Finite Group Databasebeta

← #27213955

Subgroups — #27213955

Order 13504.

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

normal subgroups only

This group has 258 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#20270025C211yesmaximal, characteristic, complemented
2#20270023C211yesmaximal, characteristic, complemented
3#20270009C212yesmaximal, complemented
4#20270005C211yesmaximal, characteristic, complemented
5#20270004C212yesmaximal, complemented
6#20270000C212yesmaximal, complemented
7#20269997C211yesmaximal, characteristic, complemented
8C2×#16809949C211yesmaximal, characteristic, complemented
9SD16×D211C212yesmaximal, complemented
10#20269985C211yesmaximal, characteristic, complemented
11#137×C211C211yesmaximal, characteristic, complemented
12#16809963—21no—
13#16809963C2212yes—
14#16809963C2212yescomplemented
15#16809962—21no—
16#16809962—22no—
17#16809962C2211yescharacteristic
18#16809962C2211yescharacteristic, complemented
19Q8×D211—21no—
20Q8×D211C2212yescomplemented
21C2×D844—21no—
22C2×D844C2211yescharacteristic
23D4×D211—22no—
24D4×D211C2212yes—
25#16809952C2211yescharacteristic, complemented
26#16809951C2212yescomplemented
27#16809950C2211yescharacteristic, complemented
28#16809949C2212yescomplemented
29#16809949C2212yescomplemented
30#16809949C2211yescharacteristic, complemented
31D1688C2212yescomplemented
32#16809947C2211yescharacteristic, complemented
33#16809945C2212yescomplemented
34C2×C211⋊C8C2211yescharacteristic, complemented
35C8×D211C2212yescomplemented
36#41×C211C2211yescharacteristic
37C2×Q8×C211C2211yescharacteristic, complemented
38Q16×C211C2212yescomplemented
39SD16×C211C2212yescomplemented
40M16×C211C2211yescharacteristic, complemented
41C22×D211—21no—
42C22×D211—22no—
43Dic422—22no—
44Dic422D411yescharacteristic
45Dic422C2311yescharacteristic
46D844—22no—
47D844—22no—
48D844—21no—
49D844D411yescharacteristic
50D844C2311yescharacteristic
51D844C2312yes—
52C22⋊D211—21no—
53C22⋊D211—22no—
54C22⋊D211D412yes—
55C4×D211—22no—
56C4×D211—21no—
57C4×D211—22no—
58C4×D211—22no—
59C4×D211C2312yes—
60C211⋊C8C2311yescharacteristic
61C211⋊C8C2311yescharacteristic
62Q8×C211—21no—
63Q8×C211C2312yes—
64Q8×C211C2311yescharacteristic
65D4×C211—21no—
66D4×C211C2311yescharacteristic
67C2×C4×C211—21no—
68C2×C4×C211—21no—
69C2×C4×C211C2311yescharacteristic
70C8×C211C2312yes—
71C22×C211—21no—
72C22×C211C2×D411yescharacteristic
73C2×D211—21no—
74C2×D211—22no—
75C2×D211—82no—
76C2×D211—81no—
77C2×D211C2×D412yes—
78C4×C211—21no—
79C4×C211—22no—
80C4×C211C2×D411yescharacteristic
81C4×C211C2411yescharacteristic
82Dic211—21no—
83Dic211C2×D412yes—
84C2×C211—21no—
85C2×C211—41no—
86C2×C211C22×D411yescharacteristic
87D211—22no—
88D211—41no—
89D211—42no—
90C211#57711yescharacteristic, complemented, Sylow
91#577—2111nomaximal, Sylow
922−1+4—2111no—
93H(5,𝔽2)—2111no—
94#137—2112no—
95#137D21111yescharacteristic, complemented
