Finite Group Databasebeta

← #18003965

Subgroups — #18003965

Order 4736.

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#14558369C211yesmaximal, characteristic, complemented
2C2×#11924562C211yesmaximal, characteristic, complemented
3#14558100C211yesmaximal, characteristic, complemented
4#14557704C214yesmaximal, complemented
5#14557699C214yesmaximal, complemented
6#14557550C211yesmaximal, characteristic, complemented
7#14557549C211yesmaximal, characteristic, complemented
8#14557515C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
10#11924584—21no—
11#11924583—21no—
12#11924583C2212yescomplemented
13#11924582—21no—
14#11924582C2212yescomplemented
15#11924581—21no—
16#11924581—22no—
17C2×#10596962—22no—
18C2×#10596962C2211yescharacteristic
19C2×#10596962C2211yescharacteristic
20#41×D37—22no—
21#41×D37—22no—
22#11924562—22no—
23#11924562C2214yescomplemented
24C2×#10596940—21no—
25C2×#10596939—21no—
26C2×#10596934C2211yescharacteristic, complemented
27#11924532C2212yescomplemented
28#11924523C2212yescomplemented
29#11924507—22no—
30#11924505—22no—
31#11924495C2211yescharacteristic, complemented
32#11924463C2214yescomplemented
33#11924462C2214yescomplemented
34#11924454C2212yescomplemented
35#11924452C2212yescomplemented
36C2×#41×C37C2211yescharacteristic
37C22×D4×C37C2211yescharacteristic, complemented
38AGL(1,ℤ/8)×C37—22no—
39AGL(1,ℤ/8)×C37C2214yescomplemented
40C2×SD16×C37—21no—
41C2×D8×C37—21no—
42C2×M16×C37C2211yescharacteristic, complemented
43C24×C37—21no—
44#41×C37—21no—
45#41×C37—21no—
46#41×C37D412yescomplemented
47#41×C37C2312yes—
48C2×Q8×C37D411yescharacteristic, complemented
49C2×D4×C37—21no—
50C2×D4×C37—21no—
51C2×D4×C37—21no—
52C2×D4×C37—21no—
53C2×D4×C37—21no—
54C2×D4×C37—21no—
55C2×D4×C37D411yescharacteristic, complemented
56C2×D4×C37C2312yes—
57C22×C4×C37—21no—
58C22×C4×C37C2311yescharacteristic
59C2×C22⋊D37—24no—
60C2×C22⋊D37—21no—
61C2×C22⋊D37—24no—
62C2×C22⋊D37D412yes—
63#10596966—22no—
64#10596966—22no—
65Q8×D37—22no—
66Q8×D37—22no—
67#10596964—24no—
68#10596964—22no—
69#10596964—22no—
70#10596964—24no—
71D4×D37—24no—
72D4×D37—22no—
73D4×D37—24no—
74D4×D37—22no—
75#10596962—24no—
76#10596962—24no—
77#10596962—21no—
78#10596962—21no—
79#10596962—21no—
80#10596962—21no—
81#10596962—21no—
82#10596962—21no—
83#10596962—22no—
84#10596962—22no—
85#10596962D412yescomplemented
86#10596962C2312yes—
87C2×D148—22no—
88C2×D148D411yescharacteristic, complemented
89C2×D148C2311yescharacteristic
90C2×C4×D37—24no—
91C2×C4×D37—21no—
92C2×C4×D37D412yes—
93C2×Dic74—22no—
94C2×Dic74D411yescharacteristic, complemented
95C2×Dic74C2311yescharacteristic
96SD16×C37—24no—
97D8×C37—24no—
98M16×C37—21no—
99M16×C37C2312yes—
100C2×C8×C37—21no—
101C22⋊Dic37—21no—
102C22⋊Dic37—24no—
103#10596940—24no—
104#10596939—24no—
105C4⋊Dic37—21no—
106C4×Dic37C2312yes—
107#10596934—21no—
108#10596934C2312yes—
109C2×#64436—21no—
110C23×C37—21no—
111C23×C37—22no—
112C23×C37—41no—
113C23×C37—41no—
114C23×C37—82no—
115C23×C37C2×D411yescharacteristic
116C22×D37—24no—
117C22×D37—21no—
118C22×D37C2×D412yes—
119Q8×C37—21no—
120Q8×C37C2×D412yes—
121D4×C37—22no—
122D4×C37—22no—
123D4×C37—21no—
124D4×C37—22no—
125D4×C37—22no—
126D4×C37—41no—
