Finite Group Databasebeta

← #31790287

Subgroups — #31790287

Order 17536.

526 conjugacy classes of subgroups.

The subgroup lattice has 424 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#22577966C211yesmaximal, characteristic, complemented
4#22577875C211yesmaximal, characteristic, complemented
5#22577619C211yesmaximal, characteristic, complemented
6#22577618C211yesmaximal, characteristic, complemented
7#22577617C211yesmaximal, characteristic, complemented
8#22577616C211yesmaximal, characteristic, complemented
9#22577597C211yesmaximal, characteristic, complemented
10#22577596C211yesmaximal, characteristic, complemented
11#22577595C211yesmaximal, characteristic, complemented
12#22577428C211yesmaximal, characteristic, complemented
13#22577406C211yesmaximal, characteristic, complemented
14#22577396C211yesmaximal, characteristic, complemented
15#459×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×D4×D137C2211yescharacteristic
18C2×D4×D137C2211yescharacteristic
19#17984112—24no—
20C2×#14555499C2211yescharacteristic, complemented
21C2×#14555498C2211yescharacteristic, complemented
22C2×D1096C2211yescharacteristic
23C2×#14555495C2211yescharacteristic
24C2×#14555495C2211yescharacteristic
25C2×#14555495C2211yescharacteristic, complemented
26D8×D137—24no—
27C2×#14555493C2211yescharacteristic
28C2×C8×D137C2211yescharacteristic
29#17984084C2211yescharacteristic, complemented
30#17984076C2211yescharacteristic, complemented
31#17984075C2211yescharacteristic, complemented
32#17984060—21no—
33#17984060—21no—
34#17984056—21no—
35#17984056—21no—
36C4×D548C2211yescharacteristic, complemented
37D4×Dic137C2211yescharacteristic, complemented
38#17984050C2211yescharacteristic, complemented
39C4×C22⋊D137—21no—
40#17984048—21no—
41#17984046C2211yescharacteristic
42#17984045C2211yescharacteristic
43C2×#14555487—21no—
44C22⋊C4×D137—21no—
45C4⋊#11764040C2211yescharacteristic, complemented
46#17984017C2211yescharacteristic, complemented
47#17984017C2211yescharacteristic, complemented
48#17984013C2211yescharacteristic, complemented
49#17984011C2211yescharacteristic, complemented
50#17984010C2211yescharacteristic, complemented
51#17984009C2211yescharacteristic, complemented
52#17984009C2211yescharacteristic, complemented
53#17984008C2211yescharacteristic, complemented
54C2×D8×C137C2211yescharacteristic
55C2×D8×C137C2211yescharacteristic, complemented
56#127×C137C2211yescharacteristic, complemented
57C4×D4×C137C2211yescharacteristic, complemented
58C4⋊C8×C137C2211yescharacteristic, complemented
59#102×C137C2211yescharacteristic, complemented
60#102×C137C2211yescharacteristic, complemented
61C23×D137—21no—
62C23×D137—21no—
63#14555509—22no—
64#14555509—22no—
65C2×Dic274C2311yescharacteristic
66C2×D548C2311yescharacteristic
67C2×D548C2311yescharacteristic
68C2×C22⋊D137—21no—
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C2×C22⋊D137—21no—
