Finite Group Databasebeta

← #31792273

Subgroups — #31792273

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×#17984115C211yesmaximal, characteristic, complemented
2C22×#14555495C211yesmaximal, characteristic, complemented
3#22578044C211yesmaximal, characteristic, complemented
4D4×C22⋊D137C211yesmaximal, characteristic, complemented
5#22577789C211yesmaximal, characteristic, complemented
6#22577655C211yesmaximal, characteristic, complemented
7#22577634C211yesmaximal, characteristic, complemented
8#22577620C211yesmaximal, characteristic, complemented
9#22577597C211yesmaximal, characteristic, complemented
10#22577547C211yesmaximal, characteristic, complemented
11#22577539C211yesmaximal, characteristic, complemented
12#22577538C211yesmaximal, characteristic, complemented
13#22577512C211yesmaximal, characteristic, complemented
14#22577471C211yesmaximal, characteristic, complemented
15#450×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic, complemented
17C22×D548C2211yescharacteristic, complemented
18C22×C22⋊D137—21no—
19C2×D4×D137C2211yescharacteristic, complemented
20#17984115—24no—
21C2×#14555497C2211yescharacteristic, complemented
22C2×D1096C2211yescharacteristic, complemented
23C2×D1096C2211yescharacteristic, complemented
24C2×#14555495—24no—
25C2×#14555495C2211yescharacteristic
26C2×#14555495C2211yescharacteristic
27C2×#14555495C2211yescharacteristic, complemented
28C2×#14555495C2211yescharacteristic, complemented
29C2×#14555493C2211yescharacteristic, complemented
30C22×#11764040C2211yescharacteristic, complemented
31#17984084C2211yescharacteristic, complemented
32#17984077C2211yescharacteristic, complemented
33#17984073—21no—
34#17984063—21no—
35#17984063C2211yescharacteristic, complemented
36#17984062C2211yescharacteristic
37#17984060—21no—
38#17984059C2211yescharacteristic, complemented
39#17984058—21no—
40#17984057—21no—
41#17984056—21no—
42D4×Dic137C2211yescharacteristic, complemented
43#17984050C2211yescharacteristic, complemented
44C4×C22⋊D137C2211yescharacteristic
45C22⋊C4×D137—21no—
46#17984019C2211yescharacteristic, complemented
47#17984017C2211yescharacteristic, complemented
48#17984012C2211yescharacteristic, complemented
49#17984010C2211yescharacteristic, complemented
50#17984010C2211yescharacteristic, complemented
51#17984008C2211yescharacteristic, complemented
52#17984000C2211yescharacteristic, complemented
53#17983998C2211yescharacteristic, complemented
54C22×D4×C137C2211yescharacteristic
55C2×SD16×C137C2211yescharacteristic, complemented
56C2×SD16×C137C2211yescharacteristic, complemented
57C22⋊Q8×C137C2211yescharacteristic, complemented
58#102×C137C2211yescharacteristic, complemented
59#102×C137C2211yescharacteristic, complemented
60C22⋊C8×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63#14555511—22no—
64#14555511—41no—
65C2×D548—22no—
66C2×D548—21no—
67C2×D548—21no—
68C2×D548—21no—
69C2×D548—41no—
70C2×D548C2311yescharacteristic
71C2×D548C2311yescharacteristic
72C2×D548C2311yescharacteristic
73C2×C22⋊D137—21no—
74C2×C22⋊D137—21no—
75C2×C22⋊D137—21no—
76C2×C22⋊D137—21no—
77C2×C22⋊D137—21no—
78C2×C22⋊D137—44no—
79C2×C22⋊D137D411yescharacteristic
80D4×D137—22no—
81D4×D137—41no—
82C22×Dic137—21no—
83C2×C4×D137—21no—
84C2×C4×D137—21no—
85C2×C4×D137C2311yescharacteristic
86#14555497—22no—
87D1096—22no—
88D1096—41no—
89#14555495—22no—
90#14555495—22no—
91#14555495—22no—
92#14555495D414yescomplemented
93#14555493—22no—
94C2×#11764040—22no—
95C2×#11764040C2311yescharacteristic
96C2×#11764040C2311yescharacteristic
97C4⋊Dic137C2311yescharacteristic
98#14555489D411yescharacteristic
99C22⋊Dic137—21no—
100C22⋊Dic137—21no—
101C22⋊Dic137D411yescharacteristic
