Finite Group Databasebeta

← #13093058

Subgroups — #13093058

Order 1472.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#67292C211yesmaximal, characteristic, complemented
2C22×D92C211yesmaximal, characteristic, complemented
3C23×C22⋊Q8C211yesmaximal, characteristic, complemented
4#10625118C211yesmaximal, characteristic, complemented
5C4×C22⋊D23C211yesmaximal, characteristic, complemented
6#10625066C211yesmaximal, characteristic, complemented
7#10625066C212yesmaximal, complemented
8#10625065C212yesmaximal, complemented
9#10625064C211yesmaximal, characteristic, complemented
10#10625053C212yesmaximal, complemented
11C22⋊C4×D23C212yesmaximal, complemented
12C23×D23C2212yes—
13C2×Q8×C23C2211yescharacteristic, complemented
14C22×C4×C23C2211yescharacteristic
15C2×C22⋊D23C2211yescharacteristic
16#67292—24no—
17C2×D92—24no—
18C2×D92C2212yes—
19C2×D92C2211yescharacteristic
20C2×D92C2211yescharacteristic
21C2×D92C2212yescomplemented
22C2×D92C2212yescomplemented
23C2×C4×D23C2212yescomplemented
24C2×C4×D23C2211yescharacteristic, complemented
25C2×C4×D23C2212yescomplemented
26C4⋊C4×C23C2211yescharacteristic, complemented
27C4⋊C4×C23C2212yescomplemented
28C22⋊C4×C23C2212yes—
29C22⋊Dic23C2211yescharacteristic
30(C2×D23)⋊C4C2212yescomplemented
31(C2×D23)⋊C4C2212yescomplemented
32(C2×D23)⋊C4C2212yescomplemented
33(C2×D23)⋊C4C2212yescomplemented
34(C2×D23)⋊C4C2211yescharacteristic, complemented
35Dic23⋊C4C2211yescharacteristic, complemented
36C4×Dic23C2211yescharacteristic, complemented
37C23×C23C2311yescharacteristic
38C22×D23—24no—
39C22×D23—24no—
40C22×D23—42no—
41C22×D23C2312yes—
42C22×D23C2311yescharacteristic
43C22×D23C2312yes—
44Q8×C23—22no—
45C2×C4×C23—22no—
46C2×C4×C23C2311yescharacteristic
47C2×C4×C23C2312yes—
48C2×C4×C23C2312yes—
49C2×C4×C23C2311yescharacteristic
50C22⋊D23—22no—
51C2×Dic23C2311yescharacteristic
52C2×Dic23C2311yescharacteristic
53C2×Dic23C2311yescharacteristic
54D92—24no—
55D92—24no—
56D92—22no—
57D92—42no—
58D92D414yescomplemented
59C4×D23—24no—
60C4×D23—22no—
61C4×D23—42no—
62C22×C23—21no—
63C22×C23—21no—
64C22×C23#4112yescomplemented
65C22×C23C2411yescharacteristic
66C2×D23—22no—
67C2×D23—22no—
68C2×D23—21no—
69C2×D23—22no—
70C2×D23—21no—
71C2×D23—22no—
72C2×D23—21no—
73C2×D23—22no—
74C2×D23—48no—
75C2×D23C2×D414yes—
76C4×C23—22no—
77C4×C23—21no—
78C4×C23—22no—
79C4×C23C2×D412yes—
80Dic23—21no—
81Dic23—21no—
82Dic23—21no—
83#546—231nomaximal, Sylow
84C2×C23—22no—
85C2×C23C22×D411yescharacteristic
86C2×C23C2×#4111yescharacteristic
87C2×C23H(5,𝔽2)11yescharacteristic
88D23—24no—
89D23—41no—
90D23—42no—
91C2×#41—231no—
92C22×D4—231no—
93#124—231no—
94#123—232no—
95C22⋊Q8D2311yescharacteristic, complemented
96#121—231no—
97#121—232no—
98#120—232no—
99C4×D4—231no—
100C4×D4—231no—
101C2×C22⋊C4—232no—
