Finite Group Databasebeta

← #30675207

Subgroups — #30675207

Order 16768.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#14523514C211yesmaximal, characteristic, complemented
2C2×#16862277C211yesmaximal, characteristic, complemented
3C2×#16862274C211yesmaximal, characteristic, complemented
4#22553824C211yesmaximal, characteristic, complemented
5#22553477C214yesmaximal, complemented
6#22553460C214yesmaximal, complemented
7#22553316C211yesmaximal, characteristic, complemented
8#22553275C211yesmaximal, characteristic, complemented
9#419×C131C211yesmaximal, characteristic, complemented
10C2×#14523514—24no—
11C2×#14523514—22no—
12C2×#14523514—24no—
13C2×#14523514C2214yes—
14C2×#14523513C2211yescharacteristic
15C22×D524—21no—
16C22×D524C2211yescharacteristic
17C22×C4×D131—21no—
18C22×C4×D131C2211yescharacteristic
19#16862277—24no—
20#16862274—24no—
21C2×#14523503C2212yescomplemented
22C2×#14523502C2212yescomplemented
23C2×#14523500C2212yescomplemented
24C2×D1048C2212yescomplemented
25C2×#14523497C2211yescharacteristic
26#16862230—21no—
27#16862226C2212yescomplemented
28#16862221C2212yescomplemented
29#16862216—21no—
30C4×D524—22no—
31#16862166C2214yescomplemented
32(C2×D131)⋊C8C2214yescomplemented
33C22×Q8×C131C2211yescharacteristic
34C2×M16×C131C2211yescharacteristic
35#117×C131C2211yescharacteristic
36#103×C131C2214yescomplemented
37C23×D131—21no—
38C23×D131D411yescharacteristic
39#14523514—216no—
40#14523514—28no—
41#14523514—28no—
42#14523513—22no—
43#14523513—22no—
44C2×Dic262C2311yescharacteristic
45C2×D524—28no—
46C2×D524—24no—
47C2×D524—22no—
48C2×D524—21no—
49C2×D524—22no—
50C2×D524D414yescomplemented
51C2×D524C2311yescharacteristic
52C2×D524C2312yes—
53C2×C22⋊D131—21no—
54C22×Dic131D411yescharacteristic
55C2×C4×D131—28no—
56C2×C4×D131—24no—
57C2×C4×D131—22no—
58C2×C4×D131—22no—
59C2×C4×D131—21no—
60C2×C4×D131D412yes—
61C2×C4×D131C2312yes—
62#14523503—24no—
63#14523502—24no—
64#14523500—24no—
65D1048—24no—
66#14523497—22no—
67C2×C131⋊C8C2312yes—
68C4⋊Dic131—22no—
69(C2×D131)⋊C4—22no—
70(C2×D131)⋊C4—22no—
71C2×Q8×C131—22no—
72C2×Q8×C131—21no—
73C2×Q8×C131D414yescomplemented
74C2×Q8×C131C2312yes—
75C22×C4×C131—21no—
76C22×C4×C131C2311yescharacteristic
77M16×C131—22no—
78C2×C8×C131C2312yes—
79C4⋊C4×C131C2312yes—
80C22⋊C4×C131—21no—
81C42×C131—21no—
82C22×D131—22no—
83C22×D131—24no—
84C22×D131—22no—
85C22×D131—21no—
86C22×D131—42no—
87C22×D131—44no—
88C22×D131C2×D412yes—
89Dic262—21no—
90Dic262—21no—
91D524—216no—
92D524—24no—
93D524—22no—
94D524—21no—
95D524—21no—
96D524—44no—
97D524C2×D414yes—
98C22⋊D131—42no—
99C2×Dic131—22no—
100C2×Dic131—21no—
101C2×Dic131C2×D412yes—
102C4×D131—216no—
103C4×D131—24no—
104C4×D131—22no—
105C4×D131—22no—
106C4×D131—42no—
107C131⋊C8—22no—
108C23×C131C2×D411yescharacteristic
109Q8×C131—24no—
110Q8×C131—22no—
111Q8×C131C2×D414yes—
112C2×C4×C131—24no—
113C2×C4×C131—22no—
114C2×C4×C131—22no—
115C2×C4×C131C2×D411yescharacteristic
116C2×C4×C131C2×D412yes—
117C2×C4×C131C2×D412yes—
118C2×C4×C131C2411yescharacteristic
119C8×C131—22no—
120C22×C131—22no—
121C22×C131#12012yes—
122C22×C131C22×D411yescharacteristic
123C2×D131—22no—
124C2×D131—24no—
125C2×D131—22no—
126C2×D131—22no—
127C2×D131—44no—
128C2×D131—48no—
129C2×D131—44no—
130C2×D131—41no—
131C2×D131—42no—
132C4×C131—24no—
133C4×C131—42no—
134C4×C131#12012yes—
135C4×C131C22×D411yescharacteristic
136C4×C131C22×D411yescharacteristic
137Dic131—22no—
138Dic131—41no—
139C2×C131—22no—
140C2×C131C2×#12011yescharacteristic
141C2×C131#57512yes—
142D131—42no—
143D131—44no—
144D131—81no—
145C131#300511yescharacteristic, complemented, Sylow
146#3005—1311nomaximal, Sylow
147C22×#41—1311no—
