Finite Group Databasebeta

← #38734909

Subgroups — #38734909

Order 24896.

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#27159431C211yesmaximal, characteristic, complemented
2#27159429C211yesmaximal, characteristic, complemented
3#27159417C211yesmaximal, characteristic, complemented
4#27159413C212yesmaximal, complemented
5#27159408C212yesmaximal, complemented
6#27159404C211yesmaximal, characteristic, complemented
7#27159403C212yesmaximal, complemented
8C2×#20261954C211yesmaximal, characteristic, complemented
9SD16×D389C212yesmaximal, complemented
10#27159392C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C389C211yesmaximal, characteristic, complemented
12#20261966—21no—
13#20261966C2211yescharacteristic
14#20261966C2211yescharacteristic
15#20261965—21no—
16#20261965—22no—
17#20261965C2212yes—
18#20261965C2212yescomplemented
19C2×Dic778—21no—
20C2×Dic778C2211yescharacteristic, complemented
21Q8×D389C2212yescomplemented
22C2×C22⋊D389—22no—
23D4×D389—21no—
24D4×D389C2212yes—
25Dic1556C2212yescomplemented
26#20261956C2211yescharacteristic, complemented
27#20261955C2212yescomplemented
28#20261954C2212yescomplemented
29#20261954C2212yescomplemented
30#20261954C2211yescharacteristic, complemented
31#20261953C2211yescharacteristic, complemented
32#20261951C2211yescharacteristic, complemented
33#20261949C2212yescomplemented
34C2×#15691005C2211yescharacteristic, complemented
35C8×D389C2212yescomplemented
36#41×C389C2211yescharacteristic, complemented
37C2×D4×C389C2211yescharacteristic
38SD16×C389C2212yescomplemented
39D8×C389C2212yescomplemented
40M16×C389C2211yescharacteristic, complemented
41C22×D389—22no—
42Dic778—21no—
43Dic778—22no—
44Dic778D411yescharacteristic
45Dic778C2312yes—
46Dic778C2311yescharacteristic
47D1556D411yescharacteristic
48D1556C2311yescharacteristic
49C22⋊D389—21no—
50C22⋊D389—22no—
51C22⋊D389—22no—
52C22⋊D389—22no—
53C22⋊D389D412yes—
54C2×Dic389—21no—
55C2×Dic389—22no—
56C2×Dic389—22no—
57C4×D389—22no—
58C4×D389—21no—
59C4×D389C2312yes—
60#15691005C2311yescharacteristic
61#15691005C2311yescharacteristic
62C23×C389—21no—
63Q8×C389C2311yescharacteristic
64D4×C389—21no—
65D4×C389—21no—
66D4×C389C2311yescharacteristic
67D4×C389C2312yes—
68C2×C4×C389—21no—
69C2×C4×C389C2311yescharacteristic
70C8×C389C2312yes—
71C22×C389—22no—
72C22×C389—21no—
73C22×C389—81no—
74C22×C389C2×D411yescharacteristic
75C2×D389—21no—
76C2×D389—82no—
77C2×D389C2×D412yes—
78C4×C389—21no—
79C4×C389C2×D411yescharacteristic
80C4×C389C2411yescharacteristic
81Dic389—21no—
82Dic389—22no—
83Dic389C2×D412yes—
84C2×C389—21no—
85C2×C389—42no—
86C2×C389—41no—
87C2×C389C22×D411yescharacteristic
88D389—22no—
89D389—41no—
90C389#57711yescharacteristic, complemented, Sylow
91#577—3891nomaximal, Sylow
922−1+4—3891no—
93H(5,𝔽2)—3891no—
94#137—3892no—
95#137—3891no—
96AGL(1,ℤ/8)—3892no—
97AGL(1,ℤ/8)D38911yescharacteristic, complemented
98#135—3891no—
