×D4×C131D411yescharacteristic
69C22×C4×C131C2312yes—
70C2×C8×C131—42no—
71C2×C8×C131C2311yescharacteristic
72C2×C8×C131C2312yes—
73C2×C8×C131C2311yescharacteristic
74C4⋊C4×C131D411yescharacteristic
75C22⋊C4×C131D412yes—
76C42×C131C2311yescharacteristic
77C22×D131—21no—
78C22×D131—22no—
79C22×D131—44no—
80C22×D131—44no—
81C22×D131—82no—
82Dic262—21no—
83Dic262—21no—
84Dic262—81no—
85D524—22no—
86D524—21no—
87D524—22no—
88D524—44no—
89D524—42no—
90D524—81no—
91D524#4112yes—
92C22⋊D131—44no—
93C2×Dic131—21no—
94C2×Dic131—21no—
95C2×Dic131—21no—
96C2×Dic131—22no—
97C2×Dic131—82no—
98C4×D131—42no—
99C23×C131C2×D412yes—
100D4×C131SD1614yescomplemented
101C2×C4×C131—22no—
102C2×C4×C131—22no—
103C2×C4×C131C2×D411yescharacteristic
104C2×C4×C131C2×D412yes—
105C2×C4×C131C2×D411yescharacteristic
106C2×C4×C131C2411yescharacteristic
107C8×C131—21no—
108C8×C131—21no—
109C8×C131—22no—
110C22×C131—22no—
111C22×C131—22no—
112C22×C131C2×SD1614yes—
113C22×C131C22×D411yescharacteristic
114C2×D131—22no—
115C2×D131—41no—
116C2×D131—41no—
117C2×D131—42no—
118C2×D131—42no—
119C2×D131—42no—
120C2×D131—88no—
121C4×C131—22no—
122C4×C131—21no—
123C4×C131C2×SD1612yes—
124C4×C131C2×#4111yescharacteristic
125C4×C131H(5,𝔽2)11yescharacteristic
126Dic131—41no—
127Dic131—41no—
128Dic131—41no—
129Dic131—42no—
130C2×C131—24no—
131C2×C131#54611yescharacteristic
132C2×C131C22×SD1611yescharacteristic
133C2×C131#57611yescharacteristic
134D131—42no—
135D131—82no—
136C131#329511yescharacteristic, complemented, Sylow
137#3295—1311nomaximal, Sylow
138#554—1311no—
139D42—1311no—
140#486—1311no—
141#481—1312no—
142#476—1311no—
143#465—1312no—
144#460—1311no—
145#450—1312no—
146C4×SD16—1311no—
147C8×D4D13111yescharacteristic, complemented
148C2×#102—1312no—
149C22×D4—2621no—
150C22×D4—1312no—
151C2×SD16—1311no—
152C2×SD16—1312no—
153C22×C8C2×D13112yescomplemented
154C4⋊Q8—1311no—
155#127—1311no—
156#125—2621no—
157C22⋊Q8—2622no—
158C22⋊Q8—1312no—
159#121—2621no—
160#121—1312no—
161#120—2622no—
162C4×Q8—1311no—
163C4×D4—2621no—
164C4×D4—1311no—
165C4×D4C2×D13111yescharacteristic
166C2×C4⋊C4—1312no—
167#106—1311no—
168#106—1312no—
169C4⋊C8C2×D13111yescharacteristic, complemented
170#103—1311no—
171#102—1311no—
172#102—1312no—
173#102—1312no—
174#102—1312no—
175#102—1312no—
176C22⋊C8C2×D13112yescomplemented
177C4×C8C2×D13111yescharacteristic, complemented
178C24—2622no—
179C2×Q8—2621no—
180C2×Q8—1311no—
181C2×D4—2624no—
182C2×D4—2622no—
183C2×D4—2622no—
184C2×D4—2621no—
185C2×D4—2621no—
186C2×D4—5242no—
187C2×D4—5244no—
188C2×D4—1312no—
189C2×D4—1311no—
190C2×D4D52411yescharacteristic
191C22×C4—2622no—
192C22×C4—2621no—
193C22×C4C22×D13112yes—
194SD16—2622no—
195SD16—5242no—
196C2×C8—42no—
197C2×C8C22×D13111yescharacteristic
198C2×C8C22×D13111yescharacteristic
