Finite Group Databasebeta

← #30676003

Subgroups — #30676003

Order 16768.

454 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 123 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#22553843C211yesmaximal, characteristic, complemented
2C2×D4×Dic131C211yesmaximal, characteristic, complemented
3#22553648C211yesmaximal, characteristic, complemented
4#22553628C211yesmaximal, characteristic, complemented
5#22553610C212yesmaximal, complemented
6#22553525C212yesmaximal, complemented
7#22553463C211yesmaximal, characteristic, complemented
8#22553403C212yesmaximal, complemented
9D8×Dic131C212yesmaximal, complemented
10#22553362C211yesmaximal, characteristic, complemented
11#447×C131C211yesmaximal, characteristic
12C23×Dic131—21no—
13C2×#14523501C2212yes—
14#16862239—21no—
15#16862237C2211yescharacteristic, complemented
16#16862233C2211yescharacteristic, complemented
17#16862227—21no—
18#16862225C2211yescharacteristic
19#16862211C2211yescharacteristic, complemented
20D4×Dic131—21no—
21D4×Dic131—21no—
22D4×Dic131—21no—
23D4×Dic131C2212yes—
24D4×Dic131C2212yes—
25D4×Dic131C2211yescharacteristic, complemented
26#16862202—21no—
27C2×C4⋊Dic131C2211yescharacteristic
28C2×C22⋊Dic131—21no—
29C2×C4×Dic131C2211yescharacteristic
30#16862180C2212yescomplemented
31#16862177C2212yescomplemented
32Dic131⋊C8C2212yescomplemented
33#16862171C2212yescomplemented
34#16862167C2212yes—
35#16862167C2212yescomplemented
36C22⋊(C131⋊C8)C2211yescharacteristic
37C8×Dic131C2212yescomplemented
38C22×D4×C131C2211yescharacteristic
39C2×D8×C131C2212yes—
40#121×C131C2211yescharacteristic
41#102×C131C2212yes—
42C22⋊C8×C131C2211yescharacteristic
43C2×Dic262C2311yescharacteristic
44C22×Dic131—22no—
45C22×Dic131—22no—
46C22×Dic131—21no—
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—
60C22⋊Dic131—22no—
61C22⋊Dic131—22no—
62C22⋊Dic131—21no—
63C22⋊Dic131—21no—
64C22⋊Dic131—21no—
65C22⋊Dic131—21no—
66C4×Dic131—21no—
67C4×Dic131D411yescharacteristic
68C4×Dic131D411yescharacteristic
69C4×Dic131C2311yescharacteristic
70C4×Dic131C2311yescharacteristic
71C24×C131—21no—
72C2×D4×C131—24no—
73C2×D4×C131—21no—
74C2×D4×C131—21no—
75C2×D4×C131C2311yescharacteristic
76C2×D4×C131C2312yes—
77C22×C4×C131C2311yescharacteristic
78D8×C131—24no—
79C2×C8×C131C2312yes—
80C4⋊C4×C131C2311yescharacteristic
81C22⋊C4×C131—21no—
82Dic262—22no—
83C2×Dic131—22no—
84C2×Dic131—22no—
85C2×Dic131—21no—
86C2×Dic131—21no—
87C2×Dic131—22no—
88C2×Dic131—44no—
89C2×Dic131—42no—
90C2×Dic131—42no—
91C2×Dic131—44no—
92C2×Dic131C2×D411yescharacteristic
93C2×Dic131C2×D411yescharacteristic
94C2×Dic131C2×D411yescharacteristic
95C2×Dic131C2×D411yescharacteristic
96C131⋊C8—22no—
97C23×C131—24no—
98C23×C131—22no—
99C23×C131—21no—
100C23×C131—42no—
101C23×C131—42no—
102C23×C131C2×D411yescharacteristic
103D4×C131—24no—
104D4×C131—22no—
105D4×C131—22no—
106D4×C131—42no—
107D4×C131#4114yes—
108C2×C4×C131—22no—
109C2×C4×C131—21no—
110C2×C4×C131C2×D411yescharacteristic
111C2×C4×C131C2411yescharacteristic
112C8×C131—22no—
113C22×C131—21no—
114C22×C131—21no—
115C22×C131—24no—
116C22×C131—48no—
117C22×C131—42no—
118C22×C131—42no—
119C22×C131—41no—
120C22×C131—41no—
121C22×C131—41no—
122C22×C131#13512yes—
123C22×C131C22×D411yescharacteristic
124C4×C131—21no—
125C4×C131—41no—
126C4×C131C2×#4112yes—
127Dic131—21no—
128Dic131—21no—
129Dic131—21no—
130Dic131—41no—
131Dic131—41no—
132Dic131—42no—
133Dic131AGL(1,ℤ/8)12yes—
134C2×C131—22no—
135C2×C131—44no—
136C2×C131—81no—
137C2×C131#52511yescharacteristic
138C2×C131C2×#13511yescharacteristic
