Finite Group Databasebeta

← #14558337

Subgroups — #14558337

Order 2368.

268 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#11924582C211yesmaximal, characteristic, complemented
2#11924581C211yesmaximal, characteristic, complemented
3#11924564C211yesmaximal, characteristic, complemented
4#11924563C212yesmaximal, complemented
5#11924560C212yesmaximal, complemented
6#11924557C212yesmaximal, complemented
7#11924556C211yesmaximal, characteristic, complemented
8C2×#10596939C211yesmaximal, characteristic, complemented
9D8×D37C212yesmaximal, complemented
10#11924544C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
12#41×C37C2211yescharacteristic, complemented
13C2×D4×C37C2211yescharacteristic, complemented
14C2×C22⋊D37—22no—
15#10596966C2212yescomplemented
16#10596964—21no—
17#10596964C2212yescomplemented
18D4×D37—22no—
19D4×D37—21no—
20D4×D37C2212yescomplemented
21D4×D37C2212yescomplemented
22#10596962—21no—
23#10596962C2211yescharacteristic
24#10596962C2211yescharacteristic, complemented
25C2×D148—21no—
26C2×D148C2211yescharacteristic, complemented
27SD16×C37C2212yescomplemented
28D8×C37C2212yescomplemented
29M16×C37C2211yescharacteristic, complemented
30#10596942C2211yescharacteristic, complemented
31#10596941C2211yescharacteristic, complemented
32#10596940C2211yescharacteristic, complemented
33#10596939C2211yescharacteristic, complemented
34#10596939C2212yescomplemented
35#10596939C2212yescomplemented
36C2×#64436C2211yescharacteristic, complemented
37D296C2212yescomplemented
38#10596930C2212yescomplemented
39#10596929C2212yescomplemented
40C8×D37C2212yescomplemented
41C23×C37—21no—
42C22×D37—21no—
43C22×D37—22no—
44C22×D37—22no—
45Q8×C37C2311yescharacteristic
46D4×C37—21no—
47D4×C37—21no—
48D4×C37C2312yes—
49D4×C37C2311yescharacteristic
50C2×C4×C37—21no—
51C2×C4×C37C2311yescharacteristic
52C22⋊D37—21no—
53C22⋊D37—22no—
54C22⋊D37—22no—
55C22⋊D37—22no—
56C22⋊D37D412yes—
57C2×Dic37—22no—
58D148—22no—
59D148—21no—
60D148D411yescharacteristic
61D148C2312yes—
62D148C2311yescharacteristic
63C4×D37—21no—
64C4×D37—22no—
65C4×D37C2312yes—
66Dic74D411yescharacteristic
67Dic74C2311yescharacteristic
68C8×C37C2312yes—
69#64436C2311yescharacteristic
70#64436C2311yescharacteristic
71C22×C37—22no—
72C22×C37—21no—
73C22×C37—81no—
74C22×C37C2×D411yescharacteristic
75C2×D37—22no—
76C2×D37—21no—
77C2×D37—82no—
78C2×D37—82no—
79C2×D37—81no—
80C2×D37C2×D412yes—
81C4×C37—21no—
82C4×C37C2×D411yescharacteristic
83C4×C37C2411yescharacteristic
84Dic37—21no—
85Dic37C2×D412yes—
86C2×C37—21no—
87C2×C37—41no—
88C2×C37—42no—
89C2×C37C22×D411yescharacteristic
90D37—22no—
91D37—42no—
92D37—41no—
93#576—371nomaximal, Sylow
94C37#57611yescharacteristic, complemented, Sylow
95H(5,𝔽2)—371no—
96H(5,𝔽2)—371no—
97AGL(1,ℤ/8)—372no—
98AGL(1,ℤ/8)—372no—
99AGL(1,ℤ/8)—371no—
100AGL(1,ℤ/8)D3711yescharacteristic, complemented
101#135—371no—
102#135—372no—
103C2×D8—371no—
