Finite Group Databasebeta

← #3291×C227 #44397247

Subgroups — #3291×C227 #44397247

Order 29056.

460 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 184 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#577×C227C214yesmaximal, complemented
2#492×C227C211yesmaximal, characteristic, complemented
3#471×C227C214yesmaximal, complemented
4#456×C227C212yesmaximal, complemented
5#453×C227C212yesmaximal, complemented
6#443×C227C212yesmaximal, complemented
72−1+4×C227C2212yes—
8H(5,𝔽2)×C227C2212yescomplemented
9#137×C227—24no—
10#137×C227C2214yescomplemented
11AGL(1,ℤ/8)×C227—24no—
12AGL(1,ℤ/8)×C227C2214yes—
13#135×C227—24no—
14#135×C227C2212yes—
15C2×SD16×C227—24no—
16C2×SD16×C227C2214yescomplemented
17#131×C227C2214yescomplemented
18C4⋊Q8×C227C2211yescharacteristic, complemented
19#127×C227C2211yescharacteristic
20#108×C227C2212yescomplemented
21#104×C227C2214yescomplemented
22#101×C227C2212yescomplemented
23#100×C227C2212yescomplemented
24C4×C8×C227C2211yescharacteristic, complemented
25#41×C227—24no—
26#41×C227—24no—
27#41×C227—42no—
28#41×C227C2314yes—
29C2×Q8×C227—24no—
30C2×Q8×C227C2312yes—
31C2×D4×C227—24no—
32C2×D4×C227—41no—
33C2×D4×C227—42no—
34C2×D4×C227C2312yes—
35Q16×C227—24no—
36Q16×C227D412yescomplemented
37SD16×C227—24no—
38SD16×C227—24no—
39SD16×C227—24no—
40SD16×C227—24no—
41SD16×C227D414yescomplemented
42D8×C227—24no—
43D8×C227D412yescomplemented
44M16×C227—24no—
45M16×C227C2314yes—
46C2×C8×C227—24no—
47C2×C8×C227C2312yes—
48C4⋊C4×C227—41no—
49C42×C227C2311yescharacteristic
50C23×C227—22no—
51C23×C227—42no—
52Q8×C227—24no—
53Q8×C227—42no—
54Q8×C227C2×D414yes—
55D4×C227—24no—
56D4×C227—24no—
57D4×C227—44no—
58D4×C227—42no—
59D4×C227—82no—
60D4×C227C2×D414yes—
61C2×C4×C227—21no—
62C2×C4×C227—24no—
63C2×C4×C227—22no—
64C2×C4×C227—42no—
65C2×C4×C227—44no—
66C2×C4×C227C2411yescharacteristic
67C8×C227—24no—
68C8×C227C2×D412yes—
69C8×C227C2×D412yes—
70C22×C227—24no—
71C22×C227—42no—
72C22×C227—82no—
73C22×C227—162no—
74C22×C227H(5,𝔽2)11yescharacteristic
75C4×C227—22no—
76C4×C227—24no—
77C4×C227—42no—
78C4×C227C22×D412yes—
79C2×C227—21no—
80C2×C227—44no—
81C2×C227—82no—
82C2×C227D4211yescharacteristic
83C227#329111yescharacteristic, direct product factor, Sylow
84#3291C22711yesmaximal, characteristic, direct product factor, Sylow
85#577C2×C22714yescomplemented
86#492C2×C22711yescharacteristic, complemented
87#471C2×C22714yescomplemented
88#456C2×C22712yescomplemented
89#453C2×C22712yescomplemented
90#443C2×C22712yescomplemented
912−1+4C22×C22712yes—
92H(5,𝔽2)C22×C22712yescomplemented
93#137—24no—
94#137C22×C22714yescomplemented
95AGL(1,ℤ/8)—24no—
96AGL(1,ℤ/8)C22×C22714yes—
97#135—24no—
98#135C22×C22712yes—
99C2×SD16—24no—
100C2×SD16C22×C22714yescomplemented
101#131C22×C22714yescomplemented
102C4⋊Q8C22×C22711yescharacteristic, complemented
103#127C22×C22711yescharacteristic
104#108C22×C22712yescomplemented
105#104C22×C22714yescomplemented
106#101C22×C22712yescomplemented
107#100C22×C22712yescomplemented
108C4×C8C22×C22711yescharacteristic, complemented
109#41—24no—
110#41—24no—
111#41—42no—
112#41C23×C22714yes—
113C2×Q8—24no—
114C2×Q8C23×C22712yes—
115C2×D4—24no—
116C2×D4—41no—
117C2×D4—42no—
118C2×D4C23×C22712yes—
119Q16—24no—
120Q16D4×C22712yescomplemented
121SD16—24no—
122SD16—24no—
123SD16—24no—
124SD16—24no—
125SD16D4×C22714yescomplemented
126D8—24no—
127D8D4×C22712yescomplemented
128M16—24no—
129M16C23×C22714yes—
130C2×C8—24no—
131C2×C8C23×C22712yes—
132C4⋊C4—41no—
133C42C23×C22711yescharacteristic
134C23—22no—
135C23—42no—
136Q8—24no—
137Q8—42no—
138Q8C2×D4×C22714yes—
139D4—24no—
140D4—24no—
141D4—44no—
142D4—42no—
143D4—82no—
144D4C2×D4×C22714yes—
145C2×C4—21no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—42no—
149C2×C4—44no—
150C2×C4C24×C22711yescharacteristic
151C8—24no—
152C8C2×D4×C22712yes—
153C8C2×D4×C22712yes—
154C22—24no—
155C22—42no—
156C22—82no—
157C22—162no—
158C22H(5,𝔽2)×C22711yescharacteristic
159C4—22no—
160C4—24no—
161C4—42no—
162C4C22×D4×C22712yes—
163C2—21no—
164C2—44no—
165C2—82no—
166C2D42×C22711yescharacteristic
subgroupG/H
center, socleC2×C227D42
commutator, Frattini, cosocleC2×C4C24×C227
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C227 ◁ C2×C4×C227 ◁ G

