Finite Group Databasebeta

← #1619×C211 #41009239

Subgroups — #1619×C211 #41009239

Order 27008.

336 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 76 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
1D42×C211C211yesmaximal, characteristic, complemented
2#493×C211C211yesmaximal, characteristic, complemented
3#459×C211C212yesmaximal, complemented
4#331×C211C211yesmaximal, characteristic, complemented
5D4⋊C8×C211C212yesmaximal, complemented
6C22×D4×C211—22no—
7C2×D8×C211—22no—
8C2×D8×C211—22no—
9#127×C211C2211yescharacteristic
10#127×C211C2211yescharacteristic, complemented
11#121×C211—22no—
12#120×C211—22no—
13C4×D4×C211C2212yes—
14C4⋊C8×C211C2212yescomplemented
15#102×C211—22no—
16C4×C8×C211C2211yescharacteristic, complemented
17C24×C211—22no—
18C2×D4×C211—24no—
19C2×D4×C211—22no—
20C2×D4×C211—22no—
21C2×D4×C211—21no—
22C2×D4×C211—44no—
23C2×D4×C211—44no—
24C2×D4×C211D412yes—
25C2×D4×C211D412yes—
26C22×C4×C211—22no—
27D8×C211—44no—
28D8×C211—44no—
29C2×C8×C211—21no—
30C2×C8×C211—22no—
31C4⋊C4×C211D412yes—
32C22⋊C4×C211—22no—
33C22⋊C4×C211—22no—
34C42×C211C2311yescharacteristic
35C23×C211—22no—
36C23×C211—22no—
37C23×C211—48no—
38C23×C211—41no—
39C23×C211—82no—
40D4×C211—22no—
41D4×C211—42no—
42D4×C211—42no—
43D4×C211—48no—
44D4×C211—42no—
45D4×C211—42no—
46D4×C211—42no—
47D4×C211—42no—
48D4×C211D814yescomplemented
49C2×C4×C211—22no—
50C2×C4×C211—42no—
51C2×C4×C211—42no—
52C2×C4×C211C2×D411yescharacteristic
53C2×C4×C211C2×D412yes—
54C8×C211—22no—
55C8×C211—42no—
56C22×C211—24no—
57C22×C211—42no—
58C22×C211—42no—
59C22×C211—44no—
60C22×C211—42no—
61C22×C211—88no—
62C22×C211—82no—
63C22×C211—81no—
64C22×C211#12011yescharacteristic
65C4×C211—21no—
66C4×C211—42no—
67C4×C211C2×D812yes—
68C4×C211AGL(1,ℤ/8)12yes—
69C2×C211—44no—
70C2×C211—82no—
71C2×C211—161no—
72C2×C211#44712yes—
73C2×C211#45311yescharacteristic
74C211#161911yescharacteristic, direct product factor, Sylow
75#1619C21111yesmaximal, characteristic, direct product factor, Sylow
76D42C2×C21111yescharacteristic, complemented
77#493C2×C21111yescharacteristic, complemented
78#459C2×C21112yescomplemented
79#331C2×C21111yescharacteristic, complemented
80D4⋊C8C2×C21112yescomplemented
81C22×D4—22no—
82C2×D8—22no—
83C2×D8—22no—
84#127C22×C21111yescharacteristic
85#127C22×C21111yescharacteristic, complemented
86#121—22no—
87#120—22no—
88C4×D4C22×C21112yes—
89C4⋊C8C22×C21112yescomplemented
90#102—22no—
91C4×C8C22×C21111yescharacteristic, complemented
92C24—22no—
93C2×D4—24no—
94C2×D4—22no—
95C2×D4—22no—
96C2×D4—21no—
97C2×D4—44no—
98C2×D4—44no—
99C2×D4D4×C21112yes—
100C2×D4D4×C21112yes—
101C22×C4—22no—
102D8—44no—
103D8—44no—
104C2×C8—21no—
105C2×C8—22no—
106C4⋊C4D4×C21112yes—
107C22⋊C4—22no—
108C22⋊C4—22no—
109C42C23×C21111yescharacteristic
110C23—22no—
111C23—22no—
112C23—48no—
113C23—41no—
114C23—82no—
115D4—22no—
116D4—42no—
117D4—42no—
118D4—48no—
119D4—42no—
120D4—42no—
121D4—42no—
122D4—42no—
123D4D8×C21114yescomplemented
124C2×C4—22no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4C2×D4×C21111yescharacteristic
128C2×C4C2×D4×C21112yes—
129C8—22no—
130C8—42no—
131C22—24no—
132C22—42no—
133C22—42no—
134C22—44no—
135C22—42no—
136C22—88no—
137C22—82no—
138C22—81no—
139C22#120×C21111yescharacteristic
140C4—21no—
141C4—42no—
142C4C2×D8×C21112yes—
143C4AGL(1,ℤ/8)×C21112yes—
144C2—44no—
145C2—82no—
146C2—161no—
147C2#447×C21112yes—
148C2#453×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC42C23×C211
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. D42×C211C2semidirect
2. #493×C211C2semidirect
3. #459×C211C2semidirect
4. #331×C211C2semidirect
5. D4⋊C8×C211C2semidirect
10. #127×C211C22semidirect
14. C4⋊C8×C211C22semidirect
16. C4×C8×C211C22semidirect
48. D4×C211D8semidirect
74. C211#1619direct (N × H)
75. #1619C211direct (N × H)
76. D42C2×C211semidirect
77. #493C2×C211semidirect
78. #459C2×C211semidirect
79. #331C2×C211semidirect
80. D4⋊C8C2×C211semidirect
85. #127C22×C211semidirect
89. C4⋊C8C22×C211semidirect
91. C4×C8C22×C211semidirect
123. D4D8×C211semidirect

Maximal subgroups

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

subgroupindex
D42×C2112
#493×C2112
#459×C2112
#331×C2112
D4⋊C8×C2112
#1619211