Finite Group Databasebeta

← #30675438

Subgroups — #30675438

Order 16768.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16862273C212yesmaximal, complemented
2#127×D131C211yesmaximal, characteristic, complemented
3C4×#14523514C211yesmaximal, characteristic, complemented
4#22553512C211yesmaximal, characteristic, complemented
5#22553510C212yesmaximal, complemented
6#22553508C211yesmaximal, characteristic, complemented
7#22553506C212yesmaximal, complemented
8#22553504C211yesmaximal, characteristic, complemented
9#22553334C211yesmaximal, characteristic, complemented
10#22553315C212yesmaximal, complemented
11#460×C131C211yesmaximal, characteristic, complemented
12C2×#14523514C2211yescharacteristic
13C2×D4×D131—21no—
14C2×D4×D131—21no—
15C2×D4×D131C2212yes—
16#16862273—28no—
17C2×#14523500C2212yescomplemented
18C2×D1048C2212yescomplemented
19C2×#14523498C2212yescomplemented
20C2×#14523496C2212yes—
21#16862243C2211yescharacteristic
22#16862242—21no—
23#16862242—21no—
24#16862242C2212yes—
25Q8×Dic131C2211yescharacteristic, complemented
26C4×D524—21no—
27C4×D524C2211yescharacteristic, complemented
28#16862208—21no—
29#16862208C2211yescharacteristic, complemented
30#16862204—21no—
31#16862204C2211yescharacteristic
32C42×D131—21no—
33C42×D131C2211yescharacteristic
34C4⋊(C131⋊C8)C2211yescharacteristic, complemented
35Dic131⋊C8C2212yescomplemented
36#16862168C2212yescomplemented
37#16862167C2212yescomplemented
38#16862166C2212yescomplemented
39C2×SD16×C131C2212yescomplemented
40#127×C131C2211yescharacteristic
41C4×Q8×C131C2211yescharacteristic, complemented
42C4⋊C8×C131C2211yescharacteristic, complemented
43#102×C131C2212yescomplemented
44C23×D131—22no—
45#14523514—22no—
46#14523514D414yescomplemented
47C2×D524—21no—
48C2×D524—21no—
49C2×D524—21no—
50C2×D524C2312yes—
51C2×D524C2311yescharacteristic
52C2×C22⋊D131—22no—
53C2×C22⋊D131—22no—
54C2×C22⋊D131—22no—
55D4×D131—24no—
56D4×D131—24no—
57D4×D131—44no—
58D4×D131—44no—
59C2×C4×D131—21no—
60C2×C4×D131—21no—
61C2×C4×D131—21no—
62C2×C4×D131D411yescharacteristic
63C2×C4×D131D411yescharacteristic
64C2×C4×D131C2311yescharacteristic
65#14523500—24no—
66D1048—24no—
67#14523498—24no—
68#14523496—24no—
69C2×C131⋊C8C2312yes—
70C4⋊Dic131—21no—
71C4⋊Dic131C2311yescharacteristic
72(C2×D131)⋊C4—21no—
73(C2×D131)⋊C4—21no—
74(C2×D131)⋊C4—21no—
75C4×Dic131—21no—
76C4×Dic131—21no—
77C4×Dic131D411yescharacteristic
78C4×Dic131D411yescharacteristic
79C4×Dic131C2311yescharacteristic
80C2×Q8×C131C2311yescharacteristic
81C2×D4×C131—21no—
82C2×D4×C131—21no—
83C2×D4×C131C2312yes—
84SD16×C131—24no—
85C2×C8×C131C2312yes—
86C4⋊C4×C131—21no—
87C4⋊C4×C131C2311yescharacteristic
88C42×C131—21no—
89C42×C131C2311yescharacteristic
90C22×D131—21no—
91C22×D131—22no—
92C22×D131—44no—
93C22×D131—44no—
94C22×D131—44no—
95C22×D131C2×D411yescharacteristic
96D524—24no—
97D524—22no—
98D524—21no—
99D524—22no—
100D524—42no—
101D524—42no—
102D524C2×D412yescomplemented
103C22⋊D131—44no—
