Finite Group Databasebeta

← #30675206

Subgroups — #30675206

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×#14523513C211yesmaximal, characteristic, complemented
2C2×#16862279C211yesmaximal, characteristic, complemented
3C2×#16862274C211yesmaximal, characteristic, complemented
4#22553823C211yesmaximal, characteristic, complemented
5#22553476C214yesmaximal, complemented
6#22553459C214yesmaximal, complemented
7#22553322C211yesmaximal, characteristic, complemented
8#22553318C211yesmaximal, characteristic, complemented
9#407×C131C211yesmaximal, characteristic, complemented
10C2×#14523513—24no—
11C2×#14523513—24no—
12C2×#14523513—22no—
13C2×#14523513C2214yes—
14C2×#14523513C2211yescharacteristic
15C22×Dic262C2211yescharacteristic
16C22×D524C2211yescharacteristic
17C22×C22⋊D131—21no—
18C22×C4×D131—21no—
19#16862279—24no—
20#16862274—24no—
21C2×Dic524C2212yescomplemented
22C2×#14523502C2212yescomplemented
23C2×#14523502C2212yescomplemented
24C2×D1048C2212yescomplemented
25#16862229—21no—
26#16862225C2212yescomplemented
27#16862220C2212yescomplemented
28#16862215—21no—
29C4×C22⋊D131—22no—
30#16862205C2211yescharacteristic
31#16862171C2214yescomplemented
32#16862168C2214yescomplemented
33C23×C4×C131C2211yescharacteristic
34C2×M16×C131C2211yescharacteristic
35C2×M16×C131C2211yescharacteristic
36C22⋊C8×C131C2214yescomplemented
37C23×D131—21no—
38#14523513—216no—
39#14523513—28no—
40#14523513—28no—
41#14523513—22no—
42#14523513—22no—
43C2×Dic262—22no—
44C2×Dic262—21no—
45C2×Dic262D414yescomplemented
46C2×Dic262C2311yescharacteristic
47C2×Dic262C2312yes—
48C2×D524—22no—
49C2×D524—21no—
50C2×D524D414yescomplemented
51C2×D524C2311yescharacteristic
52C2×D524C2312yes—
53C2×C22⋊D131—28no—
54C2×C22⋊D131—24no—
55C2×C22⋊D131—22no—
56C2×C22⋊D131—21no—
57C22×Dic131—21no—
58C2×C4×D131—28no—
59C2×C4×D131—24no—
60C2×C4×D131—21no—
61Dic524—24no—
62#14523502—24no—
63#14523502—24no—
64D1048—24no—
65C4⋊Dic131C2312yes—
66Dic131⋊C4—22no—
67C22⋊Dic131—22no—
68C22⋊Dic131—21no—
69(C2×D131)⋊C4—22no—
70C4×Dic131—21no—
71C24×C131D411yescharacteristic
72C22×C4×C131—22no—
73C22×C4×C131—22no—
74C22×C4×C131D411yescharacteristic
75C22×C4×C131D412yes—
76C22×C4×C131C2312yes—
77C22×C4×C131C2311yescharacteristic
78M16×C131—22no—
79M16×C131—22no—
80C2×C8×C131C2312yes—
81C2×C8×C131C2312yes—
82C22×D131—24no—
83C22×D131—22no—
84C22×D131—21no—
85C22×D131—44no—
86Dic262—24no—
87Dic262—22no—
88Dic262—21no—
89Dic262—21no—
90Dic262C2×D414yes—
91D524—24no—
92D524—22no—
93D524—21no—
94D524—21no—
95D524C2×D414yes—
96C22⋊D131—216no—
97C22⋊D131—44no—
98C22⋊D131—42no—
99C2×Dic131—24no—
100C2×Dic131—22no—
101C2×Dic131—22no—
102C2×Dic131—21no—
103C4×D131—216no—
104C4×D131—42no—
105C23×C131—22no—
106C23×C131—42no—
107C23×C131C2×D412yes—
108C23×C131C2×D411yescharacteristic
109C2×C4×C131—22no—
110C2×C4×C131—24no—
111C2×C4×C131—22no—
112C2×C4×C131—22no—
113C2×C4×C131C2×D412yes—
114C2×C4×C131C2×D412yes—
115C2×C4×C131C2×D412yes—
116C2×C4×C131C2×D411yescharacteristic
117C2×C4×C131C2411yescharacteristic
118C8×C131—22no—
119C8×C131—22no—
120C22×C131—22no—
121C22×C131—22no—
122C22×C131—22no—
123C22×C131—22no—
124C22×C131—44no—
125C22×C131#12012yes—
126C22×C131C22×D411yescharacteristic
127C2×D131—24no—
128C2×D131—48no—
129C2×D131—44no—
130C2×D131—41no—
131C2×D131—42no—
132C4×C131—22no—
133C4×C131#12012yes—
134C4×C131C22×D411yescharacteristic
135C4×C131C22×D411yescharacteristic
136Dic131—24no—
137Dic131—41no—
138Dic131—42no—
139C2×C131—22no—
140C2×C131—42no—
