Finite Group Databasebeta

← #20252757

Subgroups — #20252757

Order 6016.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15689959C211yesmaximal, characteristic, complemented
2C2×#13094050C211yesmaximal, characteristic, complemented
3#15689690C211yesmaximal, characteristic, complemented
4#15689294C214yesmaximal, complemented
5#15689289C214yesmaximal, complemented
6#15689140C211yesmaximal, characteristic, complemented
7#15689139C211yesmaximal, characteristic, complemented
8#15689105C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C47C211yesmaximal, characteristic, complemented
10#13094072—21no—
11#13094071—21no—
12#13094071C2212yescomplemented
13#13094070—21no—
14#13094070C2212yescomplemented
15#13094069—21no—
16#13094069—22no—
17C2×#10625350—22no—
18C2×#10625350C2211yescharacteristic
19C2×#10625350C2211yescharacteristic
20#41×D47—22no—
21#41×D47—22no—
22#13094050—22no—
23#13094050C2214yescomplemented
24C2×#10625335—21no—
25C2×#10625334—21no—
26C2×#10625329C2211yescharacteristic, complemented
27#13094020C2212yescomplemented
28#13094011C2212yescomplemented
29#13093995—22no—
30#13093993—22no—
31#13093983C2211yescharacteristic, complemented
32#13093951C2214yescomplemented
33#13093950C2214yescomplemented
34#13093942C2212yescomplemented
35#13093940C2212yescomplemented
36C2×#41×C47C2211yescharacteristic
37C22×D4×C47C2211yescharacteristic, complemented
38AGL(1,ℤ/8)×C47—22no—
39AGL(1,ℤ/8)×C47C2214yescomplemented
40C2×SD16×C47—21no—
41C2×D8×C47—21no—
42C2×M16×C47C2211yescharacteristic, complemented
43C24×C47—21no—
44#41×C47—21no—
45#41×C47—21no—
46#41×C47D412yescomplemented
47#41×C47C2312yes—
48C2×Q8×C47D411yescharacteristic, complemented
49C2×D4×C47—21no—
50C2×D4×C47—21no—
51C2×D4×C47—21no—
52C2×D4×C47—21no—
53C2×D4×C47—21no—
54C2×D4×C47—21no—
55C2×D4×C47D411yescharacteristic, complemented
56C2×D4×C47C2312yes—
57C22×C4×C47—21no—
58C22×C4×C47C2311yescharacteristic
59C2×C22⋊D47—24no—
60C2×C22⋊D47—21no—
61C2×C22⋊D47—24no—
62C2×C22⋊D47D412yes—
63#10625354—22no—
64#10625354—22no—
65Q8×D47—22no—
66Q8×D47—22no—
67#10625352—24no—
68#10625352—22no—
69#10625352—22no—
70#10625352—24no—
71D4×D47—24no—
72D4×D47—22no—
73D4×D47—24no—
74D4×D47—22no—
75#10625350—24no—
76#10625350—24no—
77#10625350—21no—
78#10625350—21no—
79#10625350—21no—
80#10625350—21no—
81#10625350—21no—
82#10625350—21no—
83#10625350—22no—
84#10625350—22no—
85#10625350D412yescomplemented
86#10625350C2312yes—
87C2×D188—22no—
88C2×D188D411yescharacteristic, complemented
89C2×D188C2311yescharacteristic
90C2×C4×D47—24no—
91C2×C4×D47—21no—
92C2×C4×D47D412yes—
93C2×Dic94—22no—
94C2×Dic94D411yescharacteristic, complemented
95C2×Dic94C2311yescharacteristic
96SD16×C47—24no—
97D8×C47—24no—
98M16×C47—21no—
99M16×C47C2312yes—
100C2×C8×C47—21no—
101C22⋊Dic47—21no—
102C22⋊Dic47—24no—
103#10625335—24no—
104#10625334—24no—
105C4⋊Dic47—21no—
106C4×Dic47C2312yes—
107#10625329—21no—
108#10625329C2312yes—
109C2×C47⋊C8—21no—
110C23×C47—21no—
111C23×C47—22no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—82no—
115C23×C47C2×D411yescharacteristic
116C22×D47—24no—
117C22×D47—21no—
118C22×D47C2×D412yes—
119Q8×C47—21no—
120Q8×C47C2×D412yes—
