Finite Group Databasebeta

← #18003899

Subgroups — #18003899

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×#11924581C211yesmaximal, characteristic, complemented
2C2×#11924556C211yesmaximal, characteristic, complemented
3#14558111C211yesmaximal, characteristic, complemented
4#14557692C212yesmaximal, complemented
5#14557682C212yesmaximal, complemented
6#14557675C212yesmaximal, complemented
7#14557665C212yesmaximal, complemented
8#14557535C211yesmaximal, characteristic, complemented
9C2×#11924457C211yesmaximal, characteristic, complemented
10#14557501C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
12#11924581—24no—
13#11924581—24no—
14C2×#10596962C2211yescharacteristic
15C2×#10596962C2211yescharacteristic
16C2×#10596964—21no—
17C2×#10596964C2212yes—
18C22×C22⋊D37—22no—
19C2×D4×D37—21no—
20C2×D4×D37C2212yes—
21#11924556—24no—
22C2×#10596942C2211yescharacteristic, complemented
23C2×#10596940C2211yescharacteristic, complemented
24C2×#10596941C2211yescharacteristic, complemented
25C2×#10596939C2211yescharacteristic, complemented
26C22×#64436C2211yescharacteristic, complemented
27#11924515—21no—
28#11924512C2212yescomplemented
29#11924510—21no—
30#11924508—22no—
31#11924507C2212yescomplemented
32C2×C4⋊Dic37C2211yescharacteristic, complemented
33#11924461C2212yescomplemented
34#11924458C2212yescomplemented
35#11924457C2212yescomplemented
36#11924457C2212yescomplemented
37#11924448C2212yescomplemented
38#11924448C2212yescomplemented
39C2×#41×C37C2211yescharacteristic, complemented
40C22×D4×C37C2211yescharacteristic
41AGL(1,ℤ/8)×C37—24no—
42C2×SD16×C37C2212yescomplemented
43C2×D8×C37C2212yescomplemented
44C2×M16×C37C2211yescharacteristic, complemented
45C24×C37—21no—
46C23×D37—22no—
47#41×C37—22no—
48#41×C37—22no—
49C2×Q8×C37C2311yescharacteristic
50C2×D4×C37—21no—
51C2×D4×C37—21no—
52C2×D4×C37—22no—
53C2×D4×C37D414yescomplemented
54C2×D4×C37C2311yescharacteristic
55C2×D4×C37C2312yes—
56C22×C4×C37—21no—
57C22×C4×C37C2311yescharacteristic
58C2×C22⋊D37—216no—
59C2×C22⋊D37—21no—
60C2×C22⋊D37—22no—
61C2×C22⋊D37—22no—
62C2×C22⋊D37—22no—
63C2×C22⋊D37D412yes—
64C22×Dic37—22no—
65C22×Dic37—21no—
66#10596964—28no—
67#10596964—24no—
68#10596964—24no—
69D4×D37—28no—
70D4×D37—24no—
71D4×D37—24no—
72#10596962—22no—
73#10596962—22no—
74#10596962—22no—
75#10596962D414yescomplemented
76C2×D148D411yescharacteristic
77C2×D148C2311yescharacteristic
78C2×C4×D37—21no—
79C2×C4×D37C2312yes—
80C2×Dic74D411yescharacteristic
81C2×Dic74C2311yescharacteristic
82SD16×C37—24no—
83D8×C37—24no—
84M16×C37—22no—
85C2×C8×C37C2312yes—
86C22⋊Dic37—22no—
87#10596942—22no—
88#10596941—22no—
89#10596940—22no—
90#10596939—22no—
91#10596938—22no—
92C4⋊Dic37—21no—
93C4⋊Dic37C2312yes—
94#10596936—22no—
95C2×#64436—22no—
96C2×#64436C2311yescharacteristic
97C2×#64436C2311yescharacteristic
98C23×C37—22no—
99C23×C37—24no—
100C23×C37—21no—
101C23×C37—81no—
102C23×C37—81no—
