Finite Group Databasebeta

← #31792276

Subgroups — #31792276

Order 17536.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984116C211yesmaximal, characteristic, complemented
2C2×D8×D137C211yesmaximal, characteristic, complemented
3#22578050C211yesmaximal, characteristic, complemented
4#22578037C211yesmaximal, characteristic, complemented
5#22577791C211yesmaximal, characteristic, complemented
6#22577633C211yesmaximal, characteristic, complemented
7#22577617C211yesmaximal, characteristic, complemented
8#22577597C211yesmaximal, characteristic, complemented
9#22577555C211yesmaximal, characteristic, complemented
10#22577546C211yesmaximal, characteristic, complemented
11#22577538C211yesmaximal, characteristic, complemented
12#22577536C211yesmaximal, characteristic, complemented
13#22577514C211yesmaximal, characteristic, complemented
14#22577472C211yesmaximal, characteristic, complemented
15#469×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic, complemented
17C22×D548C2211yescharacteristic, complemented
18C2×D4×D137—21no—
19C2×D4×D137C2211yescharacteristic
20C2×D4×D137C2211yescharacteristic, complemented
21#17984116—24no—
22C2×#14555499C2211yescharacteristic, complemented
23C2×#14555499C2211yescharacteristic, complemented
24C2×#14555498C2211yescharacteristic, complemented
25C2×D1096C2211yescharacteristic
26C2×D1096C2211yescharacteristic, complemented
27C2×#14555495C2211yescharacteristic
28C2×#14555495C2211yescharacteristic, complemented
29C2×#14555495C2211yescharacteristic, complemented
30D8×D137—24no—
31C2×C8×D137C2211yescharacteristic, complemented
32#17984084C2211yescharacteristic, complemented
33#17984075C2211yescharacteristic, complemented
34#17984070—21no—
35#17984068C2211yescharacteristic, complemented
36#17984063C2211yescharacteristic
37#17984061—21no—
38#17984059—21no—
39#17984058—21no—
40#17984056—21no—
41#17984056—21no—
42#17984050C2211yescharacteristic, complemented
43#17984050C2211yescharacteristic, complemented
44C4×C22⋊D137C2211yescharacteristic
45C22⋊C4×D137—21no—
46#17984017C2211yescharacteristic, complemented
47#17984011C2211yescharacteristic, complemented
48#17984008C2211yescharacteristic, complemented
49#17984008C2211yescharacteristic, complemented
50#17984008C2211yescharacteristic, complemented
51C22⋊#11764040C2211yescharacteristic, complemented
52#17984000C2211yescharacteristic, complemented
53#17983998C2211yescharacteristic, complemented
54C2×D8×C137C2211yescharacteristic, complemented
55C2×M16×C137C2211yescharacteristic, complemented
56#121×C137C2211yescharacteristic, complemented
57#121×C137C2211yescharacteristic, complemented
58#106×C137C2211yescharacteristic, complemented
59#102×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63C23×D137—21no—
64#14555510—22no—
65#14555510—41no—
66C2×Dic274C2311yescharacteristic
67C2×D548—22no—
68C2×D548—21no—
69C2×D548—21no—
70C2×D548—41no—
71C2×D548C2311yescharacteristic
72C2×D548C2311yescharacteristic
73C2×D548C2311yescharacteristic
74C2×C22⋊D137—21no—
75C2×C22⋊D137—21no—
76C2×C22⋊D137—21no—
77C2×C22⋊D137—21no—
78C2×C22⋊D137—21no—
