Finite Group Databasebeta

← #30675186

Subgroups — #30675186

Order 16768.

994 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 153 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×D4×D131C211yesmaximal, characteristic, complemented
2C2×#16862271C211yesmaximal, characteristic, complemented
3#22553819C211yesmaximal, characteristic, complemented
4#22553461C214yesmaximal, complemented
5#22553444C214yesmaximal, complemented
6#22553318C211yesmaximal, characteristic, complemented
7#22553317C211yesmaximal, characteristic, complemented
8#22553275C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C131C211yesmaximal, characteristic, complemented
10C24×D131—21no—
11C22×D524C2211yescharacteristic
12C22×C22⋊D131—21no—
13C2×D4×D131—24no—
14C2×D4×D131—22no—
15C2×D4×D131—24no—
16C2×D4×D131C2214yes—
17C22×C4×D131C2211yescharacteristic
18#16862271—24no—
19C2×#14523500C2212yescomplemented
20C2×#14523498C2212yescomplemented
21C2×#14523497C2211yescharacteristic, complemented
22#16862227—21no—
23#16862222C2212yescomplemented
24#16862217C2212yescomplemented
25#16862214—21no—
26C4×C22⋊D131—22no—
27#16862205C2211yescharacteristic, complemented
28#16862168C2214yescomplemented
29#16862167C2214yescomplemented
30(C2×D131)⋊C8C2214yescomplemented
31C2×#41×C131C2211yescharacteristic, complemented
32C22×D4×C131C2211yescharacteristic
33AGL(1,ℤ/8)×C131—24no—
34C2×SD16×C131C2212yescomplemented
35C2×D8×C131C2212yescomplemented
36C2×M16×C131C2211yescharacteristic, complemented
37C23×D131—28no—
38C23×D131—24no—
39C23×D131—21no—
40C23×D131—48no—
41C23×D131D411yescharacteristic
42C2×D524—21no—
43C2×D524—22no—
44C2×D524D414yescomplemented
45C2×D524C2312yes—
46C2×C22⋊D131—28no—
47C2×C22⋊D131—22no—
48C2×C22⋊D131—24no—
49D4×D131—216no—
50D4×D131—28no—
51D4×D131—28no—
52C22×Dic131D411yescharacteristic
53C2×C4×D131—22no—
54C2×C4×D131—22no—
55C2×C4×D131D412yes—
56C2×C4×D131C2312yes—
57#14523500—24no—
58#14523498—24no—
59#14523497—22no—
60C2×C131⋊C8C2312yescomplemented
61C4⋊Dic131C2312yescomplemented
62Dic131⋊C4—22no—
63C22⋊Dic131—22no—
64C22⋊Dic131—21no—
65(C2×D131)⋊C4—22no—
66C4×Dic131—21no—
67C24×C131—21no—
68#41×C131—22no—
69#41×C131—22no—
70C2×Q8×C131C2311yescharacteristic, complemented
71C2×D4×C131—21no—
72C2×D4×C131—21no—
73C2×D4×C131—22no—
74C2×D4×C131D414yescomplemented
75C2×D4×C131C2312yes—
76C2×D4×C131C2311yescharacteristic, complemented
77C22×C4×C131—21no—
78C22×C4×C131C2311yescharacteristic
79SD16×C131—24no—
80D8×C131—24no—
81M16×C131—22no—
82C2×C8×C131C2312yescomplemented
83C22×D131—22no—
84C22×D131—24no—
85C22×D131—216no—
86C22×D131—24no—
87C22×D131—22no—
88C22×D131—432no—
89C22×D131—48no—
90C22×D131—44no—
91C22×D131—48no—
92C22×D131C2×D412yes—
93D524—24no—
94D524—22no—
95D524C2×D414yes—
96C22⋊D131—216no—
97C22⋊D131—44no—
98C2×Dic131—22no—
99C2×Dic131—22no—
