Finite Group Databasebeta

← #30675185

Subgroups — #30675185

Order 16768.

994 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C22×D4×D131C211yesmaximal, characteristic, complemented
2C2×#16862270C212yesmaximal, complemented
3#22553822C211yesmaximal, characteristic, complemented
4#22553465C211yesmaximal, characteristic, complemented
5#22553464C212yesmaximal, complemented
6#22553463C211yesmaximal, characteristic, complemented
7#22553445C211yesmaximal, characteristic, complemented
8#22553444C212yesmaximal, complemented
9#22553315C212yesmaximal, complemented
10#22553274C211yesmaximal, characteristic, complemented
11#447×C131C211yesmaximal, characteristic, complemented
12C24×D131—21no—
13C2×#14523512C2211yescharacteristic, complemented
14C22×D524C2211yescharacteristic
15C22×C22⋊D131—21no—
16C2×D4×D131—28no—
17C2×D4×D131—21no—
18C2×D4×D131—21no—
19C2×D4×D131C2212yes—
20C2×D4×D131C2212yes—
21C22×C4×D131C2211yescharacteristic
22#16862270—28no—
23C2×#14523502C2212yescomplemented
24C2×#14523501C2212yescomplemented
25C2×#14523498C2212yescomplemented
26C2×#14523496C2212yescomplemented
27#16862228—21no—
28#16862226C2211yescharacteristic, complemented
29#16862225C2211yescharacteristic, complemented
30#16862217C2211yescharacteristic, complemented
31#16862214—21no—
32D4×Dic131—21no—
33(C22⋊D131)⋊C4—21no—
34#16862204C2211yescharacteristic, complemented
35#16862168C2212yescomplemented
36#16862167C2212yescomplemented
37#16862166C2212yescomplemented
38C22⋊(C131⋊C8)C2211yescharacteristic, complemented
39(C2×D131)⋊C8C2212yescomplemented
40C22×D4×C131C2211yescharacteristic
41C2×D8×C131C2212yescomplemented
42#121×C131C2211yescharacteristic, complemented
43#102×C131C2212yescomplemented
44C22⋊C8×C131C2211yescharacteristic, complemented
45C23×D131—28no—
46C23×D131—22no—
47C23×D131—22no—
48C23×D131—21no—
49C23×D131—44no—
50C23×D131—44no—
51C23×D131D411yescharacteristic
52#14523512—24no—
53C2×Dic262C2311yescharacteristic, complemented
54C2×D524—24no—
55C2×D524—21no—
56C2×D524C2312yes—
57C2×C22⋊D131—28no—
58C2×C22⋊D131—21no—
59C2×C22⋊D131—21no—
60C2×C22⋊D131—22no—
61C2×C22⋊D131—22no—
62D4×D131—28no—
63D4×D131—28no—
64D4×D131—24no—
65D4×D131—24no—
66D4×D131—24no—
67D4×D131D418yescomplemented
68C22×Dic131—21no—
69C22×Dic131D411yescharacteristic
70C2×C4×D131—24no—
71C2×C4×D131D411yescharacteristic
72C2×C4×D131D411yescharacteristic
73C2×C4×D131C2311yescharacteristic
74C2×C4×D131C2311yescharacteristic
75#14523502—24no—
76#14523501—24no—
77#14523498—24no—
78#14523496—24no—
79C2×C131⋊C8C2312yescomplemented
80C4⋊Dic131—21no—
81C4⋊Dic131C2311yescharacteristic, complemented
82Dic131⋊C4—21no—
83C22⋊Dic131—21no—
84C22⋊Dic131—21no—
85(C2×D131)⋊C4—21no—
86(C2×D131)⋊C4—21no—
87C4×Dic131—21no—
88C24×C131—21no—
89C2×D4×C131—24no—
90C2×D4×C131—21no—
91C2×D4×C131—21no—
92C2×D4×C131C2312yes—
93C2×D4×C131C2311yescharacteristic, complemented
94C22×C4×C131C2311yescharacteristic
95D8×C131—24no—
96C2×C8×C131C2312yescomplemented
97C4⋊C4×C131C2311yescharacteristic, complemented
98C22⋊C4×C131—21no—
99C22×D131—22no—
100C22×D131—22no—
101C22×D131—22no—
102C22×D131—28no—
103C22×D131—24no—
104C22×D131—22no—
105C22×D131—28no—
106C22×D131—432no—
107C22×D131—44no—
108C22×D131—44no—
109C22×D131—44no—
110C22×D131—42no—
111C22×D131—42no—
112C22×D131—44no—
113C22×D131C2×D411yescharacteristic
114C22×D131C2×D411yescharacteristic
115Dic262—22no—
116D524—24no—