127D4×C37—41no—
128D4×C37C2×D412yes—
129C2×C4×C37—21no—
130C2×C4×C37—22no—
131C2×C4×C37—22no—
132C2×C4×C37C2×D411yescharacteristic
133C2×C4×C37C2×D411yescharacteristic
134C2×C4×C37C2×D411yescharacteristic
135C2×C4×C37C2×D411yescharacteristic
136C2×C4×C37C2×D411yescharacteristic
137C2×C4×C37C2411yescharacteristic
138C22⋊D37—24no—
139C22⋊D37—24no—
140C22⋊D37—22no—
141C22⋊D37—22no—
142C22⋊D37—22no—
143C22⋊D37—22no—
144C22⋊D37—84no—
145C2×Dic37—24no—
146C2×Dic37—21no—
147C2×Dic37—22no—
148C2×Dic37C2×D412yes—
149D148—24no—
150D148—21no—
151D148—21no—
152D148—21no—
153D148C2×D412yes—
154C4×D37—24no—
155C4×D37—24no—
156C4×D37—22no—
157C4×D37—22no—
158C4×D37—22no—
159C4×D37—22no—
160Dic74—24no—
161Dic74—21no—
162Dic74—21no—
163Dic74—21no—
164Dic74C2×D412yes—
165C8×C37—22no—
166#64436—22no—
167C22×C37—22no—
168C22×C37—41no—
169C22×C37—44no—
170C22×C37—42no—
171C22×C37—81no—
172C22×C37—82no—
173C22×C37—82no—
174C22×C37#12011yescharacteristic
175C22×C37#12011yescharacteristic
176C22×C37C22×D411yescharacteristic
177C2×D37—22no—
178C2×D37—22no—
179C2×D37—42no—
180C2×D37—81no—
181C2×D37—84no—
182C4×C37—22no—
183C4×C37#12011yescharacteristic
184C4×C37#12011yescharacteristic
185C4×C37C22×D411yescharacteristic
186C4×C37C22×D411yescharacteristic
187Dic37—22no—
188Dic37—22no—
189Dic37—42no—
190#3021—371nomaximal, Sylow
191C2×C37—21no—
192C2×C37—21no—
193C2×C37—21no—
194C2×C37—42no—
195C2×C37—82no—
196C2×C37C2×#12011yescharacteristic
197D37—42no—
198D37—42no—
199#585—371no—
200C2×AGL(1,ℤ/8)—371no—
201C2×AGL(1,ℤ/8)D3711yescharacteristic, complemented
202#535—371no—
203#454—374no—
204#453—374no—
205#421—371no—
206#421—371no—
207#413—371no—
208C37#302111yescharacteristic, complemented, Sylow
2092−1+4—372no—
2102−1+4—741no—
2112−1+4—741no—
212H(5,𝔽2)—372no—
213H(5,𝔽2)—742no—
214H(5,𝔽2)—741no—
215H(5,𝔽2)—741no—
216C2×#41—371no—
217C2×#41—371no—
218C2×#41—742no—
219C2×#41—742no—
220C2×#41—742no—
221C2×#41C2×D3711yescharacteristic
222C22×D4C2×D3711yescharacteristic, complemented
223AGL(1,ℤ/8)—22no—
224AGL(1,ℤ/8)—374no—
225AGL(1,ℤ/8)—742no—
226AGL(1,ℤ/8)C2×D3714yescomplemented
227C2×SD16—21no—
228C2×SD16—741no—
229C2×D8—21no—
230C2×D8—741no—
231C2×M16—371no—
232C2×M16C2×D3711yescharacteristic, complemented
233#127—372no—
234#124—372no—
235#121—742no—
236#120—742no—
237#117—371no—
238#104—374no—
239#104—374no—
240#101—372no—
241#100—372no—
242C24—21no—
243#41—21no—
244#41—21no—
245#41—372no—
246#41—372no—
247#41—742no—
248#41—742no—
249#41—744no—
250#41—744no—
251#41—744no—
252#41—744no—
253#41—742no—
254#41—742no—
255#41—742no—
256#41—742no—
257#41—741no—
258#41—741no—
259#41—741no—
260#41—741no—
261#41—741no—
262#41—741no—
263#41C22⋊D3712yescomplemented
264#41C22×D3712yes—
265C2×Q8—371no—
266C2×Q8—371no—
267C2×Q8—742no—
268C2×Q8—742no—
269C2×Q8—742no—
270C2×Q8C22⋊D3711yescharacteristic, complemented
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—372no—
278C2×D4—371no—
279C2×D4—371no—
280C2×D4—742no—
281C2×D4—744no—
282C2×D4—744no—
283C2×D4—742no—
284C2×D4—744no—
285C2×D4—744no—
286C2×D4—741no—
287C2×D4—742no—
288C2×D4C22⋊D3711yescharacteristic, complemented
289C2×D4C22×D3712yes—
290C22×C4—21no—
291C22×C4—372no—
292C22×C4—744no—
293C22×C4—741no—
