Finite Group Databasebeta

← #30675249

Subgroups — #30675249

Order 16768.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16862293C211yesmaximal, characteristic, complemented
2C2×#16862274C211yesmaximal, characteristic, complemented
3C22×#14523502C211yesmaximal, characteristic, complemented
4#22553876C211yesmaximal, characteristic, complemented
5#22553459C214yesmaximal, complemented
6#22553453C214yesmaximal, complemented
7#22553322C211yesmaximal, characteristic, complemented
8#22553309C211yesmaximal, characteristic, complemented
9#408×C131C211yesmaximal, characteristic, complemented
10#16862293—24no—
11#16862293—24no—
12C2×#14523513C2211yescharacteristic
13C2×#14523513C2212yescomplemented
14C2×#14523512—22no—
15C2×#14523512—21no—
16C22×Dic262C2211yescharacteristic
17C22×Dic262C2212yescomplemented
18C2×Q8×D131—21no—
19C22×D524C2211yescharacteristic
20#16862274—24no—
21C2×#14523502—24no—
22C2×#14523502C2212yes—
23C2×#14523502C2214yes—
24C2×D1048C2212yes—
25#16862242—21no—
26#16862235—21no—
27#16862220C2214yes—
28#16862214—22no—
29#16862205C2211yescharacteristic
30#16862171C2212yescomplemented
31#16862171C2214yescomplemented
32#16862168C2212yescomplemented
33C2×#41×C131C2211yescharacteristic
34C2×M16×C131C2211yescharacteristic
35C22×C8×C131C2211yescharacteristic
36C22⋊C8×C131C2214yescomplemented
37C23×D131—21no—
38#14523513—24no—
39#14523513—24no—
40#14523513—22no—
41#14523513D414yescomplemented
42#14523512—216no—
43#14523512—28no—
44C2×Dic262—24no—
45C2×Dic262—24no—
46C2×Dic262—21no—
47C2×Dic262—22no—
48C2×Dic262—22no—
49C2×Dic262D414yescomplemented
50C2×Dic262C2312yes—
51C2×Dic262C2311yescharacteristic
52Q8×D131—28no—
53C2×D524—22no—
54C2×D524—22no—
55C2×D524—21no—
56C2×D524C2312yes—
57C2×D524C2311yescharacteristic
58C2×C22⋊D131—21no—
59C2×C22⋊D131—22no—
60C2×C22⋊D131—24no—
61C22×Dic131—21no—
62C22×Dic131—22no—
63C2×C4×D131—21no—
64C2×C4×D131—22no—
65#14523502—28no—
66#14523502—24no—
67D1048—24no—
68C4⋊Dic131C2312yes—
69C22⋊Dic131—21no—
70(C2×D131)⋊C4—24no—
71C4×Dic131—21no—
72#41×C131—22no—
73#41×C131—22no—
74C2×Q8×C131D411yescharacteristic
75C2×D4×C131D412yes—
76C2×D4×C131D411yescharacteristic
77C22×C4×C131C2312yes—
78C22×C4×C131C2311yescharacteristic
79M16×C131—22no—
80C2×C8×C131—22no—
81C2×C8×C131C2312yes—
82C2×C8×C131C2312yes—
83C22×D131—21no—
84C22×D131—22no—
85C22×D131—42no—
86C22×D131—82no—
87Dic262—28no—
88Dic262—24no—
89Dic262—21no—
90Dic262—22no—
91Dic262C2×D412yes—
92Dic262C2×D414yes—
93D524—21no—
94D524—22no—
95D524—22no—
96D524—42no—
97D524C2×D412yes—
98C22⋊D131—28no—
99C22⋊D131—24no—
100C22⋊D131—84no—
101C2×Dic131—28no—
102C2×Dic131—24no—
103C2×Dic131—21no—
104C2×Dic131—22no—
105C2×Dic131—22no—
106C4×D131—28no—
107C4×D131—24no—
108C23×C131C2×D412yes—
109C23×C131C2×D411yescharacteristic
110Q8×C131—22no—
111D4×C131—24no—
112D4×C131—22no—
113C2×C4×C131—22no—
114C2×C4×C131—22no—
115C2×C4×C131C2×D412yes—
116C2×C4×C131C2×D412yes—
117C2×C4×C131C2×D412yes—
118C2×C4×C131C2×D411yescharacteristic
119C2×C4×C131C2411yescharacteristic
120C8×C131—22no—
121C8×C131—22no—
122C22×C131—22no—
123C22×C131—22no—
124C22×C131—24no—
125C22×C131#12012yes—
126C22×C131C22×D411yescharacteristic
127C2×D131—22no—
128C2×D131—42no—
129C2×D131—42no—
130C2×D131—44no—
131C2×D131—84no—
132C4×C131—22no—
133C4×C131#12012yes—
134C4×C131C22×D411yescharacteristic
135C4×C131C22×D411yescharacteristic
136Dic131—24no—
137Dic131—22no—
138Dic131—42no—
139C2×C131—22no—
140C2×C131—42no—
141C2×C131C2×#12011yescharacteristic
142C2×C131#57712yes—
143D131—42no—
144D131—82no—
145C131#301011yescharacteristic, complemented, Sylow
146#3010—1311nomaximal, Sylow
