Finite Group Databasebeta

← #31792269

Subgroups — #31792269

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
2#22578047C211yesmaximal, characteristic, complemented
3D4×C22⋊D137C211yesmaximal, characteristic, complemented
4#22577791C211yesmaximal, characteristic, complemented
5#22577653C211yesmaximal, characteristic, complemented
6#22577632C211yesmaximal, characteristic, complemented
7#22577616C211yesmaximal, characteristic, complemented
8#22577596C211yesmaximal, characteristic, complemented
9#22577554C211yesmaximal, characteristic, complemented
10#22577545C211yesmaximal, characteristic, complemented
11#22577537C211yesmaximal, characteristic, complemented
12#22577535C211yesmaximal, characteristic, complemented
13#22577515C211yesmaximal, characteristic, complemented
14#22577474C211yesmaximal, characteristic, complemented
15C22×D8×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic, complemented
17C22×C22⋊D137—21no—
18C2×D4×D137C2211yescharacteristic, complemented
19#17984112—24no—
20C2×#14555499C2211yescharacteristic, complemented
21C2×#14555498C2211yescharacteristic, complemented
22C2×#14555498C2211yescharacteristic, complemented
23C2×#14555495C2211yescharacteristic, complemented
24C2×#14555495C2211yescharacteristic, complemented
25C2×#14555493C2211yescharacteristic, complemented
26#17984084C2211yescharacteristic, complemented
27#17984075C2211yescharacteristic, complemented
28#17984069—21no—
29#17984067C2211yescharacteristic, complemented
30#17984062C2211yescharacteristic
31#17984060—21no—
32#17984060—21no—
33#17984059C2211yescharacteristic, complemented
34#17984059C2211yescharacteristic, complemented
35#17984057—21no—
36#17984057—21no—
37#17984056—21no—
38D4×Dic137C2211yescharacteristic, complemented
39D4×Dic137C2211yescharacteristic, complemented
40C4×C22⋊D137C2211yescharacteristic
41C2×C22⋊Dic137—21no—
42#17984020C2211yescharacteristic, complemented
43#17984017C2211yescharacteristic, complemented
44#17984013C2211yescharacteristic, complemented
45#17984010C2211yescharacteristic, complemented
46#17984009C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic, complemented
48#17984009C2211yescharacteristic, complemented
49#17984009C2211yescharacteristic, complemented
50C22⋊#11764040C2211yescharacteristic, complemented
51#17984000C2211yescharacteristic, complemented
52#17983998C2211yescharacteristic, complemented
53C22×D4×C137C2211yescharacteristic
54C22×D4×C137C2211yescharacteristic, complemented
55C2×D8×C137—24no—
56C2×D8×C137C2211yescharacteristic
57C2×D8×C137C2211yescharacteristic
58C2×D8×C137C2211yescharacteristic, complemented
59C2×D8×C137C2211yescharacteristic, complemented
60C22×C8×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555509—22no—
63#14555509—41no—
64C2×Dic274C2311yescharacteristic
65C2×D548C2311yescharacteristic
66C2×C22⋊D137—21no—
67C2×C22⋊D137—21no—
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73C2×C22⋊D137—44no—
74C2×C22⋊D137D411yescharacteristic
75D4×D137—22no—
76D4×D137—41no—
77C22×Dic137—21no—
78C22×Dic137—21no—
79C22×Dic137—21no—
80C2×C4×D137C2311yescharacteristic
81#14555499—22no—
82#14555498—22no—
83#14555498—41no—
84#14555495—22no—
85#14555495—41no—
86#14555493—22no—
87C2×#11764040C2311yescharacteristic
88C2×#11764040C2311yescharacteristic
89C4⋊Dic137C2311yescharacteristic
90C4⋊Dic137C2311yescharacteristic
91#14555489—21no—
92#14555489D411yescharacteristic
93C22⋊Dic137—21no—
94C22⋊Dic137—21no—
95C22⋊Dic137—21no—
96C22⋊Dic137—21no—
97C22⋊Dic137—21no—
98C22⋊Dic137—21no—
99C22⋊Dic137—21no—
100C22⋊Dic137D411yescharacteristic
101#14555487—21no—
102#14555487D411yescharacteristic
103C4×Dic137C2311yescharacteristic
104C24×C137—21no—
105C24×C137—21no—
106C2×D4×C137—22no—
107C2×D4×C137—22no—
