Finite Group Databasebeta

← #38786923

Subgroups — #38786923

Order 25664.

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#27165785C211yesmaximal, characteristic, complemented
2#27165783C211yesmaximal, characteristic, complemented
3#27165769C212yesmaximal, complemented
4#27165765C211yesmaximal, characteristic, complemented
5#27165764C212yesmaximal, complemented
6#27165760C212yesmaximal, complemented
7#27165757C211yesmaximal, characteristic, complemented
8C2×#20264353C211yesmaximal, characteristic, complemented
9SD16×D401C212yesmaximal, complemented
10#27165745C211yesmaximal, characteristic, complemented
11#137×C401C211yesmaximal, characteristic, complemented
12#20264367—21no—
13#20264367C2212yes—
14#20264367C2212yescomplemented
15#20264366—21no—
16#20264366—22no—
17#20264366C2211yescharacteristic
18#20264366C2211yescharacteristic, complemented
19Q8×D401—21no—
20Q8×D401C2212yescomplemented
21C2×D1604—21no—
22C2×D1604C2211yescharacteristic
23D4×D401—22no—
24D4×D401C2212yes—
25#20264356C2211yescharacteristic, complemented
26#20264355C2212yescomplemented
27#20264354C2211yescharacteristic, complemented
28#20264353C2212yescomplemented
29#20264353C2212yescomplemented
30#20264353C2211yescharacteristic, complemented
31D3208C2212yescomplemented
32#20264351C2211yescharacteristic, complemented
33#20264349C2212yescomplemented
34C2×#15691999C2211yescharacteristic, complemented
35C8×D401C2212yescomplemented
36#41×C401C2211yescharacteristic
37C2×Q8×C401C2211yescharacteristic, complemented
38Q16×C401C2212yescomplemented
39SD16×C401C2212yescomplemented
40M16×C401C2211yescharacteristic, complemented
41C22×D401—21no—
42C22×D401—22no—
43Dic802—22no—
44Dic802D411yescharacteristic
45Dic802C2311yescharacteristic
46D1604—22no—
47D1604—22no—
48D1604—21no—
49D1604D411yescharacteristic
50D1604C2311yescharacteristic
51D1604C2312yes—
52C22⋊D401—21no—
53C22⋊D401—22no—
54C22⋊D401D412yes—
55C4×D401—22no—
56C4×D401—21no—
57C4×D401—22no—
58C4×D401—22no—
59C4×D401C2312yes—
60#15691999C2311yescharacteristic
61#15691999C2311yescharacteristic
62Q8×C401—21no—
63Q8×C401C2312yes—
64Q8×C401C2311yescharacteristic
65D4×C401—21no—
66D4×C401C2311yescharacteristic
67C2×C4×C401—21no—
68C2×C4×C401—21no—
69C2×C4×C401C2311yescharacteristic
70C8×C401C2312yes—
71C22×C401—21no—
72C22×C401C2×D411yescharacteristic
73C2×D401—21no—
74C2×D401—22no—
75C2×D401—82no—
76C2×D401—81no—
77C2×D401C2×D412yes—
78C4×C401—21no—
79C4×C401—22no—
80C4×C401C2×D411yescharacteristic
81C4×C401C2411yescharacteristic
82Dic401—21no—
83Dic401C2×D412yes—
84C2×C401—21no—
85C2×C401—41no—
86C2×C401C22×D411yescharacteristic
87D401—22no—
88D401—41no—
89D401—42no—
90C401#57711yescharacteristic, complemented, Sylow
91#577—4011nomaximal, Sylow
922−1+4—4011no—
93H(5,𝔽2)—4011no—
94#137—4012no—
95#137D40111yescharacteristic, complemented
96AGL(1,ℤ/8)—4012no—
97AGL(1,ℤ/8)—4011no—
98#135—4011no—