148C2×#137—1311no—
149C2×AGL(1,ℤ/8)—1311no—
150#525—1311no—
151#452—1314no—
152#449—1314no—
153#419D13111yescharacteristic, complemented
154#418—1311no—
155#407—1311no—
156C2×#41—2624no—
157C2×#41—2624no—
158C2×#41—2622no—
159C2×#41—1314no—
160C2×#41—1311no—
161C22×Q8C2×D13111yescharacteristic
162C22×D4—2621no—
163C22×D4—1311no—
164C23×C4—2621no—
165C23×C4—1311no—
166#137—2624no—
167AGL(1,ℤ/8)—2624no—
168C2×Q16—1312no—
169C2×SD16—1312no—
170C2×SD16—1312no—
171C2×D8—1312no—
172C2×M16—1311no—
173C2×M16C2×D13111yescharacteristic
174#123—2621no—
175C22⋊Q8—1312no—
176#121—1312no—
177#120—2621no—
178C4×D4—2622no—
179#117C2×D13111yescharacteristic
180#103C2×D13114yescomplemented
181#102—1314no—
182C22⋊C8—1314no—
183C24—2621no—
184C24—1311no—
185#41—26216no—
186#41—2628no—
187#41—2628no—
188#41—2622no—
189#41—2622no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—1311no—
193C2×Q8D52414yescomplemented
194C2×Q8C22×D13112yes—
195C2×D4—2628no—
196C2×D4—2624no—
197C2×D4—2622no—
198C2×D4—2621no—
199C2×D4—2621no—
200C2×D4—2622no—
201C2×D4—1314no—
202C2×D4—1311no—
203C2×D4—1312no—
204C22×C4—21no—
205C22×C4—2628no—
206C22×C4—2624no—
207C22×C4—2622no—
208C22×C4—2622no—
209C22×C4—2621no—
210C22×C4—1311no—
211C22×C4—1312no—
212C22×C4—1312no—
213C22×C4C22×D13111yescharacteristic
214Q16—2624no—
215SD16—2624no—
216SD16—2624no—
217D8—2624no—
218M16—22no—
219M16—2622no—
220C2×C8—1312no—
221C2×C8C22×D13112yes—
222C4⋊C4—2622no—
223C4⋊C4C22×D13112yes—
224C22⋊C4—21no—
225C22⋊C4—2622no—
226C22⋊C4—2622no—
227C42—21no—
228C23—2622no—
229C23—2624no—
230C23—2622no—
231C23—2621no—
232C23—5242no—
233C23—5244no—
234C23—1312no—
235C23D4×D13111yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—2621no—
239Q8—2621no—
240Q8C2×D52414yes—
241D4—26216no—
242D4—2624no—
243D4—2621no—
244D4—2622no—
245D4—2621no—
246D4—5244no—
247D4—5242no—
248D4—1314no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—26216no—
253C2×C4—2622no—
254C2×C4—2624no—
255C2×C4—2622no—
256C2×C4—2622no—
257C2×C4—2621no—
258C2×C4—5242no—
259C2×C4—1312no—
260C2×C4D4×D13111yescharacteristic
261C2×C4D4×D13112yes—
262C2×C4C2×D52412yes—
263C2×C4C23×D13111yescharacteristic
264C8—22no—
265C8—2622no—
266C22—22no—
267C22—2622no—
268C22—2624no—
269C22—2622no—
270C22—2622no—
271C22—5244no—
272C22—5248no—
273C22—5244no—
274C22—5241no—
275C22—5242no—
276C22#1686221612yes—
277C22C2×D4×D13111yescharacteristic
278C4—24no—
279C4—42no—
280C4—2622no—
281C4—5241no—
282C4#1686221612yes—
283C4C2×D4×D13111yescharacteristic
284C4C22×D52411yescharacteristic
285C2—22no—
286C2—5242no—
287C2—5244no—
288C2—10481no—
289C2C2×#1686221611yescharacteristic
290C2#2255411112yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#419×C131C2
socleC22×C131C22×D4
hypercenter#419D131
hypocenterC131#3005
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#14523514C2semidirect
2. C2×#16862277C2semidirect
3. C2×#16862274C2semidirect
4. #22553824C2semidirect
5. #22553477C2semidirect
6. #22553460C2semidirect
7. #22553316C2semidirect
8. #22553275C2semidirect
9. #419×C131C2semidirect
21. C2×#14523503C22semidirect
22. C2×#14523502C22semidirect
23. C2×#14523500C22semidirect
24. C2×D1048C22semidirect
27. #16862226C22semidirect
28. #16862221C22semidirect
31. #16862166C22semidirect
32. (C2×D131)⋊C8C22semidirect
36. #103×C131C22semidirect
50. C2×D524D4semidirect
73. C2×Q8×C131D4semidirect
145. C131#3005semidirect
153. #419D131semidirect
180. #103C2×D131semidirect
193. C2×Q8D524semidirect

Maximal subgroups

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

subgroupindex
C22×#145235142
C2×#168622772
C2×#168622742
#225538242
#225534772
#225534602
#225533162
#225532752
#419×C1312
#3005131