Finite Group Databasebeta

← #18004139

Subgroups — #18004139

Order 4736.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#11924556C211yesmaximal, characteristic, complemented
2#14558100C211yesmaximal, characteristic, complemented
3#14558016C211yesmaximal, characteristic, complemented
4#14557753C212yesmaximal, complemented
5#14557746C212yesmaximal, complemented
6#14557731C212yesmaximal, complemented
7#14557726C212yesmaximal, complemented
8#14557561C211yesmaximal, characteristic, complemented
9#14557535C211yesmaximal, characteristic, complemented
10C2×#11924457C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic
13#11924556—24no—
14C2×#10596942C2211yescharacteristic, complemented
15C2×#10596940C2211yescharacteristic, complemented
16C2×#10596941C2211yescharacteristic, complemented
17C2×#10596939C2211yescharacteristic, complemented
18C22×#64436C2211yescharacteristic
19#11924532C2212yes—
20#11924523C2212yes—
21#11924507—21no—
22#11924507—21no—
23#11924505—22no—
24Q8×Dic37C2212yescomplemented
25D4×Dic37—22no—
26D4×Dic37C2212yescomplemented
27#11924495—21no—
28#11924495C2211yescharacteristic
29C2×C4⋊Dic37—21no—
30C2×C4⋊Dic37C2211yescharacteristic
31#11924465C2212yescomplemented
32#11924465C2212yescomplemented
33#11924461C2212yescomplemented
34#11924458C2212yescomplemented
35#11924457C2212yes—
36#11924457C2212yes—
37C2×#41×C37C2211yescharacteristic
38C22×D4×C37C2211yescharacteristic
39AGL(1,ℤ/8)×C37—24no—
40C2×SD16×C37C2212yes—
41C2×D8×C37C2212yes—
42C2×M16×C37C2211yescharacteristic
43C24×C37—21no—
44#41×C37—22no—
45#41×C37D414yescomplemented
46C2×Q8×C37C2311yescharacteristic
47C2×D4×C37—21no—
48C2×D4×C37—22no—
49C2×D4×C37—21no—
50C2×D4×C37—22no—
51C2×D4×C37C2312yes—
52C2×D4×C37C2311yescharacteristic
53C22×C4×C37—21no—
54C22×C4×C37C2311yescharacteristic
55C2×C22⋊D37—21no—
56C2×C22⋊D37—22no—
57C2×C22⋊D37—22no—
58C22×Dic37—21no—
59C22×Dic37—22no—
60#10596962—22no—
61#10596962—22no—
62C2×D148C2311yescharacteristic
63C2×C4×D37—21no—
64C2×Dic74C2311yescharacteristic
65SD16×C37—24no—
66D8×C37—24no—
67M16×C37—22no—
68C2×C8×C37C2312yes—
69C22⋊Dic37—22no—
70C22⋊Dic37—22no—
71C22⋊Dic37—22no—
72C22⋊Dic37D412yes—
73#10596942—22no—
74#10596941—22no—
75#10596940—22no—
76#10596939—22no—
77C4⋊Dic37—21no—
78C4⋊Dic37—22no—
79C4⋊Dic37D411yescharacteristic
80C4⋊Dic37D411yescharacteristic
81C4⋊Dic37C2312yes—
82C4×Dic37—22no—
83C4×Dic37C2312yes—
84C2×#64436—22no—
85C2×#64436C2311yescharacteristic
86C2×#64436C2311yescharacteristic
87C23×C37—21no—
88C23×C37—22no—
89C23×C37—42no—
90C23×C37—81no—
91C23×C37—81no—
92C23×C37C2×D411yescharacteristic
93C22×D37—21no—
94Q8×C37—21no—
95Q8×C37C2×D412yes—
96D4×C37—24no—
97D4×C37—22no—
98D4×C37—21no—
99D4×C37—22no—
100D4×C37—42no—
101D4×C37C2×D412yes—
102C2×C4×C37—21no—
103C2×C4×C37—24no—
104C2×C4×C37C2×D411yescharacteristic
105C2×C4×C37C2×D412yes—
106C2×C4×C37#4112yes—
107C2×C4×C37C2411yescharacteristic
108C22⋊D37—42no—
109C22⋊D37—82no—
110C22⋊D37—82no—
111C2×Dic37—22no—
112C2×Dic37—21no—
113C2×Dic37—82no—
114C2×Dic37—81no—
115C2×Dic37C2×D412yes—
116C2×Dic37C2×D412yes—
117D148—21no—
118D148—21no—
119C4×D37—42no—
120Dic74—21no—
121Dic74—21no—
122C8×C37—22no—
123#64436—21no—
124#64436—21no—
125C22×C37—21no—
126C22×C37—21no—
127C22×C37—22no—
128C22×C37—41no—
129C22×C37—41no—
130C22×C37—42no—
131C22×C37—42no—
132C22×C37—42no—
133C22×C37—84no—
134C22×C37#12112yes—
135C22×C37C22×D411yescharacteristic
136C2×D37—41no—
137C2×D37—41no—
138C2×D37—41no—
139C4×C37—22no—
140C4×C37#12112yes—
141C4×C37C22×D411yescharacteristic
142C4×C37C2×#4111yescharacteristic
143Dic37—22no—
144Dic37—22no—
145Dic37—41no—
146Dic37—42no—
147#3045—371nomaximal, Sylow
148C2×C37—22no—
149C2×C37—42no—
150C2×C37—82no—
