Finite Group Databasebeta

← #38734908

Subgroups — #38734908

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#27159432C211yesmaximal, characteristic, complemented
2#27159430C211yesmaximal, characteristic, complemented
3#27159416C212yesmaximal, complemented
4#27159412C211yesmaximal, characteristic, complemented
5#27159411C212yesmaximal, complemented
6#27159407C212yesmaximal, complemented
7#27159404C211yesmaximal, characteristic, complemented
8C2×#20261953C211yesmaximal, characteristic, complemented
9SD16×D389C212yesmaximal, complemented
10#27159392C211yesmaximal, characteristic, complemented
11#137×C389C211yesmaximal, characteristic, complemented
12#20261967—21no—
13#20261967C2212yes—
14#20261967C2212yescomplemented
15#20261966—21no—
16#20261966—22no—
17#20261966C2211yescharacteristic
18#20261966C2211yescharacteristic, complemented
19Q8×D389—21no—
20Q8×D389C2212yescomplemented
21C2×D1556—21no—
22C2×D1556C2211yescharacteristic
23D4×D389—22no—
24D4×D389C2212yes—
25#20261956C2211yescharacteristic, complemented
26#20261955C2212yescomplemented
27#20261954C2211yescharacteristic, complemented
28#20261953C2212yescomplemented
29#20261953C2212yescomplemented
30#20261953C2211yescharacteristic, complemented
31D3112C2212yescomplemented
32#20261951C2211yescharacteristic, complemented
33#20261949C2212yescomplemented
34C2×#15691005C2211yescharacteristic, complemented
35C8×D389C2212yescomplemented
36#41×C389C2211yescharacteristic
37C2×Q8×C389C2211yescharacteristic, complemented
38Q16×C389C2212yescomplemented
39SD16×C389C2212yescomplemented
40M16×C389C2211yescharacteristic, complemented
41C22×D389—21no—
42C22×D389—22no—
43Dic778—22no—
44Dic778D411yescharacteristic
45Dic778C2311yescharacteristic
46D1556—22no—
47D1556—22no—
48D1556—21no—
49D1556D411yescharacteristic
50D1556C2311yescharacteristic
51D1556C2312yes—
52C22⋊D389—21no—
53C22⋊D389—22no—
54C22⋊D389D412yes—
55C4×D389—22no—
56C4×D389—21no—
57C4×D389—22no—
58C4×D389—22no—
59C4×D389C2312yes—
60#15691005C2311yescharacteristic
61#15691005C2311yescharacteristic
62Q8×C389—21no—
63Q8×C389C2312yes—
64Q8×C389C2311yescharacteristic
65D4×C389—21no—
66D4×C389C2311yescharacteristic
67C2×C4×C389—21no—
68C2×C4×C389—21no—
69C2×C4×C389C2311yescharacteristic
70C8×C389C2312yes—
71C22×C389—21no—
72C22×C389C2×D411yescharacteristic
73C2×D389—22no—
74C2×D389—21no—
75C2×D389—82no—
76C2×D389—81no—
77C2×D389C2×D412yes—
78C4×C389—21no—
79C4×C389—22no—
80C4×C389C2×D411yescharacteristic
81C4×C389C2411yescharacteristic
82Dic389—21no—
83Dic389C2×D412yes—
84C2×C389—21no—
85C2×C389—41no—
86C2×C389C22×D411yescharacteristic
87D389—22no—
88D389—41no—
89D389—42no—
90C389#57711yescharacteristic, complemented, Sylow
91#577—3891nomaximal, Sylow
922−1+4—3891no—
93H(5,𝔽2)—3891no—
94#137—3892no—
95#137D38911yescharacteristic, complemented
96AGL(1,ℤ/8)—3892no—
97AGL(1,ℤ/8)—3891no—
98#135—3891no—
