Finite Group Databasebeta

← #30675966

Subgroups — #30675966

Order 16768.

448 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16862208C211yesmaximal, characteristic
3#22553642C211yesmaximal, characteristic, complemented
4#22553601C212yesmaximal, complemented
5#22553516C212yesmaximal, complemented
6#22553465C211yesmaximal, characteristic
7#22553462C211yesmaximal, characteristic
8#22553399C212yesmaximal, complemented
9#22553397C212yesmaximal, complemented
10#22553362C211yesmaximal, characteristic, complemented
11#483×C131C211yesmaximal, characteristic, complemented
12C22×D524C2211yescharacteristic
13C22×C4×D131—21no—
14C2×#14523502C2212yes—
15C2×#14523500C2212yes—
16#16862245C2211yescharacteristic, complemented
17#16862244C2211yescharacteristic, complemented
18#16862236—21no—
19#16862225C2211yescharacteristic
20#16862223—21no—
21Q8×Dic131C2211yescharacteristic, complemented
22#16862211—21no—
23#16862211C2211yescharacteristic, complemented
24#16862208—21no—
25#16862208—21no—
26#16862208C2212yes—
27#16862208C2212yes—
28#16862202—21no—
29C2×(C2×D131)⋊C4—21no—
30C2×C4×Dic131C2211yescharacteristic
31#16862177C2212yescomplemented
32Dic131⋊C8C2212yescomplemented
33#16862169C2212yescomplemented
34#16862168C2212yes—
35#16862166C2212yes—
36C22⋊(C131⋊C8)C2211yescharacteristic
37C8×Dic131C2212yescomplemented
38C22⋊Q8×C131C2211yescharacteristic
39C2×C4⋊C4×C131C2211yescharacteristic
40#106×C131C2212yescomplemented
41#103×C131C2212yescomplemented
42C22⋊C8×C131C2211yescharacteristic
43C23×D131—21no—
44C2×Dic262—21no—
45C2×Dic262C2311yescharacteristic
46C2×D524—24no—
47C2×D524—21no—
48C2×D524C2312yes—
49C22×Dic131D411yescharacteristic
50C22×Dic131D411yescharacteristic
51C2×C4×D131—22no—
52C2×C4×D131—22no—
53C2×C4×D131—44no—
54#14523502—24no—
55#14523500—24no—
56C2×C131⋊C8C2312yes—
57C4⋊Dic131—21no—
58C4⋊Dic131C2311yescharacteristic
59Dic131⋊C4—21no—
60Dic131⋊C4—21no—
61Dic131⋊C4—21no—
62Dic131⋊C4—21no—
63Dic131⋊C4—21no—
64C22⋊Dic131—21no—
65(C2×D131)⋊C4—22no—
66(C2×D131)⋊C4—22no—
67C4×Dic131—21no—
68C4×Dic131—21no—
69C4×Dic131D411yescharacteristic
70C4×Dic131D411yescharacteristic
71C4×Dic131C2311yescharacteristic
72C4×Dic131C2311yescharacteristic
73C2×Q8×C131C2311yescharacteristic
74C22×C4×C131—21no—
75C22×C4×C131C2311yescharacteristic
76C2×C8×C131C2312yes—
77C4⋊C4×C131—21no—
78C4⋊C4×C131—21no—
79C4⋊C4×C131C2312yes—
80C4⋊C4×C131C2311yescharacteristic
81C22⋊C4×C131—21no—
82C22×D131—24no—
83C22×D131—22no—
84C22×D131—42no—
85C22×D131—42no—
86Dic262—22no—
87Dic262—42no—
88D524—24no—
89D524—22no—
90D524#4114yes—
91C2×Dic131—21no—
92C2×Dic131—21no—
93C2×Dic131—41no—
94C2×Dic131SD1614yescomplemented
95C2×Dic131C2×D411yescharacteristic
96C2×Dic131C2×D411yescharacteristic
97C2×Dic131C2×D411yescharacteristic
98C2×Dic131C2×D411yescharacteristic
99C4×D131—44no—
100C4×D131—44no—
101C131⋊C8—22no—
102C23×C131C2×D411yescharacteristic
103Q8×C131—22no—
104C2×C4×C131—22no—
105C2×C4×C131—21no—
106C2×C4×C131—21no—
107C2×C4×C131—22no—
108C2×C4×C131—42no—
109C2×C4×C131C2×D411yescharacteristic
110C2×C4×C131C2411yescharacteristic
111C8×C131—22no—
112C22×C131—21no—
113C22×C131—21no—
114C22×C131C2×SD1612yes—
115C22×C131C22×D411yescharacteristic
116C2×D131—24no—
117C2×D131—48no—
118C2×D131—42no—
119C2×D131—42no—
120C4×C131—21no—
121C4×C131—41no—
122C4×C131—42no—
123C4×C131—41no—
124C4×C131C2×#4112yes—
125Dic131—21no—
126Dic131—21no—