121D4×C47—22no—
122D4×C47—22no—
123D4×C47—21no—
124D4×C47—22no—
125D4×C47—22no—
126D4×C47—41no—
127D4×C47—41no—
128D4×C47C2×D412yes—
129C2×C4×C47—21no—
130C2×C4×C47—22no—
131C2×C4×C47—22no—
132C2×C4×C47C2×D411yescharacteristic
133C2×C4×C47C2×D411yescharacteristic
134C2×C4×C47C2×D411yescharacteristic
135C2×C4×C47C2×D411yescharacteristic
136C2×C4×C47C2×D411yescharacteristic
137C2×C4×C47C2411yescharacteristic
138C22⋊D47—24no—
139C22⋊D47—24no—
140C22⋊D47—22no—
141C22⋊D47—22no—
142C22⋊D47—22no—
143C22⋊D47—22no—
144C22⋊D47—84no—
145C2×Dic47—24no—
146C2×Dic47—21no—
147C2×Dic47—22no—
148C2×Dic47C2×D412yes—
149D188—24no—
150D188—21no—
151D188—21no—
152D188—21no—
153D188C2×D412yes—
154C4×D47—24no—
155C4×D47—24no—
156C4×D47—22no—
157C4×D47—22no—
158C4×D47—22no—
159C4×D47—22no—
160Dic94—24no—
161Dic94—21no—
162Dic94—21no—
163Dic94—21no—
164Dic94C2×D412yes—
165C8×C47—22no—
166C47⋊C8—22no—
167C22×C47—22no—
168C22×C47—41no—
169C22×C47—44no—
170C22×C47—42no—
171C22×C47—81no—
172C22×C47—82no—
173C22×C47—82no—
174C22×C47#12011yescharacteristic
175C22×C47#12011yescharacteristic
176C22×C47C22×D411yescharacteristic
177C2×D47—22no—
178C2×D47—22no—
179C2×D47—42no—
180C2×D47—81no—
181C2×D47—84no—
182C4×C47—22no—
183C4×C47#12011yescharacteristic
184C4×C47#12011yescharacteristic
185C4×C47C22×D411yescharacteristic
186C4×C47C22×D411yescharacteristic
187Dic47—22no—
188Dic47—22no—
189Dic47—42no—
190#3021—471nomaximal, Sylow
191C2×C47—21no—
192C2×C47—21no—
193C2×C47—21no—
194C2×C47—42no—
195C2×C47—82no—
196C2×C47C2×#12011yescharacteristic
197D47—42no—
198D47—42no—
199#585—471no—
200C2×AGL(1,ℤ/8)—471no—
201C2×AGL(1,ℤ/8)D4711yescharacteristic, complemented
202#535—471no—
203#454—474no—
204#453—474no—
205#421—471no—
206#421—471no—
207#413—471no—
208C47#302111yescharacteristic, complemented, Sylow
2092−1+4—472no—
2102−1+4—941no—
2112−1+4—941no—
212H(5,𝔽2)—472no—
213H(5,𝔽2)—942no—
214H(5,𝔽2)—941no—
215H(5,𝔽2)—941no—
216C2×#41—471no—
217C2×#41—471no—
218C2×#41—942no—
219C2×#41—942no—
220C2×#41—942no—
221C2×#41C2×D4711yescharacteristic
222C22×D4C2×D4711yescharacteristic, complemented
223AGL(1,ℤ/8)—22no—
224AGL(1,ℤ/8)—474no—
225AGL(1,ℤ/8)—942no—
226AGL(1,ℤ/8)C2×D4714yescomplemented
227C2×SD16—21no—
228C2×SD16—941no—
229C2×D8—21no—
230C2×D8—941no—
231C2×M16—471no—
232C2×M16C2×D4711yescharacteristic, complemented
233#127—472no—
234#124—472no—
235#121—942no—
236#120—942no—
237#117—471no—
238#104—474no—
239#104—474no—
240#101—472no—
241#100—472no—
242C24—21no—
243#41—21no—
244#41—21no—
245#41—472no—
246#41—472no—
247#41—942no—
248#41—942no—
249#41—944no—
250#41—944no—
251#41—944no—
252#41—944no—
253#41—942no—
254#41—942no—
255#41—942no—
256#41—942no—
257#41—941no—
258#41—941no—
259#41—941no—
260#41—941no—
261#41—941no—
262#41—941no—
263#41C22⋊D4712yescomplemented
264#41C22×D4712yes—
265C2×Q8—471no—
266C2×Q8—471no—
267C2×Q8—942no—
268C2×Q8—942no—
269C2×Q8—942no—
270C2×Q8C22⋊D4711yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—472no—
278C2×D4—471no—
279C2×D4—471no—
280C2×D4—942no—
281C2×D4—944no—
282C2×D4—944no—
283C2×D4—942no—
284C2×D4—944no—
285C2×D4—941no—
286C2×D4—942no—
287C2×D4—944no—
