Finite Group Databasebeta

← #31797546

Subgroups — #31797546

Order 17536.

506 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578472C211yesmaximal, characteristic, complemented
2#22578470C211yesmaximal, characteristic, complemented
3#22578444C212yesmaximal, complemented
4#22578442C212yesmaximal, complemented
5SD16×#10587739C212yesmaximal, complemented
6D8×#10587739C212yesmaximal, complemented
7#22578414C211yesmaximal, characteristic, complemented
8#22578393C211yesmaximal, characteristic, complemented
9C2×#17984150C211yesmaximal, characteristic, complemented
10#22578369C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×D137C211yesmaximal, characteristic
12C2×#14555518C2211yescharacteristic
13Q8×#10587739C2212yescomplemented
14D4×#10587739—21no—
15D4×#10587739—21no—
16D4×#10587739—22no—
17D4×#10587739C2212yes—
18D4×#10587739C2212yes—
19D4×#10587739C2212yescomplemented
20#17984167—21no—
21#17984167C2211yescharacteristic
22#17984167C2211yescharacteristic
23C2×C4⋊#10587739—21no—
24C2×C4⋊#10587739C2211yescharacteristic
25C2×C22⋊#10587739—22no—
26#17984157C2212yescomplemented
27#17984156C2212yescomplemented
28#17984151C2211yescharacteristic, complemented
29#17984151C2211yescharacteristic, complemented
30#17984150C2212yes—
31#17984150C2212yes—
32#17984150C2211yescharacteristic, complemented
33#17984150C2211yescharacteristic, complemented
34#17984142C2212yescomplemented
35C8×#10587739C2212yescomplemented
36#41×D137C2211yescharacteristic
37C2×D4×D137C2211yescharacteristic
38#17984116C411yescharacteristic, complemented
39#17984115C412yescomplemented
40#17984114C411yescharacteristic, complemented
41#17984113C411yescharacteristic, complemented
42#17984112C412yescomplemented
43SD16×D137C2212yes—
44D8×D137C2212yes—
45M16×D137C2211yescharacteristic
46AGL(1,ℤ/8)×C137C411yescharacteristic, complemented
47C22×#10587739—22no—
48C22×#10587739—22no—
49C22×#10587739—22no—
50C22×#10587739—21no—
51#14555518—21no—
52#14555518C2311yescharacteristic
53#14555518C2311yescharacteristic
54C2×#11764047—21no—
55C4⋊#10587739—22no—
56C4⋊#10587739—21no—
57C4⋊#10587739D411yescharacteristic
58C4⋊#10587739D411yescharacteristic
59C4⋊#10587739C2312yes—
60C4⋊#10587739C2311yescharacteristic
61C4⋊#10587739C2311yescharacteristic
62C22⋊#10587739—22no—
63C22⋊#10587739—22no—
64C22⋊#10587739—22no—
65C22⋊#10587739—21no—
66C22⋊#10587739D412yes—
67C4×#10587739—22no—
68C4×#10587739—21no—
69C4×#10587739C2312yes—
70C23×D137—21no—
71#14555511C2×C411yescharacteristic, complemented
72#14555510—21no—
73#14555510C2×C411yescharacteristic, complemented
74#14555509—21no—
75#14555509C2×C411yescharacteristic, complemented
76Q8×D137C2311yescharacteristic
77C2×D548C2×C411yescharacteristic, complemented
78C2×C22⋊D137—21no—
79D4×D137—21no—
80D4×D137—21no—
81D4×D137—21no—
82D4×D137C2×C412yescomplemented
83D4×D137C2311yescharacteristic
84D4×D137C2312yes—
85C2×C4×D137—21no—
86C2×C4×D137C2311yescharacteristic
87#14555499C2×C412yescomplemented
88#14555498C2×C412yescomplemented
89#14555497C2×C412yescomplemented
90D1096C2×C412yescomplemented
91#14555495C2×C412yescomplemented
92#14555495C2×C412yescomplemented
93#14555494C2×C411yescharacteristic, complemented
94#14555493C2×C412yescomplemented
95C8×D137C2312yes—
96#41×C137C2×C411yescharacteristic, complemented
97C2×D4×C137C2×C411yescharacteristic, complemented
98SD16×C137C2×C412yescomplemented
99D8×C137C2×C412yescomplemented
100M16×C137C2×C411yescharacteristic, complemented
101C2×#10587739—21no—
102C2×#10587739—21no—
103C2×#10587739—22no—
104C2×#10587739—82no—
105C2×#10587739—82no—