199C2×C8C22×D13112yes—
200C4⋊C4—2621no—
201C4⋊C4—2622no—
202C4⋊C4—2622no—
203C4⋊C4—2621no—
204C4⋊C4—1311no—
205C4⋊C4—1312no—
206C4⋊C4—1311no—
207C4⋊C4D52411yescharacteristic
208C22⋊C4—2622no—
209C22⋊C4—2621no—
210C22⋊C4—2622no—
211C22⋊C4D52412yes—
212C42—2621no—
213C42C22×D13111yescharacteristic
214C23—2621no—
215C23—2622no—
216C23—5244no—
217C23—5244no—
218C23—10482no—
219C23C2×D52412yes—
220Q8—2621no—
221Q8—2621no—
222Q8—10481no—
223D4—2622no—
224D4—2621no—
225D4—2622no—
226D4—5244no—
227D4—5244no—
228D4—5242no—
229D4—10481no—
230D4—1312no—
231D4#1452350214yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—2621no—
235C2×C4—2621no—
236C2×C4—2621no—
237C2×C4—2622no—
238C2×C4—5242no—
239C2×C4—10482no—
240C2×C4C2×D52411yescharacteristic
241C2×C4C2×D52412yes—
242C2×C4C2×D52411yescharacteristic
243C2×C4C23×D13111yescharacteristic
244C8—21no—
245C8—21no—
246C8—22no—
247C22—22no—
248C22—22no—
249C22—2622no—
250C22—5241no—
251C22—5241no—
252C22—5242no—
253C22—5242no—
254C22—5242no—
255C22—10488no—
256C22C2×#1452350214yes—
257C22C22×D52411yescharacteristic
258C4—21no—
259C4—22no—
260C4—5241no—
261C4—5242no—
262C4—5241no—
263C4—5241no—
264C4C2×#1452350212yes—
265C4#41×D13111yescharacteristic
266C4#1686229111yescharacteristic
267C2—24no—
268C2—5242no—
269C2—10482no—
270C2#2255395711yescharacteristic
271C2C22×#1452350211yescharacteristic
272C2#2255411711yescharacteristic
subgroupG/H
centerC22C22×D524
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC8×D4×C131C2
socleC22×C131C22×D4
prime radicalD4×D524C2
hypercenterC8×D4D131
hypocenterC131#3295
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #22554019C2semidirect
3. #22553659C2semidirect
4. #22553631C2semidirect
5. #22553603C2semidirect
6. #22553541C2semidirect
7. #22553513C2semidirect
8. #22553465C2semidirect
9. C4×#14523502C2semidirect
10. C2×#16862168C2semidirect
11. C8×D4×C131C2semidirect
15. C2×#14523502C22semidirect
16. #16862246C22semidirect
19. #16862225C22semidirect
24. C4×Dic262C22semidirect
27. C2×C4⋊Dic131C22semidirect
28. #16862178C22semidirect
29. #16862178C22semidirect
30. #16862171C22semidirect
33. #16862168C22semidirect
34. #16862168C22semidirect
35. #16862168C22semidirect
36. C22×C8×C131C22semidirect
38. C4⋊C8×C131C22semidirect
39. C22⋊C8×C131C22semidirect
40. C4×C8×C131C22semidirect
100. D4×C131SD16semidirect
136. C131#3295semidirect
147. C8×D4D131semidirect
153. C22×C8C2×D131semidirect
169. C4⋊C8C2×D131semidirect
176. C22⋊C8C2×D131semidirect
177. C4×C8C2×D131semidirect
231. D4#14523502semidirect

Maximal subgroups

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

subgroupindex
#225540192
D4×D5242
#225536592
#225536312
#225536032
#225535412
#225535132
#225534652
C4×#145235022
C2×#168621682
C8×D4×C1312
#3295131