288C2×D4C22⋊D4711yescharacteristic, complemented
289C2×D4C22×D4712yes—
290C22×C4—21no—
291C22×C4—472no—
292C22×C4—944no—
293C22×C4—941no—
294C22×C4C22×D4711yescharacteristic
295SD16—24no—
296SD16—944no—
297D8—24no—
298D8—944no—
299M16—21no—
300M16—472no—
301M16—941no—
302M16C22×D4712yes—
303C2×C8—21no—
304C2×C8—941no—
305C4⋊C4—941no—
306C22⋊C4—944no—
307C22⋊C4—941no—
308C42—472no—
309C23—21no—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—82no—
314C23—472no—
315C23—944no—
316C23—941no—
317C23D4×D4711yescharacteristic
318Q8—21no—
319Q8—472no—
320Q8—944no—
321Q8—941no—
322Q8—941no—
323Q8—941no—
324Q8C2×C22⋊D4712yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—22no—
329D4—22no—
330D4—41no—
331D4—41no—
332D4—472no—
333D4—944no—
334D4—944no—
335D4—944no—
336D4—942no—
337D4—942no—
338D4—942no—
339D4—942no—
340D4—941no—
341D4—941no—
342D4—941no—
343D4—3764no—
344D4C2×C22⋊D4712yes—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—472no—
349C2×C4—944no—
350C2×C4—944no—
351C2×C4—944no—
352C2×C4—941no—
353C2×C4—942no—
354C2×C4—942no—
355C2×C4—942no—
356C2×C4—942no—
357C2×C4—942no—
358C2×C4D4×D4711yescharacteristic
359C2×C4D4×D4711yescharacteristic
360C2×C4D4×D4711yescharacteristic
361C2×C4C2×C22⋊D4711yescharacteristic
362C2×C4C2×C22⋊D4711yescharacteristic
363C2×C4C23×D4711yescharacteristic
364C8—22no—
365C8—942no—
366C22—22no—
367C22—41no—
368C22—44no—
369C22—42no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—942no—
374C22—942no—
375C22—3761no—
376C22—3764no—
377C22—1882no—
378C22#1309399211yescharacteristic
379C22#1309399211yescharacteristic
380C22C2×D4×D4711yescharacteristic
381C4—22no—
382C4—942no—
383C4—942no—
384C4—1882no—
385C4#1309399211yescharacteristic
386C4#1309399211yescharacteristic
387C4C2×D4×D4711yescharacteristic
388C4C22×C22⋊D4711yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—82no—
394C2—1882no—
395C2—1882no—
396C2C2×#1309399211yescharacteristic
subgroupG/H
centerC2C2×#13093992
commutatorC2×C4×C47C24
FrattiniC2×C4C23×D47
FittingC2×AGL(1,ℤ/8)×C47C2
socleC2×C47C2×#120
hypercenterC2×AGL(1,ℤ/8)D47
hypocenterC47#3021
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #15689959C2semidirect
2. C2×#13094050C2semidirect
3. #15689690C2semidirect
4. #15689294C2semidirect
5. #15689289C2semidirect
6. #15689140C2semidirect
7. #15689139C2semidirect
8. #15689105C2semidirect
9. C2×AGL(1,ℤ/8)×C47C2semidirect
12. #13094071C22semidirect
14. #13094070C22semidirect
23. #13094050C22semidirect
26. C2×#10625329C22semidirect
27. #13094020C22semidirect
28. #13094011C22semidirect
31. #13093983C22semidirect
32. #13093951C22semidirect
33. #13093950C22semidirect
34. #13093942C22semidirect
35. #13093940C22semidirect
37. C22×D4×C47C22semidirect
39. AGL(1,ℤ/8)×C47C22semidirect
42. C2×M16×C47C22semidirect
46. #41×C47D4semidirect
48. C2×Q8×C47D4semidirect
55. C2×D4×C47D4semidirect
85. #10625350D4semidirect
88. C2×D188D4semidirect
94. C2×Dic94D4semidirect
201. C2×AGL(1,ℤ/8)D47semidirect
208. C47#3021semidirect
222. C22×D4C2×D47semidirect
226. AGL(1,ℤ/8)C2×D47semidirect
232. C2×M16C2×D47semidirect
263. #41C22⋊D47semidirect
270. C2×Q8C22⋊D47semidirect
288. C2×D4C22⋊D47semidirect