Finite Group Databasebeta

← #46849501

Subgroups — #46849501

Order 34688.

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#31774698C212yesmaximal, complemented
2#31774333C212yesmaximal, complemented
3#31774331C214yesmaximal, complemented
4#31774329C211yesmaximal, characteristic, complemented
5#31774326C212yesmaximal, complemented
6D16×D271C212yesmaximal, complemented
7#31774313C211yesmaximal, characteristic, complemented
8C2×D16×C271C211yesmaximal, characteristic, complemented
9#22576205—22no—
10#22576186—22no—
11#22576184—24no—
12#22576182C2212yescomplemented
13#22576179C2214yescomplemented
14D8×D271C2214yescomplemented
15#22576166C2211yescharacteristic
16Dic2168C2211yescharacteristic, complemented
17#22576105C2212yescomplemented
18#22576104C2214yescomplemented
19D4336C2211yescharacteristic, complemented
20#22576101C2214yescomplemented
21#22576100C2211yescharacteristic, complemented
22#22576099C2212yescomplemented
23C16×D271C2212yescomplemented
24C2×D8×C271C2212yescomplemented
25D16×C271C2212yescomplemented
26D16×C271C2212yescomplemented
27C2×C16×C271C2211yescharacteristic, complemented
28#16864654—22no—
29#16864654D411yescharacteristic
30#16864653—24no—
31#16864653—42no—
32C2×C22⋊D271—44no—
33D4×D271—24no—
34D4×D271—42no—
35Dic1084C2312yes—
36#16864643—22no—
37#16864642—24no—
38D2168C2312yes—
39#16864639—24no—
40#16864638D411yescharacteristic
41#16864637D412yes—
42C8×D271C2312yes—
43C271⋊C16C2312yes—
44C2×D4×C271—22no—
45D8×C271—22no—
46D8×C271C2314yes—
47C2×C8×C271C2311yescharacteristic
48C16×C271C2311yescharacteristic
49C16×C271C2311yescharacteristic
50C22×D271—44no—
51Dic542—22no—
52Dic542D811yescharacteristic
53D1084—22no—
54D1084D811yescharacteristic
55C22⋊D271—42no—
56C22⋊D271—44no—
57C22⋊D271—44no—
58C22⋊D271D812yes—
59C2×Dic271—44no—
60C4×D271—42no—
61C4×D271C2×D412yes—
62C271⋊C8C2×D412yes—
63C23×C271—42no—
64D4×C271—24no—
65D4×C271—42no—
66C2×C4×C271C2×D411yescharacteristic
67C8×C271C2×D411yescharacteristic
68C8×C271C2411yescharacteristic
69C22×C271—44no—
70C22×C271—162no—
71C22×C271C2×D811yescharacteristic
72C2×D271—42no—
73C2×D271—164no—
74C2×D271C2×D812yes—
75C4×C271C2×D811yescharacteristic
76C4×C271C22×D411yescharacteristic
77Dic271—42no—
78Dic271C2×D812yes—
79C2×C271—21no—
80C2×C271—84no—
81C2×C271C22×D811yescharacteristic
82D271—22no—
83D271—82no—
84C271#341511yescharacteristic, complemented, Sylow
85#3415—2711nomaximal, Sylow
86#576—2712no—
87#509—2712no—
88#509—2714no—
89#508—2712no—
90#508—2711no—
91C2×D16—2712no—
92C2×D16D27111yescharacteristic, complemented
93#504—2711no—
94H(5,𝔽2)—5422no—
95AGL(1,ℤ/8)—5424no—
96AGL(1,ℤ/8)—5422no—
97#135—2714no—
98#135—2712no—
99C2×D8—2714no—
100C2×D8C2×D27112yescomplemented
101#131—2711no—
102Q32—2711no—
103SD32—2714no—
104SD32—2712no—
105D16—2714no—
106D16—2711no—
107D16C2×D27112yescomplemented
108D16C2×D27112yescomplemented
109M32—2711no—
110M32—2712no—
111C2×C16—2712no—
112C2×C16C2×D27111yescharacteristic, complemented
113#41—2711no—
114#41—5424no—
115#41—5422no—
116#41—10842no—
117C2×D4—22no—
118C2×D4—5424no—
119C2×D4—10842no—
120C2×D4—10844no—
121Q16—2712no—
122SD16—5424no—
123SD16—5422no—
124D8—22no—
125D8—2712no—
126D8—5424no—
127D8C22×D27114yes—
128M16—2711no—
129M16—2712no—
130C2×C8—2712no—
131C2×C8C22×D27111yescharacteristic
132C16—2712no—
133C16C22×D27111yescharacteristic
134C16C22×D27111yescharacteristic
135C23—42no—
136C23—10844no—
137Q8—2711no—
138Q8—5422no—
139D4—24no—
140D4—42no—
141D4—2712no—
142D4—2711no—
143D4—5422no—
144D4—10844no—
145D4—10844no—
146D4—10842no—
147C2×C4—2712no—
148C2×C4—10844no—
149C2×C4—10842no—
150C2×C4D4×D27111yescharacteristic
151C8—2712no—
152C8D4×D27111yescharacteristic
153C8C23×D27111yescharacteristic
154C22—44no—
155C22—2712no—
156C22—162no—
157C22—10842no—
158C22—43364no—
159C22D8×D27111yescharacteristic
160C4—2712no—
161C4—10842no—
162C4D8×D27111yescharacteristic
163C4C2×D4×D27111yescharacteristic
164C2—21no—
165C2—84no—
166C2—5422no—
167C2—21682no—
168C2C2×D8×D27111yescharacteristic
subgroupG/H
centerC2C2×D8×D271
commutator, cosocleC8×C271C24
FrattiniC8C23×D271
FittingC2×D16×C271C2
socleC2×C271C22×D8
hypercenterC2×D16D271
hypocenterC271#3415
LCSG ▷ C8×C271 ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. #31774698C2semidirect
2. #31774333C2semidirect
3. #31774331C2semidirect
4. #31774329C2semidirect
5. #31774326C2semidirect
6. D16×D271C2semidirect
7. #31774313C2semidirect
8. C2×D16×C271C2semidirect
12. #22576182C22semidirect
13. #22576179C22semidirect
14. D8×D271C22semidirect
16. Dic2168C22semidirect
17. #22576105C22semidirect
18. #22576104C22semidirect
19. D4336C22semidirect
20. #22576101C22semidirect
21. #22576100C22semidirect
22. #22576099C22semidirect
23. C16×D271C22semidirect
24. C2×D8×C271C22semidirect
25. D16×C271C22semidirect
26. D16×C271C22semidirect
27. C2×C16×C271C22semidirect
84. C271#3415semidirect
92. C2×D16D271semidirect
100. C2×D8C2×D271semidirect
107. D16C2×D271semidirect
108. D16C2×D271semidirect
112. C2×C16C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317746982
#317743332
#317743312
#317743292
#317743262
D16×D2712
#317743132
C2×D16×C2712
#3415271