Finite Group Databasebeta

← #37589147

Subgroups — #37589147

Order 24512.

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

normal subgroups only

This group has 262 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
1C2×#20259456C211yesmaximal, characteristic, complemented
2#41×D383C211yesmaximal, characteristic, complemented
3#27133874C211yesmaximal, characteristic, complemented
4#27133871C211yesmaximal, characteristic, complemented
5#27133870C212yesmaximal, complemented
6#27133867C211yesmaximal, characteristic, complemented
7#27133865C212yesmaximal, complemented
8#27133862C212yesmaximal, complemented
9#27133860C212yesmaximal, complemented
10M16×D383C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C383C211yesmaximal, characteristic, complemented
12#20259458C2211yescharacteristic, complemented
13#20259457—21no—
14#20259457C2211yescharacteristic
15#20259456—21no—
16#20259456—21no—
17#20259456—21no—
18#20259456C2212yes—
19#20259456C2211yescharacteristic, complemented
20#20259456C2212yescomplemented
21C2×Dic766C2211yescharacteristic, complemented
22Q8×D383C2211yescharacteristic, complemented
23C2×C22⋊D383—21no—
24D4×D383—21no—
25D4×D383C2211yescharacteristic, complemented
26C22×Dic383—21no—
27C2×C4×D383—21no—
28C2×C4×D383C2211yescharacteristic
29Dic1532C2212yescomplemented
30#20259447C2212yescomplemented
31#20259446C2212yescomplemented
32#20259445C2212yescomplemented
33#20259445C2212yescomplemented
34#20259442C2212yescomplemented
35#20259441C2211yescharacteristic, complemented
36#20259440C2212yescomplemented
37C8×D383C2212yescomplemented
38#41×C383C2211yescharacteristic
39C2×D4×C383C2211yescharacteristic, complemented
40SD16×C383C2212yescomplemented
41D8×C383C2212yescomplemented
42M16×C383C2211yescharacteristic, complemented
43C22×D383—21no—
44C22×D383D411yescharacteristic
45Dic766—21no—
46Dic766—21no—
47Dic766C2311yescharacteristic
48Dic766C2312yes—
49D1532—21no—
50D1532C2311yescharacteristic
51C22⋊D383—21no—
52C22⋊D383—21no—
53C22⋊D383—22no—
54C22⋊D383—22no—
55C22⋊D383—21no—
56C2×Dic383—21no—
57C2×Dic383—21no—
58C2×Dic383—22no—
59C2×Dic383—22no—
60C2×Dic383D411yescharacteristic
61C4×D383—21no—
62C4×D383—21no—
63C4×D383—21no—
64C4×D383D411yescharacteristic
65C4×D383D411yescharacteristic
66C4×D383C2311yescharacteristic
67C4×D383C2311yescharacteristic
68C383⋊C8C2312yes—
69C23×C383—21no—
70Q8×C383C2311yescharacteristic
71D4×C383—21no—
72D4×C383—21no—
73D4×C383C2311yescharacteristic
74D4×C383C2312yes—
75C2×C4×C383—21no—
76C2×C4×C383C2311yescharacteristic
77C8×C383C2312yes—
78C22×C383—21no—
79C22×C383—22no—
80C22×C383—42no—
81C22×C383C2×D411yescharacteristic
82C2×D383—21no—
83C2×D383—41no—
84C2×D383—42no—
85C2×D383C2×D411yescharacteristic
86C2×D383C2×D411yescharacteristic
87C4×C383—21no—
88C4×C383C2×D411yescharacteristic
89C4×C383C2411yescharacteristic
90Dic383—21no—
91Dic383—22no—
92Dic383C2×D411yescharacteristic
93Dic383C2×D411yescharacteristic
94C2×C383—21no—
95C2×C383—41no—
96C2×C383—42no—
97C2×C383C22×D411yescharacteristic
98D383—21no—
99D383—21no—
100D383—41no—
101C383#57511yescharacteristic, complemented, Sylow
