Finite Group Databasebeta

← #3151×C229 #44420212

Subgroups — #3151×C229 #44420212

Order 29312.

656 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 224 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×AGL(1,ℤ/8)×C229C214yesmaximal, complemented
2#532×C229C211yesmaximal, characteristic, complemented
3C2×#127×C229C211yesmaximal, characteristic, complemented
4#496×C229C214yesmaximal, complemented
5#493×C229C212yesmaximal, complemented
6#492×C229C212yesmaximal, complemented
7C4×M16×C229C211yesmaximal, characteristic, complemented
8C2×#41×C229C2212yes—
9C22×D4×C229—22no—
10C22×D4×C229C2212yes—
11AGL(1,ℤ/8)×C229—216no—
12C2×SD16×C229C2218yescomplemented
13C2×D8×C229C2218yescomplemented
14C2×M16×C229C2212yes—
15C4⋊Q8×C229C2211yescharacteristic, complemented
16#127×C229—22no—
17#127×C229C2212yes—
18#127×C229C2212yes—
19#127×C229C2211yescharacteristic, complemented
20#124×C229C2212yescomplemented
21#121×C229—22no—
22C4×D4×C229—22no—
23C2×C42×C229C2211yescharacteristic
24#97×C229C2212yescomplemented
25C4×C8×C229C2212yescomplemented
26C24×C229—22no—
27#41×C229—28no—
28C2×Q8×C229C2312yes—
29C2×D4×C229—22no—
30C2×D4×C229—28no—
31C2×D4×C229—22no—
32C2×D4×C229—22no—
33C2×D4×C229—24no—
34C2×D4×C229—24no—
35C2×D4×C229—48no—
36C2×D4×C229C2312yes—
37C2×D4×C229C2314yes—
38C22×C4×C229—22no—
39C22×C4×C229D412yes—
40C22×C4×C229C2311yescharacteristic
41SD16×C229—216no—
42D8×C229—216no—
43M16×C229D418yescomplemented
44C2×C8×C229C2314yes—
45C4⋊C4×C229—22no—
46C22⋊C4×C229—24no—
47C42×C229D412yes—
48C42×C229C2311yescharacteristic
49C42×C229C2311yescharacteristic
50C23×C229—22no—
51C23×C229—24no—
52C23×C229—48no—
53C23×C229—44no—
54C23×C229C2×D411yescharacteristic
55Q8×C229—24no—
56D4×C229—24no—
57D4×C229—28no—
58D4×C229—48no—
59D4×C229—44no—
60D4×C229—44no—
61D4×C229—44no—
62D4×C229—48no—
63C2×C4×C229—24no—
64C2×C4×C229—22no—
65C2×C4×C229—44no—
66C2×C4×C229C2×D412yes—
67C2×C4×C229C2×D411yescharacteristic
68C2×C4×C229C2×D414yes—
69C2×C4×C229C2×D412yes—
70C2×C4×C229C2411yescharacteristic
71C8×C229C2×D418yescomplemented
72C22×C229—22no—
73C22×C229—42no—
74C22×C229—44no—
75C22×C229—48no—
76C22×C229—44no—
77C22×C229—88no—
78C22×C229#12712yes—
79C22×C229C22×D411yescharacteristic
80C4×C229—22no—
81C4×C229—42no—
82C4×C229#12712yes—
83C4×C229AGL(1,ℤ/8)14yescomplemented
84C4×C229C22×D412yes—
85C2×C229—22no—
86C2×C229—82no—
87C2×C229—84no—
88C2×C229C2×#12711yescharacteristic
89C2×C229C2×AGL(1,ℤ/8)12yes—
90C229#315111yescharacteristic, direct product factor, Sylow
91#3151C22911yesmaximal, characteristic, direct product factor, Sylow
92C2×AGL(1,ℤ/8)C2×C22914yescomplemented
93#532C2×C22911yescharacteristic, complemented
94C2×#127C2×C22911yescharacteristic, complemented
95#496C2×C22914yescomplemented
96#493C2×C22912yescomplemented
97#492C2×C22912yescomplemented
98C4×M16C2×C22911yescharacteristic, complemented
99C2×#41C22×C22912yes—
100C22×D4—22no—
101C22×D4C22×C22912yes—
102AGL(1,ℤ/8)—216no—
103C2×SD16C22×C22918yescomplemented
104C2×D8C22×C22918yescomplemented