79C2×C22⋊D137D411yescharacteristic
80D4×D137—22no—
81D4×D137—22no—
82D4×D137—41no—
83D4×D137—41no—
84D4×D137—44no—
85C2×C4×D137—21no—
86C2×C4×D137—21no—
87C2×C4×D137—21no—
88C2×C4×D137C2311yescharacteristic
89#14555499—22no—
90#14555499—22no—
91#14555498—41no—
92D1096—22no—
93D1096D414yescomplemented
94#14555495—22no—
95#14555495—22no—
96#14555495—41no—
97C2×#11764040C2311yescharacteristic
98C2×#11764040C2311yescharacteristic
99C8×D137—22no—
100C4⋊Dic137—21no—
101#14555489D411yescharacteristic
102C22⋊Dic137—21no—
103C22⋊Dic137—21no—
104C22⋊Dic137D411yescharacteristic
105#14555487—21no—
106#14555487—21no—
107#14555487—21no—
108#14555487—21no—
109#14555487D411yescharacteristic
110C4×Dic137C2311yescharacteristic
111C2×D4×C137—21no—
112C2×D4×C137—21no—
113C2×D4×C137C2311yescharacteristic
114C2×D4×C137C2311yescharacteristic
115C22×C4×C137C2311yescharacteristic
116D8×C137—22no—
117M16×C137—22no—
118C2×C8×C137C2311yescharacteristic
119C2×C8×C137C2311yescharacteristic
120C4⋊C4×C137C2311yescharacteristic
121C4⋊C4×C137C2311yescharacteristic
122C22⋊C4×C137—21no—
123C22⋊C4×C137—21no—
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—21no—
127C22×D137—42no—
128C22×D137—42no—
129C22×D137—42no—
130C22×D137—42no—
131C22×D137—42no—
132C22×D137—42no—
133C22×D137—81no—
134C22×D137—81no—
135C22×D137—81no—
136C22×D137C2×D411yescharacteristic
137Dic274—21no—
138Dic274—21no—
139D548—21no—
140D548—21no—
141D548—21no—
142D548—21no—
143D548—42no—
144D548—42no—
145D548C2×D412yes—
146D548C2×D412yescomplemented
147C22⋊D137—22no—
148C22⋊D137—42no—
149C22⋊D137—42no—
150C22⋊D137—42no—
151C22⋊D137—42no—
152C22⋊D137—81no—
153C2×Dic137—21no—
154C2×Dic137C2×D411yescharacteristic
155C2×Dic137C2×D411yescharacteristic
156C2×Dic137C2×D411yescharacteristic
157C4×D137—21no—
158C4×D137—21no—
159C4×D137—42no—
160C4×D137—42no—
161C4×D137—81no—
162#11764040—21no—
163#11764040—21no—
164C23×C137—21no—
165C23×C137—21no—
166C23×C137C2×D411yescharacteristic
167D4×C137—21no—
168D4×C137—21no—
169D4×C137—21no—
170D4×C137—21no—
171D4×C137—42no—
172D4×C137—81no—
173C2×C4×C137—21no—
174C2×C4×C137—21no—
175C2×C4×C137—21no—
176C2×C4×C137—21no—
177C2×C4×C137C2×D411yescharacteristic
178C2×C4×C137C2411yescharacteristic
179C8×C137—21no—
180C8×C137C2×D412yescomplemented
181C22×C137—21no—
182C22×C137—21no—
183C22×C137—21no—
184C22×C137—41no—
185C22×C137—41no—
186C22×C137—41no—
187C22×C137—41no—
188C22×C137—41no—
189C22×C137—41no—
190C22×C137C22×D411yescharacteristic
191C2×D137—21no—
192C2×D137—21no—
193C2×D137—22no—
194C2×D137—22no—
195C2×D137—41no—
196C2×D137—41no—
197C2×D137—41no—
198C2×D137—41no—
199C2×D137—41no—
200C2×D137—41no—
201C2×D137—41no—
202C2×D137—84no—
203C2×D137—84no—
204C2×D137—84no—
205C2×D137AGL(1,ℤ/8)12yescomplemented
206C4×C137—21no—
207C4×C137—41no—
208C4×C137—41no—
209C4×C137C22×D411yescharacteristic
210C4×C137H(5,𝔽2)11yescharacteristic
211Dic137—21no—
212Dic137—21no—
213Dic137—21no—
214Dic137—41no—
215C2×C137—41no—
216C2×C137—81no—
217C2×C137—81no—
218C2×C137D4211yescharacteristic
219C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C137#57611yescharacteristic