102#575—3831nomaximal, Sylow
103C2×#41—3831no—
104C2×#41—3831no—
105#137—3831no—
106#137—3832no—
107#137—3831no—
108AGL(1,ℤ/8)—3831no—
109AGL(1,ℤ/8)—3832no—
110AGL(1,ℤ/8)D38311yescharacteristic, complemented
111#135—3832no—
112#135—3832no—
113C2×M16—3831no—
114#41—3831no—
115#41—3832no—
116#41—3832no—
117#41—3831no—
118#41—3831no—
119#41—7661no—
120#41—7661no—
121#41—7661no—
122#41—7661no—
123#41C2×D38311yescharacteristic
124C2×Q8—3831no—
125C2×Q8—3831no—
126C2×D4—3831no—
127C2×D4—7661no—
128C2×D4—7661no—
129C2×D4C2×D38311yescharacteristic, complemented
130C22×C4—3831no—
131C22×C4—7661no—
132C22×C4—7661no—
133Q16—3832no—
134Q16—3832no—
135SD16—3832no—
136SD16—3832no—
137SD16—3832no—
138SD16C2×D38312yescomplemented
139D8—3832no—
140D8C2×D38312yescomplemented
141M16—3831no—
142M16—3832no—
143M16C2×D38311yescharacteristic, complemented
144C2×C8—3832no—
145C23—21no—
146C23—3831no—
147C23—7661no—
148Q8—3831no—
149Q8—3832no—
150Q8—7661no—
151Q8—7661no—
152Q8C22×D38311yescharacteristic
153D4—21no—
154D4—21no—
155D4—3831no—
156D4—7661no—
157D4—7661no—
158D4—7661no—
159D4—7662no—
160D4—7662no—
161D4—7661no—
162D4C22×D38312yes—
163D4C22×D38311yescharacteristic
164C2×C4—21no—
165C2×C4—3831no—
166C2×C4—3831no—
167C2×C4—3831no—
168C2×C4—3831no—
169C2×C4—3831no—
170C2×C4—7661no—
171C2×C4—7661no—
172C2×C4—7661no—
173C2×C4—7662no—
174C2×C4—7662no—
175C2×C4—7661no—
176C2×C4—7661no—
177C2×C4C22×D38311yescharacteristic
178C8—3832no—
179C8C22×D38312yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—3831no—
184C22—3831no—
185C22—15321no—
186C22—15322no—
187C22—7661no—
188C22D4×D38311yescharacteristic
189C4—21no—
190C4—3831no—
191C4—3831no—
192C4—7661no—
193C4—7662no—
194C4D4×D38311yescharacteristic
195C4C23×D38311yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—15321no—
200C2—7661no—
201C2—7661no—
202C2C2×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#575
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. C2×#20259456C2semidirect
2. #41×D383C2semidirect
3. #27133874C2semidirect
4. #27133871C2semidirect
5. #27133870C2semidirect
6. #27133867C2semidirect
7. #27133865C2semidirect
8. #27133862C2semidirect
9. #27133860C2semidirect
10. M16×D383C2semidirect
11. AGL(1,ℤ/8)×C383C2semidirect
12. #20259458C22semidirect
19. #20259456C22semidirect
20. #20259456C22semidirect
21. C2×Dic766C22semidirect
22. Q8×D383C22semidirect
25. D4×D383C22semidirect
29. Dic1532C22semidirect
30. #20259447C22semidirect
31. #20259446C22semidirect
32. #20259445C22semidirect
33. #20259445C22semidirect
34. #20259442C22semidirect
35. #20259441C22semidirect
36. #20259440C22semidirect
37. C8×D383C22semidirect
39. C2×D4×C383C22semidirect
40. SD16×C383C22semidirect
41. D8×C383C22semidirect
42. M16×C383C22semidirect
101. C383#575semidirect
110. AGL(1,ℤ/8)D383semidirect
129. C2×D4C2×D383semidirect
138. SD16C2×D383semidirect
140. D8C2×D383semidirect
143. M16C2×D383semidirect

Maximal subgroups

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

subgroupindex
C2×#202594562
#41×D3832
#271338742
#271338712
#271338702
#271338672
#271338652
#271338622
#271338602
M16×D3832
AGL(1,ℤ/8)×C3832
#575383