117D524—22no—
118D524C2×D414yes—
119C22⋊D131—28no—
120C22⋊D131—28no—
121C22⋊D131—42no—
122C22⋊D131—42no—
123C2×Dic131—22no—
124C2×Dic131—21no—
125C2×Dic131—21no—
126C2×Dic131—42no—
127C2×Dic131C2×D411yescharacteristic
128C2×Dic131C2×D411yescharacteristic
129C4×D131—22no—
130C4×D131—22no—
131C4×D131—22no—
132C4×D131C2×D414yes—
133C131⋊C8—22no—
134C23×C131—24no—
135C23×C131—22no—
136C23×C131—21no—
137C23×C131—42no—
138C23×C131—42no—
139C23×C131C2×D411yescharacteristic
140D4×C131—24no—
141D4×C131—22no—
142D4×C131—22no—
143D4×C131—42no—
144D4×C131C2×D414yes—
145C2×C4×C131—22no—
146C2×C4×C131—21no—
147C2×C4×C131C2×D411yescharacteristic
148C2×C4×C131C2411yescharacteristic
149C8×C131—22no—
150C22×C131—21no—
151C22×C131—21no—
152C22×C131—24no—
153C22×C131—48no—
154C22×C131—42no—
155C22×C131—42no—
156C22×C131—41no—
157C22×C131—41no—
158C22×C131—41no—
159C22×C131AGL(1,ℤ/8)12yescomplemented
160C22×C131C22×D411yescharacteristic
161C2×D131—28no—
162C2×D131—21no—
163C2×D131—21no—
164C2×D131—21no—
165C2×D131—21no—
166C2×D131—24no—
167C2×D131—416no—
168C2×D131—416no—
169C2×D131—48no—
170C2×D131—42no—
171C2×D131—42no—
172C2×D131#12012yes—
173C2×D131AGL(1,ℤ/8)12yescomplemented
174C4×C131—21no—
175C4×C131—41no—
176C4×C131C22×D412yes—
177Dic131—21no—
178Dic131—41no—
179Dic131—41no—
180Dic131#12012yes—
181C2×C131—22no—
182C2×C131—44no—
183C2×C131—81no—
184C2×C131C2×#12011yescharacteristic
185C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
186C2×C131C2×AGL(1,ℤ/8)11yescharacteristic
187D131—22no—
188D131—22no—
189D131—44no—
190C131#300311yescharacteristic, complemented, Sylow
191#3003—1311nomaximal, Sylow
192C23×D4—1311no—
193C2×AGL(1,ℤ/8)—1312no—
194#525—1311no—
195#450—1311no—
196#450—1312no—
197#450—1311no—
198#447—1312no—
199#447—1311no—
200#447D13111yescharacteristic, complemented
201#418—1312no—
202#407—1311no—
203C25—2621no—
204C2×#41—1311no—
205C22×D4—2628no—
206C22×D4—2621no—
207C22×D4—2621no—
208C22×D4—2621no—
209C22×D4—1312no—
210C22×D4—1312no—
211C22×D4—1311no—
212C22×D4C2×D13111yescharacteristic
213C23×C4—1311no—
214AGL(1,ℤ/8)—2628no—
215C2×SD16—1312no—
216C2×SD16—1312no—
217C2×D8—1312no—
218C2×D8C2×D13112yescomplemented
219C2×M16—1312no—
220#123—2621no—
221C22⋊Q8—1311no—
222C22⋊Q8—1311no—
223#121—1311no—
224#121C2×D13111yescharacteristic, complemented
225#120—2621no—
226C4×D4—2621no—
227C4×D4—2621no—
228#117—1311no—
229#102—1312no—
230#102—1312no—
231#102—1312no—
232#102C2×D13112yescomplemented
233C22⋊C8—1311no—
234C22⋊C8—1312no—
235C22⋊C8C2×D13111yescharacteristic, complemented
236C24—21no—
237C24—2628no—
238C24—2622no—
239C24—2622no—
240C24—2621no—
241C24—5244no—
242C24—5244no—
243C24—1311no—
244#41—2624no—
245C2×Q8—1311no—
246C2×D4—24no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—2624no—
250C2×D4—2628no—
251C2×D4—2628no—
252C2×D4—2624no—
253C2×D4—2628no—
254C2×D4—2624no—
255C2×D4—2621no—
256C2×D4—2621no—
257C2×D4—2624no—
258C2×D4—2622no—
259C2×D4—2622no—
260C2×D4—2621no—
261C2×D4—1318no—
262C2×D4—1312no—
263C2×D4C22×D13112yes—
264C2×D4C22×D13111yescharacteristic, complemented
265C22×C4—2624no—
266C22×C4—2621no—
267C22×C4—1311no—
268C22×C4—1311no—
269C22×C4—1311no—
270C22×C4—1311no—
271C22×C4—1311no—
272C22×C4C22×D13111yescharacteristic
273SD16—2624no—
274SD16—2624no—
275D8—24no—
276D8—2624no—
277M16—2624no—
278C2×C8—1312no—
279C2×C8C22×D13112yescomplemented
280C4⋊C4—2621no—
