Finite Group Databasebeta

← AGL(1,ℤ/8)×#87604 #27158491

Subgroups — AGL(1,ℤ/8)×#87604 #27158491

Order 12416.

548 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×#87604C211yesmaximal, characteristic, complemented
2C2×D4×#87604C211yesmaximal, characteristic, complemented
3#20261776C212yesmaximal, complemented
4#20261774C212yesmaximal, complemented
5SD16×#87604C212yesmaximal, complemented
6D8×#87604C212yesmaximal, complemented
7#20261746C211yesmaximal, characteristic, complemented
8#20261727C211yesmaximal, characteristic, complemented
9#20261726C211yesmaximal, characteristic, complemented
10M16×#87604C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×D97C211yesmaximal, characteristic
12C23×#87604—21no—
13D97⋊M16C2211yescharacteristic
14Q8×#87604C2211yescharacteristic, complemented
15Q8×#87604C2211yescharacteristic, complemented
16D4×#87604—21no—
17D4×#87604—21no—
18D4×#87604—21no—
19D4×#87604—21no—
20D4×#87604C2212yes—
21D4×#87604C2212yes—
22D4×#87604C2211yescharacteristic, complemented
23D4×#87604C2211yescharacteristic, complemented
24#15690889—21no—
25#15690889C2211yescharacteristic
26C2×C4⋊#87604C2211yescharacteristic
27C2×C22⋊#87604—21no—
28C2×C4×#87604—21no—
29C2×C4×#87604C2211yescharacteristic
30#15690879C2212yescomplemented
31#15690878C2212yescomplemented
32#15690873C2212yescomplemented
33#15690872C2212yes—
34#15690872C2212yes—
35#15690872C2212yescomplemented
36#15690864C2212yescomplemented
37C8×#87604C2212yescomplemented
38#41×D97C2211yescharacteristic
39C2×D4×D97C2211yescharacteristic
40#15690838C411yescharacteristic, complemented
41#15690837C412yescomplemented
42#15690836C411yescharacteristic, complemented
43#15690835C411yescharacteristic, complemented
44#15690834C412yescomplemented
45SD16×D97C2212yes—
46D8×D97C2212yes—
47M16×D97C2211yescharacteristic
48AGL(1,ℤ/8)×C97C411yescharacteristic, complemented
49C22×#87604—21no—
50C22×#87604—21no—
51C22×#87604—21no—
52C22×#87604—22no—
53C22×#87604—22no—
54C22×#87604—44no—
55C22×#87604D411yescharacteristic
56C22×#87604D411yescharacteristic
57#13095521—21no—
58#13095520—21no—
59#13095519C2312yes—
60C4⋊#87604—21no—
61C4⋊#87604—21no—
62C4⋊#87604—21no—
63C4⋊#87604C2311yescharacteristic
64C4⋊#87604C2311yescharacteristic
65C4⋊#87604C2312yes—
66C22⋊#87604—21no—
67C22⋊#87604—22no—
68C22⋊#87604—22no—
69C22⋊#87604—21no—
70C22⋊#87604—21no—
71C4×#87604—21no—
72C4×#87604—21no—
73C4×#87604—21no—
74C4×#87604D411yescharacteristic
75C4×#87604D411yescharacteristic
76C4×#87604C2311yescharacteristic
77C4×#87604C2311yescharacteristic
78C23×D97—21no—
79#13095512C2×C411yescharacteristic, complemented
80#13095511—21no—
81#13095511C2×C411yescharacteristic, complemented
82#13095510—21no—
83#13095510C2×C411yescharacteristic, complemented
84Q8×D97C2311yescharacteristic
85C2×D388C2×C411yescharacteristic, complemented
86C2×C22⋊D97—21no—
87D4×D97—21no—
88D4×D97—21no—
