Finite Group Databasebeta

← #18005081

Subgroups — #18005081

Order 4736.

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

normal subgroups only

This group has 596 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×#11924564C211yesmaximal, characteristic, complemented
2C2×#11924556C211yesmaximal, characteristic, complemented
3C22×#10596939C211yesmaximal, characteristic, complemented
4#14558100C211yesmaximal, characteristic, complemented
5#14558097C211yesmaximal, characteristic, complemented
6#14557921C212yesmaximal, complemented
7#14557919C212yesmaximal, complemented
8#14557914C212yesmaximal, complemented
9#14557908C212yesmaximal, complemented
10#14557491C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic
13C22×D148C2211yescharacteristic
14#11924564—24no—
15#11924556—24no—
16C2×#10596942C2211yescharacteristic, complemented
17C2×#10596930C2212yescomplemented
18C2×#10596940C2211yescharacteristic, complemented
19C2×#10596941C2211yescharacteristic, complemented
20C2×D296C2212yescomplemented
21C2×#10596939—24no—
22C2×#10596939C2212yes—
23C2×#10596939C2212yes—
24C2×#10596939C2211yescharacteristic, complemented
25C22×#64436C2211yescharacteristic
26#11924532C2212yescomplemented
27#11924532C2212yescomplemented
28#11924525C2212yescomplemented
29#11924523C2212yescomplemented
30#11924510—22no—
31#11924507—22no—
32#11924505—22no—
33#11924504—22no—
34#11924495C2211yescharacteristic
35#11924446C2212yescomplemented
36C8×Dic37C2212yescomplemented
37C2×#41×C37C2211yescharacteristic
38C22×D4×C37C2211yescharacteristic
39AGL(1,ℤ/8)×C37—24no—
40C2×SD16×C37C2212yescomplemented
41C2×D8×C37C2212yescomplemented
42C2×M16×C37C2211yescharacteristic
43C24×C37—21no—
44C23×D37—21no—
45#41×C37—22no—
46#41×C37—22no—
47C2×Q8×C37C2311yescharacteristic
48C2×D4×C37—21no—
49C2×D4×C37—22no—
50C2×D4×C37—22no—
51C2×D4×C37—21no—
52C2×D4×C37C2312yes—
53C2×D4×C37C2311yescharacteristic
54C22×C4×C37—21no—
55C22×C4×C37C2311yescharacteristic
56C2×C22⋊D37—21no—
57C2×C22⋊D37—24no—
58C2×C22⋊D37—24no—
59#10596962—22no—
60#10596962—22no—
61C2×D148—22no—
62C2×D148—22no—
63C2×D148—21no—
64C2×D148C2312yes—
65C2×D148C2311yescharacteristic
66C2×C4×D37—21no—
67C2×Dic74C2311yescharacteristic
68SD16×C37—24no—
69D8×C37—24no—
70M16×C37D414yescomplemented
71C2×C8×C37C2312yes—
72C22⋊Dic37—24no—
73C22⋊Dic37D412yes—
74#10596942—22no—
75#10596941—22no—
76#10596940—22no—
77#10596939—24no—
78#10596939—24no—
79#10596939—22no—
80#10596938—24no—
81C4⋊Dic37D412yes—
82C4×Dic37C2312yes—
83C2×#64436D414yescomplemented
84C2×#64436C2311yescharacteristic
85C2×#64436C2311yescharacteristic
86D296—24no—
87#10596930—24no—
88C23×C37—22no—
89C23×C37—21no—
90C23×C37—42no—
91C23×C37—82no—
92C23×C37C2×D411yescharacteristic
93C22×D37—22no—
94C22×D37—21no—
95C22×D37—42no—
96C22×D37—82no—
97Q8×C37—21no—
98Q8×C37—21no—
99D4×C37—22no—
100D4×C37—21no—
101D4×C37—21no—
102D4×C37—22no—
103D4×C37—42no—
104D4×C37—42no—
105C2×C4×C37—21no—
106C2×C4×C37—42no—
107C2×C4×C37C2×D411yescharacteristic
108C2×C4×C37C2×D412yes—
109C2×C4×C37C2×D412yes—
110C2×C4×C37C2411yescharacteristic
111C22⋊D37—42no—
112C22⋊D37—84no—
113C22⋊D37—84no—
114C2×Dic37—21no—
115C2×Dic37C2×D414yes—
116D148—22no—
117D148—21no—
118D148—22no—
119D148—21no—
120D148—42no—
121C4×D37—42no—
122Dic74—21no—
123Dic74—21no—
124C8×C37C2×D414yes—
125#64436C2×D412yes—
126#64436C2×D412yes—
127C22×C37—22no—
128C22×C37—41no—
129C22×C37—42no—
130C22×C37—42no—
131C22×C37—44no—
132C22×C37—84no—
133C22×C37#12712yes—
134C22×C37C22×D411yescharacteristic
135C2×D37—41no—
136C2×D37—42no—
137C2×D37—42no—
138C2×D37—44no—
139C2×D37—84no—
140C4×C37—41no—
141C4×C37#12712yes—
142C4×C37C22×D411yescharacteristic
143C4×C37C22×D411yescharacteristic
144Dic37—24no—
145Dic37—41no—
146#3155—371nomaximal, Sylow
147C2×C37—22no—
148C2×C37—81no—
149C2×C37—82no—
150C2×C37C2×#12711yescharacteristic