281C4⋊C4—2621no—
282C4⋊C4—1311no—
283C4⋊C4C22×D13111yescharacteristic, complemented
284C22⋊C4—21no—
285C22⋊C4—2621no—
286C22⋊C4—2621no—
287C22⋊C4—2621no—
288C22⋊C4—2621no—
289C42—2621no—
290C23—24no—
291C23—22no—
292C23—21no—
293C23—42no—
294C23—42no—
295C23—2622no—
296C23—2622no—
297C23—2624no—
298C23—2628no—
299C23—2628no—
300C23—2622no—
301C23—2622no—
302C23—5242no—
303C23—52432no—
304C23—5244no—
305C23—5244no—
306C23—5244no—
307C23—5244no—
308C23—5242no—
309C23—1311no—
310C23—1311no—
311C23D4×D13111yescharacteristic
312Q8—2622no—
313D4—24no—
314D4—22no—
315D4—22no—
316D4—42no—
317D4—2624no—
318D4—2628no—
319D4—2628no—
320D4—2622no—
321D4—5242no—
322D4—5242no—
323D4—1314no—
324D4D4×D13114yes—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—2622no—
328C2×C4—2622no—
329C2×C4—2622no—
330C2×C4—2622no—
331C2×C4—2621no—
332C2×C4—2621no—
333C2×C4—5242no—
334C2×C4—1311no—
335C2×C4—1311no—
336C2×C4—1314no—
337C2×C4D4×D13111yescharacteristic
338C2×C4C23×D13111yescharacteristic
339C8—22no—
340C8—2622no—
341C22—21no—
342C22—21no—
343C22—24no—
344C22—48no—
345C22—42no—
346C22—42no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—2628no—
351C22—2621no—
352C22—2621no—
353C22—2621no—
354C22—2621no—
355C22—2624no—
356C22—5248no—
357C22—52416no—
358C22—52416no—
359C22—5242no—
360C22—5242no—
361C22—1312no—
362C22—1312no—
363C22#1686227012yescomplemented
364C22C2×D4×D13111yescharacteristic
365C4—21no—
366C4—41no—
367C4—2621no—
368C4—5241no—
369C4—5241no—
370C4—1312no—
371C4C2×D4×D13112yes—
372C2—22no—
373C2—44no—
374C2—2622no—
375C2—2622no—
376C2—81no—
377C2—5244no—
378C2#120×D13111yescharacteristic
379C2AGL(1,ℤ/8)×D13111yescharacteristic
380C2C2×#1686227011yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#447×C131C2
socleC22×C131C22×D4
hypercenter#447D131
hypocenterC131#3003
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. C22×D4×D131C2semidirect
2. C2×#16862270C2semidirect
3. #22553822C2semidirect
4. #22553465C2semidirect
5. #22553464C2semidirect
6. #22553463C2semidirect
7. #22553445C2semidirect
8. #22553444C2semidirect
9. #22553315C2semidirect
10. #22553274C2semidirect
11. #447×C131C2semidirect
13. C2×#14523512C22semidirect
23. C2×#14523502C22semidirect
24. C2×#14523501C22semidirect
25. C2×#14523498C22semidirect
26. C2×#14523496C22semidirect
28. #16862226C22semidirect
29. #16862225C22semidirect
30. #16862217C22semidirect
34. #16862204C22semidirect
35. #16862168C22semidirect
36. #16862167C22semidirect
37. #16862166C22semidirect
38. C22⋊(C131⋊C8)C22semidirect
39. (C2×D131)⋊C8C22semidirect
41. C2×D8×C131C22semidirect
42. #121×C131C22semidirect
43. #102×C131C22semidirect
44. C22⋊C8×C131C22semidirect
53. C2×Dic262C23semidirect
67. D4×D131D4semidirect
79. C2×C131⋊C8C23semidirect
81. C4⋊Dic131C23semidirect
93. C2×D4×C131C23semidirect
96. C2×C8×C131C23semidirect
97. C4⋊C4×C131C23semidirect
159. C22×C131AGL(1,ℤ/8)semidirect
173. C2×D131AGL(1,ℤ/8)semidirect
190. C131#3003semidirect
200. #447D131semidirect
218. C2×D8C2×D131semidirect
224. #121C2×D131semidirect
232. #102C2×D131semidirect
235. C22⋊C8C2×D131semidirect
264. C2×D4C22×D131semidirect
279. C2×C8C22×D131semidirect
283. C4⋊C4C22×D131semidirect
363. C22#16862270semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1312
C2×#168622702
#225538222
#225534652
#225534642
#225534632
#225534452
#225534442
#225533152
#225532742
#447×C1312
#3003131