Finite Group Databasebeta

← #30675436

Subgroups — #30675436

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×#16862270C212yesmaximal, complemented
2#127×D131C211yesmaximal, characteristic, complemented
3C4×#14523512C211yesmaximal, characteristic, complemented
4#22553513C211yesmaximal, characteristic, complemented
5#22553511C212yesmaximal, complemented
6#22553509C211yesmaximal, characteristic, complemented
7#22553507C211yesmaximal, characteristic, complemented
8#22553505C212yesmaximal, complemented
9#22553334C211yesmaximal, characteristic, complemented
10#22553315C212yesmaximal, complemented
11#459×C131C211yesmaximal, characteristic, complemented
12C2×#14523512C2211yescharacteristic
13C2×D4×D131—21no—
14C2×D4×D131—21no—
15C2×D4×D131C2212yes—
16#16862270—28no—
17C2×#14523502C2212yescomplemented
18C2×#14523501C2212yescomplemented
19C2×#14523498C2212yescomplemented
20C2×#14523496C2212yes—
21#16862243C2211yescharacteristic
22#16862242—21no—
23#16862242—21no—
24#16862242C2212yes—
25C4×Dic262C2211yescharacteristic, complemented
26#16862211C2211yescharacteristic, complemented
27D4×Dic131C2211yescharacteristic, complemented
28C4×C22⋊D131—21no—
29(C22⋊D131)⋊C4—21no—
30#16862204C2211yescharacteristic
31#16862202—21no—
32C2×C4×Dic131—21no—
33C42×D131C2211yescharacteristic
34C4⋊(C131⋊C8)C2211yescharacteristic, complemented
35Dic131⋊C8C2212yescomplemented
36#16862168C2212yescomplemented
37#16862167C2212yescomplemented
38#16862166C2212yescomplemented
39C2×D8×C131C2212yescomplemented
40#127×C131C2211yescharacteristic
41C4×D4×C131C2211yescharacteristic, complemented
42C4⋊C8×C131C2211yescharacteristic, complemented
43#102×C131C2212yescomplemented
44C23×D131—22no—
45#14523512—22no—
46#14523512D414yescomplemented
47C2×Dic262C2311yescharacteristic
48C2×D524—21no—
49C2×D524—21no—
50C2×D524C2312yes—
51C2×C22⋊D131—21no—
52C2×C22⋊D131—22no—
53C2×C22⋊D131—22no—
54C2×C22⋊D131—22no—
55D4×D131—24no—
56D4×D131—24no—
57D4×D131—44no—
58D4×D131—44no—
59C22×Dic131—21no—
60C22×Dic131—21no—
61C2×C4×D131D411yescharacteristic
62C2×C4×D131D411yescharacteristic
63C2×C4×D131C2311yescharacteristic
64#14523502—24no—
65#14523501—24no—
66#14523498—24no—
67#14523496—24no—
68C2×C131⋊C8C2312yes—
69C4⋊Dic131C2311yescharacteristic
70Dic131⋊C4—21no—
71Dic131⋊C4—21no—
72C22⋊Dic131—21no—
73(C2×D131)⋊C4—21no—
74C4×Dic131—21no—
75C4×Dic131—21no—
76C4×Dic131—21no—
77C4×Dic131D411yescharacteristic
78C4×Dic131D411yescharacteristic
79C4×Dic131C2311yescharacteristic
80C2×D4×C131—21no—
81C2×D4×C131—21no—
82C2×D4×C131C2312yes—
83C2×D4×C131C2311yescharacteristic
84C22×C4×C131—21no—
85D8×C131—24no—
86C2×C8×C131C2312yes—
87C4⋊C4×C131C2311yescharacteristic
88C22⋊C4×C131—21no—
89C42×C131C2311yescharacteristic
90C22×D131—22no—
91C22×D131—44no—
92C22×D131—44no—
93C22×D131—44no—
94C22×D131C2×D411yescharacteristic
95Dic262—21no—
96Dic262C2×D412yescomplemented
97D524—22no—
98D524—22no—
99D524—42no—
100D524—42no—
101C22⋊D131—24no—
102C22⋊D131—44no—
103C22⋊D131—44no—
104C22⋊D131—44no—
105C2×Dic131—21no—
106C2×Dic131—24no—
107C2×Dic131—21no—
108C2×Dic131—42no—
109C2×Dic131C2×D411yescharacteristic
110C2×Dic131C2×D411yescharacteristic
111C2×Dic131C2×D411yescharacteristic
112C4×D131—22no—
113C4×D131—22no—
114C4×D131C2×D412yes—
115C4×D131#4112yes—
116C131⋊C8—22no—
117C23×C131—21no—
118C23×C131—22no—
119D4×C131—22no—
120D4×C131—21no—
121D4×C131—22no—
122D4×C131—42no—
123D4×C131—42no—
124D4×C131C2×D412yescomplemented
125C2×C4×C131—21no—
126C2×C4×C131—42no—
127C2×C4×C131C2×D411yescharacteristic
128C2×C4×C131C2×D411yescharacteristic
129C2×C4×C131C2411yescharacteristic
130C8×C131—22no—
131C22×C131—22no—
132C22×C131—41no—
133C22×C131—41no—
134C22×C131—42no—
135C22×C131—42no—
136C22×C131—42no—
137C22×C131C22×D411yescharacteristic
138C2×D131—21no—
139C2×D131—21no—
140C2×D131—42no—
141C2×D131—42no—
142C2×D131—42no—
143C2×D131—88no—
144C2×D131#12112yes—
145C4×C131—21no—
146C4×C131—41no—
147C4×C131AGL(1,ℤ/8)12yescomplemented
148C4×C131C22×D411yescharacteristic
149C4×C131C2×#4111yescharacteristic
150Dic131—22no—