151C2×C37#57612yes—
152D37—81no—
153D37—82no—
154C2×AGL(1,ℤ/8)—371no—
155C2×AGL(1,ℤ/8)D3711yescharacteristic, complemented
156C2×#135—371no—
157C22×D8—371no—
158#535—371no—
159#535—371no—
160#496—372no—
161#496—372no—
162#495—372no—
163#493—372no—
164#405—371no—
165C37#315511yescharacteristic, complemented, Sylow
166C2×#41—371no—
167C2×#41C2×D3711yescharacteristic
168C22×D4—371no—
169C22×D4C2×D3711yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—744no—
172#135—744no—
173C2×Q16—371no—
174C2×SD16—371no—
175C2×SD16—371no—
176C2×SD16—372no—
177C2×SD16C2×D3712yescomplemented
178C2×D8—372no—
179C2×D8—372no—
180C2×D8—371no—
181C2×D8—372no—
182C2×D8—744no—
183C2×D8C2×D3712yescomplemented
184C2×M16C2×D3711yescharacteristic
185C22×C8—371no—
186#127—372no—
187#127—372no—
188#124—372no—
189#124—372no—
190#121—742no—
191#121—742no—
192#120—742no—
193#120—742no—
194#117—371no—
195#97—372no—
196C4×C8—372no—
197C24—21no—
198C24—741no—
199#41—22no—
200#41—22no—
201#41—742no—
202#41—742no—
203C2×Q8—371no—
204C2×Q8C22×D3711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—372no—
210C2×D4—371no—
211C2×D4—744no—
212C2×D4—744no—
213C2×D4—741no—
214C2×D4—742no—
215C2×D4—741no—
216C2×D4—742no—
217C2×D4C22×D3712yes—
218C2×D4C22×D3711yescharacteristic
219C22×C4—21no—
220C22×C4—741no—
221C22×C4C22×D3711yescharacteristic
222Q16—742no—
223SD16—24no—
224SD16—742no—
225SD16—742no—
226SD16—744no—
227D8—24no—
228D8—744no—
229D8—744no—
230D8—742no—
231D8—744no—
232M16C22⋊D3714yescomplemented
233C2×C8—374no—
234C2×C8—371no—
235C2×C8—371no—
236C2×C8C22×D3712yes—
237C4⋊C4—372no—
238C22⋊C4—372no—
239C22⋊C4—744no—
240C22⋊C4—744no—
241C42—372no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—2962no—
247C23—742no—
248C23—741no—
249C23—1482no—
250C23D4×D3711yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—741no—
254Q8—741no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—2964no—
262D4—2964no—
263D4—742no—
264D4—741no—
265D4—741no—
266D4—742no—
267D4—1482no—
268D4—1482no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—374no—
272C2×C4—741no—
273C2×C4—1482no—
274C2×C4D4×D3711yescharacteristic
275C2×C4D4×D3712yes—
276C2×C4C2×C22⋊D3712yes—
277C2×C4C23×D3711yescharacteristic
278C8—372no—
279C8—372no—
280C8C2×C22⋊D3714yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—2964no—
288C22—1481no—
289C22—1482no—
290C22—1482no—
291C22—1484no—
292C22#1192453212yes—
293C22C2×D4×D3711yescharacteristic
294C4—41no—
295C4—744no—
296C4—1481no—
297C4#1192453212yes—
298C4C2×D4×D3711yescharacteristic
299C4C22×C22⋊D3711yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—2962no—
304C2—2961no—
305C2C2×#1192453211yescharacteristic
306C2#1455833712yes—
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#3155
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×#11924564C2semidirect
2. C2×#11924556C2semidirect
3. C22×#10596939C2semidirect
4. #14558100C2semidirect
5. #14558097C2semidirect
6. #14557921C2semidirect
7. #14557919C2semidirect
8. #14557914C2semidirect
9. #14557908C2semidirect
10. #14557491C2semidirect
11. C2×AGL(1,ℤ/8)×C37C2semidirect
16. C2×#10596942C22semidirect
17. C2×#10596930C22semidirect
18. C2×#10596940C22semidirect
19. C2×#10596941C22semidirect
20. C2×D296C22semidirect
24. C2×#10596939C22semidirect
26. #11924532C22semidirect
27. #11924532C22semidirect
28. #11924525C22semidirect
29. #11924523C22semidirect
35. #11924446C22semidirect
36. C8×Dic37C22semidirect
40. C2×SD16×C37C22semidirect
41. C2×D8×C37C22semidirect
70. M16×C37D4semidirect
83. C2×#64436D4semidirect
155. C2×AGL(1,ℤ/8)D37semidirect
165. C37#3155semidirect
177. C2×SD16C2×D37semidirect
183. C2×D8C2×D37semidirect
232. M16C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245642
C2×#119245562
C22×#105969392
#145581002
#145580972
#145579212
#145579192
#145579142
#145579082
#145574912
C2×AGL(1,ℤ/8)×C372
#315537