Finite Group Databasebeta

← #30675188

Subgroups — #30675188

Order 16768.

994 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#16862272C212yesmaximal, complemented
2#22553823C211yesmaximal, characteristic, complemented
3#22553463C214yesmaximal, complemented
4#22553446C214yesmaximal, complemented
5#22553317C212yesmaximal, complemented
6#22553276C211yesmaximal, characteristic, complemented
7C23×D4×C131C211yesmaximal, characteristic, complemented
8C2×#14523513C2211yescharacteristic, complemented
9#16862272—28no—
10C2×#14523501C2214yescomplemented
11C2×#14523498C2214yescomplemented
12C2×#14523497C2212yescomplemented
13#16862229—21no—
14#16862225C2212yescomplemented
15#16862220C2212yescomplemented
16#16862215—21no—
17C4×C22⋊D131—22no—
18#16862205C2211yescharacteristic, complemented
19#16862167C2218yescomplemented
20C22⋊(C131⋊C8)C2214yescomplemented
21C25×C131—21no—
22C22×D4×C131—28no—
23C22×D4×C131—21no—
24C22×D4×C131—22no—
25C22×D4×C131C2214yes—
26C22×D4×C131C2212yes—
27C23×C4×C131C2211yescharacteristic
28#14523513—24no—
29C2×Dic262C2311yescharacteristic, complemented
30C2×D524C2311yescharacteristic, complemented
31C2×C22⋊D131—22no—
32C2×C22⋊D131—21no—
33C2×C4×D131—21no—
34#14523501—28no—
35#14523498—28no—
36#14523497—24no—
37C2×C131⋊C8C2314yescomplemented
38C4⋊Dic131C2312yescomplemented
39Dic131⋊C4—22no—
40C22⋊Dic131—22no—
41C22⋊Dic131—21no—
42(C2×D131)⋊C4—22no—
43C4×Dic131—21no—
44C24×C131—28no—
45C24×C131—24no—
46C24×C131—22no—
47C24×C131—48no—
48C24×C131D411yescharacteristic
49C2×D4×C131—28no—
50C2×D4×C131—216no—
51C2×D4×C131—216no—
52C2×D4×C131—24no—
53C2×D4×C131—22no—
54C2×D4×C131—24no—
55C2×D4×C131D418yescomplemented
56C2×D4×C131C2314yes—
57C22×C4×C131—24no—
58C22×C4×C131D411yescharacteristic
59C22×C4×C131D412yes—
60C22×C4×C131C2312yes—
61C22×C4×C131C2311yescharacteristic
62C22×D131—21no—
63Dic262—22no—
64D524—22no—
65C22⋊D131—42no—
66C22⋊D131—44no—
67C2×Dic131—22no—
68C2×Dic131—21no—
69C4×D131—42no—
70C131⋊C8—24no—
71C23×C131—22no—
72C23×C131—24no—
73C23×C131—216no—
74C23×C131—28no—
75C23×C131—24no—
76C23×C131—432no—
77C23×C131—48no—
78C23×C131—48no—
79C23×C131—48no—
80C23×C131C2×D411yescharacteristic
81C23×C131C2×D412yes—
82D4×C131—216no—
83D4×C131—28no—
84D4×C131—24no—
85D4×C131C2×D418yes—
86C2×C4×C131—22no—
87C2×C4×C131—24no—
88C2×C4×C131—24no—
89C2×C4×C131C2×D412yes—
90C2×C4×C131C2×D411yescharacteristic
91C2×C4×C131C2×D414yes—
92C2×C4×C131C2411yescharacteristic
93C22×C131—28no—
94C22×C131—22no—
95C22×C131—22no—
96C22×C131—22no—
97C22×C131—28no—
98C22×C131—416no—
99C22×C131—432no—
100C22×C131—48no—
101C22×C131#12012yes—
102C22×C131AGL(1,ℤ/8)14yescomplemented
103C22×C131C22×D411yescharacteristic
104C2×D131—41no—
105C2×D131—42no—
106C4×C131—22no—
107C4×C131#12012yes—
108C4×C131C22×D412yes—
109Dic131—41no—
110Dic131—42no—
111C2×C131—22no—
112C2×C131—24no—
113C2×C131—48no—
114C2×C131C2×#12011yescharacteristic
115C2×C131C2×AGL(1,ℤ/8)12yes—
116D131—81no—
117C131#300311yescharacteristic, complemented, Sylow
118#3003—1311nomaximal, Sylow
119C23×D4D13111yescharacteristic, complemented
120C2×AGL(1,ℤ/8)—1312no—
