Finite Group Databasebeta

← #31789594

Subgroups — #31789594

Order 17536.

548 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984115C211yesmaximal, characteristic
2C4×#14555511C211yesmaximal, characteristic, complemented
3C4×D4×D137C211yesmaximal, characteristic, complemented
4#22577516C211yesmaximal, characteristic, complemented
5#22577513C211yesmaximal, characteristic, complemented
6#22577501C211yesmaximal, characteristic, complemented
7SD16×Dic137C211yesmaximal, characteristic, complemented
8#22577488C211yesmaximal, characteristic, complemented
9#22577485C211yesmaximal, characteristic, complemented
10#22577483C211yesmaximal, characteristic, complemented
11#22577444C211yesmaximal, characteristic, complemented
12#22577406C211yesmaximal, characteristic, complemented
13#22577403C211yesmaximal, characteristic, complemented
14#22577355C211yesmaximal, characteristic, complemented
15#440×C137C211yesmaximal, characteristic, complemented
16C2×#14555511C2211yescharacteristic
17C2×D4×D137C2211yescharacteristic
18C22×C4×D137—21no—
19#17984115C418yescomplemented
20C2×#14555497C2211yescharacteristic
21C2×D1096C2211yescharacteristic
22C2×#14555495C2211yescharacteristic
23C2×#14555493C2211yescharacteristic
24C2×#14555493C2211yescharacteristic
25Q8×Dic137C2211yescharacteristic, complemented
26C4×D548—21no—
27C4×D548C2211yescharacteristic
28C4×D548C2211yescharacteristic, complemented
29D4×Dic137C2211yescharacteristic
30#17984050—21no—
31#17984050C2211yescharacteristic
32#17984050C2211yescharacteristic, complemented
33C4×C22⋊D137—21no—
34#17984048—21no—
35#17984046—21no—
36#17984046C2211yescharacteristic
37C4⋊C4×D137C2211yescharacteristic
38C22⋊C4×D137—21no—
39C42×D137—21no—
40C42×D137C2211yescharacteristic
41#17984021C2211yescharacteristic, complemented
42#17984020C2211yescharacteristic, complemented
43#17984019C2211yescharacteristic, complemented
44#17984012C2211yescharacteristic, complemented
45#17984010C2211yescharacteristic
46#17984010C2211yescharacteristic, complemented
47#17984009C2211yescharacteristic
48#17984008C2211yescharacteristic
49#17984008C2211yescharacteristic, complemented
50#17983999C2211yescharacteristic, complemented
51C8×Dic137C2211yescharacteristic, complemented
52C8×Dic137C2211yescharacteristic, complemented
53C2×SD16×C137C2211yescharacteristic
54C4×Q8×C137C2211yescharacteristic, complemented
55C4×D4×C137C2211yescharacteristic
56#107×C137C2211yescharacteristic, complemented
57#103×C137C2211yescharacteristic, complemented
58#102×C137C2211yescharacteristic
59#97×C137C2211yescharacteristic, complemented
60C23×D137—21no—
61#14555511—22no—
62#14555511C2×C414yescomplemented
63C2×D548—21no—
64C2×D548C2311yescharacteristic
65C2×D548C2311yescharacteristic
66C2×C22⋊D137—21no—
67D4×D137—22no—
68D4×D137C2×C414yescomplemented
69C22×Dic137—21no—
70C2×C4×D137—21no—
