Finite Group Databasebeta

← #18004270

Subgroups — #18004270

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×#11924544C211yesmaximal, characteristic, complemented
2#14558111C211yesmaximal, characteristic, complemented
3#14558100C211yesmaximal, characteristic, complemented
4#14557783C212yesmaximal, complemented
5#14557776C212yesmaximal, complemented
6#14557766C212yesmaximal, complemented
7#14557759C212yesmaximal, complemented
8#14557575C211yesmaximal, characteristic, complemented
9#14557529C211yesmaximal, characteristic, complemented
10C2×#11924457C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic
13#11924544—24no—
14C2×#10596929C2212yescomplemented
15C22×#64436C2211yescharacteristic
16C2×C8×D37C2212yescomplemented
17#11924532—21no—
18#11924523—21no—
19#11924515—21no—
20#11924512C2212yescomplemented
21#11924510—21no—
22#11924508—22no—
23#11924507C2212yes—
24#11924507C2212yes—
25#11924507C2212yescomplemented
26#11924505—22no—
27#11924495C2211yescharacteristic
28C2×C4⋊Dic37C2211yescharacteristic
29#11924470C2212yescomplemented
30#11924468C2212yescomplemented
31#11924460C2211yescharacteristic, complemented
32#11924460C2211yescharacteristic, complemented
33#11924457C2212yes—
34#11924457C2212yes—
35#11924457C2211yescharacteristic, complemented
36#11924457C2211yescharacteristic, complemented
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×C37—22no—
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—22no—
56C2×C22⋊D37—22no—
57C2×C22⋊D37—22no—
58C2×C22⋊D37D412yes—
59C22×Dic37—21no—
60#10596962—22no—
61#10596962—22no—
62C2×D148D411yescharacteristic
63C2×C4×D37C2312yes—
64C2×Dic74D411yescharacteristic
65SD16×C37—24no—
66D8×C37—24no—
67M16×C37D414yescomplemented
68C2×C8×C37C2312yes—
69C22⋊Dic37—21no—
70C22⋊Dic37—22no—
71C22⋊Dic37—22no—
72C22⋊Dic37—22no—
73#10596938—22no—
74C4⋊Dic37—21no—
75C4⋊Dic37C2312yes—
76C4⋊Dic37C2311yescharacteristic
77C4⋊Dic37C2311yescharacteristic
78#10596936—22no—
79C4×Dic37—21no—
80C2×#64436—22no—
81C2×#64436C2311yescharacteristic
82C2×#64436C2311yescharacteristic
83#10596929—24no—
84C8×D37—24no—
85C23×C37—21no—
86C23×C37—22no—
87C23×C37—42no—
88C23×C37—81no—
89C23×C37—81no—
90C23×C37C2×D411yescharacteristic
91C22×D37C2×D412yes—
92Q8×C37—21no—
93Q8×C37—21no—
94D4×C37—21no—
95D4×C37—22no—
96D4×C37—22no—
97D4×C37—21no—
98D4×C37—42no—
99D4×C37—42no—
100C2×C4×C37—21no—
101C2×C4×C37—42no—
102C2×C4×C37C2×D411yescharacteristic
103C2×C4×C37C2×D412yes—
104C2×C4×C37#4112yes—
105C2×C4×C37C2411yescharacteristic
106C22⋊D37—24no—
107C22⋊D37—82no—
108C22⋊D37—82no—
109C22⋊D37—82no—
110C2×Dic37—21no—
111C2×Dic37—21no—
112C2×Dic37—22no—
113C2×Dic37—81no—
114C2×Dic37C2×D412yes—
115D148—22no—
116C4×D37—22no—
117C4×D37—22no—
118Dic74—22no—
119C8×C37C2×D414yes—
120#64436—21no—
121#64436—21no—
122C22×C37—21no—
123C22×C37—21no—
124C22×C37—41no—
125C22×C37—42no—
126C22×C37—41no—
127C22×C37—42no—
128C22×C37—42no—
129C22×C37—41no—
130C22×C37—84no—
131C22×C37#12112yes—
132C22×C37C22×D411yescharacteristic
133C2×D37—22no—
134C2×D37—22no—
135C2×D37—22no—
136C4×C37—41no—
137C4×C37#12112yes—
138C4×C37C22×D411yescharacteristic
139C4×C37C2×#4111yescharacteristic
140Dic37—22no—
141Dic37—42no—
142Dic37—41no—
143Dic37—41no—
144#3062—371nomaximal, Sylow
145C2×C37—22no—
146C2×C37—81no—
147C2×C37—82no—
148C2×C37C2×#12111yescharacteristic
149C2×C37#57611yescharacteristic
150C2×C37#57711yescharacteristic
151D37—42no—
152C2×AGL(1,ℤ/8)D3711yescharacteristic, complemented
153C2×#131—371no—
154#537—371no—
