Finite Group Databasebeta

← #135×D211 #27213924

Subgroups — #135×D211 #27213924

Order 13504.

266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 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×D211C212yesmaximal, complemented
2#20270000C211yesmaximal, characteristic, complemented
3#20269999C211yesmaximal, characteristic, complemented
4#20269998C212yesmaximal, complemented
5#20269997C212yesmaximal, complemented
6#20269996C211yesmaximal, characteristic, complemented
7Q16×D211C211yesmaximal, characteristic, complemented
8SD16×D211C212yesmaximal, complemented
9D8×D211C211yesmaximal, characteristic, complemented
10C2×C8×D211C211yesmaximal, characteristic, complemented
11#135×C211C211yesmaximal, characteristic, complemented
12#16809963C2212yescomplemented
13#16809962—22no—
14#16809962C2212yescomplemented
15#16809961—22no—
16#16809961C2212yescomplemented
17Q8×D211C2212yes—
18D4×D211—22no—
19D4×D211C2212yes—
20C2×C4×D211—22no—
21C2×C4×D211C2211yescharacteristic
22Dic844C2211yescharacteristic, complemented
23#16809952C2212yescomplemented
24#16809951C2212yescomplemented
25#16809950C2212yescomplemented
26#16809949C2212yescomplemented
27D1688C2211yescharacteristic, complemented
28#16809947C2212yescomplemented
29C2×C211⋊C8C2211yescharacteristic, complemented
30C8×D211C2211yescharacteristic
31C8×D211C2211yescharacteristic
32C8×D211C2211yescharacteristic, complemented
33C8×D211C2211yescharacteristic, complemented
34#41×C211C2212yescomplemented
35Q16×C211C2211yescharacteristic, complemented
36SD16×C211C2212yescomplemented
37D8×C211C2211yescharacteristic, complemented
38C2×C8×C211C2211yescharacteristic, complemented
39C22×D211—22no—
40C22×D211D411yescharacteristic
41Dic422—22no—
42Dic422C2312yes—
43D844—22no—
44D844C2312yes—
45C22⋊D211—22no—
46C22⋊D211—22no—
47C22⋊D211—22no—
48C2×Dic211—22no—
49C2×Dic211D411yescharacteristic
50C4×D211—22no—
51C4×D211—22no—
52C4×D211—22no—
53C4×D211D411yescharacteristic
54C4×D211D411yescharacteristic
55C4×D211C2311yescharacteristic
56C4×D211C2311yescharacteristic
57C211⋊C8C2311yescharacteristic
58C211⋊C8C2311yescharacteristic
59Q8×C211C2312yes—
60D4×C211—22no—
61D4×C211C2312yes—
62C2×C4×C211—22no—
63C2×C4×C211C2311yescharacteristic
64C8×C211C2311yescharacteristic
65C8×C211C2311yescharacteristic
66C22×C211—22no—
67C22×C211C2×D411yescharacteristic
68C2×D211—22no—
69C2×D211—22no—
70C2×D211—44no—
71C2×D211C2×D411yescharacteristic
72C2×D211C2×D411yescharacteristic
73C4×C211—22no—
74C4×C211C2×D411yescharacteristic
75C4×C211C2411yescharacteristic
76Dic211—22no—
77Dic211C2×D411yescharacteristic
78Dic211C2×D411yescharacteristic
79C2×C211—21no—
80C2×C211—42no—
81C2×C211C22×D411yescharacteristic
82D211—21no—
83D211—42no—
84D211#13512yesdirect product factor
85C211C2×#13511yescharacteristic, complemented, Sylow
86C2×#135—2111nomaximal, Sylow
87C2×#41—2112no—
88#135—2112no—
89#135—2111no—
90#135—2112no—
91#135—2111no—
