Finite Group Databasebeta

← #31792289

Subgroups — #31792289

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×#17984116C211yesmaximal, characteristic, complemented
2C22×#14555499C211yesmaximal, characteristic, complemented
3#22578062C211yesmaximal, characteristic, complemented
4D4×D548C211yesmaximal, characteristic, complemented
5#22577792C211yesmaximal, characteristic, complemented
6#22577649C211yesmaximal, characteristic, complemented
7#22577639C211yesmaximal, characteristic, complemented
8#22577619C211yesmaximal, characteristic, complemented
9#22577604C211yesmaximal, characteristic, complemented
10#22577556C211yesmaximal, characteristic, complemented
11#22577550C211yesmaximal, characteristic, complemented
12#22577544C211yesmaximal, characteristic, complemented
13#22577539C211yesmaximal, characteristic, complemented
14#22577505C211yesmaximal, characteristic, complemented
15#435×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic, complemented
17C22×Dic274C2211yescharacteristic, complemented
18C22×D548C2211yescharacteristic
19C22×D548C2211yescharacteristic
20C2×D4×D137—21no—
21#17984116—24no—
22C2×#14555499—24no—
23C2×#14555499C2211yescharacteristic
24C2×#14555499C2211yescharacteristic
25C2×#14555499C2211yescharacteristic, complemented
26C2×#14555499C2211yescharacteristic, complemented
27C2×#14555499C2211yescharacteristic, complemented
28C2×#14555499C2211yescharacteristic, complemented
29C2×D1096C2211yescharacteristic
30C2×D1096C2211yescharacteristic, complemented
31C2×D1096C2211yescharacteristic, complemented
32#17984085C2211yescharacteristic
33#17984076C2211yescharacteristic, complemented
34#17984074—21no—
35#17984067C2211yescharacteristic, complemented
36#17984066—21no—
37#17984063—21no—
38#17984062C2211yescharacteristic
39#17984062C2211yescharacteristic
40#17984062C2211yescharacteristic, complemented
41#17984060—21no—
42#17984058—21no—
43#17984058—21no—
44C4×Dic274C2211yescharacteristic, complemented
45C4×D548C2211yescharacteristic
46#17984048—21no—
47#17984022C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984013C2211yescharacteristic, complemented
50#17984010C2211yescharacteristic
51#17984010C2211yescharacteristic, complemented
52#17984010C2211yescharacteristic, complemented
53#17984010C2211yescharacteristic, complemented
54C2×M16×C137C2211yescharacteristic, complemented
55C22×C8×C137C2211yescharacteristic, complemented
56C4×D4×C137C2211yescharacteristic
57C4⋊C8×C137C2211yescharacteristic, complemented
58C22⋊C8×C137C2211yescharacteristic, complemented
59C22⋊C8×C137C2211yescharacteristic, complemented
60#97×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63#14555510—22no—
64#14555510—41no—
65C2×Dic274—22no—
66C2×Dic274—21no—
67C2×Dic274—41no—
68C2×Dic274C2311yescharacteristic
69C2×Dic274C2311yescharacteristic
70C2×D548—22no—
71C2×D548—22no—
72C2×D548—21no—
73C2×D548—21no—
74C2×D548—21no—
75C2×D548—21no—
76C2×D548—41no—
77C2×D548—41no—
78C2×D548C2311yescharacteristic
79C2×D548C2311yescharacteristic
80C2×D548C2311yescharacteristic
81C2×D548C2311yescharacteristic
82C2×C22⋊D137—21no—
83C2×C22⋊D137—21no—
84C2×C22⋊D137—21no—
85C2×C22⋊D137—21no—
86D4×D137—44no—
87C22×Dic137—21no—
88C2×C4×D137—21no—
89C2×C4×D137—21no—
90#14555499—22no—
91#14555499—22no—
92#14555499—22no—
93#14555499—22no—
94#14555499—41no—
95#14555499D414yescomplemented
96D1096—22no—
97D1096—22no—
98D1096—41no—
99C4⋊Dic137C2311yescharacteristic
100C4⋊Dic137C2311yescharacteristic
101#14555489—21no—
102#14555489—21no—
103C22⋊Dic137—21no—
104#14555487—21no—
105#14555487—21no—
106#14555487—21no—
107#14555487—21no—
108#14555487—21no—
109#14555487—21no—
110C4×Dic137—21no—
111C2×D4×C137D411yescharacteristic
112C22×C4×C137C2311yescharacteristic
113C22×C4×C137C2311yescharacteristic
114M16×C137—22no—
115C2×C8×C137—22no—
116C2×C8×C137C2311yescharacteristic
117C2×C8×C137C2311yescharacteristic
118C2×C8×C137C2311yescharacteristic
119C2×C8×C137C2311yescharacteristic
