Finite Group Databasebeta

← #30675190

Subgroups — #30675190

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×#16862274C211yesmaximal, characteristic, complemented
3C2×#16862272C211yesmaximal, characteristic, complemented
4#22553824C211yesmaximal, characteristic, complemented
5#22553464C214yesmaximal, complemented
6#22553447C214yesmaximal, complemented
7#22553316C211yesmaximal, characteristic, complemented
8#22553275C211yesmaximal, characteristic, complemented
9#418×C131C211yesmaximal, characteristic, complemented
10C24×D131—21no—
11C2×#14523513C2211yescharacteristic, complemented
12C22×D524C2211yescharacteristic
13C22×C22⋊D131—21no—
14C2×D4×D131—24no—
15C2×D4×D131—22no—
16C2×D4×D131—24no—
17C2×D4×D131C2214yes—
18C22×C4×D131C2211yescharacteristic
19#16862274—24no—
20#16862272—24no—
21C2×#14523502C2212yescomplemented
22C2×#14523501C2212yescomplemented
23C2×D1048C2212yescomplemented
24C2×#14523498C2212yescomplemented
25C2×#14523497C2211yescharacteristic, complemented
26#16862230—21no—
27#16862226C2212yescomplemented
28#16862221C2212yescomplemented
29#16862216—21no—
30C4×D524—22no—
31#16862166C2214yescomplemented
32(C2×D131)⋊C8C2214yescomplemented
33C22×D4×C131C2211yescharacteristic
34C2×M16×C131C2211yescharacteristic, complemented
35#117×C131C2211yescharacteristic, complemented
36#102×C131C2214yescomplemented
37C23×D131—28no—
38C23×D131—24no—
39C23×D131—21no—
40C23×D131—48no—
41C23×D131D411yescharacteristic
42#14523513—22no—
43#14523513—22no—
44C2×Dic262C2311yescharacteristic, complemented
45C2×D524—22no—
46C2×D524—21no—
47C2×D524—22no—
48C2×D524D414yescomplemented
49C2×D524C2312yes—
50C2×D524C2311yescharacteristic, complemented
51C2×C22⋊D131—28no—
52C2×C22⋊D131—24no—
53C2×C22⋊D131—21no—
54D4×D131—216no—
55D4×D131—28no—
56D4×D131—28no—
57C22×Dic131D411yescharacteristic
58C2×C4×D131—22no—
59C2×C4×D131—22no—
60C2×C4×D131—21no—
61C2×C4×D131D412yes—
62C2×C4×D131C2312yes—
63#14523502—24no—
64#14523501—24no—
65D1048—24no—
66#14523498—24no—
67#14523497—22no—
68C2×C131⋊C8C2312yescomplemented
69C4⋊Dic131—22no—
70(C2×D131)⋊C4—22no—
71(C2×D131)⋊C4—22no—
72C24×C131—21no—
73C2×D4×C131—21no—
74C2×D4×C131—22no—
75C2×D4×C131D414yescomplemented
76C2×D4×C131C2312yes—
77C22×C4×C131C2311yescharacteristic
78M16×C131—22no—
79C2×C8×C131C2312yescomplemented
80C4⋊C4×C131C2312yescomplemented
81C22⋊C4×C131—21no—
82C42×C131—21no—
83C22×D131—22no—
84C22×D131—24no—
85C22×D131—216no—
86C22×D131—24no—
87C22×D131—22no—
88C22×D131—21no—
89C22×D131—432no—
90C22×D131—48no—
91C22×D131—44no—
92C22×D131—48no—
93C22×D131C2×D412yes—
94Dic262—21no—
95Dic262—21no—
96D524—24no—
97D524—22no—
98D524—21no—
99D524—21no—
100D524—44no—
101D524C2×D414yes—
102C22⋊D131—216no—
103C22⋊D131—42no—
104C2×Dic131—22no—
105C2×Dic131—21no—
106C2×Dic131C2×D412yes—
107C4×D131—24no—
108C4×D131—22no—
109C4×D131—22no—
110C4×D131—42no—
111C131⋊C8—22no—
112C23×C131—24no—
113C23×C131—22no—
114C23×C131—44no—
115C23×C131C2×D411yescharacteristic
116D4×C131—24no—
117D4×C131—22no—
118D4×C131C2×D414yes—
119C2×C4×C131—22no—
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#12012yes—
130C22×C131C22×D411yescharacteristic
131C2×D131—28no—
132C2×D131—22no—
133C2×D131—22no—
134C2×D131—24no—
135C2×D131—432no—
136C2×D131—48no—
137C2×D131—44no—
138C2×D131—41no—
139C2×D131—42no—
140C2×D131AGL(1,ℤ/8)14yescomplemented
141C4×C131—42no—
142C4×C131#12012yes—
143C4×C131C22×D411yescharacteristic
144C4×C131C22×D411yescharacteristic
145Dic131—22no—
146Dic131—41no—
147C2×C131—22no—
148C2×C131—44no—
149C2×C131C2×#12011yescharacteristic
150C2×C131C2×AGL(1,ℤ/8)12yes—
151D131—24no—
152D131—44no—
153D131—81no—
154C131#300311yescharacteristic, complemented, Sylow
155#3003—1311nomaximal, Sylow
156C23×D4—1311no—
157C2×AGL(1,ℤ/8)—1311no—
158C2×AGL(1,ℤ/8)—1311no—
159#525—1311no—
160#450—1314no—
161#447—1314no—
162#418—1311no—
163#418D13111yescharacteristic, complemented
164#407—1311no—