102#14555487—21no—
103#14555487—21no—
104#14555487—21no—
105#14555487—21no—
106#14555487—21no—
107#14555487D411yescharacteristic
108C4×Dic137C2311yescharacteristic
109C24×C137—21no—
110C2×Q8×C137C2311yescharacteristic
111C2×D4×C137—22no—
112C2×D4×C137—21no—
113C2×D4×C137—41no—
114C2×D4×C137C2311yescharacteristic
115C2×D4×C137C2311yescharacteristic
116C22×C4×C137C2311yescharacteristic
117SD16×C137—22no—
118SD16×C137—41no—
119C2×C8×C137C2311yescharacteristic
120C2×C8×C137C2311yescharacteristic
121C4⋊C4×C137—21no—
122C4⋊C4×C137C2311yescharacteristic
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—81no—
132C22×D137—81no—
133C22×D137C2×D411yescharacteristic
134D548—21no—
135D548—21no—
136D548—21no—
137D548—21no—
138D548—21no—
139D548—42no—
140D548—42no—
141D548—81no—
142D548C2×D412yescomplemented
143C22⋊D137—22no—
144C22⋊D137—42no—
145C22⋊D137—42no—
146C22⋊D137—42no—
147C22⋊D137—42no—
148C22⋊D137—81no—
149C2×Dic137—21no—
150C2×Dic137—42no—
151C2×Dic137C2×D411yescharacteristic
152C2×Dic137C2×D411yescharacteristic
153C2×Dic137C2×D411yescharacteristic
154C4×D137—21no—
155C4×D137—21no—
156C4×D137—42no—
157C4×D137—81no—
158#11764040—21no—
159#11764040C2×D412yescomplemented
160C23×C137—21no—
161C23×C137—21no—
162C23×C137—42no—
163C23×C137—42no—
164C23×C137—81no—
165C23×C137C2×D411yescharacteristic
166Q8×C137—21no—
167Q8×C137—21no—
168D4×C137—21no—
169D4×C137—21no—
170D4×C137—21no—
171D4×C137—42no—
172D4×C137C2×D412yes—
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×C137—21no—
181C22×C137—21no—
182C22×C137—21no—
183C22×C137—22no—
184C22×C137—41no—
185C22×C137—41no—
186C22×C137—41no—
187C22×C137—41no—
188C22×C137—41no—
189C22×C137—84no—
190C22×C137AGL(1,ℤ/8)12yescomplemented
191C22×C137C22×D411yescharacteristic
192C2×D137—21no—
193C2×D137—22no—
194C2×D137—21no—
195C2×D137—21no—
196C2×D137—41no—
197C2×D137—41no—
198C2×D137—41no—
199C2×D137—41no—
200C2×D137—41no—
201C2×D137—41no—
202C2×D137—41no—
203C2×D137—41no—
204C2×D137—84no—
205C2×D137—84no—
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—22no—
216C2×C137—42no—
217C2×C137—81no—
218C2×C137D4211yescharacteristic
219C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C137#57611yescharacteristic
221D137—41no—
222D137—42no—
223D137—81no—
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#469—1371no—
233#465—1371no—
234#463—1371no—
235#459—1371no—
236#450D13711yescharacteristic, complemented
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×D4C2×D13711yescharacteristic
247AGL(1,ℤ/8)—2744no—
248C2×SD16—1371no—
249C2×SD16C2×D13711yescharacteristic, complemented
250C2×SD16C2×D13711yescharacteristic, complemented
251C2×D8—2744no—
252C2×D8—1371no—
253C2×D8—1371no—
254C2×D8—1371no—
255C2×D8—1371no—
256C2×D8—1371no—
257C2×D8—1371no—
258C2×M16—1371no—
259C22×C8—1371no—
260#127—1371no—
261#124—1371no—
262#123—2741no—
263C22⋊Q8C2×D13711yescharacteristic, complemented
264#121—2741no—
265#121—2741no—
266#121—1371no—
267#121—1371no—
268#121—1371no—
269#120—2741no—
270#120—2741no—
271#120—2741no—
272C4×D4—1371no—
273C4×D4—1371no—
274C4×D4—1371no—
275C2×C22⋊C4—2741no—
276#106—1371no—
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⋊C8C2×D13711yescharacteristic, complemented
286#97—1371no—
287C24—21no—
288C24—2741no—
289C24—2741no—
290#41—2742no—
291#41—5481no—
292C2×Q8C22×D13711yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—2742no—
297C2×D4—2741no—
298C2×D4—2741no—
299C2×D4—2742no—
300C2×D4—2741no—
301C2×D4—2741no—
302C2×D4—2741no—
303C2×D4—2741no—
304C2×D4—2741no—
305C2×D4—2741no—
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—22no—
321SD16—41no—
322SD16—2742no—
323D8—2742no—
324D8—2742no—
325D8—2742no—
