Finite Group Databasebeta

← #30676410

Subgroups — #30676410

Order 16768.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16862273C212yesmaximal, complemented
2C2×#16862270C212yesmaximal, complemented
3#22553861C211yesmaximal, characteristic, complemented
4#127×D131C211yesmaximal, characteristic, complemented
5#22553702C211yesmaximal, characteristic, complemented
6#22553701C211yesmaximal, characteristic, complemented
7#22553699C211yesmaximal, characteristic, complemented
8#22553687C212yesmaximal, complemented
9#22553683C212yesmaximal, complemented
10#22553264C211yesmaximal, characteristic, complemented
11#496×C131C211yesmaximal, characteristic, complemented
12C2×#14523514C2211yescharacteristic
13C2×#14523512C2211yescharacteristic
14C2×D4×D131—22no—
15C2×D4×D131C2212yes—
16#16862273—28no—
17#16862270—28no—
18C2×#14523502C2212yescomplemented
19C2×#14523501C2212yescomplemented
20C2×#14523500C2212yescomplemented
21C2×D1048C2212yescomplemented
22C2×#14523498C2212yescomplemented
23C2×#14523498C2212yescomplemented
24C2×#14523496C2212yes—
25#16862244C2211yescharacteristic, complemented
26#16862243C2211yescharacteristic
27#16862242—22no—
28#16862242C2212yes—
29#16862242C2211yescharacteristic, complemented
30#16862234C2211yescharacteristic, complemented
31#16862219—22no—
32(C22⋊D131)⋊C4—22no—
33C42×D131C2211yescharacteristic
34#16862157C2211yescharacteristic, complemented
35C8×Dic131C2212yescomplemented
36C2×SD16×C131C2212yescomplemented
37C2×D8×C131C2212yescomplemented
38#127×C131C2211yescharacteristic
39#124×C131C2211yescharacteristic, complemented
40#97×C131C2211yescharacteristic, complemented
41C23×D131—22no—
42#14523514—24no—
43#14523512—24no—
44C2×Dic262C2311yescharacteristic
45C2×D524—21no—
46C2×D524—22no—
47C2×D524C2311yescharacteristic
48C2×D524C2312yes—
49C2×C22⋊D131—21no—
50C2×C22⋊D131—22no—
51C2×C22⋊D131—22no—
52C2×C22⋊D131—24no—
53D4×D131—24no—
54D4×D131—24no—
55D4×D131—48no—
56C22×Dic131—21no—
57C2×C4×D131—21no—
58C2×C4×D131D412yes—
59C2×C4×D131C2311yescharacteristic
60#14523502—24no—
61#14523501—24no—
62#14523500—24no—
63D1048—24no—
64#14523498—24no—
65#14523498—24no—
66#14523496D418yescomplemented
67C2×C131⋊C8C2312yes—
68Dic131⋊C4—22no—
69(C2×D131)⋊C4—22no—
70C4×Dic131D412yes—
71C4×Dic131C2311yescharacteristic
72C2×Q8×C131C2311yescharacteristic
73C2×D4×C131—22no—
74C2×D4×C131C2312yes—
75C2×D4×C131C2311yescharacteristic
76SD16×C131—24no—
77D8×C131—24no—
78C2×C8×C131C2312yes—
79C22⋊C4×C131—22no—
80C42×C131C2311yescharacteristic
81C22×D131—21no—
82C22×D131—22no—
83C22×D131—48no—
84C22×D131—44no—
85C22×D131C2×D411yescharacteristic
86Dic262—22no—
87D524—22no—
88D524—22no—
89D524—22no—
90D524—44no—
91D524—42no—
92C22⋊D131—42no—
93C22⋊D131—44no—
94C22⋊D131—48no—
95C22⋊D131—44no—
96C2×Dic131—21no—
97C2×Dic131—42no—
98C2×Dic131C2×D411yescharacteristic
99C2×Dic131C2×D412yes—
100C4×D131—24no—
101C4×D131—42no—
102C4×D131C2×D414yes—
103C131⋊C8C2×D414yescomplemented
104C23×C131—21no—
105C23×C131—22no—
106Q8×C131—22no—
107D4×C131—22no—
108D4×C131—22no—
109D4×C131—22no—
110D4×C131—44no—
111C2×C4×C131—21no—
112C2×C4×C131C2×D412yes—
113C2×C4×C131C2411yescharacteristic
114C8×C131C2×D414yescomplemented
115C22×C131—41no—
116C22×C131—42no—
117C22×C131—44no—
118C22×C131—42no—
119C22×C131C22×D411yescharacteristic
120C2×D131—22no—
121C2×D131—41no—
122C2×D131—42no—
123C2×D131—44no—
124C2×D131—42no—
125C2×D131—88no—
126C2×D131#12712yes—
127C4×C131—22no—
128C4×C131—41no—
129C4×C131C22×D412yes—
130Dic131—41no—
131Dic131#12712yes—
132Dic131AGL(1,ℤ/8)14yescomplemented
133C2×C131—82no—
134C2×C131—81no—
135C2×C131C2×#12711yescharacteristic
136C2×C131C2×AGL(1,ℤ/8)12yes—
137D131—22no—
138D131—81no—
139D131—82no—
140C131#315111yescharacteristic, complemented, Sylow
141#3151—1311nomaximal, Sylow
142C2×AGL(1,ℤ/8)—1312no—
143C2×AGL(1,ℤ/8)—1312no—