99#135—4012no—
100C2×SD16—4011no—
101C2×SD16—4012no—
102#131—4011no—
103#41—8021no—
104#41—8022no—
105#41—8021no—
106#41—4012no—
107#41—4012no—
108#41—4011no—
109#41—4011no—
110#41C2×D40111yescharacteristic
111C2×Q8—8021no—
112C2×Q8—4012no—
113C2×Q8C2×D40111yescharacteristic, complemented
114C2×D4—8021no—
115C2×D4—8022no—
116C2×D4—4012no—
117C2×D4—4011no—
118Q16—4011no—
119Q16C2×D40112yescomplemented
120SD16—4012no—
121SD16—4012no—
122SD16—4011no—
123SD16—4011no—
124SD16—4012no—
125SD16C2×D40112yescomplemented
126D8—4011no—
127D8—4012no—
128M16—4012no—
129M16C2×D40111yescharacteristic, complemented
130C2×C8—4011no—
131C2×C8—4012no—
132C23—8022no—
133C23—8021no—
134Q8—21no—
135Q8—8022no—
136Q8—4011no—
137Q8—4011no—
138Q8C22×D40112yes—
139Q8C22×D40111yescharacteristic
140D4—21no—
141D4—8022no—
142D4—8022no—
143D4—8021no—
144D4—8022no—
145D4—8021no—
146D4—4012no—
147D4—4011no—
148D4—4011no—
149D4—4012no—
150D4C22×D40111yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—8022no—
154C2×C4—8022no—
155C2×C4—8022no—
156C2×C4—8021no—
157C2×C4—4012no—
158C2×C4C22×D40111yescharacteristic
159C8—4011no—
160C8—4011no—
161C8C22×D40112yes—
162C22—21no—
163C22—8021no—
164C22—8022no—
165C22—32081no—
166C22—32082no—
167C22—4012no—
168C22D4×D40111yescharacteristic
169C4—21no—
170C4—22no—
171C4—8021no—
172C4—4012no—
173C4D4×D40111yescharacteristic
174C4C23×D40111yescharacteristic
175C2—21no—
176C2—41no—
177C2—8022no—
178C2—16042no—
179C2—16041no—
180C2C2×D4×D40111yescharacteristic
subgroupG/H
centerC2C2×D4×D401
commutator, cosocleC4×C401C24
FrattiniC4C23×D401
Fitting#137×C401C2
socleC2×C401C22×D4
hypercenter#137D401
hypocenterC401#577
LCSG ▷ C4×C401 ▷ C2×C401 ▷ C401
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #27165785C2semidirect
2. #27165783C2semidirect
3. #27165769C2semidirect
4. #27165765C2semidirect
5. #27165764C2semidirect
6. #27165760C2semidirect
7. #27165757C2semidirect
8. C2×#20264353C2semidirect
9. SD16×D401C2semidirect
10. #27165745C2semidirect
11. #137×C401C2semidirect
14. #20264367C22semidirect
18. #20264366C22semidirect
20. Q8×D401C22semidirect
25. #20264356C22semidirect
26. #20264355C22semidirect
27. #20264354C22semidirect
28. #20264353C22semidirect
29. #20264353C22semidirect
30. #20264353C22semidirect
31. D3208C22semidirect
32. #20264351C22semidirect
33. #20264349C22semidirect
34. C2×#15691999C22semidirect
35. C8×D401C22semidirect
37. C2×Q8×C401C22semidirect
38. Q16×C401C22semidirect
39. SD16×C401C22semidirect
40. M16×C401C22semidirect
90. C401#577semidirect
95. #137D401semidirect
113. C2×Q8C2×D401semidirect
119. Q16C2×D401semidirect
125. SD16C2×D401semidirect
129. M16C2×D401semidirect

Maximal subgroups

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

subgroupindex
#271657852
#271657832
#271657692
#271657652
#271657642
#271657602
#271657572
C2×#202643532
SD16×D4012
#271657452
#137×C4012
#577401