127Dic131—41no—
128Dic131—41no—
129Dic131C2×SD1612yes—
130Dic131C2×SD1612yes—
131C2×C131—22no—
132C2×C131#52511yescharacteristic
133C2×C131C22×SD1611yescharacteristic
134C2×C131#57511yescharacteristic
135D131—44no—
136C131#310111yescharacteristic, complemented, Sylow
137#3101—1311nomaximal, Sylow
138#533—1311no—
139C2×C4×D4—1311no—
140#483—1311no—
141#483D13111yescharacteristic, complemented
142#476—1312no—
143#461—1312no—
144#450—1311no—
145#450—1311no—
146C4×SD16—1312no—
147C4×SD16—1312no—
148#431—1311no—
149C22×D4—1311no—
150C23×C4—2621no—
151C2×SD16—1312no—
152C2×SD16—1312no—
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⋊C4C2×D13111yescharacteristic
168C2×C22⋊C4—2621no—
169C2×C42—1311no—
170#106—1312no—
171#106C2×D13112yescomplemented
172C4⋊C8—1312no—
173#103—1312no—
174#103C2×D13112yescomplemented
175#102—1312no—
176#102—1312no—
177C22⋊C8—1311no—
178C22⋊C8C2×D13111yescharacteristic
179C4×C8—1312no—
180C24—2621no—
181C2×Q8—2621no—
182C2×Q8—1311no—
183C2×Q8C22×D13111yescharacteristic
184C2×D4—2624no—
185C2×D4—2621no—
186C2×D4—1312no—
187C22×C4—21no—
188C22×C4—2622no—
189C22×C4—2622no—
190C22×C4—5244no—
191C22×C4—1311no—
192C22×C4—1311no—
193C22×C4C22×D13111yescharacteristic
194SD16—2624no—
195SD16—2624no—
196C2×C8—1312no—
197C2×C8C22×D13112yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—2621no—
201C4⋊C4—2621no—
202C4⋊C4—2621no—
203C4⋊C4—2621no—
204C4⋊C4—2621no—
205C4⋊C4—2621no—
206C4⋊C4—1311no—
207C4⋊C4C22×D13111yescharacteristic
208C4⋊C4C22×D13112yes—
209C22⋊C4—21no—
210C22⋊C4—2622no—
211C22⋊C4—2622no—
212C22⋊C4—2621no—
213C42—2621no—
214C42—2621no—
215C42—1311no—
216C42—1311no—
217C42—1311no—
218C42—1311no—
219C23—2624no—
220C23—2622no—
221C23—5242no—
222C23—5242no—
223C23D4×D13111yescharacteristic
224Q8—22no—
225Q8—2622no—
226Q8—5242no—
227D4—2624no—
228D4—2622no—
229D4—1314no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—2621no—
236C2×C4—2621no—
237C2×C4—5244no—
238C2×C4—5244no—
239C2×C4—5241no—
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—2624no—
252C22—5248no—
253C22—5242no—
254C22—5242no—
255C22SD16×D13112yes—
256C22C2×D4×D13111yescharacteristic
257C4—21no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—2621no—
262C4—2621no—
263C4—5241no—
264C4—5241no—
265C4—1312no—
266C4—1312no—
267C4#41×D13112yes—
268C2—22no—
269C2—5244no—
270C2#2255382011yescharacteristic
271C2C2×SD16×D13111yescharacteristic
272C2#2255411111yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#483×C131C2
socleC22×C131C22×D4
prime radicalC22×D524C22
hypercenter#483D131
hypocenterC131#3101
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #22553864C2semidirect
3. #22553642C2semidirect
4. #22553601C2semidirect
5. #22553516C2semidirect
8. #22553399C2semidirect
9. #22553397C2semidirect
10. #22553362C2semidirect
11. #483×C131C2semidirect
16. #16862245C22semidirect
17. #16862244C22semidirect
21. Q8×Dic131C22semidirect
23. #16862211C22semidirect
31. #16862177C22semidirect
32. Dic131⋊C8C22semidirect
33. #16862169C22semidirect
37. C8×Dic131C22semidirect
40. #106×C131C22semidirect
41. #103×C131C22semidirect
94. C2×Dic131SD16semidirect
136. C131#3101semidirect
141. #483D131semidirect
171. #106C2×D131semidirect
174. #103C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225538642
C2×#168622082
#225536422
#225536012
#225535162
#225534652
#225534622
#225533992
#225533972
#225533622
#483×C1312
#3101131