Finite Group Databasebeta

← #37589155

Subgroups — #37589155

Order 24512.

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#27133887C211yesmaximal, characteristic, complemented
2#27133886C211yesmaximal, characteristic, complemented
3#27133869C211yesmaximal, characteristic, complemented
4#27133868C212yesmaximal, complemented
5#27133865C212yesmaximal, complemented
6#27133862C212yesmaximal, complemented
7#27133861C211yesmaximal, characteristic, complemented
8C2×#20259442C211yesmaximal, characteristic, complemented
9D8×D383C212yesmaximal, complemented
10#27133849C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C383C211yesmaximal, characteristic, complemented
12#20259458C2212yescomplemented
13#20259457—21no—
14#20259457C2211yescharacteristic
15#20259457C2211yescharacteristic, complemented
16#20259456—21no—
17#20259456C2212yescomplemented
18C2×D1532—21no—
19C2×D1532C2211yescharacteristic, complemented
20C2×C22⋊D383—22no—
21D4×D383—21no—
22D4×D383—22no—
23D4×D383C2212yescomplemented
24D4×D383C2212yescomplemented
25#20259447C2211yescharacteristic, complemented
26#20259446C2212yescomplemented
27#20259445C2211yescharacteristic, complemented
28#20259444C2211yescharacteristic, complemented
29D3064C2212yescomplemented
30#20259442C2211yescharacteristic, complemented
31#20259442C2212yescomplemented
32#20259442C2212yescomplemented
33#20259440C2212yescomplemented
34C2×C383⋊C8C2211yescharacteristic, complemented
35C8×D383C2212yescomplemented
36#41×C383C2211yescharacteristic, complemented
37C2×D4×C383C2211yescharacteristic, complemented
38SD16×C383C2212yescomplemented
39D8×C383C2212yescomplemented
40M16×C383C2211yescharacteristic, complemented
41C22×D383—21no—
42C22×D383—22no—
43C22×D383—22no—
44Dic766D411yescharacteristic
45Dic766C2311yescharacteristic
46D1532—22no—
47D1532—21no—
48D1532D411yescharacteristic
49D1532C2312yes—
50D1532C2311yescharacteristic
51C22⋊D383—21no—
52C22⋊D383—22no—
53C22⋊D383—22no—
54C22⋊D383—22no—
55C22⋊D383D412yes—
56C2×Dic383—22no—
57C4×D383—21no—
58C4×D383—22no—
59C4×D383C2312yes—
60C383⋊C8C2311yescharacteristic
61C383⋊C8C2311yescharacteristic
62C23×C383—21no—
63Q8×C383C2311yescharacteristic
64D4×C383—21no—
65D4×C383—21no—
66D4×C383C2311yescharacteristic
67D4×C383C2312yes—
68C2×C4×C383—21no—
69C2×C4×C383C2311yescharacteristic
70C8×C383C2312yes—
71C22×C383—21no—
72C22×C383—22no—
73C22×C383—81no—
74C22×C383C2×D411yescharacteristic
75C2×D383—21no—
76C2×D383—22no—
77C2×D383—82no—
78C2×D383—82no—
79C2×D383—81no—
80C2×D383C2×D412yes—
81C4×C383—21no—
82C4×C383C2×D411yescharacteristic
83C4×C383C2411yescharacteristic
84Dic383—21no—
85Dic383C2×D412yes—
86C2×C383—21no—
87C2×C383—41no—
88C2×C383—42no—
89C2×C383C22×D411yescharacteristic
90D383—22no—
91D383—41no—
92D383—42no—
93C383#57611yescharacteristic, complemented, Sylow
94#576—3831nomaximal, Sylow
95H(5,𝔽2)—3831no—
96H(5,𝔽2)—3831no—
97AGL(1,ℤ/8)—3832no—
98AGL(1,ℤ/8)—3832no—
99AGL(1,ℤ/8)—3831no—
100AGL(1,ℤ/8)D38311yescharacteristic, complemented
101#135—3831no—
102#135—3832no—
103C2×D8—3831no—
