Finite Group Databasebeta

← #30675896

Subgroups — #30675896

Order 16768.

462 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 121 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#121×D131C211yesmaximal, characteristic
2C2×#16862204C211yesmaximal, characteristic, complemented
3#22553639C212yesmaximal, complemented
4#22553635C212yesmaximal, complemented
5#22553631C211yesmaximal, characteristic, complemented
6#22553628C211yesmaximal, characteristic, complemented
7#22553625C211yesmaximal, characteristic, complemented
8#22553353C212yesmaximal, complemented
9#22553315C212yesmaximal, complemented
10#22553274C211yesmaximal, characteristic, complemented
11#480×C131C211yesmaximal, characteristic, complemented
12C2×D4×D131—21no—
13C2×D4×D131C2211yescharacteristic
14C22×C4×D131C2211yescharacteristic
15C2×#14523496C2212yes—
16#16862221C2211yescharacteristic
17#16862220C2211yescharacteristic
18#16862219—21no—
19#16862218—21no—
20#16862217C2211yescharacteristic
21#16862204—21no—
22#16862204—21no—
23#16862204C2212yes—
24#16862204C2212yescomplemented
25C2×C4⋊Dic131C2211yescharacteristic, complemented
26C4⋊C4×D131C2211yescharacteristic
27C2×(C2×D131)⋊C4—21no—
28C22⋊C4×D131—21no—
29C2×C4×Dic131—21no—
30#16862179C2212yescomplemented
31#16862178C2212yescomplemented
32#16862177C2212yescomplemented
33#16862168C2212yescomplemented
34#16862167C2212yescomplemented
35#16862166C2212yescomplemented
36C22⋊(C131⋊C8)C2211yescharacteristic, complemented
37(C2×D131)⋊C8C2212yescomplemented
38#121×C131C2211yescharacteristic
39C2×C4⋊C4×C131C2211yescharacteristic, complemented
40#107×C131C2212yescomplemented
41#102×C131C2212yescomplemented
42C22⋊C8×C131C2211yescharacteristic, complemented
43C23×D131—21no—
44C23×D131D411yescharacteristic
45C2×D524—21no—
46C2×D524C2311yescharacteristic
47C2×C22⋊D131—21no—
48C2×C22⋊D131—21no—
49C2×C22⋊D131—21no—
50D4×D131—24no—
51D4×D131—44no—
52C22×Dic131—21no—
53C22×Dic131D411yescharacteristic
54C2×C4×D131—24no—
55C2×C4×D131—21no—
56C2×C4×D131D411yescharacteristic
57C2×C4×D131D411yescharacteristic
58C2×C4×D131C2311yescharacteristic
59C2×C4×D131C2311yescharacteristic
60#14523496—24no—
61C2×C131⋊C8C2312yes—
62C4⋊Dic131—21no—
63C4⋊Dic131C2311yescharacteristic
64C4⋊Dic131C2312yes—
65Dic131⋊C4—21no—
66C22⋊Dic131—21no—
67(C2×D131)⋊C4—22no—
68(C2×D131)⋊C4—22no—
69(C2×D131)⋊C4—21no—
70(C2×D131)⋊C4—21no—
71C4×Dic131—22no—
72C4×Dic131—22no—
73C2×D4×C131—21no—
74C2×D4×C131C2311yescharacteristic
75C22×C4×C131—21no—
76C22×C4×C131C2311yescharacteristic
77C2×C8×C131C2312yes—
78C4⋊C4×C131—21no—
79C4⋊C4×C131C2311yescharacteristic
80C4⋊C4×C131C2312yes—
81C22⋊C4×C131—21no—
82C22×D131—22no—
83C22×D131—22no—
84C22×D131—21no—
85C22×D131—41no—
86C22×D131—42no—
87C22×D131—42no—
88C22×D131—42no—
89C22×D131C2×D411yescharacteristic
90C22×D131C2×D411yescharacteristic
91D524—22no—
92D524—42no—
93C22⋊D131—42no—
94C22⋊D131—42no—
95C22⋊D131—42no—
96C2×Dic131—22no—
97C2×Dic131—22no—
98C2×Dic131—21no—
99C2×Dic131—42no—
100C2×Dic131C2×D411yescharacteristic
101C2×Dic131C2×D411yescharacteristic
102C4×D131—22no—
103C4×D131—22no—
104C4×D131—22no—
105C4×D131—42no—
106C4×D131#4114yes—
107C131⋊C8—22no—
108C23×C131—21no—
109C23×C131C2×D411yescharacteristic
110D4×C131—22no—
111D4×C131—42no—
112C2×C4×C131—22no—
113C2×C4×C131—21no—
114C2×C4×C131—22no—
115C2×C4×C131—42no—
116C2×C4×C131C2×D411yescharacteristic
117C2×C4×C131C2411yescharacteristic
118C8×C131—22no—
119C22×C131—21no—
120C22×C131—21no—
121C22×C131—41no—
122C22×C131—41no—
123C22×C131—41no—
124C22×C131AGL(1,ℤ/8)12yes—
125C22×C131C22×D411yescharacteristic
126C2×D131—21no—
127C2×D131—21no—
128C2×D131—21no—
129C2×D131—21no—
130C2×D131—21no—
131C2×D131—44no—
132C2×D131—41no—
133C2×D131—41no—
134C2×D131—41no—
135C2×D131—84no—
136C2×D131#12312yes—
137C4×C131—21no—
138C4×C131—42no—
139C4×C131—41no—
140C4×C131C2×#4112yes—
141Dic131—21no—
142Dic131—41no—
143Dic131—42no—
144Dic131#12312yes—
145C2×C131—22no—
146C2×C131—81no—
147C2×C131C2×#12311yescharacteristic
148C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C131#57511yescharacteristic
150D131—22no—
151D131—41no—
152D131—81no—
