Finite Group Databasebeta

← #30675202

Subgroups — #30675202

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×#16862271C211yesmaximal, characteristic, complemented
3#22553819C211yesmaximal, characteristic, complemented
4#22553473C214yesmaximal, complemented
5#22553456C214yesmaximal, complemented
6#22553321C211yesmaximal, characteristic, complemented
7#22553318C211yesmaximal, characteristic, complemented
8#22553275C211yesmaximal, characteristic, complemented
9C2×#137×C131C211yesmaximal, characteristic, complemented
10C2×#14523514—24no—
11C2×#14523514—22no—
12C2×#14523514—24no—
13C2×#14523514C2214yes—
14C22×D524—21no—
15C22×D524C2211yescharacteristic
16C22×C4×D131—21no—
17C22×C4×D131C2211yescharacteristic
18#16862271—24no—
19C2×#14523500C2212yescomplemented
20C2×#14523498C2212yescomplemented
21C2×#14523497C2211yescharacteristic
22#16862227—21no—
23#16862222C2212yescomplemented
24#16862217C2212yescomplemented
25#16862214—21no—
26C4×C22⋊D131—22no—
27#16862205C2211yescharacteristic
28#16862170C2214yescomplemented
29#16862168C2214yescomplemented
30(C2×D131)⋊C8C2214yescomplemented
31C2×#41×C131C2211yescharacteristic
32C22×Q8×C131C2211yescharacteristic
33#137×C131—24no—
34C2×Q16×C131C2212yescomplemented
35C2×SD16×C131C2212yescomplemented
36C2×M16×C131C2211yescharacteristic
37C23×D131—21no—
38C23×D131D411yescharacteristic
39#14523514—216no—
40#14523514—28no—
41#14523514—28no—
42C2×D524—28no—
43C2×D524—24no—
44C2×D524—21no—
45C2×D524—22no—
46C2×D524D414yescomplemented
47C2×D524C2312yes—
48C2×C22⋊D131—22no—
49C22×Dic131D411yescharacteristic
50C2×C4×D131—28no—
51C2×C4×D131—24no—
52C2×C4×D131—22no—
53C2×C4×D131—22no—
54C2×C4×D131D412yes—
55C2×C4×D131C2312yes—
56#14523500—24no—
57#14523498—24no—
58#14523497—22no—
59C2×C131⋊C8C2312yes—
60C4⋊Dic131C2312yes—
61Dic131⋊C4—22no—
62C22⋊Dic131—22no—
63C22⋊Dic131—21no—
64(C2×D131)⋊C4—22no—
65C4×Dic131—21no—
66#41×C131—22no—
67#41×C131—22no—
68C2×Q8×C131—22no—
69C2×Q8×C131—21no—
70C2×Q8×C131D414yescomplemented
71C2×Q8×C131C2311yescharacteristic
72C2×Q8×C131C2312yes—
73C2×D4×C131—21no—
74C2×D4×C131C2311yescharacteristic
75C22×C4×C131—21no—
76C22×C4×C131—21no—
77C22×C4×C131C2311yescharacteristic
78Q16×C131—24no—
79SD16×C131—24no—
80M16×C131—22no—
81C2×C8×C131C2312yes—
82C22×D131—22no—
83C22×D131—24no—
84C22×D131—22no—
85C22×D131—42no—
86C22×D131—44no—
87C22×D131C2×D412yes—
88D524—216no—
89D524—24no—
90D524—22no—
91D524C2×D414yes—
92C22⋊D131—44no—
93C2×Dic131—22no—
94C2×Dic131—22no—
95C2×Dic131C2×D412yes—
96C4×D131—216no—
97C4×D131—24no—
98C4×D131—22no—
99C4×D131—22no—
100C131⋊C8—22no—
101C23×C131—21no—
102C23×C131C2×D411yescharacteristic
103Q8×C131—24no—
104Q8×C131—22no—
105Q8×C131—21no—
106Q8×C131—21no—
107Q8×C131C2×D414yes—
108D4×C131—21no—
109D4×C131—21no—
110D4×C131—42no—
111C2×C4×C131—24no—
112C2×C4×C131—22no—
113C2×C4×C131—21no—
114C2×C4×C131—42no—
115C2×C4×C131C2×D411yescharacteristic
116C2×C4×C131C2×D412yes—
117C2×C4×C131C2×D412yes—
118C2×C4×C131C2411yescharacteristic
119C8×C131—22no—
120C22×C131—22no—
121C22×C131—41no—
122C22×C131—42no—
123C22×C131#12012yes—
124C22×C131C22×D411yescharacteristic
125C2×D131—22no—
126C2×D131—24no—
127C2×D131—22no—
128C2×D131—22no—
129C2×D131—44no—
130C2×D131—48no—
131C2×D131—44no—
132C4×C131—24no—
133C4×C131—41no—
134C4×C131#12012yes—
135C4×C131C22×D411yescharacteristic
136C4×C131C22×D411yescharacteristic
137Dic131—22no—
138Dic131—42no—
139C2×C131—22no—
140C2×C131—81no—
141C2×C131C2×#12011yescharacteristic
142C2×C131#57512yes—
143D131—42no—
144D131—44no—
145C131#300511yescharacteristic, complemented, Sylow
146#3005—1311nomaximal, Sylow
