Finite Group Databasebeta

← #13092930

Subgroups — #13092930

Order 1472.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×C4×D23C211yesmaximal, characteristic, complemented
3C23×C22⋊Q8C211yesmaximal, characteristic, complemented
4#10625117C211yesmaximal, characteristic, complemented
5#10625101C211yesmaximal, characteristic, complemented
6#10625066C211yesmaximal, characteristic, complemented
7#10625065C212yesmaximal, complemented
8#10625064C212yesmaximal, complemented
9#10625063C211yesmaximal, characteristic, complemented
10#10625053C212yesmaximal, complemented
11(C22⋊D23)⋊C4C212yesmaximal, complemented
12C23×D23C2211yescharacteristic
13C2×Q8×C23C2211yescharacteristic, complemented
14C22×C4×C23C2211yescharacteristic
15C2×C22⋊D23C2212yes—
16C22×Dic23C2211yescharacteristic
17#67292—24no—
18C2×D92C2212yes—
19C2×D92C2212yes—
20C2×D92C2211yescharacteristic, complemented
21C2×C4×D23—24no—
22C2×C4×D23C2211yescharacteristic
23C2×C4×D23C2211yescharacteristic
24C2×C4×D23C2211yescharacteristic
25C2×C4×D23C2211yescharacteristic
26C2×C4×D23C2212yescomplemented
27C4⋊C4×C23C2211yescharacteristic, complemented
28C4⋊C4×C23C2212yescomplemented
29C22⋊C4×C23C2212yes—
30(C2×D23)⋊C4C2212yes—
31(C2×D23)⋊C4C2212yes—
32(C2×D23)⋊C4C2212yescomplemented
33(C2×D23)⋊C4C2212yescomplemented
34C4⋊Dic23C2211yescharacteristic, complemented
35Dic23⋊C4C2212yescomplemented
36C4×Dic23C2212yescomplemented
37C23×C23C2311yescharacteristic
38C22×D23—22no—
39C22×D23—22no—
40C22×D23—22no—
41C22×D23C2312yes—
42C22×D23C2311yescharacteristic
43C22×D23C2311yescharacteristic
44Q8×C23—22no—
45C2×C4×C23—22no—
46C2×C4×C23C2312yes—
47C2×C4×C23C2312yes—
48C2×C4×C23C2311yescharacteristic
49C2×C4×C23C2311yescharacteristic
50C22⋊D23—24no—
51C2×Dic23—22no—
52C2×Dic23C2312yes—
53C2×Dic23C2311yescharacteristic
54C2×Dic23C2311yescharacteristic
55D92—24no—
56D92—24no—
57D92—22no—
58C4×D23—22no—
59C4×D23—22no—
60C4×D23—22no—
61C4×D23—24no—
62C4×D23D414yescomplemented
63C22×C23—21no—
64C22×C23—21no—
65C22×C23#4112yescomplemented
66C22×C23C2411yescharacteristic
67C2×D23—28no—
68C2×D23—21no—
69C2×D23—21no—
70C2×D23—21no—
71C2×D23—21no—
72C2×D23—22no—
73C2×D23—22no—
74C2×D23—22no—
75C2×D23C2×D412yes—
76C2×D23#4112yescomplemented
77C4×C23—21no—
78C4×C23—22no—
79C4×C23—22no—
80C4×C23C2×D412yes—
81Dic23—21no—
82Dic23—22no—
83Dic23C2×D412yes—
84#525—231nomaximal, Sylow
85C2×C23—22no—
86C2×C23C22×D411yescharacteristic
87C2×C23C2×#4111yescharacteristic
88C2×C23C2×#4111yescharacteristic
89D23—22no—
90D23—22no—
91D23—42no—
92C2×#41—231no—
93C23×C4—231no—
94#123—232no—
95C22⋊Q8—231no—
96C22⋊Q8D2311yescharacteristic, complemented
97#121—231no—
98#121—231no—