120C4⋊C4×C137D411yescharacteristic
121C22⋊C4×C137D411yescharacteristic
122C22⋊C4×C137D411yescharacteristic
123C42×C137C2311yescharacteristic
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—21no—
127C22×D137—21no—
128C22×D137—42no—
129C22×D137—42no—
130C22×D137—42no—
131C22×D137—42no—
132C22×D137—81no—
133C22×D137—81no—
134Dic274—21no—
135Dic274—21no—
136Dic274—21no—
137Dic274—42no—
138Dic274C2×D412yescomplemented
139D548—21no—
140D548—21no—
141D548—21no—
142D548—21no—
143D548—21no—
144D548—21no—
145D548—21no—
146D548—42no—
147D548—42no—
148D548—42no—
149D548—81no—
150D548C2×D412yes—
151C22⋊D137—42no—
152C22⋊D137—42no—
153C22⋊D137—42no—
154C22⋊D137—42no—
155C2×Dic137—21no—
156C2×Dic137—21no—
157C2×Dic137—21no—
158C2×Dic137—21no—
159C2×Dic137—42no—
160C4×D137—42no—
161C4×D137—42no—
162C23×C137C2×D411yescharacteristic
163C23×C137C2×D411yescharacteristic
164D4×C137—22no—
165C2×C4×C137—21no—
166C2×C4×C137—21no—
167C2×C4×C137—21no—
168C2×C4×C137—21no—
169C2×C4×C137C2×D411yescharacteristic
170C2×C4×C137C2×D411yescharacteristic
171C2×C4×C137C2×D411yescharacteristic
172C2×C4×C137C2×D411yescharacteristic
173C2×C4×C137C2411yescharacteristic
174C8×C137—21no—
175C8×C137—21no—
176C8×C137—21no—
177C8×C137C2×D412yescomplemented
178C22×C137—21no—
179C22×C137—21no—
180C22×C137—21no—
181C22×C137—21no—
182C22×C137—21no—
183C22×C137AGL(1,ℤ/8)12yescomplemented
184C22×C137C22×D411yescharacteristic
185C2×D137—22no—
186C2×D137—41no—
187C2×D137—41no—
188C2×D137—41no—
189C2×D137—41no—
190C2×D137—41no—
191C2×D137—41no—
192C2×D137—41no—
193C2×D137—41no—
194C2×D137—41no—
195C2×D137—41no—
196C2×D137—41no—
197C2×D137—84no—
198C2×D137—84no—
199C4×C137—21no—
200C4×C137—21no—
201C4×C137—21no—
202C4×C137—21no—
203C4×C137C22×D411yescharacteristic
204C4×C137H(5,𝔽2)11yescharacteristic
205Dic137—22no—
206Dic137—41no—
207Dic137—41no—
208Dic137—41no—
209C2×C137—22no—
210C2×C137—41no—
211C2×C137D4211yescharacteristic
212C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C137#57711yescharacteristic
214D137—42no—
215D137—81no—
216D137—81no—
217D137—81no—
218C137#326511yescharacteristic, complemented, Sylow
219#3265—1371nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1371no—
221C22×SD16—1371no—
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#435D13711yescharacteristic, complemented
235C2×#41—1371no—
236C22×Q8—1371no—
237C22×D4—2741no—
238C22×D4—1371no—
239C22×D4—1371no—
240AGL(1,ℤ/8)—2744no—
241C2×SD16—2744no—
242C2×SD16—1371no—
243C2×SD16—1371no—
244C2×SD16—1371no—
245C2×SD16—1371no—
246C2×SD16—1371no—
247C2×SD16—1371no—
248C2×D8—1371no—
249C2×D8—1371no—
250C2×D8—1371no—
251C2×M16C2×D13711yescharacteristic, complemented
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×D4C2×D13711yescharacteristic
269#107—1371no—
270C4⋊C8C2×D13711yescharacteristic, complemented
271#103—1371no—
272#103—1371no—
273#102—1371no—
274#102—1371no—
275#102—1371no—
276#102—1371no—
277C22⋊C8C2×D13711yescharacteristic, complemented
278C22⋊C8C2×D13711yescharacteristic, complemented
279#97C2×D13711yescharacteristic, complemented
280C24—2741no—
281C24—2741no—
282#41—2742no—
283#41—5481no—
284C2×Q8—2742no—
285C2×Q8—2741no—
286C2×Q8—5481no—
287C2×Q8—1371no—
288C2×Q8—1371no—
289C2×D4—2742no—
290C2×D4—2742no—
291C2×D4—2741no—
292C2×D4—2741no—
293C2×D4—2741no—
294C2×D4—2741no—
295C2×D4—2741no—
296C2×D4—2741no—
297C2×D4—2741no—
298C2×D4—2741no—
299C2×D4—5481no—
300C2×D4—5481no—
301C2×D4—5484no—
302C2×D4—1371no—
303C2×D4—1371no—
304C2×D4—1371no—
305C2×D4—1371no—
306C2×D4D54811yescharacteristic
307C22×C4—2741no—
308C22×C4—2741no—
309C22×C4—2741no—
310C22×C4C22×D13711yescharacteristic
311C22×C4C22×D13711yescharacteristic
312SD16—2742no—
313SD16—2742no—
314SD16—2742no—
315SD16—2742no—
316SD16—5481no—
317SD16—1374no—
318D8—2742no—
319D8—2742no—
320D8—5481no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D13711yescharacteristic
324C2×C8C22×D13711yescharacteristic
