Finite Group Databasebeta

← #42119869

Subgroups — #42119869

Order 27584.

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

normal subgroups only

This group has 258 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#28352236C211yesmaximal, characteristic, complemented
2#28352234C211yesmaximal, characteristic, complemented
3#28352222C211yesmaximal, characteristic, complemented
4#28352218C212yesmaximal, complemented
5#28352213C212yesmaximal, complemented
6#28352209C211yesmaximal, characteristic, complemented
7#28352208C212yesmaximal, complemented
8C2×#20293788C211yesmaximal, characteristic, complemented
9SD16×D431C212yesmaximal, complemented
10#28352197C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C431C211yesmaximal, characteristic, complemented
12#20293800—21no—
13#20293800C2211yescharacteristic
14#20293800C2211yescharacteristic
15#20293799—21no—
16#20293799—22no—
17#20293799C2212yes—
18#20293799C2212yescomplemented
19C2×Dic862—21no—
20C2×Dic862C2211yescharacteristic, complemented
21Q8×D431C2212yescomplemented
22C2×C22⋊D431—22no—
23D4×D431—21no—
24D4×D431C2212yes—
25Dic1724C2212yescomplemented
26#20293790C2211yescharacteristic, complemented
27#20293789C2212yescomplemented
28#20293788C2212yescomplemented
29#20293788C2212yescomplemented
30#20293788C2211yescharacteristic, complemented
31#20293787C2211yescharacteristic, complemented
32#20293785C2211yescharacteristic, complemented
33#20293783C2212yescomplemented
34C2×C431⋊C8C2211yescharacteristic, complemented
35C8×D431C2212yescomplemented
36#41×C431C2211yescharacteristic, complemented
37C2×D4×C431C2211yescharacteristic
38SD16×C431C2212yescomplemented
39D8×C431C2212yescomplemented
40M16×C431C2211yescharacteristic, complemented
41C22×D431—22no—
42Dic862—21no—
43Dic862—22no—
44Dic862D411yescharacteristic
45Dic862C2312yes—
46Dic862C2311yescharacteristic
47D1724D411yescharacteristic
48D1724C2311yescharacteristic
49C22⋊D431—21no—
50C22⋊D431—22no—
51C22⋊D431—22no—
52C22⋊D431—22no—
53C22⋊D431D412yes—
54C2×Dic431—21no—
55C2×Dic431—22no—
56C2×Dic431—22no—
57C4×D431—22no—
58C4×D431—21no—
59C4×D431C2312yes—
60C431⋊C8C2311yescharacteristic
61C431⋊C8C2311yescharacteristic
62C23×C431—21no—
63Q8×C431C2311yescharacteristic
64D4×C431—21no—
65D4×C431—21no—
66D4×C431C2311yescharacteristic
67D4×C431C2312yes—
68C2×C4×C431—21no—
69C2×C4×C431C2311yescharacteristic
70C8×C431C2312yes—
71C22×C431—21no—
72C22×C431—22no—
73C22×C431—81no—
74C22×C431C2×D411yescharacteristic
75C2×D431—21no—
76C2×D431—82no—
77C2×D431C2×D412yes—
78C4×C431—21no—
79C4×C431C2×D411yescharacteristic
80C4×C431C2411yescharacteristic
81Dic431—21no—
82Dic431—22no—
83Dic431C2×D412yes—
84C2×C431—21no—
85C2×C431—42no—
86C2×C431—41no—
87C2×C431C22×D411yescharacteristic
88D431—22no—
89D431—41no—
90C431#57711yescharacteristic, complemented, Sylow
91#577—4311nomaximal, Sylow
922−1+4—4311no—
93H(5,𝔽2)—4311no—
94#137—4312no—
95#137—4311no—
96AGL(1,ℤ/8)—4312no—
