Finite Group Databasebeta

← #13093221

Subgroups — #13093221

Order 1472.

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

normal subgroups only

This group has 268 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#10625160C211yesmaximal, characteristic, complemented
2#10625155C211yesmaximal, characteristic, complemented
3C23×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4#10625121C211yesmaximal, characteristic, complemented
5C2×#67272C211yesmaximal, characteristic, complemented
6#10625087C212yesmaximal, complemented
7#10625085C212yesmaximal, complemented
8#10625082C212yesmaximal, complemented
9D8×D23C212yesmaximal, complemented
10#10625079C211yesmaximal, characteristic, complemented
11#10625078C211yesmaximal, characteristic, complemented
12#41×C23C2211yescharacteristic, complemented
13C2×D4×C23C2211yescharacteristic, complemented
14C2×C22⋊D23—22no—
15#67292C2212yescomplemented
16#67290—21no—
17#67290C2212yescomplemented
18D4×D23—22no—
19D4×D23—21no—
20D4×D23C2212yescomplemented
21D4×D23C2212yescomplemented
22#67288—21no—
23#67288C2211yescharacteristic
24#67288C2211yescharacteristic, complemented
25C2×D92—21no—
26C2×D92C2211yescharacteristic, complemented
27SD16×C23C2212yescomplemented
28D8×C23C2212yescomplemented
29M16×C23C2211yescharacteristic, complemented
30#67275C2211yescharacteristic, complemented
31#67274C2211yescharacteristic, complemented
32#67273C2211yescharacteristic, complemented
33#67272C2211yescharacteristic, complemented
34#67272C2212yescomplemented
35#67272C2212yescomplemented
36C2×C23⋊C8C2211yescharacteristic, complemented
37D184C2212yescomplemented
38#67263C2212yescomplemented
39#67262C2212yescomplemented
40C8×D23C2212yescomplemented
41C23×C23—21no—
42C22×D23—21no—
43C22×D23—22no—
44C22×D23—22no—
45Q8×C23C2311yescharacteristic
46D4×C23—21no—
47D4×C23—21no—
48D4×C23C2312yes—
49D4×C23C2311yescharacteristic
50C2×C4×C23—21no—
51C2×C4×C23C2311yescharacteristic
52C22⋊D23—21no—
53C22⋊D23—22no—
54C22⋊D23—22no—
55C22⋊D23—22no—
56C22⋊D23D412yes—
57C2×Dic23—22no—
58D92—22no—
59D92—21no—
60D92D411yescharacteristic
61D92C2312yes—
62D92C2311yescharacteristic
63C4×D23—21no—
64C4×D23—22no—
65C4×D23C2312yes—
66Dic46D411yescharacteristic
67Dic46C2311yescharacteristic
68C8×C23C2312yes—
69C23⋊C8C2311yescharacteristic
70C23⋊C8C2311yescharacteristic
71C22×C23—22no—
72C22×C23—21no—
73C22×C23—81no—
74C22×C23C2×D411yescharacteristic
75C2×D23—22no—
76C2×D23—21no—
77C2×D23—82no—
78C2×D23—82no—
79C2×D23—81no—
80C2×D23C2×D412yes—
81C4×C23—21no—
82C4×C23C2×D411yescharacteristic
83C4×C23C2411yescharacteristic
84Dic23—21no—
85Dic23C2×D412yes—
86#576—231nomaximal, Sylow
87C2×C23—21no—
88C2×C23—41no—
89C2×C23—42no—
90C2×C23C22×D411yescharacteristic
91D23—22no—
92D23—42no—
93D23—41no—
94H(5,𝔽2)—231no—
95H(5,𝔽2)—231no—
96AGL(1,ℤ/8)—232no—
97AGL(1,ℤ/8)—232no—
98AGL(1,ℤ/8)—231no—