104C2×D8—372no—
105#131—371no—
106#41—371no—
107#41—372no—
108#41—372no—
109#41—371no—
110#41—741no—
111#41—741no—
112#41C2×D3711yescharacteristic, complemented
113C2×D4—372no—
114C2×D4—372no—
115C2×D4—371no—
116C2×D4—741no—
117C2×D4—742no—
118C2×D4—741no—
119C2×D4—742no—
120C2×D4C2×D3711yescharacteristic, complemented
121Q16—371no—
122SD16—371no—
123SD16—371no—
124SD16—372no—
125SD16C2×D3712yescomplemented
126D8—372no—
127D8—371no—
128D8—372no—
129D8—372no—
130D8C2×D3712yescomplemented
131M16—372no—
132M16C2×D3711yescharacteristic, complemented
133C2×C8—371no—
134C2×C8—372no—
135C23—21no—
136C23—742no—
137C23—742no—
138C23—741no—
139Q8—371no—
140Q8—371no—
141Q8C22×D3711yescharacteristic
142D4—21no—
143D4—21no—
144D4—372no—
145D4—371no—
146D4—372no—
147D4—371no—
148D4—742no—
149D4—742no—
150D4—742no—
151D4—742no—
152D4—741no—
153D4—741no—
154D4C22×D3712yes—
155D4C22×D3711yescharacteristic
156C2×C4—21no—
157C2×C4—372no—
158C2×C4—742no—
159C2×C4—742no—
160C2×C4—741no—
161C2×C4C22×D3711yescharacteristic
162C8—371no—
163C8—371no—
164C8C22×D3712yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—372no—
169C22—2961no—
170C22—2962no—
171C22—2962no—
172C22—742no—
173C22—741no—
174C22D4×D3711yescharacteristic
175C4—21no—
176C4—372no—
177C4—741no—
178C4D4×D3711yescharacteristic
179C4C23×D3711yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—742no—
184C2—1482no—
185C2—1481no—
186C2C2×D4×D3711yescharacteristic
subgroupG/H
centerC2C2×D4×D37
commutator, cosocleC4×C37C24
FrattiniC4C23×D37
FittingAGL(1,ℤ/8)×C37C2
socleC2×C37C22×D4
hypercenterAGL(1,ℤ/8)D37
hypocenterC37#576
LCSG ▷ C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #11924582C2semidirect
2. #11924581C2semidirect
3. #11924564C2semidirect
4. #11924563C2semidirect
5. #11924560C2semidirect
6. #11924557C2semidirect
7. #11924556C2semidirect
8. C2×#10596939C2semidirect
9. D8×D37C2semidirect
10. #11924544C2semidirect
11. AGL(1,ℤ/8)×C37C2semidirect
12. #41×C37C22semidirect
13. C2×D4×C37C22semidirect
15. #10596966C22semidirect
17. #10596964C22semidirect
20. D4×D37C22semidirect
21. D4×D37C22semidirect
24. #10596962C22semidirect
26. C2×D148C22semidirect
27. SD16×C37C22semidirect
28. D8×C37C22semidirect
29. M16×C37C22semidirect
30. #10596942C22semidirect
31. #10596941C22semidirect
32. #10596940C22semidirect
33. #10596939C22semidirect
34. #10596939C22semidirect
35. #10596939C22semidirect
36. C2×#64436C22semidirect
37. D296C22semidirect
38. #10596930C22semidirect
39. #10596929C22semidirect
40. C8×D37C22semidirect
94. C37#576semidirect
100. AGL(1,ℤ/8)D37semidirect
112. #41C2×D37semidirect
120. C2×D4C2×D37semidirect
125. SD16C2×D37semidirect
130. D8C2×D37semidirect
132. M16C2×D37semidirect

Maximal subgroups

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

subgroupindex
#119245822
#119245812
#119245642
#119245632
#119245602
#119245572
#119245562
C2×#105969392
D8×D372
#119245442
AGL(1,ℤ/8)×C372
#57637