89D4×D97—21no—
90D4×D97C2×C412yescomplemented
91D4×D97C2312yes—
92D4×D97C2311yescharacteristic
93C2×C4×D97—21no—
94C2×C4×D97C2311yescharacteristic
95#13095500C2×C412yescomplemented
96#13095499C2×C412yescomplemented
97#13095498C2×C412yescomplemented
98D776C2×C412yescomplemented
99#13095496C2×C412yescomplemented
100#13095496C2×C412yescomplemented
101#13095495C2×C411yescharacteristic, complemented
102#13095494C2×C412yescomplemented
103C8×D97C2312yes—
104#41×C97C2×C411yescharacteristic, complemented
105C2×D4×C97C2×C411yescharacteristic, complemented
106SD16×C97C2×C412yescomplemented
107D8×C97C2×C412yescomplemented
108M16×C97C2×C411yescharacteristic, complemented
109C23×C97—21no—
110C22×D97—21no—
111C22×D97—22no—
112C22×D97—21no—
113C22×D97—22no—
114C22×D97—42no—
115C22×D97—42no—
116C22×D97C2×D411yescharacteristic
117C2×#87604—22no—
118C2×#87604—21no—
119C2×#87604—21no—
120C2×#87604—22no—
121C2×#87604—41no—
122C2×#87604—42no—
123C2×#87604—44no—
124C2×#87604—42no—
125C2×#87604—42no—
126C2×#87604—44no—
127C2×#87604C2×D411yescharacteristic
128C2×#87604C2×D411yescharacteristic
129C2×#87604C2×D411yescharacteristic
130C2×#87604C2×D411yescharacteristic
131Q8×C97C22×C411yescharacteristic, complemented
132D4×C97—21no—
133D4×C97—21no—
134D4×C97C22×C412yes—
135D4×C97C22×C411yescharacteristic, complemented
136C2×C4×C97—21no—
137C2×C4×C97C22×C411yescharacteristic
138C22⋊D97—21no—
139C22⋊D97—21no—
140C22⋊D97—22no—
141C22⋊D97—22no—
142C22⋊D97—21no—
143C2×Dic97—21no—
144C2×Dic97#4111yescharacteristic
145D388—21no—
146D388—21no—
147D388C22×C412yes—
148D388C22×C411yescharacteristic, complemented
149C4×D97—21no—
150C4×D97—21no—
151C4×D97—21no—
152C4×D97C22×C411yescharacteristic
153C4×D97C2×D411yescharacteristic
154C4×D97#4111yescharacteristic
155C4×D97C2411yescharacteristic
156Dic194—21no—
157Dic194C22×C411yescharacteristic, complemented
158#11715897—22no—
159C8×C97C22×C412yescomplemented
160#11715895C22×C412yescomplemented
161C22×C97—21no—
162C22×C97—22no—
163C22×C97—42no—
164C22×C97C4×D411yescharacteristic
165C2×D97—21no—
166C2×D97—21no—
167C2×D97—21no—
168C2×D97—22no—
169C2×D97—44no—
170C2×D97—42no—
171C2×D97—42no—
172C2×D97—41no—
173C2×D97—41no—
174C2×D97—42no—
175C2×D97C4×D411yescharacteristic
176C2×D97C22×D411yescharacteristic
177#87604—21no—
178#87604—21no—
179#87604—21no—
180#87604—41no—
181#87604—42no—
182#87604—41no—
183#87604AGL(1,ℤ/8)12yesdirect product factor
184C4×C97—21no—
185C4×C97C4×D411yescharacteristic
186C4×C97C23×C411yescharacteristic
187Dic97—21no—
188Dic97C4×D411yescharacteristic
189Dic97C2×#4111yescharacteristic
190C2×C97—21no—
191C2×C97—42no—
192C2×C97—41no—
193C2×C97C2×C4×D411yescharacteristic
194D97—21no—
195D97—41no—
196D97—42no—
197D97C2×AGL(1,ℤ/8)11yescharacteristic
198D97#57511yescharacteristic
199C4×AGL(1,ℤ/8)—971nomaximal, Sylow
200C97C4×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
