Finite Group Databasebeta

← #30679098

Subgroups — #30679098

Order 16768.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#22554113C212yesmaximal, complemented
2#22553748C212yesmaximal, complemented
3#22553746C214yesmaximal, complemented
4#22553744C211yesmaximal, characteristic, complemented
5#22553741C212yesmaximal, complemented
6D16×D131C212yesmaximal, complemented
7#22553728C211yesmaximal, characteristic, complemented
8C2×D16×C131C211yesmaximal, characteristic, complemented
9#16862291—22no—
10#16862272—22no—
11#16862270—24no—
12#16862268C2212yescomplemented
13#16862265C2214yescomplemented
14D8×D131C2214yescomplemented
15#16862252C2211yescharacteristic
16Dic1048C2211yescharacteristic, complemented
17#16862191C2212yescomplemented
18#16862190C2214yescomplemented
19D2096C2211yescharacteristic, complemented
20#16862187C2214yescomplemented
21#16862186C2211yescharacteristic, complemented
22#16862185C2212yescomplemented
23C16×D131C2212yescomplemented
24C2×D8×C131C2212yescomplemented
25D16×C131C2212yescomplemented
26D16×C131C2212yescomplemented
27C2×C16×C131C2211yescharacteristic, complemented
28#14523513—22no—
29#14523513D411yescharacteristic
30#14523512—24no—
31#14523512—42no—
32C2×C22⋊D131—44no—
33D4×D131—24no—
34D4×D131—42no—
35Dic524C2312yes—
36#14523502—22no—
37#14523501—24no—
38D1048C2312yes—
39#14523498—24no—
40#14523497D411yescharacteristic
41#14523496D412yes—
42C8×D131C2312yes—
43C131⋊C16C2312yes—
44C2×D4×C131—22no—
45D8×C131—22no—
46D8×C131C2314yes—
47C2×C8×C131C2311yescharacteristic
48C16×C131C2311yescharacteristic
49C16×C131C2311yescharacteristic
50C22×D131—44no—
51Dic262—22no—
52Dic262D811yescharacteristic
53D524—22no—
54D524D811yescharacteristic
55C22⋊D131—42no—
56C22⋊D131—44no—
57C22⋊D131—44no—
58C22⋊D131D812yes—
59C2×Dic131—44no—
60C4×D131—42no—
61C4×D131C2×D412yes—
62C131⋊C8C2×D412yes—
63C23×C131—42no—
64D4×C131—24no—
65D4×C131—42no—
66C2×C4×C131C2×D411yescharacteristic
67C8×C131C2×D411yescharacteristic
68C8×C131C2411yescharacteristic
69C22×C131—44no—
70C22×C131—162no—
71C22×C131C2×D811yescharacteristic
72C2×D131—42no—
73C2×D131—164no—
74C2×D131C2×D812yes—
75C4×C131C2×D811yescharacteristic
76C4×C131C22×D411yescharacteristic
77Dic131—42no—
78Dic131C2×D812yes—
79C2×C131—21no—
80C2×C131—84no—
81C2×C131C22×D811yescharacteristic
82D131—22no—
83D131—82no—
84C131#341511yescharacteristic, complemented, Sylow
85#3415—1311nomaximal, Sylow
86#576—1312no—
87#509—1312no—
88#509—1314no—
89#508—1312no—
90#508—1311no—
91C2×D16—1312no—
92C2×D16D13111yescharacteristic, complemented
93#504—1311no—
94H(5,𝔽2)—2622no—
95AGL(1,ℤ/8)—2624no—
96AGL(1,ℤ/8)—2622no—
97#135—1314no—
98#135—1312no—
99C2×D8—1314no—
100C2×D8C2×D13112yescomplemented
101#131—1311no—
102Q32—1311no—
103SD32—1314no—
104SD32—1312no—
105D16—1314no—
106D16—1311no—
107D16C2×D13112yescomplemented
108D16C2×D13112yescomplemented
109M32—1311no—
110M32—1312no—
111C2×C16—1312no—
112C2×C16C2×D13111yescharacteristic, complemented
113#41—2624no—
114#41—2622no—
115#41—5242no—
116#41—1311no—
117C2×D4—22no—
118C2×D4—2624no—
119C2×D4—5242no—
120C2×D4—5244no—
121Q16—1312no—
122SD16—2624no—
123SD16—2622no—
124D8—22no—
125D8—2624no—
126D8—1312no—
127D8C22×D13114yes—
128M16—1311no—
129M16—1312no—
130C2×C8—1312no—
131C2×C8C22×D13111yescharacteristic
132C16—1312no—
133C16C22×D13111yescharacteristic
134C16C22×D13111yescharacteristic
135C23—42no—
136C23—5244no—
137Q8—2622no—
138Q8—1311no—
139D4—24no—
140D4—42no—
141D4—2622no—
142D4—5244no—
143D4—5242no—
144D4—5244no—
145D4—1312no—
146D4—1311no—
147C2×C4—5244no—
148C2×C4—5242no—
149C2×C4—1312no—
150C2×C4D4×D13111yescharacteristic
151C8—1312no—
152C8D4×D13111yescharacteristic
153C8C23×D13111yescharacteristic
154C22—44no—
155C22—5242no—
156C22—162no—
157C22—20964no—
158C22—1312no—
159C22D8×D13111yescharacteristic
160C4—5242no—
161C4—1312no—
162C4D8×D13111yescharacteristic
163C4C2×D4×D13111yescharacteristic
164C2—21no—
165C2—2622no—
166C2—84no—
167C2—10482no—
168C2C2×D8×D13111yescharacteristic
subgroupG/H
centerC2C2×D8×D131
commutator, cosocleC8×C131C24
FrattiniC8C23×D131
FittingC2×D16×C131C2
socleC2×C131C22×D8
hypercenterC2×D16D131
hypocenterC131#3415
LCSG ▷ C8×C131 ▷ C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #22554113C2semidirect
2. #22553748C2semidirect
3. #22553746C2semidirect
4. #22553744C2semidirect
5. #22553741C2semidirect
6. D16×D131C2semidirect
7. #22553728C2semidirect
8. C2×D16×C131C2semidirect
12. #16862268C22semidirect
13. #16862265C22semidirect
14. D8×D131C22semidirect
16. Dic1048C22semidirect
17. #16862191C22semidirect
18. #16862190C22semidirect
19. D2096C22semidirect
20. #16862187C22semidirect
21. #16862186C22semidirect
22. #16862185C22semidirect
23. C16×D131C22semidirect
24. C2×D8×C131C22semidirect
25. D16×C131C22semidirect
26. D16×C131C22semidirect
27. C2×C16×C131C22semidirect
84. C131#3415semidirect
92. C2×D16D131semidirect
100. C2×D8C2×D131semidirect
107. D16C2×D131semidirect
108. D16C2×D131semidirect
112. C2×C16C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225541132
#225537482
#225537462
#225537442
#225537412
D16×D1312
#225537282
C2×D16×C1312
#3415131