99#135—3892no—
100C2×SD16—3891no—
101C2×SD16—3892no—
102#131—3891no—
103#41—7781no—
104#41—7782no—
105#41—7781no—
106#41—3892no—
107#41—3892no—
108#41—3891no—
109#41—3891no—
110#41C2×D38911yescharacteristic
111C2×Q8—7781no—
112C2×Q8—3892no—
113C2×Q8C2×D38911yescharacteristic, complemented
114C2×D4—7781no—
115C2×D4—7782no—
116C2×D4—3892no—
117C2×D4—3891no—
118Q16—3891no—
119Q16C2×D38912yescomplemented
120SD16—3892no—
121SD16—3892no—
122SD16—3891no—
123SD16—3891no—
124SD16—3892no—
125SD16C2×D38912yescomplemented
126D8—3891no—
127D8—3892no—
128M16—3892no—
129M16C2×D38911yescharacteristic, complemented
130C2×C8—3891no—
131C2×C8—3892no—
132C23—7782no—
133C23—7781no—
134Q8—21no—
135Q8—7782no—
136Q8—3891no—
137Q8—3891no—
138Q8C22×D38912yes—
139Q8C22×D38911yescharacteristic
140D4—21no—
141D4—7782no—
142D4—7782no—
143D4—7781no—
144D4—7782no—
145D4—7781no—
146D4—3892no—
147D4—3891no—
148D4—3891no—
149D4—3892no—
150D4C22×D38911yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—7782no—
154C2×C4—7782no—
155C2×C4—7782no—
156C2×C4—7781no—
157C2×C4—3892no—
158C2×C4C22×D38911yescharacteristic
159C8—3891no—
160C8—3891no—
161C8C22×D38912yes—
162C22—21no—
163C22—7781no—
164C22—7782no—
165C22—31121no—
166C22—31122no—
167C22—3892no—
168C22D4×D38911yescharacteristic
169C4—21no—
170C4—22no—
171C4—7781no—
172C4—3892no—
173C4D4×D38911yescharacteristic
174C4C23×D38911yescharacteristic
175C2—21no—
176C2—41no—
177C2—7782no—
178C2—15562no—
179C2—15561no—
180C2C2×D4×D38911yescharacteristic
subgroupG/H
centerC2C2×D4×D389
commutator, cosocleC4×C389C24
FrattiniC4C23×D389
Fitting#137×C389C2
socleC2×C389C22×D4
hypercenter#137D389
hypocenterC389#577
LCSG ▷ C4×C389 ▷ C2×C389 ▷ C389
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #27159432C2semidirect
2. #27159430C2semidirect
3. #27159416C2semidirect
4. #27159412C2semidirect
5. #27159411C2semidirect
6. #27159407C2semidirect
7. #27159404C2semidirect
8. C2×#20261953C2semidirect
9. SD16×D389C2semidirect
10. #27159392C2semidirect
11. #137×C389C2semidirect
14. #20261967C22semidirect
18. #20261966C22semidirect
20. Q8×D389C22semidirect
25. #20261956C22semidirect
26. #20261955C22semidirect
27. #20261954C22semidirect
28. #20261953C22semidirect
29. #20261953C22semidirect
30. #20261953C22semidirect
31. D3112C22semidirect
32. #20261951C22semidirect
33. #20261949C22semidirect
34. C2×#15691005C22semidirect
35. C8×D389C22semidirect
37. C2×Q8×C389C22semidirect
38. Q16×C389C22semidirect
39. SD16×C389C22semidirect
40. M16×C389C22semidirect
90. C389#577semidirect
95. #137D389semidirect
113. C2×Q8C2×D389semidirect
119. Q16C2×D389semidirect
125. SD16C2×D389semidirect
129. M16C2×D389semidirect

Maximal subgroups

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

subgroupindex
#271594322
#271594302
#271594162
#271594122
#271594112
#271594072
#271594042
C2×#202619532
SD16×D3892
#271593922
#137×C3892
#577389