201C2×AGL(1,ℤ/8)—971no—
202C4×#41—971no—
203C2×C4×D4—971no—
204#442—972no—
205#440—972no—
206C4×SD16—972no—
207C4×D8—972no—
208#428—971no—
209#418—971no—
210#417—971no—
211C4×M16—971no—
212C2×#41—971no—
213C22×D4—971no—
214C23×C4—1941no—
215AGL(1,ℤ/8)—971no—
216AGL(1,ℤ/8)—971no—
217AGL(1,ℤ/8)—972no—
218AGL(1,ℤ/8)—972no—
219AGL(1,ℤ/8)—971no—
220AGL(1,ℤ/8)#8760411yescharacteristic, direct product factor
221C2×SD16—972no—
222C2×D8—972no—
223C2×M16—971no—
224C2×M16—971no—
225C4×Q8—971no—
226C4×Q8—971no—
227C4×D4—972no—
228C4×D4—972no—
229C4×D4—971no—
230C4×D4—971no—
231C4×D4—1941no—
232C4×D4—1941no—
233C4×D4—1941no—
234C4×D4—1941no—
235#117—971no—
236#117—1941no—
237C2×C4⋊C4—971no—
238C2×C22⋊C4—1941no—
239C2×C42—971no—
240C2×C42—1941no—
241#107—972no—
242#106—972no—
243#103—972no—
244#102—972no—
245#102—972no—
246#102—972no—
247#97—972no—
248C4×C8—972no—
249C24—1941no—
250#41—971no—
251#41—971no—
252#41—971no—
253#41—1941no—
254#41—1941no—
255#41C2×#8760411yescharacteristic, complemented
256C2×Q8—971no—
257C2×D4—972no—
258C2×D4—972no—
259C2×D4—971no—
260C2×D4—971no—
261C2×D4—1941no—
262C2×D4—1941no—
263C2×D4—1941no—
264C2×D4—1941no—
265C2×D4C2×#8760411yescharacteristic, complemented
266C22×C4—971no—
267C22×C4—971no—
268C22×C4—971no—
269C22×C4—3884no—
270C22×C4—1941no—
271C22×C4—1941no—
272C22×C4—1941no—
273C22×C4—1942no—
274C22×C4—1942no—
275C22×C4—1941no—
276SD16—972no—
277SD16—972no—
278SD16—972no—
279SD16C2×#8760412yescomplemented
280D8—972no—
281D8—972no—
282D8—972no—
283D8C2×#8760412yescomplemented
284M16—971no—
285M16—972no—
286M16—1941no—
287M16—1941no—
288M16C2×#8760411yescharacteristic, complemented
289C2×C8—972no—
290C2×C8—972no—
291C4⋊C4—971no—
292C4⋊C4—971no—
293C4⋊C4—972no—
294C4⋊C4—1941no—
295C4⋊C4—1941no—
296C4⋊C4—1941no—
297C22⋊C4—1942no—
298C22⋊C4—1942no—
299C22⋊C4—1941no—
300C22⋊C4—1941no—
301C22⋊C4—1941no—
302C42—971no—
303C42—971no—
304C42—971no—
305C42—971no—
306C42—1941no—
307C42—1941no—
308C42—1941no—
309C23—21no—
310C23—971no—
311C23—3882no—
312C23—3882no—
313C23—1941no—
314C23—1942no—
315C23—1942no—
316C23—1941no—
317Q8—971no—
318Q8—1941no—
319Q8C22×#8760411yescharacteristic, complemented
320D4—21no—
321D4—21no—
322D4—971no—
323D4—972no—
324D4—1941no—
325D4—1942no—
326D4—1942no—
327D4—1941no—
328D4—1941no—
329D4—1941no—
330D4—1941no—
331D4C22×#8760412yes—
332D4C22×#8760411yescharacteristic, complemented
333C2×C4—21no—
334C2×C4—971no—
335C2×C4—971no—
336C2×C4—971no—
337C2×C4—971no—
338C2×C4—971no—
339C2×C4—971no—
340C2×C4—971no—
341C2×C4—971no—
342C2×C4—971no—
343C2×C4—3884no—
344C2×C4—3882no—
345C2×C4—3882no—
346C2×C4—3884no—
347C2×C4—3882no—
348C2×C4—3881no—
349C2×C4—1941no—
350C2×C4—1941no—
351C2×C4—1941no—
352C2×C4—1942no—
353C2×C4—1941no—
354C2×C4—1941no—
355C2×C4—1942no—
356C2×C4—1941no—
357C2×C4C22×#8760411yescharacteristic
358C8—972no—
359C8—1942no—
