Finite Group Databasebeta

← #38786918

Subgroups — #38786918

Order 25664.

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#27165783C211yesmaximal, characteristic, complemented
2#27165782C211yesmaximal, characteristic, complemented
3#27165765C211yesmaximal, characteristic, complemented
4#27165764C212yesmaximal, complemented
5#27165761C212yesmaximal, complemented
6#27165758C212yesmaximal, complemented
7#27165757C211yesmaximal, characteristic, complemented
8C2×#20264351C211yesmaximal, characteristic, complemented
9D8×D401C212yesmaximal, complemented
10#27165745C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C401C211yesmaximal, characteristic, complemented
12#20264367C2212yescomplemented
13#20264366—21no—
14#20264366C2211yescharacteristic
15#20264366C2211yescharacteristic, complemented
16#20264365—21no—
17#20264365C2212yescomplemented
18C2×D1604—21no—
19C2×D1604C2211yescharacteristic, complemented
20C2×C22⋊D401—22no—
21D4×D401—21no—
22D4×D401—22no—
23D4×D401C2212yescomplemented
24D4×D401C2212yescomplemented
25#20264356C2211yescharacteristic, complemented
26#20264355C2212yescomplemented
27#20264354C2211yescharacteristic, complemented
28#20264353C2211yescharacteristic, complemented
29D3208C2212yescomplemented
30#20264351C2211yescharacteristic, complemented
31#20264351C2212yescomplemented
32#20264351C2212yescomplemented
33#20264349C2212yescomplemented
34C2×#15691999C2211yescharacteristic, complemented
35C8×D401C2212yescomplemented
36#41×C401C2211yescharacteristic, complemented
37C2×D4×C401C2211yescharacteristic, complemented
38SD16×C401C2212yescomplemented
39D8×C401C2212yescomplemented
40M16×C401C2211yescharacteristic, complemented
41C22×D401—21no—
42C22×D401—22no—
43C22×D401—22no—
44Dic802D411yescharacteristic
45Dic802C2311yescharacteristic
46D1604—22no—
47D1604—21no—
48D1604D411yescharacteristic
49D1604C2312yes—
50D1604C2311yescharacteristic
51C22⋊D401—21no—
52C22⋊D401—22no—
53C22⋊D401—22no—
54C22⋊D401—22no—
55C22⋊D401D412yes—
56C2×Dic401—22no—
57C4×D401—21no—
58C4×D401—22no—
59C4×D401C2312yes—
60#15691999C2311yescharacteristic
61#15691999C2311yescharacteristic
62C23×C401—21no—
63Q8×C401C2311yescharacteristic
64D4×C401—21no—
65D4×C401—21no—
66D4×C401C2311yescharacteristic
67D4×C401C2312yes—
68C2×C4×C401—21no—
69C2×C4×C401C2311yescharacteristic
70C8×C401C2312yes—
71C22×C401—21no—
72C22×C401—22no—
73C22×C401—81no—
74C22×C401C2×D411yescharacteristic
75C2×D401—21no—
76C2×D401—22no—
77C2×D401—82no—
78C2×D401—82no—
79C2×D401—81no—
80C2×D401C2×D412yes—
81C4×C401—21no—
82C4×C401C2×D411yescharacteristic
83C4×C401C2411yescharacteristic
84Dic401—21no—
85Dic401C2×D412yes—
86C2×C401—21no—
87C2×C401—41no—
88C2×C401—42no—
89C2×C401C22×D411yescharacteristic
90D401—22no—
91D401—41no—
92D401—42no—
93C401#57611yescharacteristic, complemented, Sylow
94#576—4011nomaximal, Sylow
95H(5,𝔽2)—4011no—
96H(5,𝔽2)—4011no—
97AGL(1,ℤ/8)—4012no—
98AGL(1,ℤ/8)—4012no—
99AGL(1,ℤ/8)—4011no—
100AGL(1,ℤ/8)D40111yescharacteristic, complemented
101#135—4011no—
102#135—4012no—
103C2×D8—4011no—