165C25—2621no—
166C2×#41—1311no—
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)—2624no—
176AGL(1,ℤ/8)—2624no—
177C2×SD16—1312no—
178C2×SD16—1312no—
179C2×D8—1312no—
180C2×D8—1312no—
181C2×M16—1311no—
182C2×M16C2×D13111yescharacteristic, complemented
183#123—2621no—
184C22⋊Q8—1312no—
185#121—1312no—
186#120—2621no—
187C4×D4—2622no—
188#117C2×D13111yescharacteristic, complemented
189#102—1314no—
190#102C2×D13114yescomplemented
191C22⋊C8—1314no—
192C24—21no—
193C24—2628no—
194C24—2624no—
195C24—2621no—
196C24—5248no—
197C24—1311no—
198#41—2622no—
199#41—2622no—
200C2×Q8—1311no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—2622no—
204C2×D4—26216no—
205C2×D4—2628no—
206C2×D4—2628no—
207C2×D4—2628no—
208C2×D4—2621no—
209C2×D4—2624no—
210C2×D4—2621no—
211C2×D4—2622no—
212C2×D4—1314no—
213C2×D4—1311no—
214C2×D4—1312no—
215C2×D4D52414yescomplemented
216C2×D4C22×D13112yes—
217C22×C4—2622no—
218C22×C4—2622no—
219C22×C4—2621no—
220C22×C4—1311no—
221C22×C4—1312no—
222C22×C4—1312no—
223C22×C4C22×D13111yescharacteristic
224SD16—2624no—
225SD16—2624no—
226D8—2624no—
227D8—2624no—
228M16—22no—
229M16—2622no—
230C2×C8—1312no—
231C2×C8C22×D13112yescomplemented
232C4⋊C4—2622no—
233C4⋊C4C22×D13112yescomplemented
234C22⋊C4—21no—
235C22⋊C4—2622no—
236C22⋊C4—2622no—
237C42—21no—
238C23—24no—
239C23—22no—
240C23—44no—
241C23—2624no—
242C23—26216no—
243C23—2622no—
244C23—2624no—
245C23—2622no—
246C23—2621no—
247C23—5244no—
248C23—52432no—
249C23—5248no—
250C23—5248no—
251C23—1312no—
252C23D4×D13111yescharacteristic
253Q8—2621no—
254Q8—2621no—
255D4—24no—
256D4—22no—
257D4—26216no—
258D4—2624no—
259D4—2621no—
260D4—2622no—
261D4—2621no—
262D4—5244no—
263D4—5242no—
264D4—1314no—
265D4C2×D52414yes—
266C2×C4—22no—
267C2×C4—2622no—
268C2×C4—2624no—
269C2×C4—2622no—
270C2×C4—2622no—
271C2×C4—2621no—
272C2×C4—5242no—
273C2×C4—1312no—
274C2×C4D4×D13111yescharacteristic
275C2×C4D4×D13112yes—
276C2×C4C2×D52412yes—
277C2×C4C23×D13111yescharacteristic
278C8—22no—
279C8—2622no—
280C22—22no—
281C22—24no—
282C22—48no—
283C22—44no—
284C22—2628no—
285C22—2622no—
286C22—2622no—
287C22—2624no—
288C22—5248no—
289C22—52432no—
290C22—5244no—
291C22—5241no—
292C22—5242no—
293C22—1314no—
294C22#1686221612yes—
295C22C2×D4×D13111yescharacteristic
296C4—42no—
297C4—2622no—
298C4—5241no—
299C4#1686221612yes—
300C4C2×D4×D13111yescharacteristic
301C4C22×D52411yescharacteristic
302C2—22no—
303C2—44no—
304C2—2624no—
305C2—5244no—
306C2—10481no—
307C2C2×#1686221611yescharacteristic
308C2AGL(1,ℤ/8)×D13112yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#418×C131C2
socleC22×C131C22×D4
hypercenter#418D131
hypocenterC131#3003
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×D4×D131C2semidirect
2. C2×#16862274C2semidirect
3. C2×#16862272C2semidirect
4. #22553824C2semidirect
5. #22553464C2semidirect
6. #22553447C2semidirect
7. #22553316C2semidirect
8. #22553275C2semidirect
9. #418×C131C2semidirect
11. C2×#14523513C22semidirect
21. C2×#14523502C22semidirect
22. C2×#14523501C22semidirect
23. C2×D1048C22semidirect
24. C2×#14523498C22semidirect
25. C2×#14523497C22semidirect
27. #16862226C22semidirect
28. #16862221C22semidirect
31. #16862166C22semidirect
32. (C2×D131)⋊C8C22semidirect
34. C2×M16×C131C22semidirect
35. #117×C131C22semidirect
36. #102×C131C22semidirect
44. C2×Dic262C23semidirect
48. C2×D524D4semidirect
50. C2×D524C23semidirect
68. C2×C131⋊C8C23semidirect
75. C2×D4×C131D4semidirect
79. C2×C8×C131C23semidirect
80. C4⋊C4×C131C23semidirect
140. C2×D131AGL(1,ℤ/8)semidirect
154. C131#3003semidirect
163. #418D131semidirect
182. C2×M16C2×D131semidirect
188. #117C2×D131semidirect
190. #102C2×D131semidirect
215. C2×D4D524semidirect
231. C2×C8C22×D131semidirect
233. C4⋊C4C22×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×#168622742
C2×#168622722
#225538242
#225534642
#225534472
#225533162
#225532752
#418×C1312
#3003131