325C2×C8C22×D13711yescharacteristic
326C2×C8C22×D13711yescharacteristic
327C4⋊C4—2741no—
328C4⋊C4—2741no—
329C4⋊C4—1371no—
330C4⋊C4—1371no—
331C4⋊C4D54811yescharacteristic
332C22⋊C4—2741no—
333C22⋊C4—2741no—
334C22⋊C4—2741no—
335C22⋊C4—2741no—
336C22⋊C4—2741no—
337C22⋊C4—2741no—
338C22⋊C4—2741no—
339C22⋊C4D54811yescharacteristic
340C22⋊C4D54811yescharacteristic
341C42—2741no—
342C42C22×D13711yescharacteristic
343C23—2741no—
344C23—2741no—
345C23—2741no—
346C23—2741no—
347C23—5482no—
348C23—5482no—
349C23—5482no—
350C23—5482no—
351C23—10961no—
352C23—10961no—
353C23C2×D54811yescharacteristic
354C23C2×D54811yescharacteristic
355Q8—2741no—
356Q8—2741no—
357Q8—2741no—
358Q8—5482no—
359Q8—1372no—
360D4—22no—
361D4—2741no—
362D4—2741no—
363D4—2741no—
364D4—2741no—
365D4—2741no—
366D4—2741no—
367D4—2741no—
368D4—5482no—
369D4—5482no—
370D4—5482no—
371D4—5482no—
372D4—5482no—
373D4—5482no—
374D4—5482no—
375D4—10961no—
376D4—1372no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—2741no—
382C2×C4—2741no—
383C2×C4—2741no—
384C2×C4—2741no—
385C2×C4—5482no—
386C2×C4—5482no—
387C2×C4—5482no—
388C2×C4C2×D54811yescharacteristic
389C2×C4C2×D54811yescharacteristic
390C2×C4C2×D54811yescharacteristic
391C2×C4C2×D54811yescharacteristic
392C2×C4C23×D13711yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D13712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—2742no—
403C22—5481no—
404C22—5481no—
405C22—5481no—
406C22—5481no—
407C22—5481no—
408C22—5481no—
409C22—5481no—
410C22—5481no—
411C22—5481no—
412C22—5481no—
413C22—5481no—
414C22—10964no—
415C22—10964no—
416C22#1798411612yescomplemented
417C22C22×D54811yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—2742no—
423C4—5481no—
424C4—5481no—
425C4—5481no—
426C4C2×D4×D13711yescharacteristic
427C4#1798413411yescharacteristic
428C2—22no—
429C2—41no—
430C2—5482no—
431C2—10961no—
432C2—10961no—
433C2—10961no—
434C2D4×D54811yescharacteristic
435C2C2×#1798411611yescharacteristic
436C2#2257821511yescharacteristic
subgroupG/H
centerC22C22×D548
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#435×C137C2
socleC22×C137C22×D4
hypercenter#435D137
hypocenterC137#3265
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#17984116C2semidirect
2. C22×#14555499C2semidirect
3. #22578062C2semidirect
4. D4×D548C2semidirect
5. #22577792C2semidirect
6. #22577649C2semidirect
7. #22577639C2semidirect
8. #22577619C2semidirect
9. #22577604C2semidirect
10. #22577556C2semidirect
11. #22577550C2semidirect
12. #22577544C2semidirect
13. #22577539C2semidirect
14. #22577505C2semidirect
15. #435×C137C2semidirect
16. C2×#14555510C22semidirect
17. C22×Dic274C22semidirect
25. C2×#14555499C22semidirect
26. C2×#14555499C22semidirect
27. C2×#14555499C22semidirect
28. C2×#14555499C22semidirect
30. C2×D1096C22semidirect
31. C2×D1096C22semidirect
33. #17984076C22semidirect
35. #17984067C22semidirect
40. #17984062C22semidirect
44. C4×Dic274C22semidirect
47. #17984022C22semidirect
48. #17984013C22semidirect
49. #17984013C22semidirect
51. #17984010C22semidirect
52. #17984010C22semidirect
53. #17984010C22semidirect
54. C2×M16×C137C22semidirect
55. C22×C8×C137C22semidirect
57. C4⋊C8×C137C22semidirect
58. C22⋊C8×C137C22semidirect
59. C22⋊C8×C137C22semidirect
60. #97×C137C22semidirect
95. #14555499D4semidirect
138. Dic274C2×D4semidirect
177. C8×C137C2×D4semidirect
183. C22×C137AGL(1,ℤ/8)semidirect
218. C137#3265semidirect
234. #435D137semidirect
251. C2×M16C2×D137semidirect
252. C22×C8C2×D137semidirect
270. C4⋊C8C2×D137semidirect
277. C22⋊C8C2×D137semidirect
278. C22⋊C8C2×D137semidirect
279. #97C2×D137semidirect
396. C8D4×D137semidirect
416. C22#17984116semidirect

Maximal subgroups

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

subgroupindex
C2×#179841162
C22×#145554992
#225780622
D4×D5482
#225777922
#225776492
#225776392
#225776192
#225776042
#225775562
#225775502
#225775442
#225775392
#225775052
#435×C1372
#3265137