Finite Group Databasebeta

← #45691009

Subgroups — #45691009

Order 33472.

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#30660419C211yesmaximal, characteristic, complemented
2#30660418C211yesmaximal, characteristic, complemented
3#30660401C211yesmaximal, characteristic, complemented
4#30660400C212yesmaximal, complemented
5#30660397C212yesmaximal, complemented
6#30660394C212yesmaximal, complemented
7#30660393C211yesmaximal, characteristic, complemented
8C2×#22552626C211yesmaximal, characteristic, complemented
9D8×D523C212yesmaximal, complemented
10#30660381C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C523C211yesmaximal, characteristic, complemented
12#22552642C2212yescomplemented
13#22552641—21no—
14#22552641C2211yescharacteristic
15#22552641C2211yescharacteristic, complemented
16#22552640—21no—
17#22552640C2212yescomplemented
18C2×D2092—21no—
19C2×D2092C2211yescharacteristic, complemented
20C2×C22⋊D523—22no—
21D4×D523—21no—
22D4×D523—22no—
23D4×D523C2212yescomplemented
24D4×D523C2212yescomplemented
25#22552631C2211yescharacteristic, complemented
26#22552630C2212yescomplemented
27#22552629C2211yescharacteristic, complemented
28#22552628C2211yescharacteristic, complemented
29D4184C2212yescomplemented
30#22552626C2211yescharacteristic, complemented
31#22552626C2212yescomplemented
32#22552626C2212yescomplemented
33#22552624C2212yescomplemented
34C2×C523⋊C8C2211yescharacteristic, complemented
35C8×D523C2212yescomplemented
36#41×C523C2211yescharacteristic, complemented
37C2×D4×C523C2211yescharacteristic, complemented
38SD16×C523C2212yescomplemented
39D8×C523C2212yescomplemented
40M16×C523C2211yescharacteristic, complemented
41C22×D523—21no—
42C22×D523—22no—
43C22×D523—22no—
44Dic1046D411yescharacteristic
45Dic1046C2311yescharacteristic
46D2092—22no—
47D2092—21no—
48D2092D411yescharacteristic
49D2092C2312yes—
50D2092C2311yescharacteristic
51C22⋊D523—21no—
52C22⋊D523—22no—
53C22⋊D523—22no—
54C22⋊D523—22no—
55C22⋊D523D412yes—
56C2×Dic523—22no—
57C4×D523—21no—
58C4×D523—22no—
59C4×D523C2312yes—
60C523⋊C8C2311yescharacteristic
61C523⋊C8C2311yescharacteristic
62C23×C523—21no—
63Q8×C523C2311yescharacteristic
64D4×C523—21no—
65D4×C523—21no—
66D4×C523C2311yescharacteristic
67D4×C523C2312yes—
68C2×C4×C523—21no—
69C2×C4×C523C2311yescharacteristic
70C8×C523C2312yes—
71C22×C523—21no—
72C22×C523—22no—
73C22×C523—81no—
74C22×C523C2×D411yescharacteristic
75C2×D523—21no—
76C2×D523—22no—
77C2×D523—82no—
78C2×D523—82no—
79C2×D523—81no—
80C2×D523C2×D412yes—
81C4×C523—21no—
82C4×C523C2×D411yescharacteristic
83C4×C523C2411yescharacteristic
84Dic523—21no—
85Dic523C2×D412yes—
86C2×C523—21no—
87C2×C523—41no—
88C2×C523—42no—
89C2×C523C22×D411yescharacteristic
90D523—22no—
91D523—41no—
92D523—42no—
93C523#57611yescharacteristic, complemented, Sylow
94#576—5231nomaximal, Sylow
95H(5,𝔽2)—5231no—
96H(5,𝔽2)—5231no—
97AGL(1,ℤ/8)—5232no—
98AGL(1,ℤ/8)—5232no—
99AGL(1,ℤ/8)—5231no—
100AGL(1,ℤ/8)D52311yescharacteristic, complemented
101#135—5231no—
102#135—5232no—
103C2×D8—5231no—
104C2×D8—5232no—