72C2×C22⋊D137—21no—
73D4×D137—22no—
74D4×D137—22no—
75D4×D137—22no—
76D4×D137D414yescomplemented
77C22×Dic137—21no—
78C22×Dic137—21no—
79C2×C4×D137—21no—
80C2×C4×D137—21no—
81C2×C4×D137C2311yescharacteristic
82#14555499—22no—
83#14555498—22no—
84D1096—22no—
85#14555495—22no—
86#14555495—22no—
87#14555495—22no—
88#14555493—22no—
89C2×#11764040C2311yescharacteristic
90C2×#11764040C2311yescharacteristic
91C8×D137—22no—
92C4⋊Dic137C2311yescharacteristic
93#14555489D411yescharacteristic
94#14555489D411yescharacteristic
95C22⋊Dic137—21no—
96C22⋊Dic137—21no—
97C22⋊Dic137—21no—
98#14555487—21no—
99#14555487—21no—
100#14555487—21no—
101#14555487—21no—
102#14555487—21no—
103#14555487D411yescharacteristic
104#14555487D411yescharacteristic
105C4×Dic137—21no—
106C4×Dic137C2311yescharacteristic
107C2×D4×C137—21no—
108C2×D4×C137—21no—
109C2×D4×C137C2311yescharacteristic
110C2×D4×C137C2311yescharacteristic
111C2×D4×C137C2311yescharacteristic
112C22×C4×C137—21no—
113D8×C137—22no—
114D8×C137—22no—
115C2×C8×C137C2311yescharacteristic
116C2×C8×C137C2311yescharacteristic
117C4⋊C4×C137C2311yescharacteristic
118C22⋊C4×C137—21no—
119C42×C137C2311yescharacteristic
120C22×D137—21no—
121C22×D137—24no—
122C22×D137—21no—
123C22×D137—42no—
124C22×D137—81no—
125C22×D137—81no—
126C22×D137—81no—
127C22×D137—81no—
128C22×D137C2×D411yescharacteristic
129Dic274—21no—
130Dic274—21no—
131D548—21no—
132D548—21no—
133D548—21no—
134D548C2×D412yes—
135C22⋊D137—24no—
136C22⋊D137—42no—
137C22⋊D137—42no—
138C22⋊D137—81no—
139C22⋊D137—81no—
140C2×Dic137—21no—
141C2×Dic137—21no—
142C2×Dic137—42no—
143C2×Dic137—81no—
144C2×Dic137C2×D411yescharacteristic
145C2×Dic137C2×D411yescharacteristic
146C2×Dic137C2×D411yescharacteristic
147C4×D137—81no—
148C4×D137—81no—
149C4×D137C2×D412yes—
150C4×D137#4112yes—
151#11764040—21no—
152#11764040—21no—
153C23×C137—21no—
154C23×C137—21no—
155C23×C137—21no—
156D4×C137—21no—
157D4×C137—21no—
158D4×C137—21no—
159D4×C137—21no—
160D4×C137—21no—
161D4×C137—81no—
162D4×C137—81no—
163D4×C137C2×D412yes—
164C2×C4×C137—21no—
165C2×C4×C137—81no—
166C2×C4×C137C2×D411yescharacteristic
167C2×C4×C137C2×D411yescharacteristic
168C2×C4×C137C2411yescharacteristic
169C8×C137—21no—
170C8×C137—21no—
171C22×C137—22no—
172C22×C137—41no—
173C22×C137—41no—
174C22×C137—41no—
175C22×C137—41no—
176C22×C137—41no—
177C22×C137—41no—
178C22×C137—41no—
179C22×C137—41no—
180C22×C137C22×D411yescharacteristic
181C2×D137—21no—
182C2×D137—21no—
183C2×D137—22no—
184C2×D137—41no—
185C2×D137—41no—
186C2×D137—41no—
187C2×D137—41no—
188C2×D137—41no—
189C2×D137—84no—
190C2×D137—84no—
191C2×D137#12112yes—
192C4×C137—21no—
193C4×C137—21no—
194C4×C137—41no—
195C4×C137C22×D411yescharacteristic
196C4×C137C2×#4111yescharacteristic
197Dic137—21no—
198Dic137—21no—
199Dic137—41no—
200Dic137—41no—
201Dic137#12112yes—
202C2×C137—42no—
