Finite Group Databasebeta

← #30675884

Subgroups — #30675884

Order 16768.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1#22553823C211yesmaximal, characteristic
2#22553647C212yesmaximal, complemented
3#22553642C212yesmaximal, complemented
4#22553630C212yesmaximal, complemented
5#22553625C212yesmaximal, complemented
6#22553352C212yesmaximal, complemented
7#22553311C212yesmaximal, complemented
8#22553276C211yesmaximal, characteristic, complemented
9C22×C4⋊C4×C131C211yesmaximal, characteristic, complemented
10C2×#14523513C2211yescharacteristic
11C2×#14523497C2212yes—
12#16862229—21no—
13#16862225C2211yescharacteristic
14#16862225C2211yescharacteristic
15#16862220C2211yescharacteristic
16#16862220C2211yescharacteristic
17#16862215—21no—
18C4×C22⋊D131—21no—
19C4×C22⋊D131—21no—
20#16862205C2211yescharacteristic
21#16862179C2214yescomplemented
22#16862177C2214yescomplemented
23#16862169C2214yescomplemented
24#16862166C2214yescomplemented
25C22⋊(C131⋊C8)C2212yes—
26C22⋊(C131⋊C8)C2212yes—
27C23×C4×C131—21no—
28C23×C4×C131C2211yescharacteristic
29C2×C4⋊C4×C131—21no—
30C2×C4⋊C4×C131—21no—
31C2×C4⋊C4×C131—21no—
32C2×C4⋊C4×C131C2212yes—
33C2×C4⋊C4×C131C2212yes—
34C2×C4⋊C4×C131C2212yes—
35#14523513—24no—
36C2×Dic262C2311yescharacteristic
37C2×D524C2311yescharacteristic
38C2×C22⋊D131—21no—
39C2×C22⋊D131—21no—
40C2×C22⋊D131—21no—
41C2×C4×D131—21no—
42#14523497—24no—
43C2×C131⋊C8C2314yes—
44C4⋊Dic131C2311yescharacteristic
45C4⋊Dic131C2311yescharacteristic
46Dic131⋊C4—21no—
47Dic131⋊C4—21no—
48C22⋊Dic131—21no—
49C22⋊Dic131—21no—
50C22⋊Dic131—21no—
51(C2×D131)⋊C4—21no—
52(C2×D131)⋊C4—21no—
53C4×Dic131—21no—
54C24×C131D411yescharacteristic
55C22×C4×C131—24no—
56C22×C4×C131—22no—
57C22×C4×C131—22no—
58C22×C4×C131—22no—
59C22×C4×C131—44no—
60C22×C4×C131D411yescharacteristic
61C22×C4×C131D411yescharacteristic
62C22×C4×C131D411yescharacteristic
63C22×C4×C131C2311yescharacteristic
64C22×C4×C131C2311yescharacteristic
65C22×C4×C131C2311yescharacteristic
66C4⋊C4×C131—22no—
67C4⋊C4×C131—22no—
68C4⋊C4×C131—22no—
69C4⋊C4×C131C2314yes—
70C22×D131—21no—
71Dic262—22no—
72D524—22no—
73C22⋊D131—42no—
74C22⋊D131—42no—
75C22⋊D131—42no—
76C2×Dic131—21no—
77C2×Dic131—21no—
78C2×Dic131—21no—
79C4×D131—42no—
80C131⋊C8—24no—
81C23×C131—22no—
82C23×C131—22no—
83C23×C131—22no—
84C23×C131C2×D411yescharacteristic
85C23×C131C2×D411yescharacteristic
86C23×C131C2×D411yescharacteristic
87C2×C4×C131—22no—
88C2×C4×C131—22no—
89C2×C4×C131—22no—
90C2×C4×C131—22no—
91C2×C4×C131—22no—
92C2×C4×C131—24no—
93C2×C4×C131—44no—
94C2×C4×C131—44no—
95C2×C4×C131—44no—
96C2×C4×C131C2×D411yescharacteristic
97C2×C4×C131C2×D411yescharacteristic
98C2×C4×C131C2×D411yescharacteristic
99C2×C4×C131#4114yes—
100C2×C4×C131C2411yescharacteristic
101C22×C131—28no—
102C22×C131—21no—
103C22×C131—21no—
104C22×C131—21no—
105C22×C131—21no—
106C22×C131—21no—
107C22×C131—21no—
108C22×C131#12312yes—
109C22×C131AGL(1,ℤ/8)12yes—
110C22×C131#13712yes—
111C22×C131C22×D411yescharacteristic
112C2×D131—41no—
113C2×D131—41no—
114C2×D131—41no—
115C4×C131—21no—
116C4×C131—21no—
117C4×C131—44no—
118C4×C131#12312yes—
119C4×C131C2×#4112yes—
120Dic131—41no—
121Dic131—41no—
122Dic131—41no—
123C2×C131—22no—
124C2×C131—22no—
125C2×C131—22no—
126C2×C131C2×#12311yescharacteristic
127C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C131C2×#13711yescharacteristic
