Finite Group Databasebeta

← #38734903

Subgroups — #38734903

Order 24896.

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#27159430C211yesmaximal, characteristic, complemented
2#27159429C211yesmaximal, characteristic, complemented
3#27159412C211yesmaximal, characteristic, complemented
4#27159411C212yesmaximal, complemented
5#27159408C212yesmaximal, complemented
6#27159405C212yesmaximal, complemented
7#27159404C211yesmaximal, characteristic, complemented
8C2×#20261951C211yesmaximal, characteristic, complemented
9D8×D389C212yesmaximal, complemented
10#27159392C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C389C211yesmaximal, characteristic, complemented
12#20261967C2212yescomplemented
13#20261966—21no—
14#20261966C2211yescharacteristic
15#20261966C2211yescharacteristic, complemented
16#20261965—21no—
17#20261965C2212yescomplemented
18C2×D1556—21no—
19C2×D1556C2211yescharacteristic, complemented
20C2×C22⋊D389—22no—
21D4×D389—21no—
22D4×D389—22no—
23D4×D389C2212yescomplemented
24D4×D389C2212yescomplemented
25#20261956C2211yescharacteristic, complemented
26#20261955C2212yescomplemented
27#20261954C2211yescharacteristic, complemented
28#20261953C2211yescharacteristic, complemented
29D3112C2212yescomplemented
30#20261951C2211yescharacteristic, complemented
31#20261951C2212yescomplemented
32#20261951C2212yescomplemented
33#20261949C2212yescomplemented
34C2×#15691005C2211yescharacteristic, complemented
35C8×D389C2212yescomplemented
36#41×C389C2211yescharacteristic, complemented
37C2×D4×C389C2211yescharacteristic, complemented
38SD16×C389C2212yescomplemented
39D8×C389C2212yescomplemented
40M16×C389C2211yescharacteristic, complemented
41C22×D389—21no—
42C22×D389—22no—
43C22×D389—22no—
44Dic778D411yescharacteristic
45Dic778C2311yescharacteristic
46D1556—22no—
47D1556—21no—
48D1556D411yescharacteristic
49D1556C2312yes—
50D1556C2311yescharacteristic
51C22⋊D389—21no—
52C22⋊D389—22no—
53C22⋊D389—22no—
54C22⋊D389—22no—
55C22⋊D389D412yes—
56C2×Dic389—22no—
57C4×D389—21no—
58C4×D389—22no—
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—21no—
72C22×C389—22no—
73C22×C389—81no—
74C22×C389C2×D411yescharacteristic
75C2×D389—21no—
76C2×D389—22no—
77C2×D389—82no—
78C2×D389—82no—
79C2×D389—81no—
80C2×D389C2×D412yes—
81C4×C389—21no—
82C4×C389C2×D411yescharacteristic
83C4×C389C2411yescharacteristic
84Dic389—21no—
85Dic389C2×D412yes—
86C2×C389—21no—
87C2×C389—41no—
88C2×C389—42no—
89C2×C389C22×D411yescharacteristic
90D389—22no—
91D389—41no—
92D389—42no—
93C389#57611yescharacteristic, complemented, Sylow
94#576—3891nomaximal, Sylow
95H(5,𝔽2)—3891no—
96H(5,𝔽2)—3891no—
97AGL(1,ℤ/8)—3892no—
98AGL(1,ℤ/8)—3892no—
99AGL(1,ℤ/8)—3891no—
100AGL(1,ℤ/8)D38911yescharacteristic, complemented
101#135—3891no—
102#135—3892no—
103C2×D8—3891no—
104C2×D8—3892no—
105#131—3891no—
106#41—7781no—
107#41—7781no—
108#41—3892no—
109#41—3892no—
110#41—3891no—
111#41—3891no—
112#41C2×D38911yescharacteristic, complemented
113C2×D4—7781no—
114C2×D4—7782no—
115C2×D4—7781no—
116C2×D4—7782no—
117C2×D4—3892no—
118C2×D4—3892no—
119C2×D4—3891no—
120C2×D4C2×D38911yescharacteristic, complemented
121Q16—3891no—
122SD16—3891no—
123SD16—3891no—
124SD16—3892no—
125SD16C2×D38912yescomplemented
126D8—3892no—
127D8—3891no—
128D8—3892no—
129D8—3892no—
130D8C2×D38912yescomplemented
131M16—3892no—
132M16C2×D38911yescharacteristic, complemented
133C2×C8—3891no—
134C2×C8—3892no—
135C23—21no—
136C23—7782no—
137C23—7782no—
138C23—7781no—
139Q8—3891no—
140Q8—3891no—
141Q8C22×D38911yescharacteristic
142D4—21no—
143D4—21no—
144D4—7782no—
145D4—7782no—
146D4—7781no—
147D4—7782no—
148D4—7782no—
149D4—7781no—
150D4—3892no—
151D4—3891no—
152D4—3892no—
153D4—3891no—
154D4C22×D38912yes—
155D4C22×D38911yescharacteristic
156C2×C4—21no—
157C2×C4—7782no—
158C2×C4—7782no—
159C2×C4—7781no—
160C2×C4—3892no—
161C2×C4C22×D38911yescharacteristic
162C8—3891no—
163C8—3891no—
164C8C22×D38912yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—7782no—
169C22—7781no—
170C22—31121no—
171C22—31122no—
172C22—31122no—
173C22—3892no—
174C22D4×D38911yescharacteristic
175C4—21no—
176C4—7781no—
177C4—3892no—
178C4D4×D38911yescharacteristic
179C4C23×D38911yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—7782no—
184C2—15562no—
185C2—15561no—
186C2C2×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#576
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. #27159430C2semidirect
2. #27159429C2semidirect
3. #27159412C2semidirect
4. #27159411C2semidirect
5. #27159408C2semidirect
6. #27159405C2semidirect
7. #27159404C2semidirect
8. C2×#20261951C2semidirect
9. D8×D389C2semidirect
10. #27159392C2semidirect
11. AGL(1,ℤ/8)×C389C2semidirect
12. #20261967C22semidirect
15. #20261966C22semidirect
17. #20261965C22semidirect
19. C2×D1556C22semidirect
23. D4×D389C22semidirect
24. D4×D389C22semidirect
25. #20261956C22semidirect
26. #20261955C22semidirect
27. #20261954C22semidirect
28. #20261953C22semidirect
29. D3112C22semidirect
30. #20261951C22semidirect
31. #20261951C22semidirect
32. #20261951C22semidirect
33. #20261949C22semidirect
34. C2×#15691005C22semidirect
35. C8×D389C22semidirect
36. #41×C389C22semidirect
37. C2×D4×C389C22semidirect
38. SD16×C389C22semidirect
39. D8×C389C22semidirect
40. M16×C389C22semidirect
93. C389#576semidirect
100. AGL(1,ℤ/8)D389semidirect
112. #41C2×D389semidirect
120. C2×D4C2×D389semidirect
125. SD16C2×D389semidirect
130. D8C2×D389semidirect
132. M16C2×D389semidirect

Maximal subgroups

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

subgroupindex
#271594302
#271594292
#271594122
#271594112
#271594082
#271594052
#271594042
C2×#202619512
D8×D3892
#271593922
AGL(1,ℤ/8)×C3892
#576389