103C23×C37C2×D411yescharacteristic
104C22×D37—21no—
105C22×D37—28no—
106C22×D37—82no—
107C22×D37—82no—
108C22×D37C2×D412yes—
109Q8×C37—21no—
110Q8×C37—21no—
111D4×C37—24no—
112D4×C37—22no—
113D4×C37—21no—
114D4×C37—21no—
115D4×C37—42no—
116D4×C37C2×D414yes—
117C2×C4×C37—21no—
118C2×C4×C37—42no—
119C2×C4×C37C2×D411yescharacteristic
120C2×C4×C37C2×D412yes—
121C2×C4×C37C2×D412yes—
122C2×C4×C37C2411yescharacteristic
123C22⋊D37—28no—
124C22⋊D37—28no—
125C22⋊D37—24no—
126C22⋊D37—24no—
127C22⋊D37—82no—
128C2×Dic37—28no—
129C2×Dic37—21no—
130C2×Dic37—22no—
131C2×Dic37—81no—
132C2×Dic37C2×D412yes—
133D148—22no—
134D148—21no—
135D148C2×D412yes—
136C4×D37—24no—
137C4×D37—22no—
138C4×D37—22no—
139Dic74—22no—
140Dic74—21no—
141Dic74C2×D412yes—
142C8×C37—22no—
143#64436—21no—
144#64436—21no—
145C22×C37—21no—
146C22×C37—21no—
147C22×C37—24no—
148C22×C37—42no—
149C22×C37—42no—
150C22×C37—41no—
151C22×C37—41no—
152C22×C37—41no—
153C22×C37—84no—
154C22×C37#12012yes—
155C22×C37C22×D411yescharacteristic
156C2×D37—22no—
157C2×D37—22no—
158C2×D37—22no—
159C2×D37—22no—
160C2×D37—41no—
161C2×D37—41no—
162C2×D37—88no—
163C4×C37—41no—
164C4×C37#12012yes—
165C4×C37C22×D411yescharacteristic
166C4×C37C22×D411yescharacteristic
167Dic37—22no—
168Dic37—22no—
169Dic37—42no—
170#3012—371nomaximal, Sylow
171C2×C37—22no—
172C2×C37—44no—
173C2×C37—81no—
174C2×C37C2×#12011yescharacteristic
175C2×C37#57611yescharacteristic
176C2×C37#57711yescharacteristic
177D37—42no—
178D37—42no—
179C2×H(5,𝔽2)—371no—
180C2×AGL(1,ℤ/8)D3711yescharacteristic, complemented
181C2×#135—371no—
182#537—371no—
183#452—372no—
184#450—372no—
185#449—372no—
186#447—372no—
187#417—371no—
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, complemented
198C22×D4—372no—
199C22×D4—741no—
200C22×D4—742no—
201C22×D4C2×D3711yescharacteristic
202AGL(1,ℤ/8)—24no—
203#135—744no—
204C2×Q16—371no—
205C2×SD16—371no—
206C2×SD16—371no—
207C2×SD16C2×D3712yescomplemented
208C2×D8—371no—
209C2×D8C2×D3712yescomplemented
210C2×M16C2×D3711yescharacteristic, complemented
211C22×C8—371no—
212C22⋊Q8—372no—
213C22⋊Q8—741no—
214#121—372no—
215#121—741no—
216#121—742no—
217C2×C4⋊C4—371no—
218#103—372no—
219#102—372no—
220#102—372no—
221#102—372no—
222C22⋊C8—372no—
223C22⋊C8—372no—
224C24—21no—
225C24—742no—
226#41—22no—
227#41—22no—
228#41—374no—
229#41—744no—
230#41—748no—
231#41—744no—
232#41—742no—
233#41—742no—
234#41—742no—
235C2×Q8—371no—
236C2×Q8—371no—
237C2×Q8C22×D3711yescharacteristic
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—372no—
242C2×D4—371no—
243C2×D4—371no—
244C2×D4—742no—
245C2×D4—744no—
246C2×D4—748no—
247C2×D4—742no—
248C2×D4—742no—
249C2×D4—7416no—
250C2×D4—741no—
251C2×D4—744no—
252C2×D4C22⋊D3714yescomplemented
253C2×D4C22×D3711yescharacteristic
254C2×D4C22×D3712yes—
255C22×C4—21no—
256C22×C4—372no—
257C22×C4—742no—
258C22×C4—741no—
259C22×C4—741no—
260C22×C4C22×D3711yescharacteristic
261Q16—742no—
262SD16—24no—