100C2×Dic131C2×D412yes—
101C4×D131—24no—
102C4×D131—22no—
103C4×D131—22no—
104C131⋊C8—22no—
105C23×C131—24no—
106C23×C131—22no—
107C23×C131—21no—
108C23×C131—44no—
109C23×C131C2×D411yescharacteristic
110Q8×C131—21no—
111Q8×C131—21no—
112D4×C131—24no—
113D4×C131—22no—
114D4×C131—21no—
115D4×C131—21no—
116D4×C131—42no—
117D4×C131C2×D414yes—
118C2×C4×C131—21no—
119C2×C4×C131—42no—
120C2×C4×C131C2×D411yescharacteristic
121C2×C4×C131C2×D412yes—
122C2×C4×C131C2×D412yes—
123C2×C4×C131C2411yescharacteristic
124C8×C131—22no—
125C22×C131—22no—
126C22×C131—24no—
127C22×C131—48no—
128C22×C131—44no—
129C22×C131—41no—
130C22×C131—42no—
131C22×C131#12012yes—
132C22×C131C22×D411yescharacteristic
133C2×D131—28no—
134C2×D131—22no—
135C2×D131—22no—
136C2×D131—24no—
137C2×D131—432no—
138C2×D131—48no—
139C2×D131—44no—
140C2×D131AGL(1,ℤ/8)14yescomplemented
141C4×C131—41no—
142C4×C131#12012yes—
143C4×C131C22×D411yescharacteristic
144C4×C131C22×D411yescharacteristic
145Dic131—22no—
146Dic131—42no—
147C2×C131—22no—
148C2×C131—44no—
149C2×C131—81no—
150C2×C131C2×#12011yescharacteristic
151C2×C131C2×AGL(1,ℤ/8)12yes—
152D131—24no—
153D131—44no—
154C131#300311yescharacteristic, complemented, Sylow
155#3003—1311nomaximal, Sylow
156C23×D4—1311no—
157C2×AGL(1,ℤ/8)—1311no—
158C2×AGL(1,ℤ/8)D13111yescharacteristic, complemented
159#525—1311no—
160#450—1314no—
161#447—1314no—
162#418—1311no—
163#418—1311no—
164#407—1311no—
165C25—2621no—
166C2×#41C2×D13111yescharacteristic, complemented
167C22×D4—2624no—
168C22×D4—2624no—
169C22×D4—2622no—
170C22×D4—2621no—
171C22×D4—1314no—
172C22×D4—1311no—
173C22×D4C2×D13111yescharacteristic
174C23×C4—1311no—
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2624no—
177C2×SD16—1312no—
178C2×SD16C2×D13112yescomplemented
179C2×D8—1312no—
180C2×D8C2×D13112yescomplemented
181C2×M16—1311no—
182C2×M16C2×D13111yescharacteristic, complemented
183#123—2621no—
184C22⋊Q8—1312no—
185#121—1312no—
186#120—2621no—
187C4×D4—2622no—
188#117—1311no—
189#102—1314no—
190#102—1314no—
191C22⋊C8—1314no—
192C24—21no—
193C24—2628no—
194C24—2624no—
195C24—2621no—
196C24—5248no—
197C24—1311no—
198#41—22no—
199#41—22no—
200C2×Q8C22×D13111yescharacteristic, complemented
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—26216no—
205C2×D4—2628no—
206C2×D4—2628no—
207C2×D4—2628no—
208C2×D4—2622no—
209C2×D4—2624no—
210C2×D4—2621no—
211C2×D4—2622no—
212C2×D4—1314no—
213C2×D4—1312no—
214C2×D4C22⋊D13114yescomplemented
215C2×D4C22×D13112yes—
216C2×D4C22×D13111yescharacteristic, complemented
217C22×C4—21no—
218C22×C4—2622no—
219C22×C4—2622no—
220C22×C4—1311no—
221C22×C4—1312no—
222C22×C4—1312no—
223C22×C4C22×D13111yescharacteristic
224SD16—24no—
225SD16—2624no—
226D8—24no—
227D8—2624no—
228M16—22no—
229M16—2622no—
230C2×C8—1312no—
231C2×C8C22×D13112yescomplemented