151C2×C37C2×#12111yescharacteristic
152C2×C37#57611yescharacteristic
153C2×C37#57711yescharacteristic
154D37—81no—
155C2×AGL(1,ℤ/8)D3711yescharacteristic, complemented
156C2×#135—371no—
157#535—371no—
158#520—371no—
159#464—372no—
160#463—372no—
161#460—372no—
162#459—372no—
163#424—371no—
164#417—371no—
165C2×#102—371no—
166C37#304511yescharacteristic, complemented, Sylow
167C2×#41—371no—
168C2×#41C2×D3711yescharacteristic
169C22×D4C2×D3711yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—744no—
172C2×Q16—371no—
173C2×SD16—371no—
174C2×SD16—371no—
175C2×SD16C2×D3712yes—
176C2×D8—371no—
177C2×D8C2×D3712yes—
178C2×M16C2×D3711yescharacteristic
179C22×C8—371no—
180#127—372no—
181#124—372no—
182#121—741no—
183#121—741no—
184#120—742no—
185C4×Q8—372no—
186C4×D4—372no—
187C4×D4—742no—
188#117—371no—
189#117—741no—
190C2×C4⋊C4—371no—
191C2×C4⋊C4—741no—
192C4⋊C8—372no—
193C4⋊C8—372no—
194#103—372no—
195#102—372no—
196#102—372no—
197#102—372no—
198C24—21no—
199#41—22no—
200#41—742no—
201#41—742no—
202#41C22⋊D3714yescomplemented
203C2×Q8—371no—
204C2×Q8C22×D3711yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—371no—
210C2×D4—742no—
211C2×D4—742no—
212C2×D4—741no—
213C2×D4C22×D3712yes—
214C2×D4C22×D3711yescharacteristic
215C22×C4—21no—
216C22×C4—742no—
217C22×C4—741no—
218C22×C4—741no—
219C22×C4C22×D3711yescharacteristic
220Q16—742no—
221SD16—24no—
222SD16—742no—
223SD16—742no—
224D8—24no—
225D8—742no—
226M16—22no—
227C2×C8—371no—
228C2×C8—371no—
229C2×C8—742no—
230C2×C8C22×D3712yes—
231C4⋊C4—371no—
232C4⋊C4—371no—
233C4⋊C4—372no—
234C4⋊C4—742no—
235C4⋊C4—741no—
236C22⋊C4—372no—
237C22⋊C4—742no—
238C22⋊C4—742no—
239C22⋊C4—742no—
240C42—372no—
241C42—742no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—741no—
248C23D4×D3711yescharacteristic
249Q8—21no—
250Q8—741no—
251Q8—741no—
252Q8C2×C22⋊D3712yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—2962no—
259D4—2962no—
260D4—741no—
261D4—741no—
262D4—1482no—
263D4C2×C22⋊D3712yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—372no—
267C2×C4—372no—
268C2×C4—2962no—
269C2×C4—2961no—
270C2×C4—742no—
271C2×C4—741no—
272C2×C4—1482no—
273C2×C4D4×D3711yescharacteristic
274C2×C4#1059696412yes—
275C2×C4C2×C22⋊D3712yes—
276C2×C4C23×D3711yescharacteristic
277C8—22no—
278C8—741no—
279C8—741no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—1481no—
290C22—1481no—
291C22—1481no—
292C22#1192450812yes—
293C22C2×D4×D3711yescharacteristic
294C4—22no—
295C4—742no—
296C4—742no—
297C4—1481no—
298C4—1482no—
299C4#1192450812yes—
300C4C22×C22⋊D3711yescharacteristic
301C4C2×#1059696411yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—2961no—
306C2C2×#1192450811yescharacteristic
307C2#1455833711yescharacteristic
308C2#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#3045
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×#11924556C2semidirect
2. #14558100C2semidirect
3. #14558016C2semidirect
4. #14557753C2semidirect
5. #14557746C2semidirect
6. #14557731C2semidirect
7. #14557726C2semidirect
8. #14557561C2semidirect
9. #14557535C2semidirect
10. C2×#11924457C2semidirect
11. C2×AGL(1,ℤ/8)×C37C2semidirect
14. C2×#10596942C22semidirect
15. C2×#10596940C22semidirect
16. C2×#10596941C22semidirect
17. C2×#10596939C22semidirect
24. Q8×Dic37C22semidirect
26. D4×Dic37C22semidirect
31. #11924465C22semidirect
32. #11924465C22semidirect
33. #11924461C22semidirect
34. #11924458C22semidirect
45. #41×C37D4semidirect
155. C2×AGL(1,ℤ/8)D37semidirect
166. C37#3045semidirect
202. #41C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245562
#145581002
#145580162
#145577532
#145577462
#145577312
#145577262
#145575612
#145575352
C2×#119244572
C2×AGL(1,ℤ/8)×C372
#304537