92#135—2111no—
93#135D21111yescharacteristic, direct product factor
94C2×Q16—2111no—
95C2×SD16—2112no—
96C2×D8—2111no—
97C22×C8—2111no—
98#41—4222no—
99#41—4222no—
100#41—2112no—
101#41—2112no—
102#41—2112no—
103#41C2×D21112yescomplemented
104C2×Q8—2112no—
105C2×D4—4222no—
106C2×D4—2112no—
107C22×C4—4222no—
108C22×C4—2111no—
109Q16—2112no—
110Q16—2111no—
111Q16C2×D21111yescharacteristic, complemented
112SD16—2112no—
113SD16—2112no—
114SD16—2112no—
115SD16C2×D21112yescomplemented
116D8—2112no—
117D8—2111no—
118D8C2×D21111yescharacteristic, complemented
119C2×C8—2111no—
120C2×C8—2111no—
121C2×C8—2111no—
122C2×C8—2111no—
123C2×C8—2111no—
124C2×C8C2×D21111yescharacteristic, complemented
125C23—4222no—
126C23—2111no—
127Q8—4222no—
128Q8—2112no—
129Q8C22×D21112yes—
130D4—22no—
131D4—4222no—
132D4—4222no—
133D4—4222no—
134D4—4222no—
135D4—2112no—
136D4C22×D21112yes—
137C2×C4—22no—
138C2×C4—4222no—
139C2×C4—4222no—
140C2×C4—4222no—
141C2×C4—4222no—
142C2×C4—2111no—
143C2×C4—2111no—
144C2×C4—2111no—
145C2×C4—2111no—
146C2×C4—2111no—
147C2×C4C22×D21111yescharacteristic
148C8—2111no—
149C8—2111no—
150C8C22×D21111yescharacteristic
151C8C22×D21111yescharacteristic
152C22—22no—
153C22—8444no—
154C22—4222no—
155C22—4222no—
156C22—2111no—
157C22—2111no—
158C22D4×D21111yescharacteristic
159C4—22no—
160C4—4222no—
161C4—2111no—
162C4—2111no—
163C4D4×D21111yescharacteristic
164C4C23×D21111yescharacteristic
165C2—21no—
166C2—42no—
167C2—8442no—
168C2—4221no—
169C2—2112no—
170C2C2×D4×D21111yescharacteristic
subgroupG/H
centerC4D4×D211
commutator, cosocleC4×C211C24
FrattiniC4C23×D211
Fitting#135×C211C2
socleC2×C211C22×D4
hypercenter#135D211
hypocenterC211C2×#135
LCSG ▷ C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×D211C2semidirect
2. #20270000C2semidirect
3. #20269999C2semidirect
4. #20269998C2semidirect
5. #20269997C2semidirect
6. #20269996C2semidirect
7. Q16×D211C2semidirect
8. SD16×D211C2semidirect
9. D8×D211C2semidirect
10. C2×C8×D211C2semidirect
11. #135×C211C2semidirect
12. #16809963C22semidirect
14. #16809962C22semidirect
16. #16809961C22semidirect
22. Dic844C22semidirect
23. #16809952C22semidirect
24. #16809951C22semidirect
25. #16809950C22semidirect
26. #16809949C22semidirect
27. D1688C22semidirect
28. #16809947C22semidirect
29. C2×C211⋊C8C22semidirect
32. C8×D211C22semidirect
33. C8×D211C22semidirect
34. #41×C211C22semidirect
35. Q16×C211C22semidirect
36. SD16×C211C22semidirect
37. D8×C211C22semidirect
38. C2×C8×C211C22semidirect
84. D211#135direct (N × H)
85. C211C2×#135semidirect
93. #135D211direct (N × H)
103. #41C2×D211semidirect
111. Q16C2×D211semidirect
115. SD16C2×D211semidirect
118. D8C2×D211semidirect
124. C2×C8C2×D211semidirect

Maximal subgroups

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

subgroupindex
#41×D2112
#202700002
#202699992
#202699982
#202699972
#202699962
Q16×D2112
SD16×D2112
D8×D2112
C2×C8×D2112
#135×C2112
C2×#135211