232C4⋊C4—2622no—
233C4⋊C4—1312no—
234C22⋊C4—2622no—
235C22⋊C4—2621no—
236C22⋊C4—2622no—
237C42—2621no—
238C23—24no—
239C23—22no—
240C23—21no—
241C23—44no—
242C23—2624no—
243C23—26216no—
244C23—2622no—
245C23—2624no—
246C23—2622no—
247C23—5244no—
248C23—52432no—
249C23—5248no—
250C23—5248no—
251C23—1312no—
252C23D4×D13111yescharacteristic
253Q8—21no—
254Q8—21no—
255D4—24no—
256D4—22no—
257D4—21no—
258D4—21no—
259D4—42no—
260D4—26216no—
261D4—2624no—
262D4—2622no—
263D4—5244no—
264D4—1314no—
265D4C2×C22⋊D13114yes—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—2622no—
269C2×C4—2624no—
270C2×C4—2622no—
271C2×C4—2622no—
272C2×C4—2622no—
273C2×C4—1312no—
274C2×C4D4×D13111yescharacteristic
275C2×C4D4×D13112yes—
276C2×C4C2×C22⋊D13112yes—
277C2×C4C23×D13111yescharacteristic
278C8—22no—
279C8—2622no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—41no—
285C22—42no—
286C22—2628no—
287C22—2622no—
288C22—2622no—
289C22—2624no—
290C22—5248no—
291C22—52432no—
292C22—5244no—
293C22—1314no—
294C22#1686221412yes—
295C22C2×D4×D13111yescharacteristic
296C4—41no—
297C4—2622no—
298C4—5242no—
299C4#1686221412yes—
300C4C2×D4×D13111yescharacteristic
301C4C22×C22⋊D13111yescharacteristic
302C2—22no—
303C2—44no—
304C2—2624no—
305C2—81no—
306C2—5244no—
307C2C2×#1686221411yescharacteristic
308C2AGL(1,ℤ/8)×D13112yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×AGL(1,ℤ/8)×C131C2
socleC22×C131C22×D4
hypercenterC2×AGL(1,ℤ/8)D131
hypocenterC131#3003
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C22×D4×D131C2semidirect
2. C2×#16862271C2semidirect
3. #22553819C2semidirect
4. #22553461C2semidirect
5. #22553444C2semidirect
6. #22553318C2semidirect
7. #22553317C2semidirect
8. #22553275C2semidirect
9. C2×AGL(1,ℤ/8)×C131C2semidirect
19. C2×#14523500C22semidirect
20. C2×#14523498C22semidirect
21. C2×#14523497C22semidirect
23. #16862222C22semidirect
24. #16862217C22semidirect
27. #16862205C22semidirect
28. #16862168C22semidirect
29. #16862167C22semidirect
30. (C2×D131)⋊C8C22semidirect
31. C2×#41×C131C22semidirect
34. C2×SD16×C131C22semidirect
35. C2×D8×C131C22semidirect
36. C2×M16×C131C22semidirect
44. C2×D524D4semidirect
60. C2×C131⋊C8C23semidirect
61. C4⋊Dic131C23semidirect
70. C2×Q8×C131C23semidirect
74. C2×D4×C131D4semidirect
76. C2×D4×C131C23semidirect
82. C2×C8×C131C23semidirect
140. C2×D131AGL(1,ℤ/8)semidirect
154. C131#3003semidirect
158. C2×AGL(1,ℤ/8)D131semidirect
166. C2×#41C2×D131semidirect
178. C2×SD16C2×D131semidirect
180. C2×D8C2×D131semidirect
182. C2×M16C2×D131semidirect
200. C2×Q8C22×D131semidirect
214. C2×D4C22⋊D131semidirect
216. C2×D4C22×D131semidirect
231. C2×C8C22×D131semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1312
C2×#168622712
#225538192
#225534612
#225534442
#225533182
#225533172
#225532752
C2×AGL(1,ℤ/8)×C1312
#3003131