Finite Group Databasebeta

← #30675964

Subgroups — #30675964

Order 16768.

448 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 129 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#22553864C211yesmaximal, characteristic, complemented
2C2×D4×Dic131C211yesmaximal, characteristic
3#22553643C211yesmaximal, characteristic, complemented
4#22553640C211yesmaximal, characteristic, complemented
5#22553600C212yesmaximal, complemented
6#22553515C212yesmaximal, complemented
7#22553463C211yesmaximal, characteristic
8#22553403C212yesmaximal, complemented
9SD16×Dic131C212yesmaximal, complemented
10#22553362C211yesmaximal, characteristic, complemented
11#450×C131C211yesmaximal, characteristic
12C23×Dic131—21no—
13C2×#14523501C2212yes—
14#16862245C2211yescharacteristic, complemented
15#16862244C2211yescharacteristic, complemented
16#16862236—21no—
17#16862225C2211yescharacteristic
18#16862223—21no—
19Q8×Dic131C2211yescharacteristic, complemented
20#16862211—21no—
21#16862211C2211yescharacteristic, complemented
22D4×Dic131—21no—
23D4×Dic131—21no—
24D4×Dic131C2212yes—
25D4×Dic131C2212yes—
26#16862202—21no—
27C2×C4⋊Dic131C2211yescharacteristic
28C2×C22⋊Dic131—21no—
29C2×C4×Dic131C2211yescharacteristic
30#16862178C2212yescomplemented
31#16862177C2212yescomplemented
32Dic131⋊C8C2212yescomplemented
33#16862171C2212yescomplemented
34#16862170C2212yescomplemented
35#16862167C2212yes—
36C22⋊(C131⋊C8)C2211yescharacteristic
37C8×Dic131C2212yescomplemented
38C22×D4×C131C2211yescharacteristic
39C2×SD16×C131C2212yes—
40C22⋊Q8×C131C2211yescharacteristic
41#102×C131C2212yes—
42C22⋊C8×C131C2211yescharacteristic
43C2×Dic262—21no—
44C2×Dic262C2311yescharacteristic
45C22×Dic131—22no—
46C22×Dic131—22no—
47C22×Dic131—21no—
48C22×Dic131—44no—
49C22×Dic131D411yescharacteristic
50C22×Dic131D411yescharacteristic
51#14523501—24no—
52C2×C131⋊C8C2312yes—
53C4⋊Dic131—21no—
54C4⋊Dic131—21no—
55C4⋊Dic131C2312yes—
56C4⋊Dic131C2311yescharacteristic
57Dic131⋊C4—21no—
58Dic131⋊C4—21no—
59Dic131⋊C4—21no—
60Dic131⋊C4—21no—
61Dic131⋊C4—21no—
62C22⋊Dic131—22no—
63C22⋊Dic131—22no—
64C22⋊Dic131—21no—
65C4×Dic131—21no—
66C4×Dic131—21no—
67C4×Dic131D411yescharacteristic
68C4×Dic131D411yescharacteristic
69C4×Dic131C2311yescharacteristic
70C4×Dic131C2311yescharacteristic
71C24×C131—21no—
72C2×Q8×C131C2311yescharacteristic
73C2×D4×C131—24no—
74C2×D4×C131—21no—
75C2×D4×C131C2312yes—
76C22×C4×C131C2311yescharacteristic
77SD16×C131—24no—
78C2×C8×C131C2312yes—
79C4⋊C4×C131—21no—
80C4⋊C4×C131C2311yescharacteristic
81C22⋊C4×C131—21no—
82Dic262—22no—
83Dic262—42no—
84C2×Dic131—21no—
85C2×Dic131—21no—
86C2×Dic131—22no—
87C2×Dic131—41no—
88C2×Dic131—42no—
89C2×Dic131—44no—
90C2×Dic131—44no—
91C2×Dic131SD1614yescomplemented
92C2×Dic131C2×D411yescharacteristic
93C2×Dic131C2×D411yescharacteristic
94C2×Dic131C2×D411yescharacteristic
95C2×Dic131C2×D411yescharacteristic
96C131⋊C8—22no—
97C23×C131—24no—
98C23×C131—22no—
99C23×C131—42no—
100C23×C131—42no—
101C23×C131C2×D411yescharacteristic
102Q8×C131—22no—
103D4×C131—24no—
104D4×C131—22no—
105D4×C131#4114yes—
106C2×C4×C131—22no—
107C2×C4×C131—21no—
108C2×C4×C131—21no—
109C2×C4×C131C2×D411yescharacteristic
110C2×C4×C131C2411yescharacteristic
111C8×C131—22no—
112C22×C131—21no—
113C22×C131—21no—
114C22×C131—24no—
115C22×C131—48no—
116C22×C131—42no—
117C22×C131—42no—
118C22×C131C2×SD1612yes—
119C22×C131C22×D411yescharacteristic
120C4×C131—21no—
121C4×C131—41no—
122C4×C131—41no—
123C4×C131C2×#4112yes—
124Dic131—21no—
125Dic131—21no—
126Dic131—41no—
127Dic131—42no—
128Dic131—41no—
129Dic131C2×SD1612yes—
130Dic131C2×SD1612yes—
131C2×C131—22no—
132C2×C131—44no—