104C2×D8—4012no—
105#131—4011no—
106#41—8021no—
107#41—8021no—
108#41—4012no—
109#41—4012no—
110#41—4011no—
111#41—4011no—
112#41C2×D40111yescharacteristic, complemented
113C2×D4—8021no—
114C2×D4—8022no—
115C2×D4—8021no—
116C2×D4—8022no—
117C2×D4—4012no—
118C2×D4—4012no—
119C2×D4—4011no—
120C2×D4C2×D40111yescharacteristic, complemented
121Q16—4011no—
122SD16—4011no—
123SD16—4011no—
124SD16—4012no—
125SD16C2×D40112yescomplemented
126D8—4012no—
127D8—4011no—
128D8—4012no—
129D8—4012no—
130D8C2×D40112yescomplemented
131M16—4012no—
132M16C2×D40111yescharacteristic, complemented
133C2×C8—4011no—
134C2×C8—4012no—
135C23—21no—
136C23—8022no—
137C23—8022no—
138C23—8021no—
139Q8—4011no—
140Q8—4011no—
141Q8C22×D40111yescharacteristic
142D4—21no—
143D4—21no—
144D4—8022no—
145D4—8022no—
146D4—8021no—
147D4—8022no—
148D4—8022no—
149D4—8021no—
150D4—4012no—
151D4—4011no—
152D4—4012no—
153D4—4011no—
154D4C22×D40112yes—
155D4C22×D40111yescharacteristic
156C2×C4—21no—
157C2×C4—8022no—
158C2×C4—8022no—
159C2×C4—8021no—
160C2×C4—4012no—
161C2×C4C22×D40111yescharacteristic
162C8—4011no—
163C8—4011no—
164C8C22×D40112yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—8022no—
169C22—8021no—
170C22—32081no—
171C22—32082no—
172C22—32082no—
173C22—4012no—
174C22D4×D40111yescharacteristic
175C4—21no—
176C4—8021no—
177C4—4012no—
178C4D4×D40111yescharacteristic
179C4C23×D40111yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—8022no—
184C2—16042no—
185C2—16041no—
186C2C2×D4×D40111yescharacteristic
subgroupG/H
centerC2C2×D4×D401
commutator, cosocleC4×C401C24
FrattiniC4C23×D401
FittingAGL(1,ℤ/8)×C401C2
socleC2×C401C22×D4
hypercenterAGL(1,ℤ/8)D401
hypocenterC401#576
LCSG ▷ C4×C401 ▷ C2×C401 ▷ C401
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #27165783C2semidirect
2. #27165782C2semidirect
3. #27165765C2semidirect
4. #27165764C2semidirect
5. #27165761C2semidirect
6. #27165758C2semidirect
7. #27165757C2semidirect
8. C2×#20264351C2semidirect
9. D8×D401C2semidirect
10. #27165745C2semidirect
11. AGL(1,ℤ/8)×C401C2semidirect
12. #20264367C22semidirect
15. #20264366C22semidirect
17. #20264365C22semidirect
19. C2×D1604C22semidirect
23. D4×D401C22semidirect
24. D4×D401C22semidirect
25. #20264356C22semidirect
26. #20264355C22semidirect
27. #20264354C22semidirect
28. #20264353C22semidirect
29. D3208C22semidirect
30. #20264351C22semidirect
31. #20264351C22semidirect
32. #20264351C22semidirect
33. #20264349C22semidirect
34. C2×#15691999C22semidirect
35. C8×D401C22semidirect
36. #41×C401C22semidirect
37. C2×D4×C401C22semidirect
38. SD16×C401C22semidirect
39. D8×C401C22semidirect
40. M16×C401C22semidirect
93. C401#576semidirect
100. AGL(1,ℤ/8)D401semidirect
112. #41C2×D401semidirect
120. C2×D4C2×D401semidirect
125. SD16C2×D401semidirect
130. D8C2×D401semidirect
132. M16C2×D401semidirect

Maximal subgroups

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

subgroupindex
#271657832
#271657822
#271657652
#271657642
#271657612
#271657582
#271657572
C2×#202643512
D8×D4012
#271657452
AGL(1,ℤ/8)×C4012
#576401