97AGL(1,ℤ/8)D43111yescharacteristic, complemented
98#135—4311no—
99#135—4312no—
100C2×SD16—4311no—
101C2×SD16—4312no—
102#131—4311no—
103#41—8621no—
104#41—8621no—
105#41—8622no—
106#41—4312no—
107#41—4312no—
108#41—4311no—
109#41—4311no—
110#41C2×D43111yescharacteristic, complemented
111C2×Q8—8621no—
112C2×Q8—4312no—
113C2×Q8—4311no—
114C2×D4—8621no—
115C2×D4—8622no—
116C2×D4—4312no—
117C2×D4C2×D43111yescharacteristic
118Q16—4311no—
119Q16—4312no—
120SD16—4311no—
121SD16—4312no—
122SD16—4312no—
123SD16—4311no—
124SD16—4312no—
125SD16C2×D43112yescomplemented
126D8—4311no—
127D8C2×D43112yescomplemented
128M16—4312no—
129M16C2×D43111yescharacteristic, complemented
130C2×C8—4311no—
131C2×C8—4312no—
132C23—21no—
133C23—8622no—
134Q8—8622no—
135Q8—8621no—
136Q8—4312no—
137Q8—4311no—
138Q8—4311no—
139Q8C22×D43111yescharacteristic
140D4—21no—
141D4—21no—
142D4—8622no—
143D4—8622no—
144D4—8621no—
145D4—8622no—
146D4—4312no—
147D4—4311no—
148D4—4311no—
149D4C22×D43111yescharacteristic
150D4C22×D43112yes—
151C2×C4—21no—
152C2×C4—8622no—
153C2×C4—8621no—
154C2×C4—8622no—
155C2×C4—8622no—
156C2×C4—8621no—
157C2×C4—4312no—
158C2×C4C22×D43111yescharacteristic
159C8—4311no—
160C8—4311no—
161C8C22×D43112yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—8621no—
166C22—34482no—
167C22—4312no—
168C22D4×D43111yescharacteristic
169C4—21no—
170C4—8621no—
171C4—8622no—
172C4—4312no—
173C4D4×D43111yescharacteristic
174C4C23×D43111yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—8622no—
179C2—17241no—
180C2C2×D4×D43111yescharacteristic
subgroupG/H
centerC2C2×D4×D431
commutator, cosocleC4×C431C24
FrattiniC4C23×D431
FittingAGL(1,ℤ/8)×C431C2
socleC2×C431C22×D4
hypercenterAGL(1,ℤ/8)D431
hypocenterC431#577
LCSG ▷ C4×C431 ▷ C2×C431 ▷ C431
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #28352236C2semidirect
2. #28352234C2semidirect
3. #28352222C2semidirect
4. #28352218C2semidirect
5. #28352213C2semidirect
6. #28352209C2semidirect
7. #28352208C2semidirect
8. C2×#20293788C2semidirect
9. SD16×D431C2semidirect
10. #28352197C2semidirect
11. AGL(1,ℤ/8)×C431C2semidirect
18. #20293799C22semidirect
20. C2×Dic862C22semidirect
21. Q8×D431C22semidirect
25. Dic1724C22semidirect
26. #20293790C22semidirect
27. #20293789C22semidirect
28. #20293788C22semidirect
29. #20293788C22semidirect
30. #20293788C22semidirect
31. #20293787C22semidirect
32. #20293785C22semidirect
33. #20293783C22semidirect
34. C2×C431⋊C8C22semidirect
35. C8×D431C22semidirect
36. #41×C431C22semidirect
38. SD16×C431C22semidirect
39. D8×C431C22semidirect
40. M16×C431C22semidirect
90. C431#577semidirect
97. AGL(1,ℤ/8)D431semidirect
110. #41C2×D431semidirect
125. SD16C2×D431semidirect
127. D8C2×D431semidirect
129. M16C2×D431semidirect

Maximal subgroups

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

subgroupindex
#283522362
#283522342
#283522222
#283522182
#283522132
#283522092
#283522082
C2×#202937882
SD16×D4312
#283521972
AGL(1,ℤ/8)×C4312
#577431