105#131—5231no—
106#41—5231no—
107#41—5232no—
108#41—5232no—
109#41—5231no—
110#41—10461no—
111#41—10461no—
112#41C2×D52311yescharacteristic, complemented
113C2×D4—5232no—
114C2×D4—5232no—
115C2×D4—5231no—
116C2×D4—10461no—
117C2×D4—10462no—
118C2×D4—10461no—
119C2×D4—10462no—
120C2×D4C2×D52311yescharacteristic, complemented
121Q16—5231no—
122SD16—5231no—
123SD16—5231no—
124SD16—5232no—
125SD16C2×D52312yescomplemented
126D8—5232no—
127D8—5231no—
128D8—5232no—
129D8—5232no—
130D8C2×D52312yescomplemented
131M16—5232no—
132M16C2×D52311yescharacteristic, complemented
133C2×C8—5231no—
134C2×C8—5232no—
135C23—21no—
136C23—10462no—
137C23—10462no—
138C23—10461no—
139Q8—5231no—
140Q8—5231no—
141Q8C22×D52311yescharacteristic
142D4—21no—
143D4—21no—
144D4—5232no—
145D4—5231no—
146D4—5232no—
147D4—5231no—
148D4—10462no—
149D4—10462no—
150D4—10462no—
151D4—10462no—
152D4—10461no—
153D4—10461no—
154D4C22×D52312yes—
155D4C22×D52311yescharacteristic
156C2×C4—21no—
157C2×C4—5232no—
158C2×C4—10462no—
159C2×C4—10462no—
160C2×C4—10461no—
161C2×C4C22×D52311yescharacteristic
162C8—5231no—
163C8—5231no—
164C8C22×D52312yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—5232no—
169C22—10462no—
170C22—10461no—
171C22—41841no—
172C22—41842no—
173C22—41842no—
174C22D4×D52311yescharacteristic
175C4—21no—
176C4—5232no—
177C4—10461no—
178C4D4×D52311yescharacteristic
179C4C23×D52311yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—10462no—
184C2—20922no—
185C2—20921no—
186C2C2×D4×D52311yescharacteristic
subgroupG/H
centerC2C2×D4×D523
commutator, cosocleC4×C523C24
FrattiniC4C23×D523
FittingAGL(1,ℤ/8)×C523C2
socleC2×C523C22×D4
hypercenterAGL(1,ℤ/8)D523
hypocenterC523#576
LCSG ▷ C4×C523 ▷ C2×C523 ▷ C523
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #30660419C2semidirect
2. #30660418C2semidirect
3. #30660401C2semidirect
4. #30660400C2semidirect
5. #30660397C2semidirect
6. #30660394C2semidirect
7. #30660393C2semidirect
8. C2×#22552626C2semidirect
9. D8×D523C2semidirect
10. #30660381C2semidirect
11. AGL(1,ℤ/8)×C523C2semidirect
12. #22552642C22semidirect
15. #22552641C22semidirect
17. #22552640C22semidirect
19. C2×D2092C22semidirect
23. D4×D523C22semidirect
24. D4×D523C22semidirect
25. #22552631C22semidirect
26. #22552630C22semidirect
27. #22552629C22semidirect
28. #22552628C22semidirect
29. D4184C22semidirect
30. #22552626C22semidirect
31. #22552626C22semidirect
32. #22552626C22semidirect
33. #22552624C22semidirect
34. C2×C523⋊C8C22semidirect
35. C8×D523C22semidirect
36. #41×C523C22semidirect
37. C2×D4×C523C22semidirect
38. SD16×C523C22semidirect
39. D8×C523C22semidirect
40. M16×C523C22semidirect
93. C523#576semidirect
100. AGL(1,ℤ/8)D523semidirect
112. #41C2×D523semidirect
120. C2×D4C2×D523semidirect
125. SD16C2×D523semidirect
130. D8C2×D523semidirect
132. M16C2×D523semidirect

Maximal subgroups

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

subgroupindex
#306604192
#306604182
#306604012
#306604002
#306603972
#306603942
#306603932
C2×#225526262
D8×D5232
#306603812
AGL(1,ℤ/8)×C5232
#576523