360C8C22×#8760412yescomplemented
361C22—21no—
362C22—22no—
363C22—42no—
364C22—971no—
365C22—971no—
366C22—3882no—
367C22—3882no—
368C22—3881no—
369C22—3881no—
370C22—3882no—
371C22—3884no—
372C22—1941no—
373C22—1941no—
374C22—1941no—
375C22—1942no—
376C22D4×#8760411yescharacteristic
377C4—21no—
378C4—972no—
379C4—971no—
380C4—971no—
381C4—3882no—
382C4—3881no—
383C4—3881no—
384C4—1941no—
385C4—1941no—
386C4—1941no—
387C4—1941no—
388C4D4×#8760411yescharacteristic
389C4C23×#8760411yescharacteristic
390C2—21no—
391C2—42no—
392C2—41no—
393C2—971no—
394C2—971no—
395C2—3882no—
396C2—3881no—
397C2—1941no—
398C2C2×D4×#8760411yescharacteristic
subgroupG/H
centerC2C2×D4×#87604
commutatorC4×C97C23×C4
FrattiniC4C23×#87604
FittingAGL(1,ℤ/8)×C97C4
socleC2×C97C2×C4×D4
prime radicalAGL(1,ℤ/8)×D97C2
cosocleC4×D97C24
hypercenterAGL(1,ℤ/8)#87604
hypocenterC97C4×AGL(1,ℤ/8)
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×#87604C2semidirect
2. C2×D4×#87604C2semidirect
3. #20261776C2semidirect
4. #20261774C2semidirect
5. SD16×#87604C2semidirect
6. D8×#87604C2semidirect
7. #20261746C2semidirect
8. #20261727C2semidirect
9. #20261726C2semidirect
10. M16×#87604C2semidirect
14. Q8×#87604C22semidirect
15. Q8×#87604C22semidirect
22. D4×#87604C22semidirect
23. D4×#87604C22semidirect
30. #15690879C22semidirect
31. #15690878C22semidirect
32. #15690873C22semidirect
35. #15690872C22semidirect
36. #15690864C22semidirect
37. C8×#87604C22semidirect
40. #15690838C4semidirect
41. #15690837C4semidirect
42. #15690836C4semidirect
43. #15690835C4semidirect
44. #15690834C4semidirect
48. AGL(1,ℤ/8)×C97C4semidirect
79. #13095512C2×C4semidirect
81. #13095511C2×C4semidirect
83. #13095510C2×C4semidirect
85. C2×D388C2×C4semidirect
90. D4×D97C2×C4semidirect
95. #13095500C2×C4semidirect
96. #13095499C2×C4semidirect
97. #13095498C2×C4semidirect
98. D776C2×C4semidirect
99. #13095496C2×C4semidirect
100. #13095496C2×C4semidirect
101. #13095495C2×C4semidirect
102. #13095494C2×C4semidirect
104. #41×C97C2×C4semidirect
105. C2×D4×C97C2×C4semidirect
106. SD16×C97C2×C4semidirect
107. D8×C97C2×C4semidirect
108. M16×C97C2×C4semidirect
131. Q8×C97C22×C4semidirect
135. D4×C97C22×C4semidirect
148. D388C22×C4semidirect
157. Dic194C22×C4semidirect
159. C8×C97C22×C4semidirect
160. #11715895C22×C4semidirect
183. #87604AGL(1,ℤ/8)direct (N × H)
200. C97C4×AGL(1,ℤ/8)semidirect
220. AGL(1,ℤ/8)#87604direct (N × H)
255. #41C2×#87604semidirect
265. C2×D4C2×#87604semidirect
279. SD16C2×#87604semidirect
283. D8C2×#87604semidirect
288. M16C2×#87604semidirect
319. Q8C22×#87604semidirect
332. D4C22×#87604semidirect
360. C8C22×#87604semidirect

Maximal subgroups

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

subgroupindex
#41×#876042
C2×D4×#876042
#202617762
#202617742
SD16×#876042
D8×#876042
#202617462
#202617272
#202617262
M16×#876042
AGL(1,ℤ/8)×D972
C4×AGL(1,ℤ/8)97