Complements

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

NH ≅ G/Ntype
1. #577×C227C2semidirect
2. #492×C227C2semidirect
3. #471×C227C2semidirect
4. #456×C227C2semidirect
5. #453×C227C2semidirect
6. #443×C227C2semidirect
8. H(5,𝔽2)×C227C22semidirect
10. #137×C227C22semidirect
16. C2×SD16×C227C22semidirect
17. #131×C227C22semidirect
18. C4⋊Q8×C227C22semidirect
20. #108×C227C22semidirect
21. #104×C227C22semidirect
22. #101×C227C22semidirect
23. #100×C227C22semidirect
24. C4×C8×C227C22semidirect
36. Q16×C227D4semidirect
41. SD16×C227D4semidirect
43. D8×C227D4semidirect
83. C227#3291direct (N × H)
84. #3291C227direct (N × H)
85. #577C2×C227semidirect
86. #492C2×C227semidirect
87. #471C2×C227semidirect
88. #456C2×C227semidirect
89. #453C2×C227semidirect
90. #443C2×C227semidirect
92. H(5,𝔽2)C22×C227semidirect
94. #137C22×C227semidirect
100. C2×SD16C22×C227semidirect
101. #131C22×C227semidirect
102. C4⋊Q8C22×C227semidirect
104. #108C22×C227semidirect
105. #104C22×C227semidirect
106. #101C22×C227semidirect
107. #100C22×C227semidirect
108. C4×C8C22×C227semidirect
120. Q16D4×C227semidirect
125. SD16D4×C227semidirect
127. D8D4×C227semidirect

Maximal subgroups

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

subgroupindex
#577×C2272
#492×C2272
#471×C2272
#456×C2272
#453×C2272
#443×C2272
#3291227