Finite Group Databasebeta

← #18003902

Subgroups — #18003902

Order 4736.

764 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#11924582C211yesmaximal, characteristic, complemented
2C2×#11924564C211yesmaximal, characteristic, complemented
3C2×#11924556C211yesmaximal, characteristic, complemented
4#14558114C211yesmaximal, characteristic, complemented
5#14557698C212yesmaximal, complemented
6#14557685C212yesmaximal, complemented
7#14557681C212yesmaximal, complemented
8#14557668C212yesmaximal, complemented
9C2×#11924456C211yesmaximal, characteristic, complemented
10#14557501C211yesmaximal, characteristic, complemented
11C37×#417C211yesmaximal, characteristic, complemented
12#11924582—24no—
13#11924582—24no—
14C2×#10596966C2212yes—
15C2×#10596962—21no—
16C2×#10596962C2211yescharacteristic
17C2×#10596962C2211yescharacteristic, complemented
18C22×D148—21no—
19C22×D148C2211yescharacteristic
20C2×D4×D37—22no—
21C2×D4×D37C2212yes—
22#11924564—24no—
23#11924556—24no—
24C2×#10596942C2211yescharacteristic, complemented
25C2×#10596930C2212yescomplemented
26C2×#10596940C2211yescharacteristic, complemented
27C2×#10596941C2211yescharacteristic, complemented
28C2×D296C2212yescomplemented
29C2×#10596939C2211yescharacteristic, complemented
30C22×#64436C2211yescharacteristic, complemented
31#11924516—21no—
32#11924516C2212yescomplemented
33#11924511—21no—
34#11924511C2212yescomplemented
35#11924510—22no—
36#11924456C2212yescomplemented
37#11924456C2212yescomplemented
38#11924448C2212yescomplemented
39#11924448C2212yescomplemented
40C2×#41×C37C2211yescharacteristic
41C2×M16×C37C2211yescharacteristic, complemented
42C2×C4⋊C4×C37C2211yescharacteristic, complemented
43#103×C37C2212yescomplemented
44#102×C37C2212yescomplemented
45C23×D37—21no—
46C23×D37—22no—
47#41×C37—22no—
48#41×C37D414yescomplemented
49C2×Q8×C37C2311yescharacteristic
50C2×D4×C37—21no—
51C2×D4×C37C2311yescharacteristic
52C22×C4×C37—21no—
53C22×C4×C37—21no—
54C22×C4×C37C2311yescharacteristic
55C2×C22⋊D37—21no—
56C2×C22⋊D37—22no—
57C2×C22⋊D37D412yes—
58#10596966—24no—
59#10596966—24no—
60D4×D37—216no—
61D4×D37—24no—
62D4×D37—24no—
63#10596962—28no—
64#10596962—22no—
65#10596962—22no—
66#10596962—22no—
67#10596962—22no—
68C2×D148—28no—
69C2×D148—22no—
70C2×D148—21no—
71C2×D148—22no—
72C2×D148—22no—
73C2×D148D411yescharacteristic
74C2×D148D414yescomplemented
75C2×D148C2311yescharacteristic
76C2×D148C2312yes—
77C2×C4×D37—22no—
78C2×C4×D37—21no—
79C2×C4×D37C2312yes—
80C2×Dic74D411yescharacteristic
81C2×Dic74C2311yescharacteristic
82M16×C37—22no—
83C2×C8×C37C2312yes—
84C4⋊C4×C37—21no—
85C4⋊C4×C37C2312yes—
86#10596942—22no—
87#10596941—22no—
88#10596940—22no—
89#10596939—22no—
90#10596938—22no—
91#10596938—22no—
92C4⋊Dic37—22no—
93C2×#64436—22no—
94C2×#64436C2311yescharacteristic
95C2×#64436C2311yescharacteristic
96D296—24no—
97#10596930—24no—
98C23×C37—21no—
99C23×C37C2×D411yescharacteristic
100C22×D37—22no—
101C22×D37—28no—
102C22×D37—24no—
103C22×D37—21no—
104C22×D37—82no—