129D131—81no—
130C131#309211yescharacteristic, complemented, Sylow
131#3092—1311nomaximal, Sylow
132#525—1311no—
133C22×C4⋊C4D13111yescharacteristic, complemented
134#484—1312no—
135#483—1312no—
136#481—1312no—
137#480—1312no—
138#428—1312no—
139#417—1312no—
140#407—1311no—
141C2×#41—1311no—
142C23×C4—21no—
143C23×C4C2×D13111yescharacteristic
144C2×M16—1312no—
145#123—2621no—
146C22⋊Q8—1311no—
147C22⋊Q8—1311no—
148#121—1311no—
149#121—1311no—
150#120—2621no—
151C4×D4—2621no—
152C4×D4—2621no—
153#117—1311no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2×D13112yes—
158C2×C4⋊C4C2×D13112yes—
159C2×C4⋊C4C2×D13112yes—
160#107—1314no—
161#106—1314no—
162#103—1314no—
163#102—1314no—
164C22⋊C8—1312no—
165C22⋊C8—1312no—
166C24C22⋊D13111yescharacteristic
167#41—2624no—
168C2×Q8—1311no—
169C2×D4—2621no—
170C2×D4—2621no—
171C2×D4—2621no—
172C2×D4—1311no—
173C22×C4—24no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—44no—
178C22×C4—2621no—
179C22×C4C22⋊D13111yescharacteristic
180C22×C4C22⋊D13111yescharacteristic
181C22×C4C22⋊D13111yescharacteristic
182C22×C4C22×D13111yescharacteristic
183C22×C4C22×D13111yescharacteristic
184C22×C4C22×D13111yescharacteristic
185M16—2624no—
186C2×C8—1314no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—2621no—
191C4⋊C4—2621no—
192C4⋊C4—1311no—
193C4⋊C4—1311no—
194C4⋊C4C22×D13114yes—
195C22⋊C4—2621no—
196C22⋊C4—2621no—
197C22⋊C4—2621no—
198C22⋊C4—2621no—
199C22⋊C4—2621no—
200C42—2621no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—2621no—
205C23C2×C22⋊D13111yescharacteristic
206C23C2×C22⋊D13111yescharacteristic
207C23C2×C22⋊D13111yescharacteristic
208Q8—2622no—
209D4—2622no—
210D4—5242no—
211D4—5242no—
212D4—5242no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—2621no—
223C2×C4—2621no—
224C2×C4—2621no—
225C2×C4—5242no—
226C2×C4C2×C22⋊D13111yescharacteristic
227C2×C4C2×C22⋊D13111yescharacteristic
228C2×C4C2×C22⋊D13111yescharacteristic
229C2×C4#1452351314yes—
230C2×C4C23×D13111yescharacteristic
231C8—2624no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—5241no—
240C22—5241no—
241C22—5241no—
242C22#1686222912yes—
243C22#1686227212yes—
244C22#1686227712yes—
245C22C22×C22⋊D13111yescharacteristic
246C4—21no—
247C4—21no—
248C4—44no—
249C4—5241no—
250C4—5241no—
251C4—5241no—
252C4#1686222912yes—
253C4C2×#1452351312yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—10481no—
258C2C2×#1686222911yescharacteristic
259C2C2×#1686227211yescharacteristic
260C2C2×#1686227711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
FittingC22×C4⋊C4×C131C2
socleC22×C131C22×D4
prime radical#22553823C2
hypercenterC22×C4⋊C4D131
hypocenterC131#3092
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #22553647C2semidirect
3. #22553642C2semidirect
4. #22553630C2semidirect
5. #22553625C2semidirect
6. #22553352C2semidirect
7. #22553311C2semidirect
8. #22553276C2semidirect
9. C22×C4⋊C4×C131C2semidirect
21. #16862179C22semidirect
22. #16862177C22semidirect
23. #16862169C22semidirect
24. #16862166C22semidirect
130. C131#3092semidirect
133. C22×C4⋊C4D131semidirect

Maximal subgroups

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

subgroupindex
#225538232
#225536472
#225536422
#225536302
#225536252
#225533522
#225533112
#225532762
C22×C4⋊C4×C1312
#3092131