221D137—22no—
222D137—42no—
223D137—42no—
224D137—81no—
225C137#326411yescharacteristic, complemented, Sylow
226#3264—1371nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1371no—
228C22×D8—1371no—
229#546—1371no—
230D42—1371no—
231#496—1371no—
232#469D13711yescharacteristic, complemented
233#465—1371no—
234#463—1371no—
235#459—1371no—
236#450—1371no—
237#449—1371no—
238#447—1371no—
239#447—1371no—
240#442—1371no—
241#435—1371no—
242C2×#41—1371no—
243C22×D4—2741no—
244C22×D4—1371no—
245C22×D4—1371no—
246C22×D4—1371no—
247AGL(1,ℤ/8)—2744no—
248C2×SD16—1371no—
249C2×SD16—1371no—
250C2×SD16—1371no—
251C2×D8—2744no—
252C2×D8—1371no—
253C2×D8—1371no—
254C2×D8—1371no—
255C2×D8—1371no—
256C2×D8—1371no—
257C2×D8C2×D13711yescharacteristic, complemented
258C2×M16C2×D13711yescharacteristic, complemented
259C22×C8—1371no—
260#127—1371no—
261#124—1371no—
262#123—2741no—
263C22⋊Q8—1371no—
264#121—2741no—
265#121—2741no—
266#121—1371no—
267#121C2×D13711yescharacteristic, complemented
268#121C2×D13711yescharacteristic, complemented
269#120—2741no—
270#120—2741no—
271#120—2741no—
272C4×D4—1371no—
273C4×D4—1371no—
274C4×D4—1371no—
275C2×C22⋊C4—2741no—
276#106C2×D13711yescharacteristic, complemented
277C4⋊C8—1371no—
278#103—1371no—
279#102—1371no—
280#102—1371no—
281#102—1371no—
282#102C2×D13711yescharacteristic, complemented
283#102C2×D13711yescharacteristic, complemented
284C22⋊C8—1371no—
285C22⋊C8—1371no—
286#97—1371no—
287C24—2741no—
288C24—2741no—
289C24—2741no—
290#41—2742no—
291#41—5481no—
292C2×Q8—1371no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—2742no—
296C2×D4—2741no—
297C2×D4—2742no—
298C2×D4—2742no—
299C2×D4—2741no—
300C2×D4—2741no—
301C2×D4—2741no—
302C2×D4—2741no—
303C2×D4—2741no—
304C2×D4—2741no—
305C2×D4—5481no—
306C2×D4—5481no—
307C2×D4—5481no—
308C2×D4—5484no—
309C2×D4—1371no—
310C2×D4—1371no—
311C2×D4—1371no—
312C2×D4—1371no—
313C2×D4C22×D13711yescharacteristic
314C2×D4C22×D13711yescharacteristic
315C22×C4—2741no—
316C22×C4—2741no—
317C22×C4—2741no—
318C22×C4—1371no—
319C22×C4C22×D13711yescharacteristic
320SD16—2742no—
321SD16—2742no—
322SD16—5481no—
323D8—22no—
324D8—2742no—
325D8—2742no—
326D8—2742no—
327D8—5481no—
328D8—1374no—
329M16—22no—
330C2×C8—2742no—
331C2×C8—1371no—
332C2×C8—1371no—
333C2×C8C22×D13711yescharacteristic
334C2×C8C22×D13711yescharacteristic
335C4⋊C4—2741no—
336C4⋊C4—1371no—
337C4⋊C4C22×D13711yescharacteristic
338C4⋊C4C22×D13711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—2741no—
342C22⋊C4—2741no—
343C22⋊C4—2741no—
344C22⋊C4—2741no—
345C22⋊C4—2741no—
346C22⋊C4—2741no—
347C22⋊C4—1371no—
348C22⋊C4—1371no—
349C42—1371no—
350C23—21no—
351C23—21no—
352C23—2741no—
353C23—2741no—
354C23—2741no—
355C23—5482no—
356C23—5482no—
357C23—5482no—
358C23—5482no—
359C23—5482no—
360C23—5482no—
361C23—10961no—
362C23—10961no—
363C23—10961no—
364C23—1371no—
365C23D4×D13711yescharacteristic
366Q8—2741no—
367Q8—2741no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—81no—
374D4—2742no—
375D4—2741no—
376D4—2741no—
377D4—2741no—
378D4—2741no—