105C22×D37—82no—
106C22×D37—81no—
107C22×D37—81no—
108C22×D37C2×D412yes—
109Q8×C37—21no—
110Q8×C37C2×D412yes—
111D4×C37—24no—
112D4×C37—21no—
113D4×C37C2×D412yes—
114C2×C4×C37—21no—
115C2×C4×C37—24no—
116C2×C4×C37—22no—
117C2×C4×C37—81no—
118C2×C4×C37C2×D412yes—
119C2×C4×C37C2×D411yescharacteristic
120C2×C4×C37C2×D412yes—
121C2×C4×C37C2411yescharacteristic
122C22⋊D37—28no—
123C22⋊D37—24no—
124C22⋊D37—42no—
125C2×Dic37—21no—
126C2×Dic37C2×D412yes—
127D148—28no—
128D148—22no—
129D148—24no—
130D148—22no—
131D148—21no—
132D148—21no—
133D148—82no—
134D148C2×D414yes—
135C4×D37—28no—
136C4×D37—22no—
137C4×D37—22no—
138C4×D37—42no—
139Dic74—22no—
140Dic74—21no—
141Dic74—21no—
142C8×C37—22no—
143#64436—21no—
144#64436—21no—
145C22×C37—21no—
146C22×C37—21no—
147C22×C37—22no—
148C22×C37—41no—
149C22×C37—41no—
150C22×C37#12012yes—
151C22×C37C22×D411yescharacteristic
152C2×D37—22no—
153C2×D37—24no—
154C2×D37—22no—
155C2×D37—22no—
156C2×D37—42no—
157C2×D37—42no—
158C2×D37—41no—
159C2×D37—41no—
160C2×D37—41no—
161C2×D37—84no—
162C2×D37—88no—
163C4×C37—22no—
164C4×C37—42no—
165C4×C37#12012yes—
166C4×C37C22×D411yescharacteristic
167C4×C37C22×D411yescharacteristic
168Dic37—22no—
169Dic37—41no—
170#3012—371nomaximal, Sylow
171C2×C37—22no—
172C2×C37—42no—
173C2×C37C2×#12011yescharacteristic
174C2×C37#57611yescharacteristic
175C2×C37#57711yescharacteristic
176D37—42no—
177D37—44no—
178D37—81no—
179C2×H(5,𝔽2)—371no—
180C2×AGL(1,ℤ/8)—371no—
181C2×#135—371no—
182#537—371no—
183#452—372no—
184#450—372no—
185#449—372no—
186#447—372no—
187#417D3711yescharacteristic, complemented
188C2×#102—371no—
189#408—371no—
190C37#301211yescharacteristic, complemented, Sylow
191H(5,𝔽2)—744no—
192H(5,𝔽2)—744no—
193C2×#41—371no—
194C2×#41—372no—
195C2×#41—371no—
196C2×#41—741no—
197C2×#41C2×D3711yescharacteristic
198C22×D4—372no—
199C22×D4—371no—
200C22×D4—741no—
201C22×D4—742no—
202AGL(1,ℤ/8)—744no—
203#135—744no—
204C2×Q16—371no—
205C2×SD16—371no—
206C2×SD16—371no—
207C2×SD16—372no—
208C2×D8—371no—
209C2×D8—372no—
210C2×M16C2×D3711yescharacteristic, complemented
211C22×C8—371no—
212C22⋊Q8—372no—
213C22⋊Q8—741no—
214#121—372no—
215#121—741no—
216#121—742no—
217C2×C4⋊C4C2×D3711yescharacteristic, complemented
218#103C2×D3712yescomplemented
219#102—372no—
220#102—372no—
221#102C2×D3712yescomplemented
222C22⋊C8—372no—
223C22⋊C8—372no—
224C24—742no—
225C24—741no—
226#41—22no—
227#41—748no—
228#41—744no—
229#41—744no—
230#41—742no—
231#41—742no—
232#41—742no—
233#41—742no—
234#41D14814yescomplemented
235C2×Q8—371no—
236C2×Q8—371no—
237C2×Q8C22×D3711yescharacteristic
238C2×D4—21no—
239C2×D4—372no—
240C2×D4—374no—
241C2×D4—371no—
242C2×D4—372no—
243C2×D4—371no—
244C2×D4—742no—
245C2×D4—742no—
246C2×D4—748no—
247C2×D4—742no—
248C2×D4—744no—
249C2×D4—7416no—
250C2×D4—744no—
251C2×D4—741no—
252C2×D4—742no—
253C2×D4—741no—
254C2×D4C22×D3711yescharacteristic
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—372no—
258C22×C4—742no—
259C22×C4—741no—
260C22×C4C22×D3711yescharacteristic
