Finite Group Databasebeta

← #31792265

Subgroups — #31792265

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×#17984114C211yesmaximal, characteristic, complemented
2C22×#14555495C211yesmaximal, characteristic, complemented
3#22578049C211yesmaximal, characteristic, complemented
4#22577790C211yesmaximal, characteristic, complemented
5#22577654C211yesmaximal, characteristic, complemented
6#22577630C211yesmaximal, characteristic, complemented
7#22577618C211yesmaximal, characteristic, complemented
8#22577595C211yesmaximal, characteristic, complemented
9#22577554C211yesmaximal, characteristic, complemented
10#22577546C211yesmaximal, characteristic, complemented
11#22577537C211yesmaximal, characteristic, complemented
12#22577536C211yesmaximal, characteristic, complemented
13#22577513C211yesmaximal, characteristic, complemented
14#22577475C211yesmaximal, characteristic, complemented
15D42×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic, complemented
17C22×D548C2211yescharacteristic, complemented
18#17984114—24no—
19C2×#14555498C2211yescharacteristic, complemented
20C2×#14555498C2211yescharacteristic, complemented
21C2×#14555498C2211yescharacteristic, complemented
22C2×#14555495—24no—
23C2×#14555495C2211yescharacteristic
24C2×#14555495C2211yescharacteristic
25C2×#14555495C2211yescharacteristic, complemented
26C2×#14555495C2211yescharacteristic, complemented
27C2×#14555495C2211yescharacteristic, complemented
28C2×#14555495C2211yescharacteristic, complemented
29C2×#14555494C2211yescharacteristic, complemented
30C22×#11764040C2211yescharacteristic, complemented
31#17984076C2211yescharacteristic, complemented
32#17984073—21no—
33#17984067C2211yescharacteristic, complemented
34#17984062C2211yescharacteristic, complemented
35#17984061—21no—
36#17984056—21no—
37C4×D548C2211yescharacteristic, complemented
38C22⋊C4×D137—21no—
39#17984019C2211yescharacteristic, complemented
40C4⋊#11764040C2211yescharacteristic, complemented
41#17984011C2211yescharacteristic, complemented
42#17984009C2211yescharacteristic, complemented
43#17984009C2211yescharacteristic, complemented
44#17984009C2211yescharacteristic, complemented
45#17984008C2211yescharacteristic, complemented
46#17984008C2211yescharacteristic, complemented
47C22⋊#11764040C2211yescharacteristic, complemented
48C22⋊#11764040C2211yescharacteristic, complemented
49#17983999C2211yescharacteristic, complemented
50C22×D4×C137—21no—
51C22×D4×C137C2211yescharacteristic
52C22×D4×C137C2211yescharacteristic, complemented
53#127×C137C2211yescharacteristic, complemented
54#121×C137—21no—
55#121×C137C2211yescharacteristic
56#121×C137C2211yescharacteristic, complemented
57#120×C137—21no—
58#120×C137—21no—
59C4×D4×C137C2211yescharacteristic
60C4×D4×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62#14555510—22no—
63#14555510—41no—
64C2×Dic274C2311yescharacteristic
65C2×D548—22no—
66C2×D548—21no—
67C2×D548—41no—
68C2×D548C2311yescharacteristic
69C2×D548C2311yescharacteristic
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74#14555498—22no—
75#14555498—22no—
76#14555498—41no—
77#14555495—22no—
78#14555495—22no—
79#14555495—22no—
80#14555495—22no—
81#14555495—41no—
82#14555495D414yescomplemented
83#14555494—22no—
84C2×#11764040—22no—
85C2×#11764040C2311yescharacteristic
86C2×#11764040C2311yescharacteristic
87C2×#11764040C2311yescharacteristic
88C2×#11764040C2311yescharacteristic
89C4⋊Dic137C2311yescharacteristic
90#14555489—21no—
91C22⋊Dic137—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96C24×C137—21no—
97C24×C137—21no—
98C2×D4×C137—22no—
99C2×D4×C137—22no—
100C2×D4×C137—21no—
101C2×D4×C137—21no—
102C2×D4×C137—21no—
103C2×D4×C137—21no—
104C2×D4×C137—21no—
105C2×D4×C137—21no—
106C2×D4×C137—41no—