71C2×C4×D137—21no—
72C2×C4×D137—21no—
73C2×C4×D137—21no—
74C2×C4×D137—21no—
75C2×C4×D137—21no—
76C2×C4×D137—44no—
77C2×C4×D137D411yescharacteristic
78C2×C4×D137D411yescharacteristic
79C2×C4×D137C2311yescharacteristic
80#14555497C2×C414yescomplemented
81D1096C2×C414yescomplemented
82#14555495C2×C414yescomplemented
83#14555493—22no—
84#14555493C2×C414yescomplemented
85C2×#11764040C2311yescharacteristic
86C2×#11764040C2311yescharacteristic
87C4⋊Dic137—21no—
88C4⋊Dic137C2311yescharacteristic
89C4⋊Dic137C2311yescharacteristic
90#14555489—21no—
91C22⋊Dic137—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95#14555487—21no—
96#14555487—21no—
97C4×Dic137—21no—
98C4×Dic137—21no—
99C4×Dic137D411yescharacteristic
100C4×Dic137D411yescharacteristic
101C4×Dic137C2311yescharacteristic
102C2×Q8×C137C2311yescharacteristic
103C2×D4×C137C2311yescharacteristic
104C22×C4×C137—21no—
105SD16×C137C2×C414yescomplemented
106C2×C8×C137C2311yescharacteristic
107C2×C8×C137C2311yescharacteristic
108C4⋊C4×C137—21no—
109C4⋊C4×C137C2311yescharacteristic
110C4⋊C4×C137C2311yescharacteristic
111C22⋊C4×C137—21no—
112C42×C137—21no—
113C42×C137C2311yescharacteristic
114C22×D137—21no—
115C22×D137—21no—
116C22×D137—24no—
117C22×D137—42no—
118C22×D137—42no—
119C22×D137C2×D411yescharacteristic
120D548—24no—
121D548—21no—
122D548—21no—
123D548C22×C412yes—
124D548C22×C412yescomplemented
125C22⋊D137—24no—
126C2×Dic137—21no—
127C2×Dic137—21no—
128C2×Dic137—42no—
129C2×Dic137C2×D411yescharacteristic
130C2×Dic137C2×D411yescharacteristic
131C2×Dic137C2×D411yescharacteristic
132C4×D137—24no—
133C4×D137—22no—
134C4×D137—41no—
135C4×D137—42no—
136C4×D137—42no—
137C4×D137—42no—
138C4×D137—42no—
139C4×D137—42no—
140C4×D137C22×C412yes—
141C4×D137#4112yes—
142#11764040—21no—
143#11764040C22×C412yescomplemented
144C23×C137—21no—
145Q8×C137—21no—
146Q8×C137C22×C412yescomplemented
147D4×C137—21no—
148D4×C137C22×C412yes—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—42no—
153C2×C4×C137C2×D411yescharacteristic
154C2×C4×C137C2×D411yescharacteristic
155C2×C4×C137C2411yescharacteristic
156C8×C137—21no—
157C8×C137C22×C412yescomplemented
158C22×C137—22no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137C22×D411yescharacteristic
162C2×D137—21no—
163C2×D137—21no—
164C2×D137—22no—
165C2×D137—22no—
166C2×D137—48no—
167C2×D137—41no—
168C2×D137—41no—
169C2×D137—41no—
170C2×D137—41no—
171C2×D137C4×D412yes—
172C4×C137—22no—
173C4×C137—21no—
174C4×C137—21no—
175C4×C137—41no—
176C4×C137—41no—
177C4×C137C23×C411yescharacteristic
178C4×C137C2×#4111yescharacteristic
179Dic137—21no—
180Dic137—41no—
181Dic137—41no—
182Dic137C4×D412yes—
183Dic137AGL(1,ℤ/8)12yescomplemented
184C2×C137—42no—
185C2×C137C2×C4×D411yescharacteristic
186C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
187C2×C137#57511yescharacteristic
188D137—22no—
189D137—42no—