261Q16—742no—
262SD16—744no—
263SD16—742no—
264SD16—742no—
265D8—744no—
266D8—742no—
267M16—22no—
268C2×C8—371no—
269C2×C8—371no—
270C2×C8—742no—
271C2×C8C22×D3712yes—
272C4⋊C4—21no—
273C4⋊C4—742no—
274C4⋊C4C22×D3712yes—
275C22⋊C4—742no—
276C22⋊C4—742no—
277C23—21no—
278C23—372no—
279C23—2961no—
280C23—2961no—
281C23—2962no—
282C23—2962no—
283C23—748no—
284C23—744no—
285C23—742no—
286C23—741no—
287C23D4×D3711yescharacteristic
288Q8—21no—
289Q8—742no—
290Q8—741no—
291Q8—741no—
292Q8C2×D14812yes—
293D4—24no—
294D4—21no—
295D4—374no—
296D4—2962no—
297D4—748no—
298D4—748no—
299D4—744no—
300D4—744no—
301D4—742no—
302D4—741no—
303D4—742no—
304D4—741no—
305D4—1482no—
306D4C2×D14812yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—81no—
311C2×C4—372no—
312C2×C4—748no—
313C2×C4—742no—
314C2×C4—742no—
315C2×C4—741no—
316C2×C4—1482no—
317C2×C4C2×D14812yes—
318C2×C4D4×D3711yescharacteristic
319C2×C4D4×D3712yes—
320C2×C4C23×D3711yescharacteristic
321C8—22no—
322C8—741no—
323C8—741no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—2964no—
330C22—2968no—
331C22—744no—
332C22—742no—
333C22—742no—
334C22—742no—
335C22—1481no—
336C22—1481no—
337C22—1481no—
338C22—1482no—
339C22—1482no—
340C22#1192450612yes—
341C22C2×D4×D3711yescharacteristic
342C4—22no—
343C4—42no—
344C4—742no—
345C4—1481no—
346C4#1192450612yes—
347C4C2×D4×D3711yescharacteristic
348C4C22×D14811yescharacteristic
349C2—22no—
350C2—42no—
351C2—2961no—
352C2—1482no—
353C2—1484no—
354C2C2×#1192450611yescharacteristic
355C2#1455833711yescharacteristic
356C2#1455834211yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#417C2
socleC22×C37C22×D4
hypercenter#417D37
hypocenterC37#3012
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#11924582C2semidirect
2. C2×#11924564C2semidirect
3. C2×#11924556C2semidirect
4. #14558114C2semidirect
5. #14557698C2semidirect
6. #14557685C2semidirect
7. #14557681C2semidirect
8. #14557668C2semidirect
9. C2×#11924456C2semidirect
10. #14557501C2semidirect
11. C37×#417C2semidirect
17. C2×#10596962C22semidirect
24. C2×#10596942C22semidirect
25. C2×#10596930C22semidirect
26. C2×#10596940C22semidirect
27. C2×#10596941C22semidirect
28. C2×D296C22semidirect
29. C2×#10596939C22semidirect
30. C22×#64436C22semidirect
32. #11924516C22semidirect
34. #11924511C22semidirect
36. #11924456C22semidirect
37. #11924456C22semidirect
38. #11924448C22semidirect
39. #11924448C22semidirect
41. C2×M16×C37C22semidirect
42. C2×C4⋊C4×C37C22semidirect
43. #103×C37C22semidirect
44. #102×C37C22semidirect
48. #41×C37D4semidirect
74. C2×D148D4semidirect
187. #417D37semidirect
190. C37#3012semidirect
210. C2×M16C2×D37semidirect
217. C2×C4⋊C4C2×D37semidirect
218. #103C2×D37semidirect
221. #102C2×D37semidirect
234. #41D148semidirect

Maximal subgroups

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

subgroupindex
C2×#119245822
C2×#119245642
C2×#119245562
#145581142
#145576982
#145576852
#145576812
#145576682
C2×#119244562
#145575012
C37×#4172
#301237