Finite Group Databasebeta

← #31792284

Subgroups — #31792284

Order 17536.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984115C211yesmaximal, characteristic, complemented
2#22578064C211yesmaximal, characteristic, complemented
3D4×C22⋊D137C211yesmaximal, characteristic, complemented
4#22577789C211yesmaximal, characteristic, complemented
5#22577646C211yesmaximal, characteristic, complemented
6#22577639C211yesmaximal, characteristic, complemented
7#22577620C211yesmaximal, characteristic, complemented
8#22577601C211yesmaximal, characteristic, complemented
9#22577552C211yesmaximal, characteristic, complemented
10#22577547C211yesmaximal, characteristic, complemented
11#22577542C211yesmaximal, characteristic, complemented
12#22577537C211yesmaximal, characteristic, complemented
13#22577504C211yesmaximal, characteristic, complemented
14#22577474C211yesmaximal, characteristic, complemented
15C22×SD16×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic, complemented
17C22×C22⋊D137—21no—
18C2×D4×D137C2211yescharacteristic
19#17984115—24no—
20C2×#14555497C2211yescharacteristic, complemented
21C2×#14555497C2211yescharacteristic, complemented
22C2×D1096C2211yescharacteristic, complemented
23C2×#14555495C2211yescharacteristic
24C2×#14555495C2211yescharacteristic, complemented
25C2×#14555493C2211yescharacteristic, complemented
26#17984084C2211yescharacteristic
27#17984077—21no—
28#17984077C2211yescharacteristic, complemented
29#17984064—21no—
30#17984064C2211yescharacteristic, complemented
31#17984062—21no—
32#17984062C2211yescharacteristic
33#17984062C2211yescharacteristic, complemented
34#17984060—21no—
35#17984059C2211yescharacteristic
36#17984057—21no—
37#17984056—21no—
38Q8×Dic137C2211yescharacteristic, complemented
39D4×Dic137C2211yescharacteristic
40C4×C22⋊D137—21no—
41C4×C22⋊D137C2211yescharacteristic
42#17984022C2211yescharacteristic, complemented
43#17984017C2211yescharacteristic, complemented
44#17984012C2211yescharacteristic, complemented
45#17984012C2211yescharacteristic, complemented
46#17984010C2211yescharacteristic, complemented
47#17984010C2211yescharacteristic, complemented
48#17984009C2211yescharacteristic
49#17984009C2211yescharacteristic, complemented
50C22⋊#11764040C2211yescharacteristic, complemented
51#17984000C2211yescharacteristic, complemented
52#17983998C2211yescharacteristic, complemented
53C22×Q8×C137C2211yescharacteristic, complemented
54C22×D4×C137C2211yescharacteristic
55C2×SD16×C137—24no—
56C2×SD16×C137C2211yescharacteristic
57C2×SD16×C137C2211yescharacteristic
58C2×SD16×C137C2211yescharacteristic, complemented
59C2×SD16×C137C2211yescharacteristic, complemented
60C22×C8×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555511—22no—
63#14555511—41no—
64C2×D548—21no—
65C2×D548C2311yescharacteristic
66C2×D548C2311yescharacteristic
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—
78C2×C4×D137—21no—
79C2×C4×D137C2311yescharacteristic
80#14555497—22no—
81#14555497—41no—