139C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
140C131#310511yescharacteristic, complemented, Sylow
141#3105—1311nomaximal, Sylow
142#529—1311no—
143C2×C4×D4—1311no—
144#484—1311no—
145#481—1311no—
146#478—1312no—
147#463—1312no—
148#450—1311no—
149#447D13111yescharacteristic
150C4×SD16—1312no—
151C4×D8—1312no—
152#431—1311no—
153C22×D4C2×D13111yescharacteristic
154C23×C4—2621no—
155C2×SD16—1312no—
156C2×D8C2×D13112yes—
157#126—2621no—
158#125—1311no—
159#124—1311no—
160#123—2621no—
161C22⋊Q8—1311no—
162#121C2×D13111yescharacteristic
163C4×Q8—1311no—
164C4×D4—2621no—
165C4×D4—2621no—
166C4×D4—2621no—
167C4×D4—1312no—
168C4×D4—1312no—
169C4×D4—1311no—
170#117—2621no—
171C2×C4⋊C4—1311no—
172C2×C22⋊C4—2621no—
173C2×C42—1311no—
174#107—1312no—
175#106—1312no—
176C4⋊C8—1312no—
177#103—1312no—
178#102—1312no—
179#102—1312no—
180#102C2×D13112yes—
181C22⋊C8—1311no—
182C22⋊C8C2×D13111yescharacteristic
183C4×C8—1312no—
184C24—21no—
185C2×Q8—1311no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4C22×D13111yescharacteristic
190C2×D4C22×D13112yes—
191C22×C4—2621no—
192C22×C4—2621no—
193C22×C4—2622no—
194C22×C4—2622no—
195C22×C4—5244no—
196C22×C4—1311no—
197C22×C4—1311no—
198C22×C4C22×D13111yescharacteristic
199SD16—2624no—
200D8—24no—
201C2×C8—1312no—
202C2×C8C22×D13112yes—
203C4⋊C4—2621no—
204C4⋊C4—2621no—
205C4⋊C4—2621no—
206C4⋊C4—2621no—
207C4⋊C4—2621no—
208C4⋊C4—1311no—
209C4⋊C4—1312no—
210C4⋊C4C22×D13111yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—2621no—
213C22⋊C4—2622no—
214C22⋊C4—2622no—
215C22⋊C4—2621no—
216C22⋊C4—2621no—
217C22⋊C4—2621no—
218C42—2621no—
219C42—1311no—
220C42—1311no—
221C42—1311no—
222C42—1311no—
223C23—24no—
224C23—22no—
225C23—21no—
226C23—42no—
227C23—42no—
228C23D4×D13111yescharacteristic
229Q8—2622no—
230D4—24no—
231D4—22no—
232D4—22no—
233D4—42no—
234D4#1452351214yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—2622no—
238C2×C4—2622no—
239C2×C4—2621no—
240C2×C4—2621no—
241C2×C4—2622no—
242C2×C4—5244no—
243C2×C4—5242no—
244C2×C4—5244no—
245C2×C4—5242no—
246C2×C4—1311no—
247C2×C4—1311no—
248C2×C4—1311no—
249C2×C4—1311no—
250C2×C4D4×D13111yescharacteristic
251C2×C4C23×D13111yescharacteristic
252C8—22no—
253C8—2622no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22#1686226512yes—
264C22C2×D4×D13111yescharacteristic
265C4—21no—
266C4—41no—
267C4—2621no—
268C4—2621no—
269C4—2621no—
270C4—5241no—
271C4—5242no—
272C4—5241no—
273C4—1312no—
274C4C2×#1452351212yes—
275C2—22no—
276C2—44no—
277C2—81no—
278C2#2255381811yescharacteristic
279C2C2×#1686226511yescharacteristic
280C2AGL(1,ℤ/8)×D13111yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting, prime radical#447×C131C2
socleC22×C131C22×D4
hypercenter#447D131
hypocenterC131#3105
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #22553843C2semidirect
2. C2×D4×Dic131C2semidirect
3. #22553648C2semidirect
4. #22553628C2semidirect
5. #22553610C2semidirect
6. #22553525C2semidirect
7. #22553463C2semidirect
8. #22553403C2semidirect
9. D8×Dic131C2semidirect
10. #22553362C2semidirect
15. #16862237C22semidirect
16. #16862233C22semidirect
19. #16862211C22semidirect
25. D4×Dic131C22semidirect
30. #16862180C22semidirect
31. #16862177C22semidirect
32. Dic131⋊C8C22semidirect
33. #16862171C22semidirect
35. #16862167C22semidirect
37. C8×Dic131C22semidirect
140. C131#3105semidirect

Maximal subgroups

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

subgroupindex
#225538432
C2×D4×Dic1312
#225536482
#225536282
#225536102
#225535252
#225534632
#225534032
D8×Dic1312
#225533622
#447×C1312
#3105131