190D137—42no—
191C137C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
192C4×AGL(1,ℤ/8)—1371nomaximal, Sylow
193C2×AGL(1,ℤ/8)—1371no—
194C4×#41—1371no—
195C2×C4×D4—1371no—
196#442—1371no—
197#442—1371no—
198#440—1371no—
199#440D13711yescharacteristic, complemented
200C4×SD16—1371no—
201C4×SD16—1371no—
202C4×D8—1371no—
203C4×D8—1371no—
204#428—1371no—
205#418—1371no—
206#417—1371no—
207C4×M16—1371no—
208C2×#41—1371no—
209C22×D4—1371no—
210C23×C4—2741no—
211AGL(1,ℤ/8)—1378no—
212C2×SD16—1371no—
213C2×SD16C2×D13711yescharacteristic
214C2×D8—1371no—
215C2×D8—1371no—
216C2×M16—1371no—
217C2×M16—1371no—
218C4×Q8—1371no—
219C4×Q8C2×D13711yescharacteristic, complemented
220C4×D4—2741no—
221C4×D4—2741no—
222C4×D4—2741no—
223C4×D4—2741no—
224C4×D4—1371no—
225C4×D4—1371no—
226C4×D4—1371no—
227C4×D4—1371no—
228C4×D4—1371no—
229C4×D4C2×D13711yescharacteristic
230#117—2741no—
231#117—1371no—
232C2×C4⋊C4—1371no—
233C2×C22⋊C4—2741no—
234C2×C42—2741no—
235C2×C42—1371no—
236#107—1371no—
237#107C2×D13711yescharacteristic, complemented
238#106—1371no—
239#106—1371no—
240#103—1371no—
241#103C2×D13711yescharacteristic, complemented
242#102—1371no—
243#102—1371no—
244#102—1371no—
245#102—1371no—
246#102—1371no—
247#102C2×D13711yescharacteristic
248#97—1371no—
249#97C2×D13711yescharacteristic, complemented
250C4×C8—1371no—
251C4×C8—1371no—
252C24—2741no—
253#41—2742no—
254#41—1374no—
255C2×Q8C22×D13711yescharacteristic
256C2×D4—2741no—
257C2×D4—2742no—
258C2×D4—2741no—
259C2×D4—1374no—
260C2×D4—1371no—
261C2×D4—1371no—
262C2×D4C22×D13711yescharacteristic
263C22×C4—21no—
264C22×C4—2741no—
265C22×C4—2741no—
266C22×C4—2741no—
267C22×C4—2741no—
268C22×C4—2741no—
269C22×C4—2741no—
270C22×C4—2741no—
271C22×C4—5484no—
272C22×C4—1371no—
273C22×C4—1371no—
274C22×C4—1371no—
275SD16—1374no—
276SD16C4×D13714yescomplemented
277D8—1374no—
278D8—1374no—
279M16—2742no—
280M16—1374no—
281C2×C8—1371no—
282C2×C8—1371no—
283C2×C8C22×D13711yescharacteristic
284C2×C8C22×D13711yescharacteristic
285C4⋊C4—21no—
286C4⋊C4—2741no—
287C4⋊C4—2741no—
288C4⋊C4—1371no—
289C4⋊C4—1371no—
290C4⋊C4C22×D13711yescharacteristic
291C4⋊C4C22×D13711yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—2741no—
294C22⋊C4—2741no—
295C22⋊C4—2741no—
296C22⋊C4—2741no—
297C22⋊C4—2741no—
298C22⋊C4—2741no—
299C42—21no—
300C42—2741no—
301C42—2741no—
302C42—1371no—
303C42—1371no—
304C42—1371no—
305C42C22×D13711yescharacteristic
306C23—21no—
307C23—2744no—
308C23—2741no—
309C23—2741no—
310C23—5482no—
311C23—5482no—
312C23—1371no—
313Q8—21no—
314Q8C2×C4×D13712yescomplemented
315D4—21no—
316D4—2744no—
317D4—2744no—
318D4—2741no—
319D4—2741no—
320D4—1372no—
321D4—1372no—
322D4C2×C4×D13712yes—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—2744no—
328C2×C4—2742no—
329C2×C4—2741no—
330C2×C4—2741no—
331C2×C4—5481no—
332C2×C4—5482no—
