Finite Group Databasebeta

← #18004074

Subgroups — #18004074

Order 4736.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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×#11924561C211yesmaximal, characteristic, complemented
2C2×#11924532C211yesmaximal, characteristic, complemented
3#41×Dic37C211yesmaximal, characteristic, complemented
4#14557731C214yesmaximal, complemented
5#14557726C214yesmaximal, complemented
6#14557556C211yesmaximal, characteristic, complemented
7#14557539C211yesmaximal, characteristic, complemented
8#14557538C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
10C22×D148C2211yescharacteristic
11C22×C22⋊D37—22no—
12#11924561—24no—
13C2×#10596941C2212yescomplemented
14C2×#10596939C2212yescomplemented
15C2×#10596934C2211yescharacteristic
16#11924532—21no—
17#11924532—21no—
18#11924532C2214yes—
19Q8×Dic37C2212yescomplemented
20D4×Dic37—22no—
21D4×Dic37C2212yescomplemented
22#11924495—21no—
23#11924495C2211yescharacteristic
24C2×C4×Dic37—21no—
25C2×C4×Dic37C2211yescharacteristic
26#11924465C2214yescomplemented
27#11924458C2214yescomplemented
28#11924457C2214yescomplemented
29C2×#41×C37C2211yescharacteristic
30C22×D4×C37C2211yescharacteristic
31AGL(1,ℤ/8)×C37—24no—
32C2×SD16×C37C2212yescomplemented
33C2×D8×C37C2212yescomplemented
34C2×M16×C37C2211yescharacteristic
35C24×C37—21no—
36C23×D37—21no—
37#41×C37—22no—
38#41×C37D414yescomplemented
39C2×Q8×C37C2311yescharacteristic
40C2×D4×C37—21no—
41C2×D4×C37—22no—
42C2×D4×C37—21no—
43C2×D4×C37—22no—
44C2×D4×C37C2312yes—
45C2×D4×C37C2311yescharacteristic
46C22×C4×C37—21no—
47C22×C4×C37C2311yescharacteristic
48C2×C22⋊D37—24no—
49C2×C22⋊D37—24no—
50C2×C22⋊D37—48no—
51C22×Dic37—22no—
52C22×Dic37D412yes—
53C2×D148—22no—
54C2×D148—22no—
55C2×D148—21no—
56C2×D148C2312yes—
57SD16×C37—24no—
58D8×C37—24no—
59M16×C37—22no—
60C2×C8×C37C2312yes—
61C22⋊Dic37—21no—
62C22⋊Dic37—22no—
63#10596941—24no—
64#10596939—24no—
65C4⋊Dic37—22no—
66C4⋊Dic37C2312yes—
67C4×Dic37—21no—
68C4×Dic37—22no—
69C4×Dic37D411yescharacteristic
70C4×Dic37D411yescharacteristic
71C4×Dic37C2312yes—
72#10596934—22no—
73C2×#64436C2312yes—
74C23×C37—21no—
75C23×C37—22no—
76C23×C37—44no—
77C23×C37—42no—
78C23×C37C2×D411yescharacteristic
79C22×D37—22no—
80C22×D37—44no—
81C22×D37—42no—
82Q8×C37—21no—
83Q8×C37C2×D412yescomplemented
84D4×C37—24no—
85D4×C37—22no—
86D4×C37—21no—
87D4×C37—22no—
88D4×C37—42no—
89D4×C37C2×D412yescomplemented
90C2×C4×C37—21no—
91C2×C4×C37—24no—
92C2×C4×C37C2×D411yescharacteristic
93C2×C4×C37C2×D412yes—
94C2×C4×C37#4112yes—
95C2×C4×C37C2411yescharacteristic
96C22⋊D37—48no—
97C22⋊D37—48no—
98C2×Dic37—24no—
99C2×Dic37—22no—
100C2×Dic37—44no—
101C2×Dic37C2×D412yes—
102C2×Dic37C2×D412yes—
103D148—22no—
104D148—22no—
105D148—42no—
106C8×C37—22no—
107#64436—22no—
108C22×C37—22no—
109C22×C37—22no—
110C22×C37—42no—
111C22×C37—42no—
112C22×C37—44no—
113C22×C37—84no—
114C22×C37#12112yes—
115C22×C37C22×D411yescharacteristic
116C2×D37—42no—
117C2×D37—44no—
118C2×D37—84no—
119C4×C37—22no—
120C4×C37#12112yes—
121C4×C37C22×D411yescharacteristic
122C4×C37C2×#4111yescharacteristic
123Dic37—22no—
124Dic37—42no—
125Dic37AGL(1,ℤ/8)14yescomplemented
126#3038—371nomaximal, Sylow
127C2×C37—22no—
128C2×C37—42no—
129C2×C37—82no—
130C2×C37C2×#12111yescharacteristic
131C2×C37C2×AGL(1,ℤ/8)12yes—
132D37—82no—
133C2×AGL(1,ℤ/8)—371no—
134C2×AGL(1,ℤ/8)D3711yescharacteristic, complemented