82D1096—22no—
83#14555495—22no—
84#14555495—41no—
85#14555493—22no—
86C2×#11764040C2311yescharacteristic
87C2×#11764040C2311yescharacteristic
88C4⋊Dic137—21no—
89C4⋊Dic137C2311yescharacteristic
90C4⋊Dic137C2311yescharacteristic
91#14555489—21no—
92#14555489D411yescharacteristic
93C22⋊Dic137—21no—
94C22⋊Dic137—21no—
95C22⋊Dic137D411yescharacteristic
96#14555487—21no—
97#14555487—21no—
98#14555487—21no—
99#14555487—21no—
100#14555487—21no—
101#14555487D411yescharacteristic
102C4×Dic137—21no—
103C4×Dic137C2311yescharacteristic
104C24×C137—21no—
105C2×Q8×C137—21no—
106C2×Q8×C137—22no—
107C2×Q8×C137—41no—
108C2×Q8×C137C2311yescharacteristic
109C2×Q8×C137C2311yescharacteristic
110C2×D4×C137—22no—
111C2×D4×C137—21no—
112C2×D4×C137—41no—
113C2×D4×C137C2311yescharacteristic
114C2×D4×C137C2311yescharacteristic
115C22×C4×C137—21no—
116C22×C4×C137C2311yescharacteristic
117SD16×C137—22no—
118SD16×C137—22no—
119SD16×C137—22no—
120SD16×C137D414yescomplemented
121C2×C8×C137—22no—
122C2×C8×C137C2311yescharacteristic
123C2×C8×C137C2311yescharacteristic
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—42no—
127C22×D137—42no—
128C22×D137—81no—
129C22×D137C2×D411yescharacteristic
130D548—21no—
131D548—21no—
132D548—21no—
133D548—21no—
134D548—42no—
135C22⋊D137—22no—
136C22⋊D137—42no—
137C22⋊D137—42no—
138C22⋊D137—42no—
139C22⋊D137—42no—
140C22⋊D137—42no—
141C22⋊D137—81no—
142C2×Dic137—21no—
143C2×Dic137—21no—
144C2×Dic137—42no—
145C2×Dic137C2×D411yescharacteristic
146C2×Dic137C2×D411yescharacteristic
147C2×Dic137C2×D411yescharacteristic
148C4×D137—21no—
149C4×D137—21no—
150C4×D137—42no—
151#11764040—21no—
152#11764040—21no—
153C23×C137—21no—
154C23×C137—21no—
155C23×C137—42no—
156C23×C137—42no—
157C23×C137—81no—
158C23×C137C2×D411yescharacteristic
159Q8×C137—21no—
160Q8×C137—21no—
161Q8×C137—21no—
162Q8×C137—42no—
163Q8×C137C2×D412yescomplemented
164D4×C137—21no—
165D4×C137—21no—
166D4×C137—21no—
167D4×C137—42no—
168D4×C137C2×D412yes—
169C2×C4×C137—21no—
170C2×C4×C137—21no—
171C2×C4×C137—21no—
172C2×C4×C137—21no—
173C2×C4×C137—42no—
174C2×C4×C137C2×D411yescharacteristic
175C2×C4×C137C2411yescharacteristic
176C8×C137—21no—
177C8×C137C2×D412yescomplemented
178C22×C137—21no—
179C22×C137—21no—
180C22×C137—22no—
181C22×C137—41no—
182C22×C137—41no—
183C22×C137—41no—
184C22×C137—41no—
185C22×C137—41no—
186C22×C137—84no—
187C22×C137AGL(1,ℤ/8)12yescomplemented
188C22×C137C22×D411yescharacteristic
189C2×D137—21no—
190C2×D137—21no—
191C2×D137—21no—
192C2×D137—41no—
193C2×D137—41no—
194C2×D137—41no—
195C2×D137—41no—
196C2×D137—41no—
197C2×D137—41no—
198C2×D137—84no—
199C4×C137—21no—
200C4×C137—22no—
201C4×C137—41no—
202C4×C137C22×D411yescharacteristic
203C4×C137H(5,𝔽2)11yescharacteristic
204Dic137—21no—
205Dic137—21no—
206Dic137—21no—
207Dic137—41no—
208Dic137—41no—
209C2×C137—22no—
210C2×C137—42no—
211C2×C137—81no—
212C2×C137D4211yescharacteristic
213C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C137#57711yescharacteristic
215D137—41no—