151Dic131—21no—
152Dic131—41no—
153Dic131#12112yes—
154Dic131AGL(1,ℤ/8)12yescomplemented
155C2×C131—42no—
156C2×C131—82no—
157C2×C131C2×#12111yescharacteristic
158C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
160D131—22no—
161D131—82no—
162C131#303811yescharacteristic, complemented, Sylow
163#3038—1311nomaximal, Sylow
164C2×AGL(1,ℤ/8)—1312no—
165C2×#127—1311no—
166C4×#41—1311no—
167#460—1311no—
168#460—1311no—
169#460—1312no—
170#459—1312no—
171#459—1311no—
172#459D13111yescharacteristic, complemented
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×SD16—1312no—
182C2×D8—1312no—
183C2×D8C2×D13112yescomplemented
184C2×M16—1312no—
185#127—2621no—
186#127—2621no—
187#127—1312no—
188#127—1311no—
189#127C2×D13111yescharacteristic
190C4×Q8—1311no—
191C4×Q8—1311no—
192C4×D4—2621no—
193C4×D4—2621no—
194C4×D4—1311no—
195C4×D4C2×D13111yescharacteristic, complemented
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×Q8—1311no—
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×D4C22×D13112yes—
225C2×D4C22×D13111yescharacteristic
226C22×C4—21no—
227C22×C4—2621no—
228C22×C4—2621no—
229C22×C4—1311no—
230C22×C4—1311no—
231C22×C4—1311no—
232SD16—2624no—
233SD16—2624no—
234D8—24no—
235D8—2624no—
236M16—2624no—
237C2×C8—1312no—
238C2×C8C22×D13112yes—
239C4⋊C4—2621no—
240C4⋊C4—2621no—
241C4⋊C4—1311no—
242C4⋊C4C22×D13111yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—2621no—
245C22⋊C4—2621no—
246C42—2621no—
247C42—2621no—
248C42—2621no—
249C42—1311no—
250C42—1311no—
251C42—1311no—
252C42C22×D13111yescharacteristic
253C23—21no—
254C23—22no—
255C23—2622no—
256C23—5244no—
257C23—5244no—
258C23—5244no—
259C23—1311no—
260Q8—2621no—
261Q8—1312no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—2624no—
268D4—2622no—
269D4—2622no—
270D4—5242no—
271D4—5244no—
272D4—5244no—
273D4—5244no—
274D4—5242no—
275D4D4×D13112yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—2622no—
279C2×C4—2621no—
280C2×C4—2624no—
281C2×C4—2622no—
282C2×C4—2621no—
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—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—2621no—
301C22—2621no—
302C22—5242no—
303C22—5242no—
304C22—5242no—
305C22—10488no—
306C22—1312no—
307C22C2×D4×D13111yescharacteristic
308C4—21no—
309C4—41no—
310C4—2622no—
311C4—2621no—
312C4—5241no—
313C4—1312no—
314C4—1312no—
315C4#1686227012yescomplemented
316C4C2×D4×D13111yescharacteristic
317C4#41×D13111yescharacteristic
318C2—42no—
319C2—2622no—
320C2—82no—
321C2—10482no—
322C2#121×D13111yescharacteristic
323C2AGL(1,ℤ/8)×D13111yescharacteristic
324C2C2×#1686227011yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#459×C131C2
socleC22×C131C22×D4
hypercenter#459D131
hypocenterC131#3038
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16862270C2semidirect
2. #127×D131C2semidirect
3. C4×#14523512C2semidirect
4. #22553513C2semidirect
5. #22553511C2semidirect
6. #22553509C2semidirect
7. #22553507C2semidirect
8. #22553505C2semidirect
9. #22553334C2semidirect
10. #22553315C2semidirect
11. #459×C131C2semidirect
17. C2×#14523502C22semidirect
18. C2×#14523501C22semidirect
19. C2×#14523498C22semidirect
25. C4×Dic262C22semidirect
26. #16862211C22semidirect
27. D4×Dic131C22semidirect
34. C4⋊(C131⋊C8)C22semidirect
35. Dic131⋊C8C22semidirect
36. #16862168C22semidirect
37. #16862167C22semidirect
38. #16862166C22semidirect
39. C2×D8×C131C22semidirect
41. C4×D4×C131C22semidirect
42. C4⋊C8×C131C22semidirect
43. #102×C131C22semidirect
46. #14523512D4semidirect
96. Dic262C2×D4semidirect
124. D4×C131C2×D4semidirect
147. C4×C131AGL(1,ℤ/8)semidirect
154. Dic131AGL(1,ℤ/8)semidirect
162. C131#3038semidirect
172. #459D131semidirect
183. C2×D8C2×D131semidirect
195. C4×D4C2×D131semidirect
202. C4⋊C8C2×D131semidirect
206. #102C2×D131semidirect
275. D4D4×D131semidirect
315. C4#16862270semidirect

Maximal subgroups

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

subgroupindex
C2×#168622702
#127×D1312
C4×#145235122
#225535132
#225535112
#225535092
#225535072
#225535052
#225533342
#225533152
#459×C1312
#3038131