Finite Group Databasebeta

← #1885×C211 #41009505

Subgroups — #1885×C211 #41009505

Order 27008.

360 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 112 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×#124×C211C211yesmaximal, characteristic
2#518×C211C211yesmaximal, characteristic
3#419×C211C212yesmaximal, complemented
4C2×#100×C211C211yesmaximal, characteristic
5#339×C211C212yesmaximal, complemented
6C22×Q8×C211C2211yescharacteristic
7C22×D4×C211C2211yescharacteristic
8C2×M16×C211C2212yes—
9#124×C211—22no—
10#124×C211C414yescomplemented
11C4×D4×C211—24no—
12#117×C211C2212yes—
13C2×C22⋊C4×C211—22no—
14C2×C22⋊C4×C211—22no—
15C2×C42×C211C2211yescharacteristic
16#103×C211—24no—
17#100×C211—22no—
18C24×C211D412yes—
19C2×Q8×C211—21no—
20C2×Q8×C211—22no—
21C2×Q8×C211—22no—
22C2×Q8×C211C2×C412yescomplemented
23C2×D4×C211—22no—
24C2×D4×C211—22no—
25C2×D4×C211C2×C412yes—
26C2×D4×C211D412yes—
27C22×C4×C211—21no—
28C22×C4×C211—21no—
29C22×C4×C211—24no—
30C22×C4×C211C2311yescharacteristic
31M16×C211—22no—
32M16×C211—22no—
33C2×C8×C211—22no—
34C4⋊C4×C211D414yes—
35C22⋊C4×C211—24no—
36C22⋊C4×C211—28no—
37C22⋊C4×C211—22no—
38C42×C211—21no—
39C42×C211—22no—
40C42×C211C2×C412yescomplemented
41C23×C211—22no—
42C23×C211—22no—
43C23×C211—44no—
44C23×C211C22⋊C414yes—
45C23×C211C2×D411yescharacteristic
46Q8×C211—24no—
47Q8×C211—42no—
48D4×C211—24no—
49D4×C211—24no—
50C2×C4×C211—22no—
51C2×C4×C211—22no—
52C2×C4×C211—24no—
53C2×C4×C211—42no—
54C2×C4×C211—41no—
55C2×C4×C211—41no—
56C2×C4×C211—84no—
57C2×C4×C211C22×C411yescharacteristic
58C2×C4×C211C2×D412yes—
59C2×C4×C211C2×D411yescharacteristic
60C2×C4×C211#4112yes—
61C8×C211—42no—
62C22×C211—22no—
63C22×C211—24no—
64C22×C211—44no—
65C22×C211—44no—
66C22×C211—82no—
67C22×C211C2×C22⋊C411yescharacteristic
68C22×C211#12011yescharacteristic
69C22×C211#12311yescharacteristic
70C4×C211—42no—
71C4×C211—42no—
72C4×C211—44no—
73C4×C211C4×D412yes—
74C4×C211#12112yes—
75C2×C211—21no—
76C2×C211—21no—
77C2×C211—44no—
78C2×C211#38611yescharacteristic
79C2×C211#45512yes—
80C211#188511yescharacteristic, direct product factor, Sylow
81#1885C21111yesmaximal, characteristic, direct product factor, Sylow
82C2×#124C2×C21111yescharacteristic
83#518C2×C21111yescharacteristic
84#419C2×C21112yescomplemented
85C2×#100C2×C21111yescharacteristic
86#339C2×C21112yescomplemented
87C22×Q8C22×C21111yescharacteristic
88C22×D4C22×C21111yescharacteristic
89C2×M16C22×C21112yes—
90#124—22no—
91#124C4×C21114yescomplemented
92C4×D4—24no—
93#117C22×C21112yes—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C42C22×C21111yescharacteristic
97#103—24no—
98#100—22no—
99C24D4×C21112yes—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8C2×C4×C21112yescomplemented
104C2×D4—22no—
105C2×D4—22no—
106C2×D4C2×C4×C21112yes—
107C2×D4D4×C21112yes—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—24no—
111C22×C4C23×C21111yescharacteristic
112M16—22no—
113M16—22no—
114C2×C8—22no—
115C4⋊C4D4×C21114yes—
116C22⋊C4—24no—
117C22⋊C4—28no—
118C22⋊C4—22no—
119C42—21no—
120C42—22no—
121C42C2×C4×C21112yescomplemented
122C23—22no—
123C23—22no—
124C23—44no—
125C23C22⋊C4×C21114yes—
126C23C2×D4×C21111yescharacteristic
127Q8—24no—
128Q8—42no—
129D4—24no—
130D4—24no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—24no—
134C2×C4—42no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—84no—
138C2×C4C22×C4×C21111yescharacteristic
139C2×C4C2×D4×C21112yes—
140C2×C4C2×D4×C21111yescharacteristic
141C2×C4#41×C21112yes—
142C8—42no—
143C22—22no—
144C22—24no—
145C22—44no—
146C22—44no—
147C22—82no—
148C22C2×C22⋊C4×C21111yescharacteristic
149C22#120×C21111yescharacteristic
150C22#123×C21111yescharacteristic
151C4—42no—
152C4—42no—
153C4—44no—
154C4C4×D4×C21112yes—
155C4#121×C21112yes—
156C2—21no—
157C2—21no—
158C2—44no—
159C2#386×C21111yescharacteristic
160C2#455×C21112yes—
subgroupG/H
center, socleC22×C211#120
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC22×D4×C211C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
3. #419×C211C2semidirect
5. #339×C211C2semidirect
10. #124×C211C4semidirect
22. C2×Q8×C211C2×C4semidirect
40. C42×C211C2×C4semidirect
80. C211#1885direct (N × H)
81. #1885C211direct (N × H)
84. #419C2×C211semidirect
86. #339C2×C211semidirect
91. #124C4×C211semidirect
103. C2×Q8C2×C4×C211semidirect
121. C42C2×C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#124×C2112
#518×C2112
#419×C2112
C2×#100×C2112
#339×C2112
#1885211