99#120—232no—
100C4×D4—232no—
101C4×D4—232no—
102#117—231no—
103C23#52511yescharacteristic, complemented, Sylow
104C24—231no—
105#41—464no—
106C2×Q8C2×D2311yescharacteristic, complemented
107C2×D4—232no—
108C2×D4—232no—
109C2×D4—232no—
110C2×D4—231no—
111C22×C4—231no—
112C22×C4—231no—
113C22×C4—231no—
114C22×C4—232no—
115C22×C4—231no—
116C22×C4—231no—
117C22×C4—464no—
118C22×C4C2×D2311yescharacteristic
119C4⋊C4—231no—
120C4⋊C4—232no—
121C4⋊C4C2×D2311yescharacteristic, complemented
122C4⋊C4C2×D2312yescomplemented
123C22⋊C4—232no—
124C22⋊C4—232no—
125C22⋊C4—232no—
126C22⋊C4—232no—
127C22⋊C4C2×D2312yes—
128C42—232no—
129C23—231no—
130C23—231no—
131C23—232no—
132C23—462no—
133C23—462no—
134C23—462no—
135C23C22×D2311yescharacteristic
136Q8—22no—
137D4—464no—
138D4—464no—
139D4—464no—
140D4—462no—
141C2×C4—22no—
142C2×C4—231no—
143C2×C4—234no—
144C2×C4—232no—
145C2×C4—231no—
146C2×C4—462no—
147C2×C4—464no—
148C2×C4—462no—
149C2×C4—462no—
150C2×C4—462no—
151C2×C4C22×D2312yes—
152C2×C4C22×D2311yescharacteristic
153C2×C4C22×D2312yes—
154C2×C4C22×D2311yescharacteristic
155C22—21no—
156C22—21no—
157C22—232no—
158C22—232no—
159C22—468no—
160C22—461no—
161C22—461no—
162C22—461no—
163C22—461no—
164C22—462no—
165C22—462no—
166C22—462no—
167C22#6729212yescomplemented
168C22C23×D2311yescharacteristic
169C4—21no—
170C4—22no—
171C4—22no—
172C4—232no—
173C4—461no—
174C4—462no—
175C4D4×D2312yes—
176C2—22no—
177C2—462no—
178C2—462no—
179C2—922no—
180C2C2×D4×D2311yescharacteristic
181C2C2×#6729211yescharacteristic
182C2#41×D2311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D23
commutator, socle, cosocleC22×C23C24
FittingC23×C22⋊Q8C2
hypercenterC22⋊Q8D23
hypocenterC23#525
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×C4×D23C2semidirect
3. C23×C22⋊Q8C2semidirect
4. #10625117C2semidirect
5. #10625101C2semidirect
6. #10625066C2semidirect
7. #10625065C2semidirect
8. #10625064C2semidirect
9. #10625063C2semidirect
10. #10625053C2semidirect
11. (C22⋊D23)⋊C4C2semidirect
13. C2×Q8×C23C22semidirect
20. C2×D92C22semidirect
26. C2×C4×D23C22semidirect
27. C4⋊C4×C23C22semidirect
28. C4⋊C4×C23C22semidirect
32. (C2×D23)⋊C4C22semidirect
33. (C2×D23)⋊C4C22semidirect
34. C4⋊Dic23C22semidirect
35. Dic23⋊C4C22semidirect
36. C4×Dic23C22semidirect
62. C4×D23D4semidirect
65. C22×C23#41semidirect
76. C2×D23#41semidirect
96. C22⋊Q8D23semidirect
103. C23#525semidirect
106. C2×Q8C2×D23semidirect
121. C4⋊C4C2×D23semidirect
122. C4⋊C4C2×D23semidirect
167. 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×C4×D232
C23×C22⋊Q82
#106251172
#106251012
#106250662
#106250652
#106250642
#106250632
#106250532
(C22⋊D23)⋊C42
#52523