333C2×C4—5482no—
334C2×C4—5482no—
335C2×C4—5482no—
336C2×C4—5482no—
337C2×C4—5482no—
338C2×C4—1371no—
339C2×C4—1371no—
340C2×C4—1371no—
341C2×C4—1372no—
342C2×C4—1372no—
343C2×C4D4×D13711yescharacteristic
344C2×C4D4×D13711yescharacteristic
345C2×C4C23×D13711yescharacteristic
346C8—21no—
347C8—2741no—
348C8—1372no—
349C8C2×C4×D13712yescomplemented
350C22—22no—
351C22—41no—
352C22—41no—
353C22—2741no—
354C22—2741no—
355C22—2742no—
356C22—2742no—
357C22—5488no—
358C22—5481no—
359C22—5481no—
360C22—5481no—
361C22—5481no—
362C22—1372no—
363C22C2×D4×D13711yescharacteristic
364C4—21no—
365C4—21no—
366C4—22no—
367C4—41no—
368C4—41no—
369C4—2741no—
370C4—5481no—
371C4—5481no—
372C4—1372no—
373C4—1372no—
374C4C22×C4×D13711yescharacteristic
375C4#41×D13711yescharacteristic
376C2—42no—
377C2—2742no—
378C2—5482no—
379C2—5482no—
380C2C4×D4×D13711yescharacteristic
381C2AGL(1,ℤ/8)×D13711yescharacteristic
382C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutatorC4×C137C23×C4
FrattiniC2×C4C23×D137
Fitting#440×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984115C2
cosocleC2×C4×C137C24
hypercenter#440D137
hypocenterC137C4×AGL(1,ℤ/8)
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×#14555511C2semidirect
3. C4×D4×D137C2semidirect
4. #22577516C2semidirect
5. #22577513C2semidirect
6. #22577501C2semidirect
7. SD16×Dic137C2semidirect
8. #22577488C2semidirect
9. #22577485C2semidirect
10. #22577483C2semidirect
11. #22577444C2semidirect
12. #22577406C2semidirect
13. #22577403C2semidirect
14. #22577355C2semidirect
15. #440×C137C2semidirect
19. #17984115C4semidirect
25. Q8×Dic137C22semidirect
28. C4×D548C22semidirect
32. #17984050C22semidirect
41. #17984021C22semidirect
42. #17984020C22semidirect
43. #17984019C22semidirect
44. #17984012C22semidirect
46. #17984010C22semidirect
49. #17984008C22semidirect
50. #17983999C22semidirect
51. C8×Dic137C22semidirect
52. C8×Dic137C22semidirect
54. C4×Q8×C137C22semidirect
56. #107×C137C22semidirect
57. #103×C137C22semidirect
59. #97×C137C22semidirect
62. #14555511C2×C4semidirect
68. D4×D137C2×C4semidirect
80. #14555497C2×C4semidirect
81. D1096C2×C4semidirect
82. #14555495C2×C4semidirect
84. #14555493C2×C4semidirect
105. SD16×C137C2×C4semidirect
124. D548C22×C4semidirect
143. #11764040C22×C4semidirect
146. Q8×C137C22×C4semidirect
157. C8×C137C22×C4semidirect
183. Dic137AGL(1,ℤ/8)semidirect
191. C137C4×AGL(1,ℤ/8)semidirect
199. #440D137semidirect
219. C4×Q8C2×D137semidirect
237. #107C2×D137semidirect
241. #103C2×D137semidirect
249. #97C2×D137semidirect
276. SD16C4×D137semidirect
314. Q8C2×C4×D137semidirect
349. C8C2×C4×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841152
C4×#145555112
C4×D4×D1372
#225775162
#225775132
#225775012
SD16×Dic1372
#225774882
#225774852
#225774832
#225774442
#225774062
#225774032
#225773552
#440×C1372
C4×AGL(1,ℤ/8)137