Finite Group Databasebeta

← #13093213

Subgroups — #13093213

Order 1472.

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

normal subgroups only

This group has 262 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#41×D23C211yesmaximal, characteristic, complemented
2C2×#67290C211yesmaximal, characteristic, complemented
3C23×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4#10625122C211yesmaximal, characteristic, complemented
5#10625103C211yesmaximal, characteristic, complemented
6#10625087C212yesmaximal, complemented
7#10625086C212yesmaximal, complemented
8#10625083C212yesmaximal, complemented
9#10625082C212yesmaximal, complemented
10#10625080C211yesmaximal, characteristic, complemented
11M16×D23C211yesmaximal, characteristic, complemented
12#41×C23C2211yescharacteristic
13C2×D4×C23C2211yescharacteristic, complemented
14C2×C22⋊D23—21no—
15C22×Dic23—21no—
16#67292C2211yescharacteristic, complemented
17Q8×D23C2211yescharacteristic, complemented
18#67290—21no—
19#67290—21no—
20#67290—21no—
21#67290C2212yes—
22#67290C2211yescharacteristic, complemented
23#67290C2212yescomplemented
24D4×D23—21no—
25D4×D23C2211yescharacteristic, complemented
26#67288—21no—
27#67288C2211yescharacteristic
28C2×C4×D23—21no—
29C2×C4×D23C2211yescharacteristic
30C2×Dic46C2211yescharacteristic, complemented
31SD16×C23C2212yescomplemented
32D8×C23C2212yescomplemented
33M16×C23C2211yescharacteristic, complemented
34#67275C2212yescomplemented
35#67273C2212yescomplemented
36#67273C2212yescomplemented
37#67272C2212yescomplemented
38#67267C2211yescharacteristic, complemented
39Dic92C2212yescomplemented
40#67263C2212yescomplemented
41#67262C2212yescomplemented
42C8×D23C2212yescomplemented
43C23×C23—21no—
44C22×D23—21no—
45C22×D23D411yescharacteristic
46Q8×C23C2311yescharacteristic
47D4×C23—21no—
48D4×C23—21no—
49D4×C23C2312yes—
50D4×C23C2311yescharacteristic
51C2×C4×C23—21no—
52C2×C4×C23C2311yescharacteristic
53C22⋊D23—21no—
54C22⋊D23—21no—
55C22⋊D23—22no—
56C22⋊D23—22no—
57C22⋊D23—21no—
58C2×Dic23—21no—
59C2×Dic23—21no—
60C2×Dic23—22no—
61C2×Dic23—22no—
62C2×Dic23D411yescharacteristic
63D92—21no—
64D92C2311yescharacteristic
65C4×D23—21no—
66C4×D23—21no—
67C4×D23—21no—
68C4×D23D411yescharacteristic
69C4×D23D411yescharacteristic
70C4×D23C2311yescharacteristic
71C4×D23C2311yescharacteristic
72Dic46—21no—
73Dic46—21no—
74Dic46C2312yes—
75Dic46C2311yescharacteristic
76C8×C23C2312yes—
77C23⋊C8C2312yes—
78C22×C23—22no—
79C22×C23—21no—
80C22×C23—42no—
81C22×C23C2×D411yescharacteristic
82C2×D23—21no—
83C2×D23—41no—
84C2×D23—42no—
85C2×D23C2×D411yescharacteristic
86C2×D23C2×D411yescharacteristic
87C4×C23—21no—
88C4×C23C2×D411yescharacteristic
89C4×C23C2411yescharacteristic
90Dic23—21no—
91Dic23—22no—
92Dic23C2×D411yescharacteristic
93Dic23C2×D411yescharacteristic
94#575—231nomaximal, Sylow
95C2×C23—21no—
96C2×C23—41no—
97C2×C23—42no—
98C2×C23C22×D411yescharacteristic
99D23—21no—
100D23—21no—
101D23—41no—
102C2×#41—231no—
103C2×#41—231no—
104#137—231no—
105#137—232no—
106#137—231no—
107AGL(1,ℤ/8)—231no—