263SD16—742no—
264SD16—742no—
265D8—24no—
266D8—742no—
267M16—22no—
268C2×C8—371no—
269C2×C8—371no—
270C2×C8—742no—
271C2×C8C22×D3712yes—
272C4⋊C4—372no—
273C4⋊C4—742no—
274C4⋊C4—741no—
275C22⋊C4—742no—
276C22⋊C4—742no—
277C23—22no—
278C23—24no—
279C23—21no—
280C23—81no—
281C23—81no—
282C23—372no—
283C23—2962no—
284C23—2962no—
285C23—748no—
286C23—741no—
287C23D4×D3711yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—372no—
291Q8—742no—
292Q8—741no—
293D4—24no—
294D4—22no—
295D4—21no—
296D4—21no—
297D4—42no—
298D4—372no—
299D4—2962no—
300D4—748no—
301D4—748no—
302D4—744no—
303D4—744no—
304D4—742no—
305D4—741no—
306D4C2×C22⋊D3714yes—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—372no—
310C2×C4—2961no—
311C2×C4—748no—
312C2×C4—741no—
313C2×C4—744no—
314C2×C4—742no—
315C2×C4—742no—
316C2×C4—742no—
317C2×C4D4×D3711yescharacteristic
318C2×C4D4×D3712yes—
319C2×C4C2×C22⋊D3712yes—
320C2×C4C23×D3711yescharacteristic
321C8—22no—
322C8—741no—
323C8—741no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—84no—
333C22—2968no—
334C22—742no—
335C22—742no—
336C22—742no—
337C22—742no—
338C22—1481no—
339C22—1481no—
340C22#1192450412yes—
341C22C2×D4×D3711yescharacteristic
342C4—41no—
343C4—742no—
344C4—742no—
345C4—1482no—
346C4#1192450412yes—
347C4C2×D4×D3711yescharacteristic
348C4C22×C22⋊D3711yescharacteristic
349C2—22no—
350C2—44no—
351C2—81no—
352C2—1482no—
353C2—1482no—
354C2C2×#1192450411yescharacteristic
355C2#1455833711yescharacteristic
356C2#1455834311yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×AGL(1,ℤ/8)×C37C2
socleC22×C37C22×D4
hypercenterC2×AGL(1,ℤ/8)D37
hypocenterC37#3012
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#11924581C2semidirect
2. C2×#11924556C2semidirect
3. #14558111C2semidirect
4. #14557692C2semidirect
5. #14557682C2semidirect
6. #14557675C2semidirect
7. #14557665C2semidirect
8. #14557535C2semidirect
9. C2×#11924457C2semidirect
10. #14557501C2semidirect
11. C2×AGL(1,ℤ/8)×C37C2semidirect
22. C2×#10596942C22semidirect
23. C2×#10596940C22semidirect
24. C2×#10596941C22semidirect
25. C2×#10596939C22semidirect
26. C22×#64436C22semidirect
28. #11924512C22semidirect
31. #11924507C22semidirect
32. C2×C4⋊Dic37C22semidirect
33. #11924461C22semidirect
34. #11924458C22semidirect
35. #11924457C22semidirect
36. #11924457C22semidirect
37. #11924448C22semidirect
38. #11924448C22semidirect
39. C2×#41×C37C22semidirect
42. C2×SD16×C37C22semidirect
43. C2×D8×C37C22semidirect
44. C2×M16×C37C22semidirect
53. C2×D4×C37D4semidirect
75. #10596962D4semidirect
180. C2×AGL(1,ℤ/8)D37semidirect
190. C37#3012semidirect
197. C2×#41C2×D37semidirect
207. C2×SD16C2×D37semidirect
209. C2×D8C2×D37semidirect
210. C2×M16C2×D37semidirect
252. C2×D4C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245812
C2×#119245562
#145581112
#145576922
#145576822
#145576752
#145576652
#145575352
C2×#119244572
#145575012
C2×AGL(1,ℤ/8)×C372
#301237