104C2×D8—3832no—
105#131—3831no—
106#41—3831no—
107#41—3832no—
108#41—3832no—
109#41—3831no—
110#41—7661no—
111#41—7661no—
112#41C2×D38311yescharacteristic, complemented
113C2×D4—3832no—
114C2×D4—3832no—
115C2×D4—3831no—
116C2×D4—7661no—
117C2×D4—7662no—
118C2×D4—7661no—
119C2×D4—7662no—
120C2×D4C2×D38311yescharacteristic, complemented
121Q16—3831no—
122SD16—3831no—
123SD16—3831no—
124SD16—3832no—
125SD16C2×D38312yescomplemented
126D8—3832no—
127D8—3831no—
128D8—3832no—
129D8—3832no—
130D8C2×D38312yescomplemented
131M16—3832no—
132M16C2×D38311yescharacteristic, complemented
133C2×C8—3831no—
134C2×C8—3832no—
135C23—21no—
136C23—7662no—
137C23—7662no—
138C23—7661no—
139Q8—3831no—
140Q8—3831no—
141Q8C22×D38311yescharacteristic
142D4—21no—
143D4—21no—
144D4—3832no—
145D4—3831no—
146D4—3832no—
147D4—3831no—
148D4—7662no—
149D4—7662no—
150D4—7662no—
151D4—7662no—
152D4—7661no—
153D4—7661no—
154D4C22×D38312yes—
155D4C22×D38311yescharacteristic
156C2×C4—21no—
157C2×C4—3832no—
158C2×C4—7662no—
159C2×C4—7662no—
160C2×C4—7661no—
161C2×C4C22×D38311yescharacteristic
162C8—3831no—
163C8—3831no—
164C8C22×D38312yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—3832no—
169C22—30641no—
170C22—30642no—
171C22—30642no—
172C22—7662no—
173C22—7661no—
174C22D4×D38311yescharacteristic
175C4—21no—
176C4—3832no—
177C4—7661no—
178C4D4×D38311yescharacteristic
179C4C23×D38311yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—15322no—
184C2—15321no—
185C2—7662no—
186C2C2×D4×D38311yescharacteristic
subgroupG/H
centerC2C2×D4×D383
commutator, cosocleC4×C383C24
FrattiniC4C23×D383
FittingAGL(1,ℤ/8)×C383C2
socleC2×C383C22×D4
hypercenterAGL(1,ℤ/8)D383
hypocenterC383#576
LCSG ▷ C4×C383 ▷ C2×C383 ▷ C383
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27133887C2semidirect
2. #27133886C2semidirect
3. #27133869C2semidirect
4. #27133868C2semidirect
5. #27133865C2semidirect
6. #27133862C2semidirect
7. #27133861C2semidirect
8. C2×#20259442C2semidirect
9. D8×D383C2semidirect
10. #27133849C2semidirect
11. AGL(1,ℤ/8)×C383C2semidirect
12. #20259458C22semidirect
15. #20259457C22semidirect
17. #20259456C22semidirect
19. C2×D1532C22semidirect
23. D4×D383C22semidirect
24. D4×D383C22semidirect
25. #20259447C22semidirect
26. #20259446C22semidirect
27. #20259445C22semidirect
28. #20259444C22semidirect
29. D3064C22semidirect
30. #20259442C22semidirect
31. #20259442C22semidirect
32. #20259442C22semidirect
33. #20259440C22semidirect
34. C2×C383⋊C8C22semidirect
35. C8×D383C22semidirect
36. #41×C383C22semidirect
37. C2×D4×C383C22semidirect
38. SD16×C383C22semidirect
39. D8×C383C22semidirect
40. M16×C383C22semidirect
93. C383#576semidirect
100. AGL(1,ℤ/8)D383semidirect
112. #41C2×D383semidirect
120. C2×D4C2×D383semidirect
125. SD16C2×D383semidirect
130. D8C2×D383semidirect
132. M16C2×D383semidirect

Maximal subgroups

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

subgroupindex
#271338872
#271338862
#271338692
#271338682
#271338652
#271338622
#271338612
C2×#202594422
D8×D3832
#271338492
AGL(1,ℤ/8)×C3832
#576383