106C2×#10587739—82no—
107C2×#10587739—81no—
108C2×#10587739C2×D412yes—
109C2×#10587739C2×D412yes—
110#11764047—21no—
111#11764047—21no—
112C22×D137—21no—
113C22×D137—21no—
114C22×D137—22no—
115C22×D137—22no—
116C22×D137—81no—
117C22×D137—81no—
118C22×D137C2×D411yescharacteristic
119Dic274—21no—
120Dic274C22×C411yescharacteristic
121D548—21no—
122D548—21no—
123D548C22×C411yescharacteristic
124D548C22×C412yes—
125C22⋊D137—21no—
126C22⋊D137—21no—
127C22⋊D137—22no—
128C22⋊D137—22no—
129C22⋊D137—21no—
130C2×Dic137—21no—
131C2×Dic137#4111yescharacteristic
132C4×D137—21no—
133C4×D137—21no—
134C4×D137—21no—
135C4×D137C22×C411yescharacteristic
136C4×D137C2×D411yescharacteristic
137C4×D137#4111yescharacteristic
138C4×D137C2411yescharacteristic
139#11764040C22×C412yes—
140C23×C137—21no—
141Q8×C137C22×C411yescharacteristic
142D4×C137—21no—
143D4×C137—21no—
144D4×C137C22×C411yescharacteristic
145D4×C137C22×C412yes—
146C2×C4×C137—21no—
147C2×C4×C137C22×C411yescharacteristic
148C8×C137C22×C412yes—
149C22×C137—21no—
150C22×C137—22no—
151C22×C137—81no—
152C22×C137C4×D411yescharacteristic
153C2×D137—21no—
154C2×D137—21no—
155C2×D137—21no—
156C2×D137—22no—
157C2×D137—42no—
158C2×D137—41no—
159C2×D137—41no—
160C2×D137—42no—
161C2×D137—82no—
162C2×D137—81no—
163C2×D137C4×D411yescharacteristic
164C2×D137C22×D411yescharacteristic
165#10587739—22no—
166#10587739—22no—
167#10587739—41no—
168#10587739—42no—
169#10587739—41no—
170C4×C137—21no—
171C4×C137C4×D411yescharacteristic
172C4×C137C23×C411yescharacteristic
173Dic137—21no—
174Dic137C4×D411yescharacteristic
175Dic137C2×#4111yescharacteristic
176C2×C137—21no—
177C2×C137—42no—
178C2×C137—41no—
179C2×C137C2×C4×D411yescharacteristic
180D137—21no—
181D137—41no—
182D137—42no—
183D137#57611yescharacteristic
184D137#57711yescharacteristic
185C137#294611yescharacteristic, complemented, Sylow
186#2946—1371nomaximal, Sylow
187C2×AGL(1,ℤ/8)—1371no—
188#520—1371no—
189#518—1371no—
190#442—1372no—
191#440—1372no—
192C4×SD16—1372no—
193C4×D8—1372no—
194#428—1371no—
195#416—1371no—
196C2×#102—1371no—
197#405—1371no—
198C2×#41—1371no—
199C22×D4—1371no—
200AGL(1,ℤ/8)—1371no—
201AGL(1,ℤ/8)—1371no—
202AGL(1,ℤ/8)—1372no—
203AGL(1,ℤ/8)—1372no—
204AGL(1,ℤ/8)—1371no—
205AGL(1,ℤ/8)#1058773911yescharacteristic, complemented
206C2×SD16—1372no—
207C2×D8—1372no—
208C2×M16—1371no—
209C22×C8—1371no—
210C4×Q8—1372no—
211C4×D4—2741no—
212C4×D4—2741no—
213C4×D4—2742no—
214C4×D4—1372no—
215C4×D4—1372no—
216C4×D4—1372no—
217#117—2741no—
218#117—1371no—
219#117—1371no—
220C2×C4⋊C4—2741no—
221C2×C4⋊C4—1371no—
222C2×C22⋊C4—2742no—
223#107—1372no—
224#106—1372no—
225#103—1371no—
226#103—1371no—
227#102—1372no—
228#102—1372no—
229#102—1371no—
230#102—1371no—
231#97—1372no—
232C4×C8—1372no—
233C24—2741no—
234#41—2741no—
235#41—2741no—
236#41—1371no—
237#41—1371no—
238#41—1371no—
239#41C2×#1058773911yescharacteristic, complemented
240C2×Q8—1371no—
241C2×D4—2741no—
242C2×D4—2741no—
243C2×D4—2741no—
244C2×D4—2741no—
245C2×D4—1371no—
246C2×D4—1372no—
247C2×D4—1372no—
248C2×D4—1371no—
249C2×D4C2×#1058773911yescharacteristic, complemented
250C22×C4—2741no—
251C22×C4—2741no—
252C22×C4—2742no—
253C22×C4—2742no—
254C22×C4—2742no—
255C22×C4—1371no—
256SD16—1372no—
257SD16—1372no—
258SD16—1372no—
259SD16C2×#1058773912yescomplemented
260D8—1372no—
261D8—1372no—
262D8—1372no—
263D8C2×#1058773912yescomplemented
264M16—1371no—
265M16—1372no—
266M16C2×#1058773911yescharacteristic, complemented
267C2×C8—2741no—
268C2×C8—2741no—
269C2×C8—1371no—
270C2×C8—1371no—
271C2×C8—1372no—
272C4⋊C4—2741no—
273C4⋊C4—2742no—