203C2×C137—81no—
204C2×C137—81no—
205C2×C137C2×#12111yescharacteristic
206C2×C137#57611yescharacteristic
207C2×C137#57611yescharacteristic
208D137—22no—
209D137—42no—
210D137—81no—
211C137#304111yescharacteristic, complemented, Sylow
212#3041—1371nomaximal, Sylow
213C2×AGL(1,ℤ/8)—1371no—
214C22×D8—1371no—
215#535—1371no—
216#518—1371no—
217#463—1371no—
218#463—1371no—
219#463—1371no—
220#463—1371no—
221#459—1371no—
222#459—1371no—
223#459—1371no—
224#459D13711yescharacteristic, complemented
225#424—1371no—
226#418—1371no—
227#416—1371no—
228C2×#41—1371no—
229C22×D4—1371no—
230C22×D4—1371no—
231AGL(1,ℤ/8)—2744no—
232C2×SD16—1371no—
233C2×SD16—1371no—
234C2×D8—2744no—
235C2×D8—1371no—
236C2×D8—1371no—
237C2×D8—1371no—
238C2×D8—1371no—
239C2×D8C2×D13711yescharacteristic
240C2×D8C2×D13711yescharacteristic, complemented
241C2×M16—1371no—
242C22×C8—1371no—
243#127—1371no—
244#127C2×D13711yescharacteristic, complemented
245#124—1371no—
246#124—1371no—
247#121—2741no—
248#121—2741no—
249#120—2741no—
250#120—2741no—
251C4×D4—2741no—
252C4×D4—2741no—
253C4×D4—1371no—
254C4×D4—1371no—
255C4×D4—1371no—
256C4×D4C2×D13711yescharacteristic, complemented
257#117—1371no—
258#117—1371no—
259C2×C22⋊C4—2741no—
260C2×C22⋊C4—2741no—
261C4⋊C8—1371no—
262C4⋊C8—1371no—
263C4⋊C8—1371no—
264C4⋊C8C2×D13711yescharacteristic, complemented
265#103—1371no—
266#103—1371no—
267#102—1371no—
268#102—1371no—
269#102—1371no—
270#102—1371no—
271#102C2×D13711yescharacteristic, complemented
272#102C2×D13711yescharacteristic, complemented
273C24—2741no—
274C24—2741no—
275#41—2742no—
276#41—2742no—
277C2×Q8—1371no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—2741no—
281C2×D4—2742no—
282C2×D4—2742no—
283C2×D4—2742no—
284C2×D4—2741no—
285C2×D4—2741no—
286C2×D4—2741no—
287C2×D4—2741no—
288C2×D4—1374no—
289C2×D4—1371no—
290C2×D4—1371no—
291C2×D4C22×D13711yescharacteristic
292C2×D4C22×D13711yescharacteristic
293C2×D4C22×D13711yescharacteristic
294C22×C4—21no—
295C22×C4—2741no—
296C22×C4—2741no—
297C22×C4—2741no—
298C22×C4—2741no—
299C22×C4—1371no—
300SD16—2742no—
301SD16—2742no—
302D8—22no—
303D8—22no—
304D8—2742no—
305D8—2742no—
306D8—2742no—
307D8—2742no—
308M16—2742no—
309C2×C8—2742no—
310C2×C8—1371no—
311C2×C8—1371no—
312C2×C8C22×D13711yescharacteristic
313C2×C8C22×D13711yescharacteristic
314C4⋊C4—1371no—
315C4⋊C4—1371no—
316C4⋊C4—1371no—
317C4⋊C4C22×D13711yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—2741no—
320C22⋊C4—2741no—
321C22⋊C4—2741no—
322C22⋊C4—2741no—
323C22⋊C4—2741no—
324C22⋊C4—2741no—
325C22⋊C4—2741no—
326C22⋊C4—2741no—
327C22⋊C4—1371no—
328C22⋊C4—1371no—
329C42—2741no—
330C42—1371no—
331C42C22×D13711yescharacteristic
332C23—21no—
333C23—21no—
334C23—21no—
335C23—2744no—
336C23—2741no—
337C23—2741no—
338C23—5482no—
339C23—10961no—
340C23—10961no—
341C23—10961no—
342C23—10961no—
343C23—1371no—
344Q8—2741no—
