Finite Group Databasebeta

← #41003637

Subgroups — #41003637

Order 26944.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 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#41×D421C212yesmaximal, complemented
2#27212398C212yesmaximal, complemented
3#27212395C212yesmaximal, complemented
4#27212393C212yesmaximal, complemented
5#27212390C212yesmaximal, complemented
6#27212388C211yesmaximal, characteristic, complemented
7#27212386C212yesmaximal, complemented
8C2×#20269754C211yesmaximal, characteristic, complemented
9#135×C421C211yesmaximal, characteristic, complemented
10#20269772C2212yes—
11#20269771—22no—
12#20269771C2212yescomplemented
13#20269770—22no—
14#20269770C2212yes—
15Q8×D421C2212yescomplemented
16D4×D421—22no—
17D4×D421C2212yescomplemented
18C2×C4×D421—22no—
19C2×C4×D421C2211yescharacteristic
20Dic1684C2211yescharacteristic, complemented
21#20269761C2212yescomplemented
22#20269760C2212yescomplemented
23#20269759C2212yescomplemented
24#20269758C2212yescomplemented
25D3368C2211yescharacteristic, complemented
26#20269756C2212yescomplemented
27#20269754C2212yescomplemented
28#20269754C2212yescomplemented
29C2×#16809890C2211yescharacteristic, complemented
30#41×C421C2212yescomplemented
31Q16×C421C2211yescharacteristic, complemented
32SD16×C421C2212yescomplemented
33D8×C421C2211yescharacteristic, complemented
34C2×C8×C421C2211yescharacteristic, complemented
35C22×D421—22no—
36C22×D421D411yescharacteristic
37Dic842—22no—
38Dic842C2312yes—
39D1684—22no—
40D1684C2312yes—
41C22⋊D421—22no—
42C22⋊D421—22no—
43C22⋊D421—22no—
44C2×Dic421—22no—
45C2×Dic421D411yescharacteristic
46C4×D421—22no—
47C4×D421—22no—
48C4×D421—22no—
49C4×D421D412yes—
50C4×D421C2312yes—
51#16809890C2312yes—
52Q8×C421C2312yes—
53D4×C421—22no—
54D4×C421C2312yes—
55C2×C4×C421—22no—
56C2×C4×C421C2311yescharacteristic
57C8×C421C2311yescharacteristic
58C8×C421C2311yescharacteristic
59C22×C421—22no—
60C22×C421C2×D411yescharacteristic
61C2×D421—22no—
62C2×D421—41no—
63C2×D421—44no—
64C2×D421C2×D412yes—
65C4×C421—22no—
66C4×C421C2×D411yescharacteristic
67C4×C421C2411yescharacteristic
68Dic421—22no—
69Dic421C2×D412yes—
70C2×C421—21no—
71C2×C421—42no—
72C2×C421C22×D411yescharacteristic
73D421—22no—
74D421—42no—
75C421#57511yescharacteristic, complemented, Sylow
76#575—4211nomaximal, Sylow
77C2×#41—4212no—
78#137—4212no—
79#137—4212no—
80AGL(1,ℤ/8)—4212no—
81AGL(1,ℤ/8)—4212no—
82#135—4212no—
83#135—4211no—
84#135D42111yescharacteristic, complemented
85C2×M16—4211no—
86#41—8422no—
87#41—8422no—
88#41—4212no—
89#41—4212no—
90#41—4212no—
91#41C2×D42112yescomplemented
92C2×Q8—4212no—
93C2×D4—8422no—
94C2×D4—4212no—
95C22×C4—8422no—
96C22×C4—4211no—
97Q16—4212no—
98Q16—4211no—
99Q16C2×D42111yescharacteristic, complemented
100SD16—4212no—
101SD16—4212no—
102SD16—4212no—
103SD16C2×D42112yescomplemented
104D8—4212no—
105D8—4211no—
106D8C2×D42111yescharacteristic, complemented
107M16—4212no—
108M16—4212no—
109C2×C8—4211no—
110C2×C8C2×D42111yescharacteristic, complemented
111C23—8422no—
112C23—4211no—
113Q8—8422no—
114Q8—4212no—
115Q8C22×D42112yes—
116D4—22no—
117D4—8422no—
118D4—8422no—
119D4—8422no—
120D4—8422no—
121D4—4212no—
122D4C22×D42112yes—
123C2×C4—22no—
124C2×C4—8422no—
125C2×C4—8422no—
126C2×C4—8422no—
127C2×C4—8422no—
128C2×C4—4211no—
129C2×C4—4212no—
130C2×C4—4212no—
131C2×C4C22×D42111yescharacteristic
132C8—4212no—
133C8C22×D42111yescharacteristic
134C8C22×D42111yescharacteristic
135C22—22no—
136C22—8422no—
137C22—16841no—
138C22—16844no—
139C22—4212no—
140C22D4×D42111yescharacteristic
141C4—22no—
142C4—8422no—
143C4—4212no—
144C4D4×D42111yescharacteristic
145C4C23×D42111yescharacteristic
146C2—21no—
147C2—42no—
148C2—8422no—
149C2—16842no—
150C2C2×D4×D42111yescharacteristic
subgroupG/H
centerC4D4×D421
commutator, cosocleC4×C421C24
FrattiniC4C23×D421
Fitting#135×C421C2
socleC2×C421C22×D4
hypercenter#135D421
hypocenterC421#575
LCSG ▷ C4×C421 ▷ C2×C421 ▷ C421
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D421C2semidirect
2. #27212398C2semidirect
3. #27212395C2semidirect
4. #27212393C2semidirect
5. #27212390C2semidirect
6. #27212388C2semidirect
7. #27212386C2semidirect
8. C2×#20269754C2semidirect
9. #135×C421C2semidirect
12. #20269771C22semidirect
15. Q8×D421C22semidirect
17. D4×D421C22semidirect
20. Dic1684C22semidirect
21. #20269761C22semidirect
22. #20269760C22semidirect
23. #20269759C22semidirect
24. #20269758C22semidirect
25. D3368C22semidirect
26. #20269756C22semidirect
27. #20269754C22semidirect
28. #20269754C22semidirect
29. C2×#16809890C22semidirect
30. #41×C421C22semidirect
31. Q16×C421C22semidirect
32. SD16×C421C22semidirect
33. D8×C421C22semidirect
34. C2×C8×C421C22semidirect
75. C421#575semidirect
84. #135D421semidirect
91. #41C2×D421semidirect
99. Q16C2×D421semidirect
103. SD16C2×D421semidirect
106. D8C2×D421semidirect
110. C2×C8C2×D421semidirect

Maximal subgroups

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

subgroupindex
#41×D4212
#272123982
#272123952
#272123932
#272123902
#272123882
#272123862
C2×#202697542
#135×C4212
#575421