108C2×D4×C137—21no—
109C2×D4×C137—21no—
110C2×D4×C137—41no—
111C2×D4×C137—41no—
112C2×D4×C137C2311yescharacteristic
113C2×D4×C137C2311yescharacteristic
114C2×D4×C137C2311yescharacteristic
115C2×D4×C137C2311yescharacteristic
116C22×C4×C137C2311yescharacteristic
117D8×C137—22no—
118D8×C137—22no—
119D8×C137—22no—
120D8×C137D414yescomplemented
121C2×C8×C137—22no—
122C2×C8×C137C2311yescharacteristic
123C2×C8×C137C2311yescharacteristic
124C22×D137—21no—
125C22×D137—42no—
126C22×D137—42no—
127C22×D137—81no—
128C22×D137C2×D411yescharacteristic
129Dic274—21no—
130Dic274—21no—
131D548—21no—
132D548—21no—
133C22⋊D137—22no—
134C22⋊D137—42no—
135C22⋊D137—42no—
136C22⋊D137—42no—
137C22⋊D137—42no—
138C22⋊D137—42no—
139C22⋊D137—81no—
140C22⋊D137—81no—
141C2×Dic137—21no—
142C2×Dic137—21no—
143C2×Dic137—21no—
144C2×Dic137—42no—
145C2×Dic137—42no—
146C2×Dic137—81no—
147C2×Dic137C2×D411yescharacteristic
148C2×Dic137C2×D411yescharacteristic
149C2×Dic137C2×D411yescharacteristic
150C4×D137—21no—
151C4×D137—21no—
152#11764040—21no—
153#11764040—21no—
154C23×C137—21no—
155C23×C137—21no—
156C23×C137—21no—
157C23×C137—21no—
158C23×C137—42no—
159C23×C137—42no—
160C23×C137—42no—
161C23×C137—42no—
162C23×C137—81no—
163C23×C137—81no—
164C23×C137C2×D411yescharacteristic
165D4×C137—21no—
166D4×C137—21no—
167D4×C137—21no—
168D4×C137—21no—
169D4×C137—21no—
170D4×C137—21no—
171D4×C137—42no—
172D4×C137—42no—
173D4×C137C2×D412yes—
174D4×C137C2×D412yescomplemented
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—22no—
184C22×C137—22no—
185C22×C137—41no—
186C22×C137—41no—
187C22×C137—41no—
188C22×C137—41no—
189C22×C137—41no—
190C22×C137—41no—
191C22×C137—41no—
192C22×C137—41no—
193C22×C137—41no—
194C22×C137—41no—
195C22×C137—84no—
196C22×C137—84no—
197C22×C137AGL(1,ℤ/8)12yescomplemented
198C22×C137C22×D411yescharacteristic
199C2×D137—21no—
200C2×D137—21no—
201C2×D137—21no—
202C2×D137—41no—
203C2×D137—41no—
204C2×D137—41no—
205C2×D137—84no—
206C4×C137—21no—
207C4×C137C22×D411yescharacteristic
208C4×C137H(5,𝔽2)11yescharacteristic
209Dic137—21no—
210Dic137—21no—
211Dic137—21no—
212Dic137—41no—
213Dic137—41no—
214Dic137—41no—
215C2×C137—22no—
216C2×C137—42no—
217C2×C137—42no—
218C2×C137—81no—
219C2×C137—81no—
220C2×C137D4211yescharacteristic
221C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C137#57611yescharacteristic
223D137—41no—
224D137—81no—
225C137#326411yescharacteristic, complemented, Sylow
226#3264—1371nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1371no—
228C22×D8D13711yescharacteristic, complemented
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#442—1371no—
241#435—1371no—
242C2×#41—1371no—
243C22×D4—2741no—
244C22×D4—1371no—
245C22×D4C2×D13711yescharacteristic
246C22×D4C2×D13711yescharacteristic, complemented
247AGL(1,ℤ/8)—2744no—
248C2×SD16—1371no—
249C2×SD16—1371no—
250C2×SD16—1371no—
251C2×D8—24no—
252C2×D8—1371no—
253C2×D8—1371no—
254C2×D8C2×D13711yescharacteristic
255C2×D8C2×D13711yescharacteristic
256C2×D8C2×D13711yescharacteristic, complemented
257C2×D8C2×D13711yescharacteristic, complemented
258C2×M16—1371no—
259C22×C8C2×D13711yescharacteristic, complemented
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×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#102—1371no—
283#102—1371no—
284C22⋊C8—1371no—
285C22⋊C8—1371no—
286#97—1371no—
287C24—21no—
288C24—21no—
289C24—2741no—
290#41—2742no—
291#41—5481no—
292C2×Q8—1371no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—2742no—
300C2×D4—2741no—
301C2×D4—2741no—
302C2×D4—2741no—
303C2×D4—2741no—
304C2×D4—2741no—
305C2×D4—2741no—
306C2×D4—2741no—
307C2×D4—5481no—
308C2×D4—5484no—
309C2×D4—1371no—
310C2×D4—1371no—
311C2×D4C22×D13711yescharacteristic
312C2×D4C22×D13711yescharacteristic