345Q8—2741no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—81no—
352D4—81no—
353D4—2744no—
354D4—2741no—
355D4—2741no—
356D4—2741no—
357D4—5482no—
358D4—5482no—
359D4—10961no—
360D4—10961no—
361D4—1372no—
362D4D4×D13712yes—
363C2×C4—21no—
364C2×C4—81no—
365C2×C4—2741no—
366C2×C4—2741no—
367C2×C4—5482no—
368C2×C4—10961no—
369C2×C4—10961no—
370C2×C4—10961no—
371C2×C4—1371no—
372C2×C4—1371no—
373C2×C4—1371no—
374C2×C4—1372no—
375C2×C4—1372no—
376C2×C4D4×D13711yescharacteristic
377C2×C4D4×D13711yescharacteristic
378C2×C4C23×D13711yescharacteristic
379C8—21no—
380C8—21no—
381C8—2741no—
382C8—2741no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—2741no—
393C22—2741no—
394C22—2742no—
395C22—5481no—
396C22—5481no—
397C22—5481no—
398C22—5481no—
399C22—5481no—
400C22—10964no—
401C22—10964no—
402C22—1372no—
403C22C2×D4×D13711yescharacteristic
404C4—21no—
405C4—21no—
406C4—41no—
407C4—2741no—
408C4—2741no—
409C4—5481no—
410C4—5481no—
411C4—1372no—
412C4C2×D4×D13711yescharacteristic
413C4#41×D13711yescharacteristic
414C2—42no—
415C2—81no—
416C2—81no—
417C2—2742no—
418C2—5482no—
419C2—10961no—
420C2#121×D13711yescharacteristic
421C2#2257820411yescharacteristic
422C2#2257820711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#459×C137C2
socleC22×C137C22×D4
hypercenter#459D137
hypocenterC137#3041
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#17984112C2semidirect
2. C2×D8×D137C2semidirect
3. #22577966C2semidirect
4. #22577875C2semidirect
5. #22577619C2semidirect
6. #22577618C2semidirect
7. #22577617C2semidirect
8. #22577616C2semidirect
9. #22577597C2semidirect
10. #22577596C2semidirect
11. #22577595C2semidirect
12. #22577428C2semidirect
13. #22577406C2semidirect
14. #22577396C2semidirect
15. #459×C137C2semidirect
20. C2×#14555499C22semidirect
21. C2×#14555498C22semidirect
25. C2×#14555495C22semidirect
29. #17984084C22semidirect
30. #17984076C22semidirect
31. #17984075C22semidirect
36. C4×D548C22semidirect
37. D4×Dic137C22semidirect
38. #17984050C22semidirect
45. C4⋊#11764040C22semidirect
46. #17984017C22semidirect
47. #17984017C22semidirect
48. #17984013C22semidirect
49. #17984011C22semidirect
50. #17984010C22semidirect
51. #17984009C22semidirect
52. #17984009C22semidirect
53. #17984008C22semidirect
55. C2×D8×C137C22semidirect
56. #127×C137C22semidirect
57. C4×D4×C137C22semidirect
58. C4⋊C8×C137C22semidirect
59. #102×C137C22semidirect
60. #102×C137C22semidirect
76. D4×D137D4semidirect
211. C137#3041semidirect
224. #459D137semidirect
240. C2×D8C2×D137semidirect
244. #127C2×D137semidirect
256. C4×D4C2×D137semidirect
264. C4⋊C8C2×D137semidirect
271. #102C2×D137semidirect
272. #102C2×D137semidirect

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
#225779662
#225778752
#225776192
#225776182
#225776172
#225776162
#225775972
#225775962
#225775952
#225774282
#225774062
#225773962
#459×C1372
#3041137