133C2×C131#52511yescharacteristic
134C2×C131C22×SD1611yescharacteristic
135C2×C131#57511yescharacteristic
136C131#310111yescharacteristic, complemented, Sylow
137#3101—1311nomaximal, Sylow
138#533—1311no—
139C2×C4×D4—1311no—
140#483—1311no—
141#483—1311no—
142#476—1312no—
143#461—1312no—
144#450—1311no—
145#450D13111yescharacteristic
146C4×SD16—1312no—
147C4×SD16—1312no—
148#431—1311no—
149C22×D4C2×D13111yescharacteristic
150C23×C4—2621no—
151C2×SD16—1312no—
152C2×SD16C2×D13112yes—
153C4⋊Q8—1311no—
154C4⋊Q8—1311no—
155#125—2621no—
156C22⋊Q8—2621no—
157C22⋊Q8—1311no—
158C22⋊Q8C2×D13111yescharacteristic
159C4×Q8—2621no—
160C4×Q8—1311no—
161C4×Q8—1311no—
162C4×D4—2621no—
163C4×D4—2621no—
164C4×D4—1312no—
165C4×D4—1312no—
166#117—2621no—
167C2×C4⋊C4—1311no—
168C2×C22⋊C4—2621no—
169C2×C42—1311no—
170#106—1312no—
171#106—1312no—
172C4⋊C8—1312no—
173#103—1312no—
174#103—1312no—
175#102—1312no—
176#102C2×D13112yes—
177C22⋊C8—1311no—
178C22⋊C8C2×D13111yescharacteristic
179C4×C8—1312no—
180C24—21no—
181C2×Q8—2621no—
182C2×Q8—1311no—
183C2×Q8C22×D13111yescharacteristic
184C2×D4—24no—
185C2×D4—21no—
186C2×D4C22×D13112yes—
187C22×C4—2621no—
188C22×C4—2622no—
189C22×C4—2622no—
190C22×C4—5244no—
191C22×C4—1311no—
192C22×C4—1311no—
193C22×C4C22×D13111yescharacteristic
194SD16—24no—
195SD16—2624no—
196C2×C8—1312no—
197C2×C8C22×D13112yes—
198C4⋊C4—21no—
199C4⋊C4—2621no—
200C4⋊C4—2621no—
201C4⋊C4—2621no—
202C4⋊C4—2621no—
203C4⋊C4—2621no—
204C4⋊C4—2621no—
205C4⋊C4—2621no—
206C4⋊C4—1311no—
207C4⋊C4—1312no—
208C4⋊C4C22×D13111yescharacteristic
209C22⋊C4—21no—
210C22⋊C4—2621no—
211C22⋊C4—2622no—
212C22⋊C4—2622no—
213C42—2621no—
214C42—2621no—
215C42—1311no—
216C42—1311no—
217C42—1311no—
218C42—1311no—
219C23—24no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23D4×D13111yescharacteristic
224Q8—22no—
225Q8—2622no—
226Q8—5242no—
227D4—24no—
228D4—22no—
229D4#1452351214yes—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—2621no—
234C2×C4—2621no—
235C2×C4—2622no—
236C2×C4—5241no—
237C2×C4—5242no—
238C2×C4—5244no—
239C2×C4—5244no—
240C2×C4—1311no—
241C2×C4—1311no—
242C2×C4—1314no—
243C2×C4—1311no—
244C2×C4—1311no—
245C2×C4D4×D13111yescharacteristic
246C2×C4C23×D13111yescharacteristic
247C8—22no—
248C8—2622no—
249C22—21no—
250C22—21no—
251C22—24no—
252C22—48no—
253C22—42no—
254C22—42no—
255C22SD16×D13112yes—
256C22C2×D4×D13111yescharacteristic
257C4—21no—
258C4—41no—
259C4—41no—
260C4—2621no—
261C4—2621no—
262C4—5241no—
263C4—5242no—
264C4—5241no—
265C4—1312no—
266C4—1312no—
267C4C2×#1452351212yes—
268C2—22no—
269C2—44no—
270C2#2255381811yescharacteristic
271C2C2×SD16×D13111yescharacteristic
272C2#2255410811yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#450×C131C2
socleC22×C131C22×D4
prime radicalC22×D4×C131C22
hypercenter#450D131
hypocenterC131#3101
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #22553864C2semidirect
3. #22553643C2semidirect
4. #22553640C2semidirect
5. #22553600C2semidirect
6. #22553515C2semidirect
8. #22553403C2semidirect
9. SD16×Dic131C2semidirect
10. #22553362C2semidirect
14. #16862245C22semidirect
15. #16862244C22semidirect
19. Q8×Dic131C22semidirect
21. #16862211C22semidirect
30. #16862178C22semidirect
31. #16862177C22semidirect
32. Dic131⋊C8C22semidirect
33. #16862171C22semidirect
34. #16862170C22semidirect
37. C8×Dic131C22semidirect
91. C2×Dic131SD16semidirect
136. C131#3101semidirect

Maximal subgroups

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

subgroupindex
#225538642
C2×D4×Dic1312
#225536432
#225536402
#225536002
#225535152
#225534632
#225534032
SD16×Dic1312
#225533622
#450×C1312
#3101131