274C4⋊C4—1372no—
275C4⋊C4—1371no—
276C4⋊C4—1371no—
277C4⋊C4—1371no—
278C4⋊C4—1371no—
279C22⋊C4—2741no—
280C22⋊C4—2742no—
281C22⋊C4—2742no—
282C22⋊C4—2742no—
283C22⋊C4—1372no—
284C42—2742no—
285C42—2741no—
286C42—1372no—
287C23—21no—
288C23—2741no—
289C23—2742no—
290C23—2742no—
291C23—2741no—
292C23—10961no—
293C23—10961no—
294C23—1371no—
295Q8—2741no—
296Q8—1371no—
297Q8C22×#1058773911yescharacteristic
298D4—21no—
299D4—21no—
300D4—2741no—
301D4—2741no—
302D4—2741no—
303D4—2742no—
304D4—2742no—
305D4—2741no—
306D4—2741no—
307D4—1371no—
308D4—1372no—
309D4C22×#1058773911yescharacteristic
310D4C22×#1058773912yes—
311C2×C4—21no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—2741no—
316C2×C4—2741no—
317C2×C4—2741no—
318C2×C4—2742no—
319C2×C4—10962no—
320C2×C4—10962no—
321C2×C4—10962no—
322C2×C4—10961no—
323C2×C4—1372no—
324C2×C4—1372no—
325C2×C4—1371no—
326C2×C4—1371no—
327C2×C4—1371no—
328C2×C4—1371no—
329C2×C4—1371no—
330C2×C4C22×#1058773911yescharacteristic
331C8—2741no—
332C8—2741no—
333C8—1372no—
334C8C22×#1058773912yes—
335C22—21no—
336C22—22no—
337C22—81no—
338C22—2741no—
339C22—2741no—
340C22—2741no—
341C22—2742no—
342C22—5481no—
343C22—5482no—
344C22—5482no—
345C22—5481no—
346C22—10961no—
347C22—10962no—
348C22—1371no—
349C22—1371no—
350C22D4×#1058773911yescharacteristic
351C4—21no—
352C4—2742no—
353C4—2742no—
354C4—2741no—
355C4—5481no—
356C4—5482no—
357C4—5481no—
358C4—1371no—
359C4—1371no—
360C4D4×#1058773911yescharacteristic
361C4C23×#1058773911yescharacteristic
362C2—21no—
363C2—42no—
364C2—41no—
365C2—2741no—
366C2—5482no—
367C2—5481no—
368C2—1371no—
369C2—1371no—
370C2C2×D4×#1058773911yescharacteristic
subgroupG/H
centerC2C2×D4×#10587739
commutatorC4×C137C23×C4
FrattiniC4C23×#10587739
FittingAGL(1,ℤ/8)×C137C4
socleC2×C137C2×C4×D4
prime radicalAGL(1,ℤ/8)×D137C2
cosocleC4×D137C24
hypercenterAGL(1,ℤ/8)#10587739
hypocenterC137#2946
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #22578472C2semidirect
2. #22578470C2semidirect
3. #22578444C2semidirect
4. #22578442C2semidirect
5. SD16×#10587739C2semidirect
6. D8×#10587739C2semidirect
7. #22578414C2semidirect
8. #22578393C2semidirect
9. C2×#17984150C2semidirect
10. #22578369C2semidirect
13. Q8×#10587739C22semidirect
19. D4×#10587739C22semidirect
26. #17984157C22semidirect
27. #17984156C22semidirect
28. #17984151C22semidirect
29. #17984151C22semidirect
32. #17984150C22semidirect
33. #17984150C22semidirect
34. #17984142C22semidirect
35. C8×#10587739C22semidirect
38. #17984116C4semidirect
39. #17984115C4semidirect
40. #17984114C4semidirect
41. #17984113C4semidirect
42. #17984112C4semidirect
46. AGL(1,ℤ/8)×C137C4semidirect
71. #14555511C2×C4semidirect
73. #14555510C2×C4semidirect
75. #14555509C2×C4semidirect
77. C2×D548C2×C4semidirect
82. D4×D137C2×C4semidirect
87. #14555499C2×C4semidirect
88. #14555498C2×C4semidirect
89. #14555497C2×C4semidirect
90. D1096C2×C4semidirect
91. #14555495C2×C4semidirect
92. #14555495C2×C4semidirect
93. #14555494C2×C4semidirect
94. #14555493C2×C4semidirect
96. #41×C137C2×C4semidirect
97. C2×D4×C137C2×C4semidirect
98. SD16×C137C2×C4semidirect
99. D8×C137C2×C4semidirect
100. M16×C137C2×C4semidirect
185. C137#2946semidirect
205. AGL(1,ℤ/8)#10587739semidirect
239. #41C2×#10587739semidirect
249. C2×D4C2×#10587739semidirect
259. SD16C2×#10587739semidirect
263. D8C2×#10587739semidirect
266. M16C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784722
#225784702
#225784442
#225784422
SD16×#105877392
D8×#105877392
#225784142
#225783932
C2×#179841502
#225783692
AGL(1,ℤ/8)×D1372
#2946137