155#535—371no—
156#469—372no—
157#468—372no—
158#466—372no—
159#465—372no—
160#428—371no—
161#416—371no—
162C2×#102—371no—
163C37#306211yescharacteristic, complemented, Sylow
164C2×#41—371no—
165C2×#41C2×D3711yescharacteristic
166C22×D4C2×D3711yescharacteristic
167AGL(1,ℤ/8)—24no—
168C2×SD16C2×D3712yes—
169C2×D8C2×D3712yes—
170#131—744no—
171C2×M16—372no—
172C2×M16C2×D3711yescharacteristic
173C22×C8—371no—
174C22×C8—372no—
175#127—741no—
176#124—741no—
177C22⋊Q8—372no—
178C22⋊Q8—741no—
179#121—372no—
180#121—372no—
181#121—372no—
182#121—741no—
183#121—742no—
184#120—742no—
185#117—371no—
186C2×C4⋊C4—371no—
187#107—372no—
188#106—372no—
189#103—371no—
190#103—371no—
191#102—372no—
192#102—372no—
193#102—371no—
194#102—371no—
195C24—21no—
196#41—22no—
197#41—22no—
198#41—742no—
199#41—742no—
200C2×Q8—371no—
201C2×Q8C22×D3711yescharacteristic
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—372no—
207C2×D4—371no—
208C2×D4—742no—
209C2×D4—742no—
210C2×D4—742no—
211C2×D4C22×D3711yescharacteristic
212C2×D4C22×D3712yes—
213C22×C4—21no—
214C22×C4—372no—
215C22×C4—741no—
216C22×C4C22×D3711yescharacteristic
217SD16—24no—
218D8—24no—
219M16—744no—
220M16C22⋊D3714yescomplemented
221C2×C8—371no—
222C2×C8—371no—
223C2×C8—742no—
224C2×C8—744no—
225C2×C8C22×D3712yes—
226C4⋊C4—372no—
227C4⋊C4—371no—
228C4⋊C4—371no—
229C4⋊C4—742no—
230C4⋊C4—741no—
231C22⋊C4—742no—
232C22⋊C4—742no—
233C22⋊C4—742no—
234C22⋊C4—742no—
235C22⋊C4—741no—
236C42—741no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—81no—
241C23—81no—
242C23—372no—
243C23D4×D3711yescharacteristic
244Q8—21no—
245Q8—21no—
246Q8—742no—
247D4—21no—
248D4—22no—
249D4—22no—
250D4—21no—
251D4—42no—
252D4—42no—
253D4—2962no—
254D4—2962no—
255D4—2962no—
256D4—744no—
257D4—742no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—372no—
261C2×C4—2961no—
262C2×C4—742no—
263C2×C4—742no—
264C2×C4—741no—
265C2×C4—741no—
266C2×C4—742no—
267C2×C4D4×D3711yescharacteristic
268C2×C4#1059696412yes—
269C2×C4C2×C22⋊D3712yes—
270C2×C4C23×D3711yescharacteristic
271C8—741no—
272C8—741no—
273C8C2×C22⋊D3714yes—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—84no—
283C22—742no—
284C22—742no—
285C22—742no—
286C22#1192450712yes—
287C22C2×D4×D3711yescharacteristic
288C4—41no—
289C4—742no—
290C4—1482no—
291C4—1481no—
292C4—1481no—
293C4#1192450712yes—
294C4C22×C22⋊D3711yescharacteristic
295C4C2×#1059696411yescharacteristic
296C2—22no—
297C2—82no—
298C2—81no—
299C2—1482no—
300C2C2×#1192450711yescharacteristic
301C2#1455833711yescharacteristic
302C2#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#3062
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×#11924544C2semidirect
2. #14558111C2semidirect
3. #14558100C2semidirect
4. #14557783C2semidirect
5. #14557776C2semidirect
6. #14557766C2semidirect
7. #14557759C2semidirect
8. #14557575C2semidirect
9. #14557529C2semidirect
10. C2×#11924457C2semidirect
11. C2×AGL(1,ℤ/8)×C37C2semidirect
14. C2×#10596929C22semidirect
16. C2×C8×D37C22semidirect
20. #11924512C22semidirect
25. #11924507C22semidirect
29. #11924470C22semidirect
30. #11924468C22semidirect
31. #11924460C22semidirect
32. #11924460C22semidirect
35. #11924457C22semidirect
36. #11924457C22semidirect
67. M16×C37D4semidirect
152. C2×AGL(1,ℤ/8)D37semidirect
163. C37#3062semidirect
220. M16C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245442
#145581112
#145581002
#145577832
#145577762
#145577662
#145577592
#145575752
#145575292
C2×#119244572
C2×AGL(1,ℤ/8)×C372
#306237