144#532—1311no—
145C2×#127—1311no—
146#496—1311no—
147#496—1311no—
148#496—1311no—
149#496D13111yescharacteristic, complemented
150#493—1312no—
151#492—1312no—
152C4×M16—1311no—
153C2×#41—1311no—
154C2×#41—1311no—
155C22×D4—2622no—
156C22×D4—1312no—
157AGL(1,ℤ/8)—2628no—
158AGL(1,ℤ/8)—2628no—
159C2×SD16—1312no—
160C2×SD16—1312no—
161C2×SD16—1312no—
162C2×SD16C2×D13112yescomplemented
163C2×D8—1312no—
164C2×D8—1312no—
165C2×D8—1312no—
166C2×D8C2×D13112yescomplemented
167C2×M16—1312no—
168C4⋊Q8—1311no—
169#127—2622no—
170#127—1312no—
171#127—1311no—
172#127—1311no—
173#127C2×D13111yescharacteristic
174#124—1311no—
175#124C2×D13111yescharacteristic, complemented
176#121—2622no—
177C4×D4—2622no—
178C2×C42—1311no—
179#97—1311no—
180#97C2×D13111yescharacteristic, complemented
181C4×C8—1312no—
182C24—2622no—
183#41—2624no—
184#41—2624no—
185C2×Q8—1311no—
186C2×Q8C22×D13111yescharacteristic
187C2×D4—22no—
188C2×D4—2621no—
189C2×D4—2621no—
190C2×D4—2624no—
191C2×D4—2624no—
192C2×D4—2622no—
193C2×D4—2622no—
194C2×D4—2624no—
195C2×D4—2622no—
196C2×D4—5248no—
197C2×D4—1311no—
198C2×D4—1312no—
199C2×D4C22×D13111yescharacteristic
200C2×D4C22×D13112yes—
201C22×C4—2621no—
202C22×C4—2621no—
203C22×C4—1312no—
204C22×C4—1311no—
205SD16—24no—
206SD16—2624no—
207SD16—2624no—
208SD16—2624no—
209D8—24no—
210D8—2624no—
211D8—2624no—
212D8—2624no—
213M16—1318no—
214C2×C8—1312no—
215C2×C8C22×D13112yes—
216C4⋊C4—2622no—
217C22⋊C4—22no—
218C22⋊C4—2622no—
219C42—1312no—
220C42—1311no—
221C42C22×D13111yescharacteristic
222C23—21no—
223C23—22no—
224C23—2621no—
225C23—2622no—
226C23—5248no—
227C23—5244no—
228C23—1311no—
229Q8—22no—
230Q8—2622no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—2622no—
236D4—2622no—
237D4—2622no—
238D4—5244no—
239D4—5242no—
240D4—5242no—
241D4—5244no—
242D4—5248no—
243D4—5244no—
244C2×C4—21no—
245C2×C4—2624no—
246C2×C4—2621no—
247C2×C4—5242no—
248C2×C4—5242no—
249C2×C4—1312no—
250C2×C4—1311no—
251C2×C4—1314no—
252C2×C4D4×D13112yes—
253C2×C4C23×D13111yescharacteristic
254C8—1314no—
255C8D4×D13114yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—2622no—
261C22—5244no—
262C22—5242no—
263C22—5241no—
264C22—5242no—
265C22—10488no—
266C22—1312no—
267C22C2×D4×D13111yescharacteristic
268C4—22no—
269C4—41no—
270C4—5241no—
271C4—1314no—
272C4—1312no—
273C4C2×D4×D13112yes—
274C2—2622no—
275C2—82no—
276C2—81no—
277C2—10482no—
278C2—10481no—
279C2#127×D13111yescharacteristic
280C2AGL(1,ℤ/8)×D13112yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#496×C131C2
socleC22×C131C22×D4
hypercenter#496D131
hypocenterC131#3151
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16862273C2semidirect
2. C2×#16862270C2semidirect
3. #22553861C2semidirect
4. #127×D131C2semidirect
5. #22553702C2semidirect
6. #22553701C2semidirect
7. #22553699C2semidirect
8. #22553687C2semidirect
9. #22553683C2semidirect
10. #22553264C2semidirect
11. #496×C131C2semidirect
18. C2×#14523502C22semidirect
19. C2×#14523501C22semidirect
20. C2×#14523500C22semidirect
21. C2×D1048C22semidirect
22. C2×#14523498C22semidirect
23. C2×#14523498C22semidirect
25. #16862244C22semidirect
29. #16862242C22semidirect
30. #16862234C22semidirect
34. #16862157C22semidirect
35. C8×Dic131C22semidirect
36. C2×SD16×C131C22semidirect
37. C2×D8×C131C22semidirect
39. #124×C131C22semidirect
40. #97×C131C22semidirect
66. #14523496D4semidirect
103. C131⋊C8C2×D4semidirect
114. C8×C131C2×D4semidirect
132. Dic131AGL(1,ℤ/8)semidirect
140. C131#3151semidirect
149. #496D131semidirect
162. C2×SD16C2×D131semidirect
166. C2×D8C2×D131semidirect
175. #124C2×D131semidirect
180. #97C2×D131semidirect
255. C8D4×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622732
C2×#168622702
#225538612
#127×D1312
#225537022
#225537012
#225536992
#225536872
#225536832
#225532642
#496×C1312
#3151131