Finite Group Databasebeta

← #30679099

Subgroups — #30679099

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#22554114C212yesmaximal, complemented
2#22553749C214yesmaximal, complemented
3#22553747C212yesmaximal, complemented
4#22553745C212yesmaximal, complemented
5C2×D2096C211yesmaximal, characteristic, complemented
6D16×D131C212yesmaximal, complemented
7#22553728C211yesmaximal, characteristic, complemented
8#508×C131C211yesmaximal, characteristic, complemented
9#16862292—22no—
10#16862273—24no—
11#16862271—22no—
12#16862269C2214yescomplemented
13C2×D1048C2212yescomplemented
14D8×D131C2214yescomplemented
15#16862252C2211yescharacteristic
16#16862189C2214yescomplemented
17D2096C2212yescomplemented
18D2096C2212yescomplemented
19#16862187C2214yescomplemented
20#16862186C2211yescharacteristic, complemented
21#16862185C2212yescomplemented
22C16×D131C2212yescomplemented
23#135×C131C2212yescomplemented
24Q32×C131C2211yescharacteristic, complemented
25SD32×C131C2212yescomplemented
26D16×C131C2211yescharacteristic, complemented
27C2×C16×C131C2211yescharacteristic, complemented
28#14523514—24no—
29#14523513—42no—
30#14523513D411yescharacteristic
31C2×D524—22no—
32C2×D524—42no—
33D4×D131—24no—
34D4×D131—44no—
35#14523500—24no—
36D1048—22no—
37D1048C2314yes—
38#14523498—24no—
39#14523497D411yescharacteristic
40#14523496D412yes—
41C8×D131C2312yes—
42C131⋊C16C2312yes—
43#41×C131—22no—
44Q16×C131C2312yes—
45SD16×C131—22no—
46D8×C131C2312yes—
47C2×C8×C131C2311yescharacteristic
48C16×C131C2311yescharacteristic
49C16×C131C2311yescharacteristic
50C22×D131—42no—
51C22×D131—44no—
52Dic262D811yescharacteristic
53D524—24no—
54D524—44no—
55D524—42no—
56D524D811yescharacteristic
57C22⋊D131—44no—
58C22⋊D131D812yes—
59C4×D131—44no—
60C4×D131C2×D412yes—
61C131⋊C8C2×D412yes—
62Q8×C131—22no—
63D4×C131—22no—
64D4×C131—42no—
65C2×C4×C131—42no—
66C2×C4×C131C2×D411yescharacteristic
67C8×C131C2×D411yescharacteristic
68C8×C131C2411yescharacteristic
69C22×C131—42no—
70C22×C131C2×D811yescharacteristic
71C2×D131—44no—
72C2×D131—164no—
73C2×D131—162no—
74C2×D131C2×D812yes—
75C4×C131—42no—
76C4×C131C2×D811yescharacteristic
77C4×C131C22×D411yescharacteristic
78Dic131C2×D812yes—
79C2×C131—21no—
80C2×C131—82no—
81C2×C131C22×D811yescharacteristic
82D131—22no—
83D131—84no—
84C131#341511yescharacteristic, complemented, Sylow
85#3415—1311nomaximal, Sylow
86#576—1312no—
87#509—1312no—
88#509—1314no—
89#508—1312no—
90#508D13111yescharacteristic, complemented
91C2×D16—1312no—
92C2×D16—1311no—
93#504—1311no—
94H(5,𝔽2)—2622no—
95AGL(1,ℤ/8)—2624no—
96AGL(1,ℤ/8)—2622no—
97#135—1314no—
98#135C2×D13112yescomplemented
99C2×D8—1314no—
100C2×D8—1312no—
101#131—1311no—
102Q32C2×D13111yescharacteristic, complemented
103SD32—1314no—
104SD32C2×D13112yescomplemented
105D16—1314no—
106D16—1312no—
107D16—1312no—
108D16C2×D13111yescharacteristic, complemented
109M32—1311no—
110M32—1312no—
111C2×C16—1312no—
112C2×C16C2×D13111yescharacteristic, complemented
113#41—22no—
114#41—2624no—
115#41—5242no—
116#41—1311no—
117C2×D4—2624no—
118C2×D4—2622no—
119C2×D4—5242no—
120C2×D4—5244no—
121Q16C22×D13112yes—
122SD16—22no—
123SD16—2624no—
124D8—2624no—
125D8—2622no—
126D8—1314no—
127D8C22×D13112yes—
128M16—1311no—
129M16—1312no—
130C2×C8—1312no—
131C2×C8C22×D13111yescharacteristic
132C16—1312no—
133C16C22×D13111yescharacteristic
134C16C22×D13111yescharacteristic
135C23—5244no—
136C23—5242no—
137Q8—22no—
138Q8—1311no—
139D4—22no—
140D4—42no—
141D4—2624no—
142D4—5244no—
143D4—5244no—
144D4—5242no—
145D4—1312no—
146D4—1311no—
147C2×C4—42no—
148C2×C4—5244no—
149C2×C4—1312no—
150C2×C4D4×D13111yescharacteristic
151C8—1312no—
152C8D4×D13111yescharacteristic
153C8C23×D13111yescharacteristic
154C22—42no—
155C22—5244no—
156C22—20962no—
157C22—20964no—
158C22—1312no—
159C22D8×D13111yescharacteristic
160C4—42no—
161C4—1312no—
162C4D8×D13111yescharacteristic
163C4C2×D4×D13111yescharacteristic
164C2—21no—
165C2—2622no—
166C2—82no—
167C2—10484no—
168C2C2×D8×D13111yescharacteristic
subgroupG/H
centerC2C2×D8×D131
commutator, cosocleC8×C131C24
FrattiniC8C23×D131
Fitting#508×C131C2
socleC2×C131C22×D8
hypercenter#508D131
hypocenterC131#3415
LCSG ▷ C8×C131 ▷ C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #22554114C2semidirect
2. #22553749C2semidirect
3. #22553747C2semidirect
4. #22553745C2semidirect
5. C2×D2096C2semidirect
6. D16×D131C2semidirect
7. #22553728C2semidirect
8. #508×C131C2semidirect
12. #16862269C22semidirect
13. C2×D1048C22semidirect
14. D8×D131C22semidirect
16. #16862189C22semidirect
17. D2096C22semidirect
18. D2096C22semidirect
19. #16862187C22semidirect
20. #16862186C22semidirect
21. #16862185C22semidirect
22. C16×D131C22semidirect
23. #135×C131C22semidirect
24. Q32×C131C22semidirect
25. SD32×C131C22semidirect
26. D16×C131C22semidirect
27. C2×C16×C131C22semidirect
84. C131#3415semidirect
90. #508D131semidirect
98. #135C2×D131semidirect
102. Q32C2×D131semidirect
104. SD32C2×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
#225541142
#225537492
#225537472
#225537452
C2×D20962
D16×D1312
#225537282
#508×C1312
#3415131