99#135—3892no—
100C2×SD16—3891no—
101C2×SD16—3892no—
102#131—3891no—
103#41—7781no—
104#41—7781no—
105#41—7782no—
106#41—3892no—
107#41—3892no—
108#41—3891no—
109#41—3891no—
110#41C2×D38911yescharacteristic, complemented
111C2×Q8—7781no—
112C2×Q8—3892no—
113C2×Q8—3891no—
114C2×D4—7781no—
115C2×D4—7782no—
116C2×D4—3892no—
117C2×D4C2×D38911yescharacteristic
118Q16—3891no—
119Q16—3892no—
120SD16—3891no—
121SD16—3892no—
122SD16—3892no—
123SD16—3891no—
124SD16—3892no—
125SD16C2×D38912yescomplemented
126D8—3891no—
127D8C2×D38912yescomplemented
128M16—3892no—
129M16C2×D38911yescharacteristic, complemented
130C2×C8—3891no—
131C2×C8—3892no—
132C23—21no—
133C23—7782no—
134Q8—7782no—
135Q8—7781no—
136Q8—3892no—
137Q8—3891no—
138Q8—3891no—
139Q8C22×D38911yescharacteristic
140D4—21no—
141D4—21no—
142D4—7782no—
143D4—7782no—
144D4—7781no—
145D4—7782no—
146D4—3892no—
147D4—3891no—
148D4—3891no—
149D4C22×D38911yescharacteristic
150D4C22×D38912yes—
151C2×C4—21no—
152C2×C4—7782no—
153C2×C4—7781no—
154C2×C4—7782no—
155C2×C4—7782no—
156C2×C4—7781no—
157C2×C4—3892no—
158C2×C4C22×D38911yescharacteristic
159C8—3891no—
160C8—3891no—
161C8C22×D38912yes—
162C22—21no—
163C22—22no—
164C22—81no—
165C22—7781no—
166C22—31122no—
167C22—3892no—
168C22D4×D38911yescharacteristic
169C4—21no—
170C4—7781no—
171C4—7782no—
172C4—3892no—
173C4D4×D38911yescharacteristic
174C4C23×D38911yescharacteristic
175C2—21no—
176C2—42no—
177C2—41no—
178C2—7782no—
179C2—15561no—
180C2C2×D4×D38911yescharacteristic
subgroupG/H
centerC2C2×D4×D389
commutator, cosocleC4×C389C24
FrattiniC4C23×D389
FittingAGL(1,ℤ/8)×C389C2
socleC2×C389C22×D4
hypercenterAGL(1,ℤ/8)D389
hypocenterC389#577
LCSG ▷ C4×C389 ▷ C2×C389 ▷ C389
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27159431C2semidirect
2. #27159429C2semidirect
3. #27159417C2semidirect
4. #27159413C2semidirect
5. #27159408C2semidirect
6. #27159404C2semidirect
7. #27159403C2semidirect
8. C2×#20261954C2semidirect
9. SD16×D389C2semidirect
10. #27159392C2semidirect
11. AGL(1,ℤ/8)×C389C2semidirect
18. #20261965C22semidirect
20. C2×Dic778C22semidirect
21. Q8×D389C22semidirect
25. Dic1556C22semidirect
26. #20261956C22semidirect
27. #20261955C22semidirect
28. #20261954C22semidirect
29. #20261954C22semidirect
30. #20261954C22semidirect
31. #20261953C22semidirect
32. #20261951C22semidirect
33. #20261949C22semidirect
34. C2×#15691005C22semidirect
35. C8×D389C22semidirect
36. #41×C389C22semidirect
38. SD16×C389C22semidirect
39. D8×C389C22semidirect
40. M16×C389C22semidirect
90. C389#577semidirect
97. AGL(1,ℤ/8)D389semidirect
110. #41C2×D389semidirect
125. SD16C2×D389semidirect
127. D8C2×D389semidirect
129. M16C2×D389semidirect

Maximal subgroups

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

subgroupindex
#271594312
#271594292
#271594172
#271594132
#271594082
#271594042
#271594032
C2×#202619542
SD16×D3892
#271593922
AGL(1,ℤ/8)×C3892
#577389