141C2×C131C2×#12011yescharacteristic
142C2×C131#57512yes—
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#419—1311no—
154#418—1311no—
155#407D13111yescharacteristic, complemented
156C2×#41—2624no—
157C2×#41—2624no—
158C2×#41—2622no—
159C2×#41—1311no—
160C2×#41—1314no—
161C22×Q8—1311no—
162C22×D4—2621no—
163C22×D4—1311no—
164C23×C4—2621no—
165C23×C4C2×D13111yescharacteristic
166#137—2624no—
167AGL(1,ℤ/8)—2624no—
168C2×Q16—1312no—
169C2×SD16—1312no—
170C2×SD16—1312no—
171C2×D8—1312no—
172C2×M16C2×D13111yescharacteristic
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⋊C8C2×D13114yescomplemented
183C24—2621no—
184C24D52411yescharacteristic
185#41—26216no—
186#41—2628no—
187#41—2628no—
188#41—2622no—
189#41—2622no—
190C2×Q8—2622no—
191C2×Q8—2621no—
192C2×Q8—1314no—
193C2×Q8—1311no—
194C2×Q8—1312no—
195C2×D4—2628no—
196C2×D4—2624no—
197C2×D4—2622no—
198C2×D4—2621no—
199C2×D4—2622no—
200C2×D4—2621no—
201C2×D4—1314no—
202C2×D4—1311no—
203C2×D4—1312no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—2628no—
207C22×C4—2621no—
208C22×C4—2624no—
209C22×C4—2621no—
210C22×C4D52412yes—
211C22×C4D52411yescharacteristic
212C22×C4C22×D13112yes—
213C22×C4C22×D13111yescharacteristic
214Q16—2624no—
215SD16—2624no—
216SD16—2624no—
217D8—2624no—
218M16—22no—
219M16—22no—
220C2×C8C22×D13112yes—
221C2×C8C22×D13112yes—
222C4⋊C4—2622no—
223C4⋊C4—1312no—
224C22⋊C4—2621no—
225C22⋊C4—2622no—
226C22⋊C4—2622no—
227C42—2621no—
228C23—22no—
229C23—42no—
230C23—2624no—
231C23—2622no—
232C23—2621no—
233C23—5244no—
234C23C2×D52412yes—
235C23C2×D52411yescharacteristic
236Q8—2624no—
237Q8—2622no—
238Q8—2621no—
239Q8—2621no—
240Q8—1314no—
241D4—26216no—
242D4—2624no—
243D4—2622no—
244D4—2621no—
245D4—2621no—
246D4—5244no—
247D4—5242no—
248D4—1314no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—26216no—
254C2×C4—2624no—
255C2×C4—2622no—
256C2×C4—2622no—
257C2×C4—2621no—
258C2×C4—5242no—
259C2×C4D4×D13112yes—
260C2×C4D4×D13112yes—
261C2×C4C2×D52412yes—
262C2×C4C2×D52411yescharacteristic
263C2×C4C23×D13111yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—2624no—
272C22—5248no—
273C22—5244no—
274C22—5242no—
275C22—5241no—
276C22#1686221612yes—
277C22C22×D52411yescharacteristic
278C4—22no—
279C4—2624no—
280C4—5242no—
281C4—5241no—
282C4#1686221612yes—
283C4C2×D4×D13111yescharacteristic
284C4C2×D4×D13111yescharacteristic
285C2—22no—
286C2—42no—
287C2—5244no—
288C2—10481no—
289C2C2×#1686221611yescharacteristic
290C2#2255411212yes—
subgroupG/H
centerC22C22×D524
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#407×C131C2
socleC22×C131C22×D4
hypercenter#407D131
hypocenterC131#3005
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#14523513C2semidirect
2. C2×#16862279C2semidirect
3. C2×#16862274C2semidirect
4. #22553823C2semidirect
5. #22553476C2semidirect
6. #22553459C2semidirect
7. #22553322C2semidirect
8. #22553318C2semidirect
9. #407×C131C2semidirect
21. C2×Dic524C22semidirect
22. C2×#14523502C22semidirect
23. C2×#14523502C22semidirect
24. C2×D1048C22semidirect
26. #16862225C22semidirect
27. #16862220C22semidirect
31. #16862171C22semidirect
32. #16862168C22semidirect
36. C22⋊C8×C131C22semidirect
45. C2×Dic262D4semidirect
50. C2×D524D4semidirect
145. C131#3005semidirect
155. #407D131semidirect
182. C22⋊C8C2×D131semidirect

Maximal subgroups

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

subgroupindex
C22×#145235132
C2×#168622792
C2×#168622742
#225538232
#225534762
#225534592
#225533222
#225533182
#407×C1312
#3005131