147C2×2−1+4—1311no—
148C2×AGL(1,ℤ/8)—1311no—
149C22×SD16—1311no—
150#535—1311no—
151#449—1314no—
152#448—1314no—
153#419—1311no—
154#416—1311no—
155#408D13111yescharacteristic, complemented
1562−1+4—2624no—
1572−1+4—2624no—
158C2×#41—2622no—
159C2×#41—2621no—
160C2×#41—1312no—
161C2×#41—1311no—
162C2×#41C2×D13111yescharacteristic
163C22×Q8—2621no—
164C22×Q8—1312no—
165C22×Q8—1311no—
166C22×D4—1311no—
167AGL(1,ℤ/8)—2624no—
168C2×SD16—2624no—
169C2×SD16—1314no—
170C2×SD16—1312no—
171C2×D8—1312no—
172C2×M16C2×D13111yescharacteristic
173C22×C8C2×D13111yescharacteristic
174#127—2621no—
175#124—2621no—
176#121—1314no—
177#120—2622no—
178#117—1311no—
179#103—1312no—
180#103—1314no—
181#102—1312no—
182C22⋊C8C2×D13114yescomplemented
183C24—2621no—
184#41—22no—
185#41—22no—
186#41—2624no—
187#41—2624no—
188#41—26216no—
189#41—2628no—
190#41—2622no—
191#41—1314no—
192C2×Q8—2624no—
193C2×Q8—2624no—
194C2×Q8—2628no—
195C2×Q8—2622no—
196C2×Q8—2622no—
197C2×Q8—2621no—
198C2×Q8—1314no—
199C2×Q8—1311no—
200C2×Q8—1312no—
201C2×Q8D52411yescharacteristic
202C2×D4—2622no—
203C2×D4—2621no—
204C2×D4—2624no—
205C2×D4—2622no—
206C2×D4—2622no—
207C2×D4—2621no—
208C2×D4—1312no—
209C2×D4—1311no—
210C2×D4D52412yes—
211C2×D4D52411yescharacteristic
212C22×C4—2622no—
213C22×C4—2621no—
214C22×C4—2622no—
215C22×C4—2621no—
216C22×C4C22×D13112yes—
217C22×C4C22×D13111yescharacteristic
218SD16—2628no—
219SD16—2624no—
220D8—2624no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D13112yes—
224C2×C8C22×D13112yes—
225C4⋊C4—1312no—
226C22⋊C4—2621no—
227C22⋊C4—2624no—
228C42—2621no—
229C23—2621no—
230C23—2622no—
231C23—5242no—
232C23—10482no—
233C23C2×D52412yes—
234C23C2×D52411yescharacteristic
235Q8—22no—
236Q8—2628no—
237Q8—2624no—
238Q8—2621no—
239Q8—2622no—
240Q8—1312no—
241Q8—1314no—
242D4—24no—
243D4—22no—
244D4—2628no—
245D4—2624no—
246D4—2621no—
247D4—2622no—
248D4—2622no—
249D4—5242no—
250D4—10484no—
251D4—1312no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—2628no—
255C2×C4—2628no—
256C2×C4—2624no—
257C2×C4—2622no—
258C2×C4—2621no—
259C2×C4—2624no—
260C2×C4—2622no—
261C2×C4D4×D13112yes—
262C2×C4D4×D13112yes—
263C2×C4C2×D52412yes—
264C2×C4C2×D52411yescharacteristic
265C2×C4C23×D13111yescharacteristic
266C8—22no—
267C8—22no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—2622no—
272C22—5242no—
273C22—5244no—
274C22—5242no—
275C22—10484no—
276C22#1686221612yes—
277C22C22×D52411yescharacteristic
278C4—22no—
279C4—2624no—
280C4—2622no—
281C4—5242no—
282C4#1686221612yes—
283C4C2×D4×D13111yescharacteristic
284C4C2×D4×D13111yescharacteristic
285C2—22no—
286C2—42no—
287C2—5242no—
288C2—10482no—
289C2C2×#1686221611yescharacteristic
290C2#2255412412yes—
subgroupG/H
centerC22C22×D524
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#408×C131C2
socleC22×C131C22×D4
hypercenter#408D131
hypocenterC131#3010
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. C2×#16862293C2semidirect
2. C2×#16862274C2semidirect
3. C22×#14523502C2semidirect
4. #22553876C2semidirect
5. #22553459C2semidirect
6. #22553453C2semidirect
7. #22553322C2semidirect
8. #22553309C2semidirect
9. #408×C131C2semidirect
13. C2×#14523513C22semidirect
17. C22×Dic262C22semidirect
30. #16862171C22semidirect
31. #16862171C22semidirect
32. #16862168C22semidirect
36. C22⋊C8×C131C22semidirect
41. #14523513D4semidirect
49. C2×Dic262D4semidirect
145. C131#3010semidirect
155. #408D131semidirect
182. C22⋊C8C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622932
C2×#168622742
C22×#145235022
#225538762
#225534592
#225534532
#225533222
#225533092
#408×C1312
#3010131