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—22no—
325D8—22no—
326D8—2742no—
327D8—5481no—
328D8C22⋊D13714yescomplemented
329M16—2742no—
330C2×C8—22no—
331C2×C8—1371no—
332C2×C8—1371no—
333C2×C8C22×D13711yescharacteristic
334C2×C8C22×D13711yescharacteristic
335C4⋊C4—2741no—
336C4⋊C4—1371no—
337C4⋊C4—1371no—
338C4⋊C4—1371no—
339C22⋊C4—2741no—
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—21no—
353C23—21no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—2741no—
361C23—5482no—
362C23—5482no—
363C23—10961no—
364C23—1371no—
365C23D4×D13711yescharacteristic
366Q8—2741no—
367Q8—2741no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—2742no—
377D4—2741no—
378D4—2741no—
379D4—5482no—
380D4—5482no—
381D4—5482no—
382D4—5482no—
383D4—5482no—
384D4—10961no—
385D4—10961no—
386D4C2×C22⋊D13712yes—
387D4C2×C22⋊D13712yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—2741no—
391C2×C4—2741no—
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—
406C8C2×C22⋊D13712yescomplemented
407C22—21no—
408C22—21no—
409C22—22no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—2741no—
424C22—2741no—
425C22—2741no—
426C22—5481no—
427C22—5481no—
428C22—5481no—
429C22—10964no—
430C22#1798411212yescomplemented
431C22C2×D4×D13711yescharacteristic
432C4—21no—
433C4—2741no—
434C4—2741no—
435C4—2741no—
436C4—5481no—
437C4—5481no—
438C4—5481no—
439C4C22×C22⋊D13711yescharacteristic
440C4#1798413311yescharacteristic
441C2—22no—
442C2—42no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—5481no—
447C2—10961no—
448C2D4×C22⋊D13711yescharacteristic
449C2C2×#1798411211yescharacteristic
450C2#2257820411yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC22×D8×C137C2
socleC22×C137C22×D4
hypercenterC22×D8D137
hypocenterC137#3264
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#17984112C2semidirect
2. #22578047C2semidirect
3. D4×C22⋊D137C2semidirect
4. #22577791C2semidirect
5. #22577653C2semidirect
6. #22577632C2semidirect
7. #22577616C2semidirect
8. #22577596C2semidirect
9. #22577554C2semidirect
10. #22577545C2semidirect
11. #22577537C2semidirect
12. #22577535C2semidirect
13. #22577515C2semidirect
14. #22577474C2semidirect
15. C22×D8×C137C2semidirect
16. C2×#14555509C22semidirect
18. C2×D4×D137C22semidirect
20. C2×#14555499C22semidirect
21. C2×#14555498C22semidirect
22. C2×#14555498C22semidirect
23. C2×#14555495C22semidirect
24. C2×#14555495C22semidirect
25. C2×#14555493C22semidirect
26. #17984084C22semidirect
27. #17984075C22semidirect
29. #17984067C22semidirect
33. #17984059C22semidirect
34. #17984059C22semidirect
38. D4×Dic137C22semidirect
39. D4×Dic137C22semidirect
42. #17984020C22semidirect
43. #17984017C22semidirect
44. #17984013C22semidirect
45. #17984010C22semidirect
46. #17984009C22semidirect
47. #17984009C22semidirect
48. #17984009C22semidirect
49. #17984009C22semidirect
50. C22⋊#11764040C22semidirect
51. #17984000C22semidirect
52. #17983998C22semidirect
54. C22×D4×C137C22semidirect
58. C2×D8×C137C22semidirect
59. C2×D8×C137C22semidirect
60. C22×C8×C137C22semidirect
120. D8×C137D4semidirect
174. D4×C137C2×D4semidirect
180. C8×C137C2×D4semidirect
197. C22×C137AGL(1,ℤ/8)semidirect
225. C137#3264semidirect
228. C22×D8D137semidirect
246. C22×D4C2×D137semidirect
256. C2×D8C2×D137semidirect
257. C2×D8C2×D137semidirect
259. C22×C8C2×D137semidirect
328. D8C22⋊D137semidirect
387. D4C2×C22⋊D137semidirect
406. C8C2×C22⋊D137semidirect
430. C22#17984112semidirect

Maximal subgroups

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

subgroupindex
C2×#179841122
#225780472
D4×C22⋊D1372
#225777912
#225776532
#225776322
#225776162
#225775962
#225775542
#225775452
#225775372
#225775352
#225775152
#225774742
C22×D8×C1372
#3264137