108AGL(1,ℤ/8)—232no—
109AGL(1,ℤ/8)D2311yescharacteristic, complemented
110#135—232no—
111#135—232no—
112C2×M16—231no—
113C23#57511yescharacteristic, complemented, Sylow
114#41—231no—
115#41—232no—
116#41—231no—
117#41—232no—
118#41—231no—
119#41—461no—
120#41—461no—
121#41—461no—
122#41—461no—
123#41C2×D2311yescharacteristic
124C2×Q8—231no—
125C2×Q8—231no—
126C2×D4—231no—
127C2×D4—461no—
128C2×D4—461no—
129C2×D4C2×D2311yescharacteristic, complemented
130C22×C4—231no—
131C22×C4—461no—
132C22×C4—461no—
133Q16—232no—
134Q16—232no—
135SD16—232no—
136SD16—232no—
137SD16—232no—
138SD16C2×D2312yescomplemented
139D8—232no—
140D8C2×D2312yescomplemented
141M16—231no—
142M16—232no—
143M16C2×D2311yescharacteristic, complemented
144C2×C8—232no—
145C23—21no—
146C23—231no—
147C23—461no—
148Q8—231no—
149Q8—232no—
150Q8—461no—
151Q8—461no—
152Q8C22×D2311yescharacteristic
153D4—21no—
154D4—21no—
155D4—231no—
156D4—461no—
157D4—461no—
158D4—461no—
159D4—462no—
160D4—462no—
161D4—461no—
162D4C22×D2312yes—
163D4C22×D2311yescharacteristic
164C2×C4—21no—
165C2×C4—231no—
166C2×C4—231no—
167C2×C4—231no—
168C2×C4—231no—
169C2×C4—231no—
170C2×C4—461no—
171C2×C4—461no—
172C2×C4—461no—
173C2×C4—462no—
174C2×C4—462no—
175C2×C4—461no—
176C2×C4—461no—
177C2×C4C22×D2311yescharacteristic
178C8—232no—
179C8C22×D2312yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—231no—
184C22—231no—
185C22—461no—
186C22—921no—
187C22—922no—
188C22D4×D2311yescharacteristic
189C4—21no—
190C4—231no—
191C4—231no—
192C4—461no—
193C4—462no—
194C4D4×D2311yescharacteristic
195C4C23×D2311yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—461no—
200C2—461no—
201C2—921no—
202C2C2×D4×D2311yescharacteristic
subgroupG/H
centerC2C2×D4×D23
commutator, cosocleC4×C23C24
FrattiniC4C23×D23
FittingC23×AGL(1,ℤ/8)C2
socleC2×C23C22×D4
hypercenterAGL(1,ℤ/8)D23
hypocenterC23#575
LCSG ▷ C4×C23 ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #41×D23C2semidirect
2. C2×#67290C2semidirect
3. C23×AGL(1,ℤ/8)C2semidirect
4. #10625122C2semidirect
5. #10625103C2semidirect
6. #10625087C2semidirect
7. #10625086C2semidirect
8. #10625083C2semidirect
9. #10625082C2semidirect
10. #10625080C2semidirect
11. M16×D23C2semidirect
13. C2×D4×C23C22semidirect
16. #67292C22semidirect
17. Q8×D23C22semidirect
22. #67290C22semidirect
23. #67290C22semidirect
25. D4×D23C22semidirect
30. C2×Dic46C22semidirect
31. SD16×C23C22semidirect
32. D8×C23C22semidirect
33. M16×C23C22semidirect
34. #67275C22semidirect
35. #67273C22semidirect
36. #67273C22semidirect
37. #67272C22semidirect
38. #67267C22semidirect
39. Dic92C22semidirect
40. #67263C22semidirect
41. #67262C22semidirect
42. C8×D23C22semidirect
109. AGL(1,ℤ/8)D23semidirect
113. C23#575semidirect
129. C2×D4C2×D23semidirect
138. SD16C2×D23semidirect
140. D8C2×D23semidirect
143. M16C2×D23semidirect

Maximal subgroups

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

subgroupindex
#41×D232
C2×#672902
C23×AGL(1,ℤ/8)2
#106251222
#106251032
#106250872
#106250862
#106250832
#106250822
#106250802
M16×D232
#57523