Finite Group Databasebeta

← #31792271

Subgroups — #31792271

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×#17984112C211yesmaximal, characteristic, complemented
2C2×D8×D137C211yesmaximal, characteristic, complemented
3#22578048C211yesmaximal, characteristic, complemented
4D4×D548C211yesmaximal, characteristic, complemented
5#22577792C211yesmaximal, characteristic, complemented
6#22577657C211yesmaximal, characteristic, complemented
7#22577633C211yesmaximal, characteristic, complemented
8#22577618C211yesmaximal, characteristic, complemented
9#22577598C211yesmaximal, characteristic, complemented
10#22577555C211yesmaximal, characteristic, complemented
11#22577549C211yesmaximal, characteristic, complemented
12#22577538C211yesmaximal, characteristic, complemented
13#22577538C211yesmaximal, characteristic, complemented
14#22577476C211yesmaximal, characteristic, complemented
15#442×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic, complemented
17C22×D548—21no—
18C2×D4×D137C2211yescharacteristic
19C2×D4×D137C2211yescharacteristic, complemented
20C2×D4×D137C2211yescharacteristic, complemented
21#17984112—24no—
22C2×#14555499C2211yescharacteristic, complemented
23C2×#14555499C2211yescharacteristic, complemented
24C2×#14555498C2211yescharacteristic, complemented
25C2×D1096C2211yescharacteristic, complemented
26C2×D1096C2211yescharacteristic, complemented
27C2×#14555495C2211yescharacteristic
28C2×#14555495C2211yescharacteristic, complemented
29C2×#14555495C2211yescharacteristic, complemented
30D8×D137—24no—
31C2×#14555493C2211yescharacteristic, complemented
32C2×C8×D137C2211yescharacteristic, complemented
33#17984085C2211yescharacteristic, complemented
34#17984076C2211yescharacteristic, complemented
35#17984072—21no—
36#17984068C2211yescharacteristic, complemented
37#17984063C2211yescharacteristic
38#17984063C2211yescharacteristic, complemented
39#17984063C2211yescharacteristic, complemented
40#17984062—21no—
41#17984062—21no—
42#17984058—21no—
43#17984058—21no—
44#17984058—21no—
45C4×D548C2211yescharacteristic
46C2×#14555487—21no—
47C4⋊#11764040C2211yescharacteristic, complemented
48#17984011C2211yescharacteristic, complemented
49#17984008C2211yescharacteristic, complemented
50#17984008C2211yescharacteristic, complemented
51#17984008C2211yescharacteristic, complemented
52#17984000C2211yescharacteristic, complemented
53#17984000C2211yescharacteristic, complemented
54C2×D8×C137C2211yescharacteristic
55C4×D4×C137C2211yescharacteristic, complemented
56C4×D4×C137C2211yescharacteristic, complemented
57#106×C137C2211yescharacteristic, complemented
58#102×C137C2211yescharacteristic, complemented
59#102×C137C2211yescharacteristic, complemented
60#97×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63C23×D137—21no—
64#14555509—22no—
65#14555509—41no—
66C2×Dic274C2311yescharacteristic
67C2×D548—21no—
68C2×D548—21no—
69C2×D548—21no—
70C2×D548—21no—
71C2×D548—21no—
72C2×D548—44no—
73C2×D548D411yescharacteristic
74C2×D548C2311yescharacteristic
75C2×D548C2311yescharacteristic
76C2×D548C2311yescharacteristic
77C2×C22⋊D137—21no—
78C2×C22⋊D137—21no—
79C2×C22⋊D137—21no—
80C2×C22⋊D137—21no—
81D4×D137—22no—
82D4×D137—22no—
83D4×D137—22no—
84D4×D137—41no—
85D4×D137—41no—
86D4×D137—41no—
87C22×Dic137—21no—
88C2×C4×D137C2311yescharacteristic
89C2×C4×D137C2311yescharacteristic
90#14555499—22no—
91#14555499—41no—
92#14555498—22no—
93D1096—22no—
94D1096—41no—
95#14555495—22no—
96#14555495—22no—
97#14555495—22no—