135C2×#127—371no—
136C4×#41—371no—
137#460—374no—
138#459—374no—
139#423—371no—
140#418—371no—
141#418—371no—
142C37#303811yescharacteristic, complemented, Sylow
143C2×#41C2×D3711yescharacteristic
144C22×D4—371no—
145C22×D4—742no—
146C22×D4C2×D3711yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—744no—
149C2×SD16—372no—
150C2×SD16C2×D3712yescomplemented
151C2×D8—372no—
152C2×D8C2×D3712yescomplemented
153C2×M16—371no—
154C2×M16C2×D3711yescharacteristic
155#127—374no—
156#127—741no—
157#127—741no—
158C4×Q8—372no—
159C4×D4—372no—
160C4×D4—742no—
161#117—371no—
162#117—741no—
163C2×C42—371no—
164C2×C42—741no—
165C4⋊C8—374no—
166#102—374no—
167#102—374no—
168C24—21no—
169C24—741no—
170#41—22no—
171#41C22⋊D3714yescomplemented
172C2×Q8C22×D3711yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—372no—
178C2×D4—744no—
179C2×D4—744no—
180C2×D4—741no—
181C2×D4—742no—
182C2×D4—742no—
183C2×D4—1488no—
184C2×D4C22×D3712yes—
185C2×D4C22×D3711yescharacteristic
186C22×C4—21no—
187C22×C4—372no—
188C22×C4—742no—
189C22×C4C22×D3711yescharacteristic
190SD16—24no—
191SD16—744no—
192D8—24no—
193D8—744no—
194M16—22no—
195M16—742no—
196C2×C8—372no—
197C2×C8C22×D3712yes—
198C4⋊C4—372no—
199C4⋊C4—742no—
200C22⋊C4—742no—
201C22⋊C4—741no—
202C42—371no—
203C42—371no—
204C42—372no—
205C42—742no—
206C42—741no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—742no—
212C23—1484no—
213C23—1482no—
214C23D4×D3711yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D3712yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—742no—
223D4—742no—
224D4—1488no—
225D4—1488no—
226D4—1482no—
227D4C2×C22⋊D3712yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—372no—
231C2×C4—372no—
232C2×C4—744no—
233C2×C4—742no—
234C2×C4—1484no—
235C2×C4D4×D3711yescharacteristic
236C2×C4#1059696412yes—
237C2×C4C2×C22⋊D3712yes—
238C2×C4C23×D3711yescharacteristic
239C8—22no—
240C8—742no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—2964no—
248C22—1482no—
249C22—1484no—
250C22#1192450812yes—
251C22C2×D4×D3711yescharacteristic
252C4—22no—
253C4—374no—
254C4—742no—
255C4—1482no—
256C4#1192450812yes—
257C4C22×C22⋊D3711yescharacteristic
258C4C2×#1059696411yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—2962no—
263C2C2×#1192450811yescharacteristic
264C2AGL(1,ℤ/8)×D3712yes—
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#3038
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×#11924561C2semidirect
2. C2×#11924532C2semidirect
3. #41×Dic37C2semidirect
4. #14557731C2semidirect
5. #14557726C2semidirect
6. #14557556C2semidirect
7. #14557539C2semidirect
8. #14557538C2semidirect
9. C2×AGL(1,ℤ/8)×C37C2semidirect
13. C2×#10596941C22semidirect
14. C2×#10596939C22semidirect
19. Q8×Dic37C22semidirect
21. D4×Dic37C22semidirect
26. #11924465C22semidirect
27. #11924458C22semidirect
28. #11924457C22semidirect
32. C2×SD16×C37C22semidirect
33. C2×D8×C37C22semidirect
38. #41×C37D4semidirect
83. Q8×C37C2×D4semidirect
89. D4×C37C2×D4semidirect
125. Dic37AGL(1,ℤ/8)semidirect
134. C2×AGL(1,ℤ/8)D37semidirect
142. C37#3038semidirect
150. C2×SD16C2×D37semidirect
152. C2×D8C2×D37semidirect
171. #41C22⋊D37semidirect
216. Q8C2×C22⋊D37semidirect
227. D4C2×C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245612
C2×#119245322
#41×Dic372
#145577312
#145577262
#145575562
#145575392
#145575382
C2×AGL(1,ℤ/8)×C372
#303837