216D137—81no—
217D137—81no—
218C137#326511yescharacteristic, complemented, Sylow
219#3265—1371nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1371no—
221C22×SD16D13711yescharacteristic, complemented
222#548—1371no—
223D42—1371no—
224#496—1371no—
225#468—1371no—
226#466—1371no—
227#463—1371no—
228#460—1371no—
229#450—1371no—
230#449—1371no—
231#448—1371no—
232#447—1371no—
233#440—1371no—
234#435—1371no—
235C2×#41—1371no—
236C22×Q8C2×D13711yescharacteristic, complemented
237C22×D4—2741no—
238C22×D4—1371no—
239C22×D4C2×D13711yescharacteristic
240AGL(1,ℤ/8)—2744no—
241C2×SD16—24no—
242C2×SD16—1371no—
243C2×SD16—1371no—
244C2×SD16C2×D13711yescharacteristic
245C2×SD16C2×D13711yescharacteristic
246C2×SD16C2×D13711yescharacteristic, complemented
247C2×SD16C2×D13711yescharacteristic, complemented
248C2×D8—1371no—
249C2×D8—1371no—
250C2×D8—1371no—
251C2×M16—1371no—
252C22×C8C2×D13711yescharacteristic, complemented
253#127—1371no—
254#124—2741no—
255#124—1371no—
256C22⋊Q8—2741no—
257C22⋊Q8—1371no—
258#121—2741no—
259#121—2741no—
260#121—1371no—
261#121—1371no—
262#121—1371no—
263#120—2741no—
264#120—2741no—
265C4×Q8—1371no—
266C4×D4—2741no—
267C4×D4—1371no—
268C4×D4—1371no—
269#107—1371no—
270C4⋊C8—1371no—
271#103—1371no—
272#103—1371no—
273#102—1371no—
274#102—1371no—
275#102—1371no—
276#102—1371no—
277C22⋊C8—1371no—
278C22⋊C8—1371no—
279#97—1371no—
280C24—21no—
281C24—2741no—
282#41—2742no—
283#41—5481no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D13711yescharacteristic
288C2×Q8C22×D13711yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—2741no—
293C2×D4—2742no—
294C2×D4—2741no—
295C2×D4—2741no—
296C2×D4—2741no—
297C2×D4—2741no—
298C2×D4—2741no—
299C2×D4—2741no—
300C2×D4—5481no—
301C2×D4—5484no—
302C2×D4—1371no—
303C2×D4—1371no—
304C2×D4—1371no—
305C2×D4C22×D13711yescharacteristic
306C2×D4C22×D13711yescharacteristic
307C22×C4—21no—
308C22×C4—2741no—
309C22×C4—2741no—
310C22×C4—1371no—
311C22×C4C22×D13711yescharacteristic
312SD16—22no—
313SD16—22no—
314SD16—22no—
315SD16—2742no—
316SD16—5481no—
317SD16C22⋊D13714yescomplemented
318D8—2742no—
319D8—2742no—
320D8—5481no—
321M16—2742no—
322C2×C8—22no—
323C2×C8—1371no—
324C2×C8—1371no—
325C2×C8C22×D13711yescharacteristic
326C2×C8C22×D13711yescharacteristic
327C4⋊C4—2741no—
328C4⋊C4—2741no—
329C4⋊C4—1371no—
330C4⋊C4—1371no—
331C4⋊C4—1371no—
332C22⋊C4—2741no—
333C22⋊C4—2741no—
334C22⋊C4—2741no—
335C22⋊C4—2741no—
336C22⋊C4—2741no—
337C22⋊C4—2741no—
338C22⋊C4—2741no—
339C22⋊C4—1371no—
340C22⋊C4—1371no—
341C42—2741no—
342C42—1371no—
343C23—21no—
344C23—21no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—2741no—
349C23—2741no—
350C23—5482no—
351C23—5482no—
352C23—10961no—
353C23—1371no—
354C23D4×D13711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8C2×C22⋊D13712yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—2742no—
365D4—2741no—
366D4—2741no—
367D4—2741no—
368D4—2741no—
369D4—5482no—
370D4—5482no—
371D4—5482no—
372D4—5482no—
373D4—5482no—