153C131#309411yescharacteristic, complemented, Sylow
154#3094—1311nomaximal, Sylow
155C2×#121—1311no—
156C2×#117—1311no—
157#482—1312no—
158#482—1312no—
159#481—1311no—
160#481—1311no—
161#480—1311no—
162#480D13111yescharacteristic, complemented
163#428—1312no—
164#418—1312no—
165#407—1311no—
166C22×D4—2621no—
167C22×D4—1311no—
168C23×C4—1311no—
169C2×M16—1312no—
170#121—2621no—
171#121—2621no—
172#121—1311no—
173#121—1311no—
174#121—1311no—
175#121C2×D13111yescharacteristic
176#117—2621no—
177#117—2621no—
178#117—1312no—
179#117—1312no—
180C2×C4⋊C4—1311no—
181C2×C4⋊C4—1311no—
182C2×C4⋊C4C2×D13111yescharacteristic, complemented
183C2×C22⋊C4—2621no—
184C2×C22⋊C4—2621no—
185C2×C42—2621no—
186#107—1312no—
187#107C2×D13112yescomplemented
188#106—1312no—
189#106—1312no—
190#102—1312no—
191#102—1312no—
192#102—1312no—
193#102C2×D13112yescomplemented
194C22⋊C8—1311no—
195C22⋊C8—1312no—
196C22⋊C8C2×D13111yescharacteristic, complemented
197C24—2621no—
198C24—1311no—
199C2×D4—21no—
200C2×D4—2621no—
201C2×D4—2621no—
202C2×D4—2621no—
203C2×D4—2624no—
204C2×D4—2621no—
205C2×D4—5244no—
206C2×D4—1311no—
207C2×D4C22×D13111yescharacteristic
208C22×C4—21no—
209C22×C4—2624no—
210C22×C4—2621no—
211C22×C4—2621no—
212C22×C4—1311no—
213C22×C4—1311no—
214C22×C4—1311no—
215C22×C4—1311no—
216C22×C4—1311no—
217C22×C4C22×D13111yescharacteristic
218M16—2624no—
219C2×C8—1312no—
220C2×C8C22×D13112yes—
221C4⋊C4—21no—
222C4⋊C4—2621no—
223C4⋊C4—2621no—
224C4⋊C4—1312no—
225C4⋊C4—1311no—
226C4⋊C4C22×D13112yes—
227C4⋊C4C22×D13111yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—2622no—
230C22⋊C4—2622no—
231C22⋊C4—2621no—
232C22⋊C4—2621no—
233C22⋊C4—2621no—
234C42—2622no—
235C42—2622no—
236C23—21no—
237C23—2622no—
238C23—2622no—
239C23—2621no—
240C23—5241no—
241C23—5242no—
242C23—5242no—
243C23—5242no—
244C23—1311no—
245C23—1311no—
246C23D4×D13111yescharacteristic
247D4—22no—
248D4—42no—
249D4—2622no—
250D4—5242no—
251D4—5242no—
252D4—5242no—
253D4—5242no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—2622no—
259C2×C4—2622no—
260C2×C4—2622no—
261C2×C4—2622no—
262C2×C4—2622no—
263C2×C4—2621no—
264C2×C4—5242no—
265C2×C4—5242no—
266C2×C4—1311no—
267C2×C4—1311no—
268C2×C4—1314no—
269C2×C4D4×D13111yescharacteristic
270C2×C4C23×D13111yescharacteristic
271C8—22no—
272C8—2622no—
273C22—21no—
274C22—21no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—2621no—
279C22—2621no—
280C22—2621no—
281C22—2621no—
282C22—2621no—
283C22—5244no—
284C22—5241no—
285C22—5241no—
286C22—5241no—
287C22—10484no—
288C22—1312no—
289C22#1686227012yes—
290C22C2×D4×D13111yescharacteristic
291C4—21no—
292C4—42no—
293C4—41no—
294C4—2621no—
295C4—5241no—
296C4—5242no—
297C4—1312no—
298C4#41×D13112yes—
299C2—22no—
300C2—2622no—
301C2—81no—
302C2—5241no—
303C2—10481no—
304C2#123×D13111yescharacteristic
305C2C2×#1686227011yescharacteristic
306C2#2255411111yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#480×C131C2
socleC22×C131C22×D4
prime radical#121×D131C2
hypercenter#480D131
hypocenterC131#3094
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×#16862204C2semidirect
3. #22553639C2semidirect
4. #22553635C2semidirect
5. #22553631C2semidirect
6. #22553628C2semidirect
7. #22553625C2semidirect
8. #22553353C2semidirect
9. #22553315C2semidirect
10. #22553274C2semidirect
11. #480×C131C2semidirect
24. #16862204C22semidirect
25. C2×C4⋊Dic131C22semidirect
30. #16862179C22semidirect
31. #16862178C22semidirect
32. #16862177C22semidirect
33. #16862168C22semidirect
34. #16862167C22semidirect
35. #16862166C22semidirect
36. C22⋊(C131⋊C8)C22semidirect
37. (C2×D131)⋊C8C22semidirect
39. C2×C4⋊C4×C131C22semidirect
40. #107×C131C22semidirect
41. #102×C131C22semidirect
42. C22⋊C8×C131C22semidirect
153. C131#3094semidirect
162. #480D131semidirect
182. C2×C4⋊C4C2×D131semidirect
187. #107C2×D131semidirect
193. #102C2×D131semidirect
196. C22⋊C8C2×D131semidirect

Maximal subgroups

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

subgroupindex
#121×D1312
C2×#168622042
#225536392
#225536352
#225536312
#225536282
#225536252
#225533532
#225533152
#225532742
#480×C1312
#3094131