98#14555493—22no—
99C2×#11764040C2311yescharacteristic
100C2×#11764040C2311yescharacteristic
101C8×D137—22no—
102C4⋊Dic137—21no—
103C4⋊Dic137D411yescharacteristic
104#14555487—21no—
105#14555487—21no—
106#14555487—21no—
107#14555487—21no—
108#14555487—21no—
109#14555487—21no—
110#14555487D411yescharacteristic
111#14555487D411yescharacteristic
112C2×D4×C137C2311yescharacteristic
113C2×D4×C137C2311yescharacteristic
114C22×C4×C137—21no—
115C22×C4×C137—21no—
116D8×C137D414yescomplemented
117C2×C8×C137C2311yescharacteristic
118C2×C8×C137C2311yescharacteristic
119C4⋊C4×C137C2311yescharacteristic
120C4⋊C4×C137C2311yescharacteristic
121C22⋊C4×C137—21no—
122C22⋊C4×C137—21no—
123C42×C137C2311yescharacteristic
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
137C22×D137C2×D411yescharacteristic
138Dic274—21no—
139Dic274—21no—
140D548—22no—
141D548—21no—
142D548—21no—
143D548—21no—
144D548—21no—
145D548—21no—
146D548—21no—
147D548—42no—
148D548—42no—
149D548—42no—
150D548—81no—
151D548—81no—
152C22⋊D137—42no—
153C22⋊D137—42no—
154C22⋊D137—42no—
155C22⋊D137—42no—
156C2×Dic137—21no—
157C2×Dic137—42no—
158C2×Dic137C2×D411yescharacteristic
159C2×Dic137C2×D411yescharacteristic
160C4×D137—21no—
161C4×D137—21no—
162C4×D137—21no—
163C4×D137—21no—
164#11764040—21no—
165#11764040—21no—
166C23×C137—21no—
167C23×C137—21no—
168D4×C137—21no—
169D4×C137—21no—
170D4×C137C2×D412yes—
171D4×C137C2×D412yescomplemented
172C2×C4×C137—21no—
173C2×C4×C137—21no—
174C2×C4×C137—42no—
175C2×C4×C137—81no—
176C2×C4×C137C2×D411yescharacteristic
177C2×C4×C137C2×D411yescharacteristic
178C2×C4×C137C2411yescharacteristic
179C8×C137—21no—
180C8×C137C2×D412yescomplemented
181C22×C137—22no—
182C22×C137—22no—
183C22×C137—41no—
184C22×C137—41no—
185C22×C137—41no—
186C22×C137—41no—
187C22×C137C22×D411yescharacteristic
188C2×D137—21no—
189C2×D137—21no—
190C2×D137—21no—
191C2×D137—21no—
192C2×D137—21no—
193C2×D137—41no—
194C2×D137—41no—
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—21no—
208C4×C137—41no—
209C4×C137—41no—
210C4×C137C22×D411yescharacteristic
211C4×C137H(5,𝔽2)11yescharacteristic
212Dic137—21no—
213Dic137—21no—
214Dic137—41no—
215C2×C137—42no—
216C2×C137—42no—
217C2×C137D4211yescharacteristic
218C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C137#57611yescharacteristic
220D137—22no—
221D137—41no—
222D137—81no—
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#450—1371no—
237#449—1371no—
238#447—1371no—
239#447—1371no—
240#442D13711yescharacteristic, complemented
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
258C2×M16—1371no—
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#121—1371no—
268#121—1371no—
269#120—2741no—
270#120—2741no—
271#120—2741no—
272C4×D4—1371no—
273C4×D4C2×D13711yescharacteristic, complemented
274C4×D4C2×D13711yescharacteristic, complemented
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#97C2×D13711yescharacteristic, complemented
287C24—2741no—
288C24—2741no—
289C24—2741no—
290#41—2742no—
291#41—5481no—
292C2×Q8—1371no—
293C2×D4—2741no—
294C2×D4—2741no—
295C2×D4—2741no—
296C2×D4—2742no—
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—5484no—
306C2×D4—5481no—
307C2×D4—5481no—
308C2×D4—5481no—
309C2×D4—1371no—
310C2×D4—1371no—
311C2×D4—1371no—
312C2×D4—1371no—
313C2×D4C22×D13711yescharacteristic
314C2×D4C22×D13711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—2741no—
318C22×C4—1371no—
319C22×C4—1371no—
320SD16—2742no—
321SD16—2742no—
322SD16—5481no—
323D8—2742no—
324D8—2742no—