99AGL(1,ℤ/8)D2311yescharacteristic, complemented
100#135—231no—
101#135—232no—
102C2×D8—231no—
103C2×D8—232no—
104#131—231no—
105C23#57611yescharacteristic, complemented, Sylow
106#41—231no—
107#41—232no—
108#41—232no—
109#41—231no—
110#41—461no—
111#41—461no—
112#41C2×D2311yescharacteristic, complemented
113C2×D4—232no—
114C2×D4—232no—
115C2×D4—231no—
116C2×D4—461no—
117C2×D4—462no—
118C2×D4—461no—
119C2×D4—462no—
120C2×D4C2×D2311yescharacteristic, complemented
121Q16—231no—
122SD16—231no—
123SD16—231no—
124SD16—232no—
125SD16C2×D2312yescomplemented
126D8—232no—
127D8—231no—
128D8—232no—
129D8—232no—
130D8C2×D2312yescomplemented
131M16—232no—
132M16C2×D2311yescharacteristic, complemented
133C2×C8—231no—
134C2×C8—232no—
135C23—21no—
136C23—462no—
137C23—462no—
138C23—461no—
139Q8—231no—
140Q8—231no—
141Q8C22×D2311yescharacteristic
142D4—21no—
143D4—21no—
144D4—232no—
145D4—231no—
146D4—232no—
147D4—231no—
148D4—462no—
149D4—462no—
150D4—462no—
151D4—462no—
152D4—461no—
153D4—461no—
154D4C22×D2312yes—
155D4C22×D2311yescharacteristic
156C2×C4—21no—
157C2×C4—232no—
158C2×C4—462no—
159C2×C4—462no—
160C2×C4—461no—
161C2×C4C22×D2311yescharacteristic
162C8—231no—
163C8—231no—
164C8C22×D2312yes—
165C22—21no—
166C22—22no—
167C22—81no—
168C22—232no—
169C22—462no—
170C22—461no—
171C22—1841no—
172C22—1842no—
173C22—1842no—
174C22D4×D2311yescharacteristic
175C4—21no—
176C4—232no—
177C4—461no—
178C4D4×D2311yescharacteristic
179C4C23×D2311yescharacteristic
180C2—21no—
181C2—42no—
182C2—41no—
183C2—462no—
184C2—922no—
185C2—921no—
186C2C2×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#576
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. #10625160C2semidirect
2. #10625155C2semidirect
3. C23×AGL(1,ℤ/8)C2semidirect
4. #10625121C2semidirect
5. C2×#67272C2semidirect
6. #10625087C2semidirect
7. #10625085C2semidirect
8. #10625082C2semidirect
9. D8×D23C2semidirect
10. #10625079C2semidirect
11. #10625078C2semidirect
12. #41×C23C22semidirect
13. C2×D4×C23C22semidirect
15. #67292C22semidirect
17. #67290C22semidirect
20. D4×D23C22semidirect
21. D4×D23C22semidirect
24. #67288C22semidirect
26. C2×D92C22semidirect
27. SD16×C23C22semidirect
28. D8×C23C22semidirect
29. M16×C23C22semidirect
30. #67275C22semidirect
31. #67274C22semidirect
32. #67273C22semidirect
33. #67272C22semidirect
34. #67272C22semidirect
35. #67272C22semidirect
36. C2×C23⋊C8C22semidirect
37. D184C22semidirect
38. #67263C22semidirect
39. #67262C22semidirect
40. C8×D23C22semidirect
99. AGL(1,ℤ/8)D23semidirect
105. C23#576semidirect
112. #41C2×D23semidirect
120. C2×D4C2×D23semidirect
125. SD16C2×D23semidirect
130. D8C2×D23semidirect
132. M16C2×D23semidirect

Maximal subgroups

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

subgroupindex
#106251602
#106251552
C23×AGL(1,ℤ/8)2
#106251212
C2×#672722
#106250872
#106250852
#106250822
D8×D232
#106250792
#106250782
#57623