379D4—5482no—
380D4—5482no—
381D4—5482no—
382D4—5482no—
383D4—5482no—
384D4—5482no—
385D4—10961no—
386D4—1372no—
387D4—1372no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—2741no—
393C2×C4—2741no—
394C2×C4—2741no—
395C2×C4—5482no—
396C2×C4—5482no—
397C2×C4—10961no—
398C2×C4—1371no—
399C2×C4—1371no—
400C2×C4—1371no—
401C2×C4D4×D13711yescharacteristic
402C2×C4C23×D13711yescharacteristic
403C8—21no—
404C8—2741no—
405C8—2741no—
406C8D4×D13712yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—2741no—
417C22—2741no—
418C22—2742no—
419C22—2742no—
420C22—5481no—
421C22—5481no—
422C22—5481no—
423C22—5481no—
424C22—5481no—
425C22—5481no—
426C22—5481no—
427C22—10964no—
428C22—10964no—
429C22—10964no—
430C22—1372no—
431C22C2×D4×D13711yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—2741no—
436C4—2741no—
437C4—2741no—
438C4—5481no—
439C4C2×D4×D13711yescharacteristic
440C4#1798413411yescharacteristic
441C2—41no—
442C2—81no—
443C2—81no—
444C2—2742no—
445C2—5482no—
446C2—5482no—
447C2—10961no—
448C2#2257803711yescharacteristic
449C2AGL(1,ℤ/8)×D13711yescharacteristic
450C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#469×C137C2
socleC22×C137C22×D4
hypercenter#469D137
hypocenterC137#3264
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#17984116C2semidirect
2. C2×D8×D137C2semidirect
3. #22578050C2semidirect
4. #22578037C2semidirect
5. #22577791C2semidirect
6. #22577633C2semidirect
7. #22577617C2semidirect
8. #22577597C2semidirect
9. #22577555C2semidirect
10. #22577546C2semidirect
11. #22577538C2semidirect
12. #22577536C2semidirect
13. #22577514C2semidirect
14. #22577472C2semidirect
15. #469×C137C2semidirect
16. C2×#14555510C22semidirect
17. C22×D548C22semidirect
20. C2×D4×D137C22semidirect
22. C2×#14555499C22semidirect
23. C2×#14555499C22semidirect
24. C2×#14555498C22semidirect
26. C2×D1096C22semidirect
28. C2×#14555495C22semidirect
29. C2×#14555495C22semidirect
31. C2×C8×D137C22semidirect
32. #17984084C22semidirect
33. #17984075C22semidirect
35. #17984068C22semidirect
42. #17984050C22semidirect
43. #17984050C22semidirect
46. #17984017C22semidirect
47. #17984011C22semidirect
48. #17984008C22semidirect
49. #17984008C22semidirect
50. #17984008C22semidirect
51. C22⋊#11764040C22semidirect
52. #17984000C22semidirect
53. #17983998C22semidirect
54. C2×D8×C137C22semidirect
55. C2×M16×C137C22semidirect
56. #121×C137C22semidirect
57. #121×C137C22semidirect
58. #106×C137C22semidirect
59. #102×C137C22semidirect
60. #102×C137C22semidirect
93. D1096D4semidirect
146. D548C2×D4semidirect
180. C8×C137C2×D4semidirect
205. C2×D137AGL(1,ℤ/8)semidirect
225. C137#3264semidirect
232. #469D137semidirect
257. C2×D8C2×D137semidirect
258. C2×M16C2×D137semidirect
267. #121C2×D137semidirect
268. #121C2×D137semidirect
276. #106C2×D137semidirect
282. #102C2×D137semidirect
283. #102C2×D137semidirect
406. C8D4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841162
C2×D8×D1372
#225780502
#225780372
#225777912
#225776332
#225776172
#225775972
#225775552
#225775462
#225775382
#225775362
#225775142
#225774722
#469×C1372
#3264137