107C2×D4×C137—41no—
108C2×D4×C137—44no—
109C2×D4×C137D411yescharacteristic
110C2×D4×C137C2311yescharacteristic
111C2×D4×C137C2311yescharacteristic
112C2×D4×C137C2311yescharacteristic
113C22×C4×C137—21no—
114C22×C4×C137C2311yescharacteristic
115C22×C4×C137C2311yescharacteristic
116C4⋊C4×C137D411yescharacteristic
117C4⋊C4×C137C2311yescharacteristic
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C22⋊C4×C137—21no—
121C22⋊C4×C137D411yescharacteristic
122C22⋊C4×C137D411yescharacteristic
123C42×C137C2311yescharacteristic
124C22×D137—21no—
125C22×D137—21no—
126C22×D137—42no—
127C22×D137—42no—
128C22×D137—81no—
129Dic274—21no—
130Dic274—21no—
131D548—21no—
132D548—21no—
133D548—21no—
134D548—42no—
135D548C2×D412yescomplemented
136C22⋊D137—42no—
137C22⋊D137—81no—
138C2×Dic137—21no—
139C2×Dic137—21no—
140C4×D137—42no—
141C4×D137—81no—
142#11764040—21no—
143#11764040—21no—
144#11764040—21no—
145#11764040C2×D412yescomplemented
146C23×C137—21no—
147C23×C137—21no—
148C23×C137—21no—
149C23×C137—42no—
150C23×C137—42no—
151C23×C137—42no—
152C23×C137—42no—
153C23×C137—81no—
154C23×C137—81no—
155C23×C137C2×D411yescharacteristic
156C23×C137C2×D411yescharacteristic
157D4×C137—22no—
158D4×C137—21no—
159D4×C137—21no—
160D4×C137—21no—
161D4×C137—21no—
162D4×C137—21no—
163D4×C137—42no—
164D4×C137—42no—
165D4×C137—42no—
166D4×C137—42no—
167D4×C137—42no—
168D4×C137—81no—
169D4×C137C2×D412yes—
170C2×C4×C137—21no—
171C2×C4×C137—21no—
172C2×C4×C137—21no—
173C2×C4×C137—21no—
174C2×C4×C137—21no—
175C2×C4×C137—42no—
176C2×C4×C137C2×D411yescharacteristic
177C2×C4×C137C2×D411yescharacteristic
178C2×C4×C137C2×D411yescharacteristic
179C2×C4×C137C2×D411yescharacteristic
180C2×C4×C137C2411yescharacteristic
181C22×C137—21no—
182C22×C137—21no—
183C22×C137—21no—
184C22×C137—22no—
185C22×C137—21no—
186C22×C137—21no—
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—22no—
200C2×D137—41no—
201C2×D137—41no—
202C2×D137—41no—
203C2×D137—41no—
204C2×D137—41no—
205C2×D137—84no—
206C4×C137—21no—
207C4×C137—21no—
208C4×C137—21no—
209C4×C137—21no—
210C4×C137—41no—
211C4×C137C22×D411yescharacteristic
212C4×C137H(5,𝔽2)11yescharacteristic
213Dic137—41no—
214Dic137—41no—
215C2×C137—22no—
216C2×C137—41no—
217C2×C137—42no—
218C2×C137—81no—
219C2×C137—81no—
220C2×C137D4211yescharacteristic
221C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C137#57611yescharacteristic
223D137—42no—
224D137—81no—
225C137#326411yescharacteristic, complemented, Sylow
226#3264—1371nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1371no—
228C22×D8—1371no—
229#546—1371no—
230D42D13711yescharacteristic, complemented
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—21no—
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—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#127C2×D13711yescharacteristic, complemented
261#124—1371no—
262#123—2741no—
263C22⋊Q8—1371no—
264#121—21no—
265#121—2741no—
266#121—1371no—
267#121C2×D13711yescharacteristic
268#121C2×D13711yescharacteristic, complemented
269#120—21no—
270#120—21no—
271#120—2741no—
272C4×D4—1371no—
273C4×D4C2×D13711yescharacteristic
274C4×D4C2×D13711yescharacteristic, complemented
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—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—44no—
304C2×D4—2742no—
305C2×D4—2741no—
306C2×D4—2741no—
307C2×D4—2741no—
308C2×D4—5481no—
309C2×D4—1371no—
310C2×D4—1371no—
311C2×D4C22⋊D13711yescharacteristic
312C2×D4C22×D13711yescharacteristic
313C2×D4C22×D13711yescharacteristic
314C2×D4C22×D13711yescharacteristic
315C22×C4—21no—
316C22×C4—2741no—
317C22×C4—2741no—
318C22×C4C22×D13711yescharacteristic
319C22×C4C22×D13711yescharacteristic
320SD16—2742no—
321SD16—2742no—
322SD16—5481no—
323D8—2742no—