147C22×#41—1311no—
148C2×#137D13111yescharacteristic, complemented
149C2×AGL(1,ℤ/8)—1311no—
150#525—1311no—
151#452—1314no—
152#449—1314no—
153#419—1311no—
154#418—1311no—
155#407—1311no—
156C2×#41—2624no—
157C2×#41—2624no—
158C2×#41—2622no—
159C2×#41—1314no—
160C2×#41C2×D13111yescharacteristic
161C22×Q8C2×D13111yescharacteristic
162C22×D4—2621no—
163C22×D4—1311no—
164C23×C4—2621no—
165C23×C4—1311no—
166#137—24no—
167AGL(1,ℤ/8)—2624no—
168C2×Q16C2×D13112yescomplemented
169C2×SD16—1312no—
170C2×SD16C2×D13112yescomplemented
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#117—1311no—
180#103—1314no—
181#102—1314no—
182C22⋊C8—1314no—
183C24—2621no—
184C24—1311no—
185#41—22no—
186#41—22no—
187#41—26216no—
188#41—2628no—
189#41—2628no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8C22⋊D13114yescomplemented
193C2×Q8C22×D13111yescharacteristic
194C2×Q8C22×D13112yes—
195C2×D4—21no—
196C2×D4—2628no—
197C2×D4—2624no—
198C2×D4—2622no—
199C2×D4—2621no—
200C2×D4—2622no—
201C2×D4—1314no—
202C2×D4—1312no—
203C2×D4C22×D13111yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—2628no—
207C22×C4—2624no—
208C22×C4—2622no—
209C22×C4—2622no—
210C22×C4—1311no—
211C22×C4—1312no—
212C22×C4—1312no—
213C22×C4C22×D13111yescharacteristic
214Q16—24no—
215SD16—24no—
216SD16—2624no—
217D8—2624no—
218M16—22no—
219M16—2622no—
220C2×C8—1312no—
221C2×C8C22×D13112yes—
222C4⋊C4—2622no—
223C4⋊C4—1312no—
224C22⋊C4—2622no—
225C22⋊C4—2621no—
226C22⋊C4—2622no—
227C42—2621no—
228C23—21no—
229C23—2622no—
230C23—2624no—
231C23—2622no—
232C23—5242no—
233C23—5244no—
234C23—1312no—
235C23D4×D13111yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—21no—
239Q8—21no—
240Q8C2×C22⋊D13114yes—
241D4—21no—
242D4—21no—
243D4—42no—
244D4—26216no—
245D4—2624no—
246D4—2622no—
247D4—5244no—
248D4—1314no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—42no—
253C2×C4—26216no—
254C2×C4—2622no—
255C2×C4—2624no—
256C2×C4—2622no—
257C2×C4—2622no—
258C2×C4—2622no—
259C2×C4—1312no—
260C2×C4D4×D13111yescharacteristic
261C2×C4D4×D13112yes—
262C2×C4C2×C22⋊D13112yes—
263C2×C4C23×D13111yescharacteristic
264C8—22no—
265C8—2622no—
266C22—22no—
267C22—41no—
268C22—42no—
269C22—2622no—
270C22—2624no—
271C22—2622no—
272C22—2622no—
273C22—5244no—
274C22—5248no—
275C22—5244no—
276C22#1686221412yes—
277C22C2×D4×D13111yescharacteristic
278C4—24no—
279C4—41no—
280C4—2622no—
281C4—5242no—
282C4#1686221412yes—
283C4C2×D4×D13111yescharacteristic
284C4C22×C22⋊D13111yescharacteristic
285C2—22no—
286C2—81no—
287C2—5242no—
288C2—5244no—
289C2C2×#1686221411yescharacteristic
290C2#2255411112yes—
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC2×#137×C131C2
socleC22×C131C22×D4
hypercenterC2×#137D131
hypocenterC131#3005
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#14523514C2semidirect
2. C2×#16862271C2semidirect
3. #22553819C2semidirect
4. #22553473C2semidirect
5. #22553456C2semidirect
6. #22553321C2semidirect
7. #22553318C2semidirect
8. #22553275C2semidirect
9. C2×#137×C131C2semidirect
19. C2×#14523500C22semidirect
20. C2×#14523498C22semidirect
23. #16862222C22semidirect
24. #16862217C22semidirect
28. #16862170C22semidirect
29. #16862168C22semidirect
30. (C2×D131)⋊C8C22semidirect
34. C2×Q16×C131C22semidirect
35. C2×SD16×C131C22semidirect
46. C2×D524D4semidirect
70. C2×Q8×C131D4semidirect
145. C131#3005semidirect
148. C2×#137D131semidirect
168. C2×Q16C2×D131semidirect
170. C2×SD16C2×D131semidirect
192. C2×Q8C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C22×#145235142
C2×#168622712
#225538192
#225534732
#225534562
#225533212
#225533182
#225532752
C2×#137×C1312
#3005131