102C23#54611yescharacteristic, complemented, Sylow
103C24—232no—
104#41—464no—
105C2×Q8C2×D2311yescharacteristic, complemented
106C2×D4—231no—
107C2×D4—231no—
108C2×D4—232no—
109C2×D4—232no—
110C2×D4—231no—
111C2×D4—232no—
112C2×D4—464no—
113C22×C4—232no—
114C22×C4—231no—
115C22×C4—232no—
116C22×C4C2×D2311yescharacteristic
117C4⋊C4—231no—
118C4⋊C4C2×D2311yescharacteristic, complemented
119C4⋊C4C2×D2312yescomplemented
120C22⋊C4—231no—
121C22⋊C4—232no—
122C22⋊C4—232no—
123C22⋊C4—232no—
124C22⋊C4—231no—
125C22⋊C4—232no—
126C22⋊C4C2×D2312yes—
127C42—231no—
128C23—232no—
129C23—232no—
130C23—231no—
131C23—464no—
132C23—464no—
133C23—922no—
134C23C22×D2311yescharacteristic
135Q8—22no—
136D4—234no—
137D4—464no—
138D4—464no—
139D4—462no—
140D4—462no—
141D4—922no—
142C2×C4—22no—
143C2×C4—231no—
144C2×C4—231no—
145C2×C4—231no—
146C2×C4—464no—
147C2×C4—462no—
148C2×C4—922no—
149C2×C4C22×D2312yes—
150C2×C4C22×D2312yes—
151C2×C4C22×D2311yescharacteristic
152C2×C4C22×D2311yescharacteristic
153C22—21no—
154C22—21no—
155C22—234no—
156C22—462no—
157C22—462no—
158C22—461no—
159C22—462no—
160C22—461no—
161C22—462no—
162C22—461no—
163C22—462no—
164C22—928no—
165C22#6729212yescomplemented
166C22C23×D2311yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—461no—
171C4—461no—
172C4—461no—
173C4D4×D2312yes—
174C2—22no—
175C2—464no—
176C2—922no—
177C2—921no—
178C2C2×D4×D2311yescharacteristic
179C2C2×#6729211yescharacteristic
180C2#1062516011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D23
commutator, socle, cosocleC22×C23C24
FittingC23×C22⋊Q8C2
hypercenterC22⋊Q8D23
hypocenterC23#546
LCSG ▷ C22×C23 ▷ C23
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#67292C2semidirect
2. C22×D92C2semidirect
3. C23×C22⋊Q8C2semidirect
4. #10625118C2semidirect
5. C4×C22⋊D23C2semidirect
6. #10625066C2semidirect
7. #10625066C2semidirect
8. #10625065C2semidirect
9. #10625064C2semidirect
10. #10625053C2semidirect
11. C22⋊C4×D23C2semidirect
13. C2×Q8×C23C22semidirect
21. C2×D92C22semidirect
22. C2×D92C22semidirect
23. C2×C4×D23C22semidirect
24. C2×C4×D23C22semidirect
25. C2×C4×D23C22semidirect
26. C4⋊C4×C23C22semidirect
27. C4⋊C4×C23C22semidirect
30. (C2×D23)⋊C4C22semidirect
31. (C2×D23)⋊C4C22semidirect
32. (C2×D23)⋊C4C22semidirect
33. (C2×D23)⋊C4C22semidirect
34. (C2×D23)⋊C4C22semidirect
35. Dic23⋊C4C22semidirect
36. C4×Dic23C22semidirect
58. D92D4semidirect
64. C22×C23#41semidirect
95. C22⋊Q8D23semidirect
102. C23#546semidirect
105. C2×Q8C2×D23semidirect
118. C4⋊C4C2×D23semidirect
119. C4⋊C4C2×D23semidirect
165. C22#67292semidirect

Maximal subgroups

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

subgroupindex
C2×#672922
C22×D922
C23×C22⋊Q82
#106251182
C4×C22⋊D232
#106250662
#106250652
#106250642
#106250532
C22⋊C4×D232
#54623