121#525—1311no—
122#450—1314no—
123#447—1314no—
124#418—1312no—
125#407—1311no—
126C25—21no—
127C2×#41—1311no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C2×D13114yes—
132C22×D4C2×D13112yes—
133C23×C4C2×D13111yescharacteristic
134AGL(1,ℤ/8)—2628no—
135C2×SD16—1314no—
136C2×D8—1314no—
137C2×M16—1312no—
138#123—2621no—
139C22⋊Q8—1312no—
140#121—1312no—
141#120—2621no—
142C4×D4—2622no—
143#117—1311no—
144#102—1318no—
145C22⋊C8—1314no—
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24C22⋊D13111yescharacteristic
151#41—2624no—
152C2×Q8—1311no—
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—24no—
159C2×D4—2622no—
160C2×D4—2621no—
161C2×D4—1311no—
162C2×D4C22⋊D13118yescomplemented
163C2×D4C22×D13114yes—
164C22×C4—24no—
165C22×C4—2621no—
166C22×C4C22⋊D13112yes—
167C22×C4C22⋊D13111yescharacteristic
168C22×C4C22×D13112yes—
169C22×C4C22×D13111yescharacteristic
170SD16—2628no—
171D8—2628no—
172M16—2624no—
173C2×C8—1314no—
174C4⋊C4—2622no—
175C4⋊C4—1312no—
176C22⋊C4—2621no—
177C22⋊C4—2622no—
178C22⋊C4—2622no—
179C42—2621no—
180C23—22no—
181C23—24no—
182C23—216no—
183C23—28no—
184C23—24no—
185C23—432no—
186C23—48no—
187C23—48no—
188C23—48no—
189C23—2621no—
190C23C2×C22⋊D13111yescharacteristic
191C23C2×C22⋊D13112yes—
192Q8—2622no—
193D4—216no—
194D4—28no—
195D4—24no—
196D4—2622no—
197D4—5242no—
198D4—5244no—
199D4C2×C22⋊D13118yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—2622no—
204C2×C4—2621no—
205C2×C4—5242no—
206C2×C4C2×C22⋊D13112yes—
207C2×C4C2×C22⋊D13111yescharacteristic
208C2×C4C2×C22⋊D13114yes—
209C2×C4C23×D13111yescharacteristic
210C8—2624no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—416no—
217C22—432no—
218C22—48no—
219C22—5241no—
220C22—5242no—
221C22#1686221512yes—
222C22#1686227214yescomplemented
223C22C22×C22⋊D13111yescharacteristic
224C4—22no—
225C4—5241no—
226C4—5242no—
227C4#1686221512yes—
228C4C22×C22⋊D13112yes—
229C2—22no—
230C2—24no—
231C2—48no—
232C2—10481no—
233C2C2×#1686221511yescharacteristic
234C2C2×#1686227212yes—
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC23×D4×C131C2
socleC22×C131C22×D4
hypercenterC23×D4D131
hypocenterC131#3003
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16862272C2semidirect
2. #22553823C2semidirect
3. #22553463C2semidirect
4. #22553446C2semidirect
5. #22553317C2semidirect
6. #22553276C2semidirect
7. C23×D4×C131C2semidirect
8. C2×#14523513C22semidirect
10. C2×#14523501C22semidirect
11. C2×#14523498C22semidirect
12. C2×#14523497C22semidirect
14. #16862225C22semidirect
15. #16862220C22semidirect
18. #16862205C22semidirect
19. #16862167C22semidirect
20. C22⋊(C131⋊C8)C22semidirect
29. C2×Dic262C23semidirect
30. C2×D524C23semidirect
37. C2×C131⋊C8C23semidirect
38. C4⋊Dic131C23semidirect
55. C2×D4×C131D4semidirect
102. C22×C131AGL(1,ℤ/8)semidirect
117. C131#3003semidirect
119. C23×D4D131semidirect
162. C2×D4C22⋊D131semidirect
222. C22#16862272semidirect

Maximal subgroups

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

subgroupindex
C2×#168622722
#225538232
#225534632
#225534462
#225533172
#225532762
C23×D4×C1312
#3003131