374D4—5482no—
375D4—10961no—
376D4C2×C22⋊D13712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—2741no—
383C2×C4—2741no—
384C2×C4—2741no—
385C2×C4—2741no—
386C2×C4—5482no—
387C2×C4—5482no—
388C2×C4—1371no—
389C2×C4—1371no—
390C2×C4—1371no—
391C2×C4D4×D13711yescharacteristic
392C2×C4C23×D13711yescharacteristic
393C8—21no—
394C8—2741no—
395C8—2741no—
396C8C2×C22⋊D13712yescomplemented
397C22—21no—
398C22—21no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—84no—
406C22—2741no—
407C22—2741no—
408C22—2741no—
409C22—5481no—
410C22—5481no—
411C22—5481no—
412C22—5481no—
413C22—5481no—
414C22—5481no—
415C22—10964no—
416C22#1798411512yescomplemented
417C22C2×D4×D13711yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—2741no—
422C4—2741no—
423C4—2741no—
424C4—5481no—
425C4—5481no—
426C4C22×C22⋊D13711yescharacteristic
427C4#1798413311yescharacteristic
428C2—22no—
429C2—42no—
430C2—81no—
431C2—5481no—
432C2—10961no—
433C2—10961no—
434C2D4×C22⋊D13711yescharacteristic
435C2C2×#1798411511yescharacteristic
436C2#2257820911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC22×SD16×C137C2
socleC22×C137C22×D4
hypercenterC22×SD16D137
hypocenterC137#3265
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#17984115C2semidirect
2. #22578064C2semidirect
3. D4×C22⋊D137C2semidirect
4. #22577789C2semidirect
5. #22577646C2semidirect
6. #22577639C2semidirect
7. #22577620C2semidirect
8. #22577601C2semidirect
9. #22577552C2semidirect
10. #22577547C2semidirect
11. #22577542C2semidirect
12. #22577537C2semidirect
13. #22577504C2semidirect
14. #22577474C2semidirect
15. C22×SD16×C137C2semidirect
16. C2×#14555511C22semidirect
20. C2×#14555497C22semidirect
21. C2×#14555497C22semidirect
22. C2×D1096C22semidirect
24. C2×#14555495C22semidirect
25. C2×#14555493C22semidirect
28. #17984077C22semidirect
30. #17984064C22semidirect
33. #17984062C22semidirect
38. Q8×Dic137C22semidirect
42. #17984022C22semidirect
43. #17984017C22semidirect
44. #17984012C22semidirect
45. #17984012C22semidirect
46. #17984010C22semidirect
47. #17984010C22semidirect
49. #17984009C22semidirect
50. C22⋊#11764040C22semidirect
51. #17984000C22semidirect
52. #17983998C22semidirect
53. C22×Q8×C137C22semidirect
58. C2×SD16×C137C22semidirect
59. C2×SD16×C137C22semidirect
60. C22×C8×C137C22semidirect
120. SD16×C137D4semidirect
163. Q8×C137C2×D4semidirect
177. C8×C137C2×D4semidirect
187. C22×C137AGL(1,ℤ/8)semidirect
218. C137#3265semidirect
221. C22×SD16D137semidirect
236. C22×Q8C2×D137semidirect
246. C2×SD16C2×D137semidirect
247. C2×SD16C2×D137semidirect
252. C22×C8C2×D137semidirect
317. SD16C22⋊D137semidirect
359. Q8C2×C22⋊D137semidirect
396. C8C2×C22⋊D137semidirect
416. C22#17984115semidirect

Maximal subgroups

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

subgroupindex
C2×#179841152
#225780642
D4×C22⋊D1372
#225777892
#225776462
#225776392
#225776202
#225776012
#225775522
#225775472
#225775422
#225775372
#225775042
#225774742
C22×SD16×C1372
#3265137