294C22×C4C22×D3711yescharacteristic
295SD16—24no—
296SD16—744no—
297D8—24no—
298D8—744no—
299M16—21no—
300M16—372no—
301M16—741no—
302M16C22×D3712yes—
303C2×C8—21no—
304C2×C8—741no—
305C4⋊C4—741no—
306C22⋊C4—744no—
307C22⋊C4—741no—
308C42—372no—
309C23—21no—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—82no—
314C23—372no—
315C23—744no—
316C23—741no—
317C23D4×D3711yescharacteristic
318Q8—21no—
319Q8—372no—
320Q8—744no—
321Q8—741no—
322Q8—741no—
323Q8—741no—
324Q8C2×C22⋊D3712yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—22no—
329D4—22no—
330D4—41no—
331D4—41no—
332D4—372no—
333D4—2964no—
334D4—744no—
335D4—744no—
336D4—744no—
337D4—742no—
338D4—742no—
339D4—742no—
340D4—742no—
341D4—741no—
342D4—741no—
343D4—741no—
344D4C2×C22⋊D3712yes—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—372no—
349C2×C4—744no—
350C2×C4—744no—
351C2×C4—744no—
352C2×C4—741no—
353C2×C4—742no—
354C2×C4—742no—
355C2×C4—742no—
356C2×C4—742no—
357C2×C4—742no—
358C2×C4D4×D3711yescharacteristic
359C2×C4D4×D3711yescharacteristic
360C2×C4D4×D3711yescharacteristic
361C2×C4C2×C22⋊D3711yescharacteristic
362C2×C4C2×C22⋊D3711yescharacteristic
363C2×C4C23×D3711yescharacteristic
364C8—22no—
365C8—742no—
366C22—22no—
367C22—41no—
368C22—44no—
369C22—42no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—2961no—
374C22—2964no—
375C22—742no—
376C22—742no—
377C22—1482no—
378C22#1192450411yescharacteristic
379C22#1192450411yescharacteristic
380C22C2×D4×D3711yescharacteristic
381C4—22no—
382C4—742no—
383C4—742no—
384C4—1482no—
385C4#1192450411yescharacteristic
386C4#1192450411yescharacteristic
387C4C2×D4×D3711yescharacteristic
388C4C22×C22⋊D3711yescharacteristic
389C2—21no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—82no—
394C2—1482no—
395C2—1482no—
396C2C2×#1192450411yescharacteristic
subgroupG/H
centerC2C2×#11924504
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×AGL(1,ℤ/8)×C37C2
socleC2×C37C2×#120
hypercenterC2×AGL(1,ℤ/8)D37
hypocenterC37#3021
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #14558369C2semidirect
2. C2×#11924562C2semidirect
3. #14558100C2semidirect
4. #14557704C2semidirect
5. #14557699C2semidirect
6. #14557550C2semidirect
7. #14557549C2semidirect
8. #14557515C2semidirect
9. C2×AGL(1,ℤ/8)×C37C2semidirect
12. #11924583C22semidirect
14. #11924582C22semidirect
23. #11924562C22semidirect
26. C2×#10596934C22semidirect
27. #11924532C22semidirect
28. #11924523C22semidirect
31. #11924495C22semidirect
32. #11924463C22semidirect
33. #11924462C22semidirect
34. #11924454C22semidirect
35. #11924452C22semidirect
37. C22×D4×C37C22semidirect
39. AGL(1,ℤ/8)×C37C22semidirect
42. C2×M16×C37C22semidirect
46. #41×C37D4semidirect
48. C2×Q8×C37D4semidirect
55. C2×D4×C37D4semidirect
85. #10596962D4semidirect
88. C2×D148D4semidirect
94. C2×Dic74D4semidirect
201. C2×AGL(1,ℤ/8)D37semidirect
208. C37#3021semidirect
222. C22×D4C2×D37semidirect
226. AGL(1,ℤ/8)C2×D37semidirect
232. C2×M16C2×D37semidirect
263. #41C22⋊D37semidirect
270. C2×Q8C22⋊D37semidirect
288. C2×D4C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
#145583692
C2×#119245622
#145581002
#145577042
#145576992
#145575502
#145575492
#145575152
C2×AGL(1,ℤ/8)×C372
#302137