324D8—2742no—
325D8—2742no—
326D8—2742no—
327D8—5481no—
328D8—1374no—
329M16—2742no—
330C2×C8—2742no—
331C2×C8—1371no—
332C2×C8—1371no—
333C2×C8—1371no—
334C2×C8—1371no—
335C4⋊C4—2741no—
336C4⋊C4—1371no—
337C4⋊C4C22⋊D13711yescharacteristic
338C4⋊C4C22×D13711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—2741no—
343C22⋊C4—2741no—
344C22⋊C4—2741no—
345C22⋊C4—2741no—
346C22⋊C4—2741no—
347C22⋊C4C22⋊D13711yescharacteristic
348C22⋊C4C22⋊D13711yescharacteristic
349C42C22×D13711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—2741no—
360C23—2741no—
361C23—5482no—
362C23—5482no—
363C23—10961no—
364C23C2×C22⋊D13711yescharacteristic
365C23C2×C22⋊D13711yescharacteristic
366Q8—2741no—
367Q8—2741no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—2741no—
381D4—2741no—
382D4—2741no—
383D4—5482no—
384D4—5482no—
385D4—10961no—
386D4—1372no—
387D4D4×D13712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—2741no—
395C2×C4—2741no—
396C2×C4—5482no—
397C2×C4—10961no—
398C2×C4C2×C22⋊D13711yescharacteristic
399C2×C4C2×C22⋊D13711yescharacteristic
400C2×C4C2×C22⋊D13711yescharacteristic
401C2×C4C2×C22⋊D13711yescharacteristic
402C2×C4C23×D13711yescharacteristic
403C8—2741no—
404C8—2741no—
405C8—2741no—
406C8—1372no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—2742no—
424C22—5481no—
425C22—5481no—
426C22—5481no—
427C22—5481no—
428C22—5481no—
429C22—10964no—
430C22#1798411412yescomplemented
431C22C22×C22⋊D13711yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—5481no—
438C4—5481no—
439C4C2×D4×D13711yescharacteristic
440C4#1798413311yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—5482no—
447C2—10961no—
448C2D4×C22⋊D13711yescharacteristic
449C2C2×#1798411411yescharacteristic
450C2#2257820611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingD42×C137C2
socleC22×C137C22×D4
hypercenterD42D137
hypocenterC137#3264
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#17984114C2semidirect
2. C22×#14555495C2semidirect
3. #22578049C2semidirect
4. #22577790C2semidirect
5. #22577654C2semidirect
6. #22577630C2semidirect
7. #22577618C2semidirect
8. #22577595C2semidirect
9. #22577554C2semidirect
10. #22577546C2semidirect
11. #22577537C2semidirect
12. #22577536C2semidirect
13. #22577513C2semidirect
14. #22577475C2semidirect
15. D42×C137C2semidirect
16. C2×#14555510C22semidirect
17. C22×D548C22semidirect
19. C2×#14555498C22semidirect
20. C2×#14555498C22semidirect
21. C2×#14555498C22semidirect
25. C2×#14555495C22semidirect
26. C2×#14555495C22semidirect
27. C2×#14555495C22semidirect
28. C2×#14555495C22semidirect
29. C2×#14555494C22semidirect
30. C22×#11764040C22semidirect
31. #17984076C22semidirect
33. #17984067C22semidirect
34. #17984062C22semidirect
37. C4×D548C22semidirect
39. #17984019C22semidirect
40. C4⋊#11764040C22semidirect
41. #17984011C22semidirect
42. #17984009C22semidirect
43. #17984009C22semidirect
44. #17984009C22semidirect
45. #17984008C22semidirect
46. #17984008C22semidirect
47. C22⋊#11764040C22semidirect
48. C22⋊#11764040C22semidirect
49. #17983999C22semidirect
52. C22×D4×C137C22semidirect
53. #127×C137C22semidirect
56. #121×C137C22semidirect
60. C4×D4×C137C22semidirect
82. #14555495D4semidirect
135. D548C2×D4semidirect
145. #11764040C2×D4semidirect
197. C22×C137AGL(1,ℤ/8)semidirect
225. C137#3264semidirect
230. D42D137semidirect
246. C22×D4C2×D137semidirect
260. #127C2×D137semidirect
268. #121C2×D137semidirect
274. C4×D4C2×D137semidirect
430. C22#17984114semidirect

Maximal subgroups

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

subgroupindex
C2×#179841142
C22×#145554952
#225780492
#225777902
#225776542
#225776302
#225776182
#225775952
#225775542
#225775462
#225775372
#225775362
#225775132
#225774752
D42×C1372
#3264137