105C2×M16C22×C22912yes—
106C4⋊Q8C22×C22911yescharacteristic, complemented
107#127—22no—
108#127C22×C22912yes—
109#127C22×C22912yes—
110#127C22×C22911yescharacteristic, complemented
111#124C22×C22912yescomplemented
112#121—22no—
113C4×D4—22no—
114C2×C42C22×C22911yescharacteristic
115#97C22×C22912yescomplemented
116C4×C8C22×C22912yescomplemented
117C24—22no—
118#41—28no—
119C2×Q8C23×C22912yes—
120C2×D4—22no—
121C2×D4—28no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—24no—
125C2×D4—24no—
126C2×D4—48no—
127C2×D4C23×C22912yes—
128C2×D4C23×C22914yes—
129C22×C4—22no—
130C22×C4D4×C22912yes—
131C22×C4C23×C22911yescharacteristic
132SD16—216no—
133D8—216no—
134M16D4×C22918yescomplemented
135C2×C8C23×C22914yes—
136C4⋊C4—22no—
137C22⋊C4—24no—
138C42D4×C22912yes—
139C42C23×C22911yescharacteristic
140C42C23×C22911yescharacteristic
141C23—22no—
142C23—24no—
143C23—48no—
144C23—44no—
145C23C2×D4×C22911yescharacteristic
146Q8—24no—
147D4—24no—
148D4—28no—
149D4—48no—
150D4—44no—
151D4—44no—
152D4—48no—
153D4—44no—
154C2×C4—24no—
155C2×C4—22no—
156C2×C4—44no—
157C2×C4C2×D4×C22912yes—
158C2×C4C2×D4×C22911yescharacteristic
159C2×C4C2×D4×C22914yes—
160C2×C4C2×D4×C22912yes—
161C2×C4C24×C22911yescharacteristic
162C8C2×D4×C22918yescomplemented
163C22—22no—
164C22—42no—
165C22—44no—
166C22—48no—
167C22—44no—
168C22—88no—
169C22#127×C22912yes—
170C22C22×D4×C22911yescharacteristic
171C4—22no—
172C4—42no—
173C4#127×C22912yes—
174C4AGL(1,ℤ/8)×C22914yescomplemented
175C4C22×D4×C22912yes—
176C2—22no—
177C2—84no—
178C2—82no—
179C2C2×#127×C22911yescharacteristic
180C2C2×AGL(1,ℤ/8)×C22912yes—
subgroupG/H
center, socleC22×C229C22×D4
commutator, Frattini, cosocleC2×C4C24×C229
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C229 ◁ C2×C42×C229 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×AGL(1,ℤ/8)×C229C2semidirect
2. #532×C229C2semidirect
3. C2×#127×C229C2semidirect
4. #496×C229C2semidirect
5. #493×C229C2semidirect
6. #492×C229C2semidirect
7. C4×M16×C229C2semidirect
12. C2×SD16×C229C22semidirect
13. C2×D8×C229C22semidirect
15. C4⋊Q8×C229C22semidirect
19. #127×C229C22semidirect
20. #124×C229C22semidirect
24. #97×C229C22semidirect
25. C4×C8×C229C22semidirect
43. M16×C229D4semidirect
71. C8×C229C2×D4semidirect
83. C4×C229AGL(1,ℤ/8)semidirect
90. C229#3151direct (N × H)
91. #3151C229direct (N × H)
92. C2×AGL(1,ℤ/8)C2×C229semidirect
93. #532C2×C229semidirect
94. C2×#127C2×C229semidirect
95. #496C2×C229semidirect
96. #493C2×C229semidirect
97. #492C2×C229semidirect
98. C4×M16C2×C229semidirect
103. C2×SD16C22×C229semidirect
104. C2×D8C22×C229semidirect
106. C4⋊Q8C22×C229semidirect
110. #127C22×C229semidirect
111. #124C22×C229semidirect
115. #97C22×C229semidirect
116. C4×C8C22×C229semidirect
134. M16D4×C229semidirect
162. C8C2×D4×C229semidirect
174. C4AGL(1,ℤ/8)×C229semidirect

Maximal subgroups

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

subgroupindex
C2×AGL(1,ℤ/8)×C2292
#532×C2292
C2×#127×C2292
#496×C2292
#493×C2292
#492×C2292
C4×M16×C2292
#3151229