104C22⋊D131—44no—
105C22⋊D131—44no—
106C2×Dic131—21no—
107C2×Dic131C2×D411yescharacteristic
108C2×Dic131C2×D411yescharacteristic
109C2×Dic131C2×D411yescharacteristic
110C4×D131—22no—
111C4×D131—24no—
112C4×D131—22no—
113C4×D131—42no—
114C4×D131—42no—
115C4×D131C2×D412yes—
116C4×D131#4112yes—
117C131⋊C8—22no—
118C23×C131—22no—
119Q8×C131—21no—
120Q8×C131C2×D412yescomplemented
121D4×C131—22no—
122D4×C131—22no—
123D4×C131—42no—
124D4×C131—42no—
125C2×C4×C131—21no—
126C2×C4×C131—21no—
127C2×C4×C131C2×D411yescharacteristic
128C2×C4×C131C2×D411yescharacteristic
129C2×C4×C131C2411yescharacteristic
130C8×C131—22no—
131C22×C131—42no—
132C22×C131—42no—
133C22×C131—42no—
134C22×C131C22×D411yescharacteristic
135C2×D131—21no—
136C2×D131—21no—
137C2×D131—22no—
138C2×D131—41no—
139C2×D131—41no—
140C2×D131—42no—
141C2×D131—42no—
142C2×D131—42no—
143C2×D131—88no—
144C2×D131#12112yes—
145C4×C131—22no—
146C4×C131—21no—
147C4×C131—41no—
148C4×C131AGL(1,ℤ/8)12yescomplemented
149C4×C131C22×D411yescharacteristic
150C4×C131C2×#4111yescharacteristic
151Dic131—21no—
152Dic131—41no—
153Dic131#12112yes—
154Dic131AGL(1,ℤ/8)12yescomplemented
155C2×C131—82no—
156C2×C131C2×#12111yescharacteristic
157C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
158C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
159D131—22no—
160D131—42no—
161D131—82no—
162C131#303811yescharacteristic, complemented, Sylow
163#3038—1311nomaximal, Sylow
164C2×AGL(1,ℤ/8)—1312no—
165C2×#127—1311no—
166C4×#41—1311no—
167#460—1312no—
168#460—1311no—
169#460D13111yescharacteristic, complemented
170#459—1311no—
171#459—1311no—
172#459—1312no—
173#423—1311no—
174#418—1312no—
175C2×#41—1311no—
176C22×D4—2621no—
177C22×D4—2621no—
178C22×D4—1312no—
179AGL(1,ℤ/8)—2628no—
180C2×SD16—1312no—
181C2×SD16C2×D13112yescomplemented
182C2×D8—1312no—
183C2×D8—1312no—
184C2×M16—1312no—
185#127—2621no—
186#127—2621no—
187#127—1312no—
188#127—1311no—
189#127C2×D13111yescharacteristic
190C4×Q8—1311no—
191C4×Q8C2×D13111yescharacteristic, complemented
192C4×D4—2621no—
193C4×D4—2621no—
194C4×D4—1311no—
195C4×D4—1311no—
196#117—2621no—
197#117—1311no—
198C2×C42—2621no—
199C2×C42—1311no—
200C4⋊C8—1312no—
201C4⋊C8—1311no—
202C4⋊C8C2×D13111yescharacteristic, complemented
203#102—1312no—
204#102—1312no—
205#102—1312no—
206#102C2×D13112yescomplemented
207C24—2622no—
208#41—2622no—
209#41—1314no—
210C2×Q8C22×D13111yescharacteristic
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—2621no—
214C2×D4—2624no—
215C2×D4—2624no—
216C2×D4—2622no—
217C2×D4—2622no—
218C2×D4—2622no—
219C2×D4—2621no—
220C2×D4—2621no—
221C2×D4—5244no—
222C2×D4—5244no—
223C2×D4—1312no—
224C2×D4—1311no—
225C2×D4C22×D13112yes—
226C22×C4—2621no—
227C22×C4—2621no—
228C22×C4—2621no—
229C22×C4—1311no—
230C22×C4—1311no—
231C22×C4—1311no—
232SD16—24no—
233SD16—2624no—
234D8—2624no—
235D8—2624no—
236M16—2624no—
237C2×C8—1312no—
238C2×C8C22×D13112yes—
239C4⋊C4—21no—
240C4⋊C4—2621no—
241C4⋊C4—1311no—