96AGL(1,ℤ/8)—2112no—
97AGL(1,ℤ/8)—2111no—
98#135—2111no—
99#135—2112no—
100C2×SD16—2111no—
101C2×SD16—2112no—
102#131—2111no—
103#41—4221no—
104#41—4222no—
105#41—4221no—
106#41—2112no—
107#41—2112no—
108#41—2111no—
109#41—2111no—
110#41C2×D21111yescharacteristic
111C2×Q8—4221no—
112C2×Q8—2112no—
113C2×Q8C2×D21111yescharacteristic, complemented
114C2×D4—4221no—
115C2×D4—4222no—
116C2×D4—2112no—
117C2×D4—2111no—
118Q16—2111no—
119Q16C2×D21112yescomplemented
120SD16—2112no—
121SD16—2112no—
122SD16—2111no—
123SD16—2111no—
124SD16—2112no—
125SD16C2×D21112yescomplemented
126D8—2111no—
127D8—2112no—
128M16—2112no—
129M16C2×D21111yescharacteristic, complemented
130C2×C8—2111no—
131C2×C8—2112no—
132C23—4222no—
133C23—4221no—
134Q8—21no—
135Q8—4222no—
136Q8—2111no—
137Q8—2111no—
138Q8C22×D21112yes—
139Q8C22×D21111yescharacteristic
140D4—21no—
141D4—4222no—
142D4—4222no—
143D4—4221no—
144D4—4222no—
145D4—4221no—
146D4—2112no—
147D4—2111no—
148D4—2111no—
149D4—2112no—
150D4C22×D21111yescharacteristic
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—4222no—
154C2×C4—4222no—
155C2×C4—4222no—
156C2×C4—4221no—
157C2×C4—2112no—
158C2×C4C22×D21111yescharacteristic
159C8—2111no—
160C8—2111no—
161C8C22×D21112yes—
162C22—21no—
163C22—16881no—
164C22—16882no—
165C22—4221no—
166C22—4222no—
167C22—2112no—
168C22D4×D21111yescharacteristic
169C4—21no—
170C4—22no—
171C4—4221no—
172C4—2112no—
173C4D4×D21111yescharacteristic
174C4C23×D21111yescharacteristic
175C2—21no—
176C2—41no—
177C2—8442no—
178C2—8441no—
179C2—4222no—
180C2C2×D4×D21111yescharacteristic
subgroupG/H
centerC2C2×D4×D211
commutator, cosocleC4×C211C24
FrattiniC4C23×D211
Fitting#137×C211C2
socleC2×C211C22×D4
hypercenter#137D211
hypocenterC211#577
LCSG ▷ C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #20270025C2semidirect
2. #20270023C2semidirect
3. #20270009C2semidirect
4. #20270005C2semidirect
5. #20270004C2semidirect
6. #20270000C2semidirect
7. #20269997C2semidirect
8. C2×#16809949C2semidirect
9. SD16×D211C2semidirect
10. #20269985C2semidirect
11. #137×C211C2semidirect
14. #16809963C22semidirect
18. #16809962C22semidirect
20. Q8×D211C22semidirect
25. #16809952C22semidirect
26. #16809951C22semidirect
27. #16809950C22semidirect
28. #16809949C22semidirect
29. #16809949C22semidirect
30. #16809949C22semidirect
31. D1688C22semidirect
32. #16809947C22semidirect
33. #16809945C22semidirect
34. C2×C211⋊C8C22semidirect
35. C8×D211C22semidirect
37. C2×Q8×C211C22semidirect
38. Q16×C211C22semidirect
39. SD16×C211C22semidirect
40. M16×C211C22semidirect
90. C211#577semidirect
95. #137D211semidirect
113. C2×Q8C2×D211semidirect
119. Q16C2×D211semidirect
125. SD16C2×D211semidirect
129. M16C2×D211semidirect

Maximal subgroups

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

subgroupindex
#202700252
#202700232
#202700092
#202700052
#202700042
#202700002
#202699972
C2×#168099492
SD16×D2112
#202699852
#137×C2112
#577211