326D8—2742no—
327D8—5481no—
328D8—1374no—
329M16—2742no—
330C2×C8—2742no—
331C2×C8—1371no—
332C2×C8—1371no—
333C2×C8C22×D13711yescharacteristic
334C2×C8C22×D13711yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—1371no—
337C4⋊C4—1371no—
338C4⋊C4C22×D13711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—2741no—
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—42no—
353C23—42no—
354C23—81no—
355C23—2741no—
356C23—2741no—
357C23—2741no—
358C23—5482no—
359C23—5482no—
360C23—5482no—
361C23—5482no—
362C23—10961no—
363C23—10961no—
364C23—1371no—
365C23D4×D13711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—2742no—
373D4—2741no—
374D4—2741no—
375D4—2741no—
376D4—2741no—
377D4—2741no—
378D4—5482no—
379D4—5482no—
380D4—5482no—
381D4—5482no—
382D4—5482no—
383D4—5482no—
384D4—10961no—
385D4—10961no—
386D4—1372no—
387D4D4×D13712yes—
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—21no—
405C8—2741no—
406C8—1372no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—84no—
416C22—2741no—
417C22—2741no—
418C22—2741no—
419C22—2742no—
420C22—5481no—
421C22—5481no—
422C22—5481no—
423C22—5481no—
424C22—5481no—
425C22—5481no—
426C22—5481no—
427C22—5481no—
428C22—10964no—
429C22—10964no—
430C22#1798411512yescomplemented
431C22C2×D4×D13711yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—2741no—
436C4—2741no—
437C4—2741no—
438C4—5481no—
439C4C2×D4×D13711yescharacteristic
440C4#1798413311yescharacteristic
441C2—22no—
442C2—42no—
443C2—81no—
444C2—5481no—
445C2—5482no—
446C2—10961no—
447C2—10961no—
448C2#2257803511yescharacteristic
449C2C2×#1798411511yescharacteristic
450C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#450×C137C2
socleC22×C137C22×D4
hypercenter#450D137
hypocenterC137#3264
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#17984115C2semidirect
2. C22×#14555495C2semidirect
3. #22578044C2semidirect
4. D4×C22⋊D137C2semidirect
5. #22577789C2semidirect
6. #22577655C2semidirect
7. #22577634C2semidirect
8. #22577620C2semidirect
9. #22577597C2semidirect
10. #22577547C2semidirect
11. #22577539C2semidirect
12. #22577538C2semidirect
13. #22577512C2semidirect
14. #22577471C2semidirect
15. #450×C137C2semidirect
16. C2×#14555511C22semidirect
17. C22×D548C22semidirect
19. C2×D4×D137C22semidirect
21. C2×#14555497C22semidirect
22. C2×D1096C22semidirect
23. C2×D1096C22semidirect
27. C2×#14555495C22semidirect
28. C2×#14555495C22semidirect
29. C2×#14555493C22semidirect
30. C22×#11764040C22semidirect
31. #17984084C22semidirect
32. #17984077C22semidirect
35. #17984063C22semidirect
38. #17984059C22semidirect
42. D4×Dic137C22semidirect
43. #17984050C22semidirect
46. #17984019C22semidirect
47. #17984017C22semidirect
48. #17984012C22semidirect
49. #17984010C22semidirect
50. #17984010C22semidirect
51. #17984008C22semidirect
52. #17984000C22semidirect
53. #17983998C22semidirect
55. C2×SD16×C137C22semidirect
56. C2×SD16×C137C22semidirect
57. C22⋊Q8×C137C22semidirect
58. #102×C137C22semidirect
59. #102×C137C22semidirect
60. C22⋊C8×C137C22semidirect
92. #14555495D4semidirect
142. D548C2×D4semidirect
159. #11764040C2×D4semidirect
190. C22×C137AGL(1,ℤ/8)semidirect
225. C137#3264semidirect
236. #450D137semidirect
249. C2×SD16C2×D137semidirect
250. C2×SD16C2×D137semidirect
263. C22⋊Q8C2×D137semidirect
282. #102C2×D137semidirect
283. #102C2×D137semidirect
285. C22⋊C8C2×D137semidirect
430. C22#17984115semidirect

Maximal subgroups

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

subgroupindex
C2×#179841152
C22×#145554952
#225780442
D4×C22⋊D1372
#225777892
#225776552
#225776342
#225776202
#225775972
#225775472
#225775392
#225775382
#225775122
#225774712
#450×C1372
#3264137