242C4⋊C4C22×D13111yescharacteristic
243C22⋊C4—2621no—
244C22⋊C4—2621no—
245C22⋊C4—2621no—
246C42—21no—
247C42—2621no—
248C42—2621no—
249C42—1311no—
250C42—1311no—
251C42—1311no—
252C42C22×D13111yescharacteristic
253C23—22no—
254C23—2621no—
255C23—2622no—
256C23—5244no—
257C23—5244no—
258C23—5244no—
259C23—1311no—
260Q8—21no—
261Q8D4×D13112yescomplemented
262D4—22no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—2624no—
267D4—2622no—
268D4—2621no—
269D4—2622no—
270D4—5242no—
271D4—5244no—
272D4—5244no—
273D4—5244no—
274D4—5242no—
275D4—1312no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—2624no—
279C2×C4—2622no—
280C2×C4—2622no—
281C2×C4—2621no—
282C2×C4—5242no—
283C2×C4—5242no—
284C2×C4—1311no—
285C2×C4—1311no—
286C2×C4—1311no—
287C2×C4—1312no—
288C2×C4—1312no—
289C2×C4D4×D13111yescharacteristic
290C2×C4D4×D13111yescharacteristic
291C2×C4C23×D13111yescharacteristic
292C8—22no—
293C8—2622no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—2621no—
298C22—2621no—
299C22—2622no—
300C22—5241no—
301C22—5241no—
302C22—5242no—
303C22—5242no—
304C22—5242no—
305C22—10488no—
306C22—1312no—
307C22C2×D4×D13111yescharacteristic
308C4—22no—
309C4—21no—
310C4—41no—
311C4—2621no—
312C4—5241no—
313C4—1312no—
314C4—1312no—
315C4#1686227312yescomplemented
316C4C2×D4×D13111yescharacteristic
317C4#41×D13111yescharacteristic
318C2—2622no—
319C2—82no—
320C2—5242no—
321C2—10482no—
322C2#121×D13111yescharacteristic
323C2AGL(1,ℤ/8)×D13111yescharacteristic
324C2C2×#1686227311yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#460×C131C2
socleC22×C131C22×D4
hypercenter#460D131
hypocenterC131#3038
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×#16862273C2semidirect
2. #127×D131C2semidirect
3. C4×#14523514C2semidirect
4. #22553512C2semidirect
5. #22553510C2semidirect
6. #22553508C2semidirect
7. #22553506C2semidirect
8. #22553504C2semidirect
9. #22553334C2semidirect
10. #22553315C2semidirect
11. #460×C131C2semidirect
17. C2×#14523500C22semidirect
18. C2×D1048C22semidirect
19. C2×#14523498C22semidirect
25. Q8×Dic131C22semidirect
27. C4×D524C22semidirect
29. #16862208C22semidirect
34. C4⋊(C131⋊C8)C22semidirect
35. Dic131⋊C8C22semidirect
36. #16862168C22semidirect
37. #16862167C22semidirect
38. #16862166C22semidirect
39. C2×SD16×C131C22semidirect
41. C4×Q8×C131C22semidirect
42. C4⋊C8×C131C22semidirect
43. #102×C131C22semidirect
46. #14523514D4semidirect
102. D524C2×D4semidirect
120. Q8×C131C2×D4semidirect
148. C4×C131AGL(1,ℤ/8)semidirect
154. Dic131AGL(1,ℤ/8)semidirect
162. C131#3038semidirect
169. #460D131semidirect
181. C2×SD16C2×D131semidirect
191. C4×Q8C2×D131semidirect
202. C4⋊C8C2×D131semidirect
206. #102C2×D131semidirect
261. Q8D4×D131semidirect
315. C4#16862273semidirect

Maximal subgroups

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

subgroupindex
C2×#168622732
#127×D1312
C4×#145235142
#225535122
#225535102
#225535082
#225535062
#225535042
#225533342
#225533152
#460×C1312
#3038131