325D8—2742no—
326D8—2742no—
327D8—5481no—
328D8D54814yescomplemented
329M16—2742no—
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—
349C42C22×D13711yescharacteristic
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—
365C23—1371no—
366Q8—2741no—
367Q8—2741no—
368D4—21no—
369D4—21no—
370D4—2742no—
371D4—2741no—
372D4—2741no—
373D4—2741no—
374D4—2741no—
375D4—2741no—
376D4—2741no—
377D4—5482no—
378D4—5482no—
379D4—5482no—
380D4—5482no—
381D4—5482no—
382D4—5482no—
383D4—5482no—
384D4—10961no—
385D4—10961no—
386D4C2×D54812yes—
387D4C2×D54812yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—2741no—
393C2×C4—2741no—
394C2×C4—2741no—
395C2×C4—2741no—
396C2×C4—2741no—
397C2×C4—5482no—
398C2×C4—1371no—
399C2×C4—1371no—
400C2×C4D4×D13711yescharacteristic
401C2×C4D4×D13711yescharacteristic
402C2×C4C23×D13711yescharacteristic
403C8—21no—
404C8—2741no—
405C8—2741no—
406C8C2×D54812yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—2741no—
414C22—2741no—
415C22—2741no—
416C22—2741no—
417C22—2741no—
418C22—5481no—
419C22—5481no—
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—21no—
434C4—41no—
435C4—41no—
436C4—2741no—
437C4—2741no—
438C4—5481no—
439C4C22×D54811yescharacteristic
440C4#1798413411yescharacteristic
441C2—42no—
442C2—42no—
443C2—2742no—
444C2—5481no—
445C2—10961no—
446C2—10961no—
447C2—10961no—
448C2D4×D54811yescharacteristic
449C2AGL(1,ℤ/8)×D13711yescharacteristic
450C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#442×C137C2
socleC22×C137C22×D4
hypercenter#442D137
hypocenterC137#3264
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#17984112C2semidirect
2. C2×D8×D137C2semidirect
3. #22578048C2semidirect
4. D4×D548C2semidirect
5. #22577792C2semidirect
6. #22577657C2semidirect
7. #22577633C2semidirect
8. #22577618C2semidirect
9. #22577598C2semidirect
10. #22577555C2semidirect
11. #22577549C2semidirect
12. #22577538C2semidirect
13. #22577538C2semidirect
14. #22577476C2semidirect
15. #442×C137C2semidirect
16. C2×#14555509C22semidirect
19. C2×D4×D137C22semidirect
20. C2×D4×D137C22semidirect
22. C2×#14555499C22semidirect
23. C2×#14555499C22semidirect
24. C2×#14555498C22semidirect
25. C2×D1096C22semidirect
26. C2×D1096C22semidirect
28. C2×#14555495C22semidirect
29. C2×#14555495C22semidirect
31. C2×#14555493C22semidirect
32. C2×C8×D137C22semidirect
33. #17984085C22semidirect
34. #17984076C22semidirect
36. #17984068C22semidirect
38. #17984063C22semidirect
39. #17984063C22semidirect
47. C4⋊#11764040C22semidirect
48. #17984011C22semidirect
49. #17984008C22semidirect
50. #17984008C22semidirect
51. #17984008C22semidirect
52. #17984000C22semidirect
53. #17984000C22semidirect
55. C4×D4×C137C22semidirect
56. C4×D4×C137C22semidirect
57. #106×C137C22semidirect
58. #102×C137C22semidirect
59. #102×C137C22semidirect
60. #97×C137C22semidirect
116. D8×C137D4semidirect
171. D4×C137C2×D4semidirect
180. C8×C137C2×D4semidirect
205. C2×D137AGL(1,ℤ/8)semidirect
225. C137#3264semidirect
240. #442D137semidirect
273. C4×D4C2×D137semidirect
274. C4×D4C2×D137semidirect
276. #106C2×D137semidirect
282. #102C2×D137semidirect
283. #102C2×D137semidirect
286. #97C2×D137semidirect
328. D8D548semidirect
387. D4C2×D548semidirect
406. C8C2×D548semidirect

Maximal subgroups

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

subgroupindex
C2×#179841122
C2×D8×D1372
#225780482
D4×D5482
#225777922
#225776572
#225776332
#225776182
#225775982
#225775552
#225775492
#225775382
#225774762
#442×C1372
#3264137