Finite Group Databasebeta

← #1909×C211 #41009529

Subgroups — #1909×C211 #41009529

Order 27008.

312 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 108 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
1#568×C211C211yesmaximal, characteristic
2#536×C211C211yesmaximal, characteristic
3#419×C211C212yesmaximal, complemented
4#410×C211C211yesmaximal, characteristic
5H(3,ℤ/4)×C211C212yesmaximal, complemented
6C2×#41×C211C2211yescharacteristic
7C22×Q8×C211C2211yescharacteristic
8#131×C211—24no—
9C2×M16×C211—22no—
10C2×M16×C211C2212yes—
11C4⋊Q8×C211C412yescomplemented
12#124×C211C412yescomplemented
13#123×C211—22no—
14C22⋊Q8×C211—22no—
15#117×C211C2212yes—
16#117×C211C2211yescharacteristic
17#103×C211—24no—
18#99×C211—22no—
19#41×C211—22no—
20#41×C211—22no—
21C2×Q8×C211—21no—
22C2×Q8×C211—22no—
23C2×Q8×C211—22no—
24C2×Q8×C211C2×C412yes—
25C2×Q8×C211C2×C411yescharacteristic
26C2×D4×C211C2×C411yescharacteristic
27C2×D4×C211D412yes—
28C22×C4×C211—21no—
29C22×C4×C211D412yes—
30C22×C4×C211C2311yescharacteristic
31M16×C211—24no—
32M16×C211—22no—
33M16×C211—22no—
34C2×C8×C211—24no—
35C2×C8×C211D414yes—
36C4⋊C4×C211—24no—
37C4⋊C4×C211—21no—
38C4⋊C4×C211—22no—
39C22⋊C4×C211—24no—
40C22⋊C4×C211—21no—
41C22⋊C4×C211—22no—
42C42×C211—22no—
43C42×C211C2×C412yes—
44C23×C211C22⋊C412yes—
45C23×C211C2×D411yescharacteristic
46Q8×C211—22no—
47Q8×C211—24no—
48Q8×C211—41no—
49Q8×C211—41no—
50D4×C211—24no—
51D4×C211—22no—
52C2×C4×C211—22no—
53C2×C4×C211—22no—
54C2×C4×C211—22no—
55C2×C4×C211—22no—
56C2×C4×C211—42no—
57C2×C4×C211—42no—
58C2×C4×C211C22⋊C412yes—
59C2×C4×C211C22×C411yescharacteristic
60C2×C4×C211C2×D412yes—
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211#4112yes—
63C8×C211—24no—
64C22×C211—22no—
65C22×C211—41no—
66C22×C211—42no—
67C22×C211C2×C22⋊C411yescharacteristic
68C22×C211#12112yes—
69C4×C211—22no—
70C4×C211—42no—
71C4×C211—42no—
72C4×C211—82no—
73C4×C211C4×D412yes—
74C4×C211#12011yescharacteristic
75C4×C211#12311yescharacteristic
76C2×C211—21no—
77C2×C211—22no—
78C2×C211—42no—
79C2×C211#38611yescharacteristic
80C211#190911yescharacteristic, direct product factor, Sylow
81#1909C21111yesmaximal, characteristic, direct product factor, Sylow
82#568C2×C21111yescharacteristic
83#536C2×C21111yescharacteristic
84#419C2×C21112yescomplemented
85#410C2×C21111yescharacteristic
86H(3,ℤ/4)C2×C21112yescomplemented
87C2×#41C22×C21111yescharacteristic
88C22×Q8C22×C21111yescharacteristic
89#131—24no—
90C2×M16—22no—
91C2×M16C22×C21112yes—
92C4⋊Q8C4×C21112yescomplemented
93#124C4×C21112yescomplemented
94#123—22no—
95C22⋊Q8—22no—
96#117C22×C21112yes—
97#117C22×C21111yescharacteristic
98#103—24no—
99#99—22no—
100#41—22no—
101#41—22no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×C4×C21112yes—
106C2×Q8C2×C4×C21111yescharacteristic
107C2×D4C2×C4×C21111yescharacteristic
108C2×D4D4×C21112yes—
109C22×C4—21no—
110C22×C4D4×C21112yes—
111C22×C4C23×C21111yescharacteristic
112M16—24no—
113M16—22no—
114M16—22no—
115C2×C8—24no—
116C2×C8D4×C21114yes—
117C4⋊C4—24no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C22⋊C4—24no—
121C22⋊C4—21no—
122C22⋊C4—22no—
123C42—22no—
124C42C2×C4×C21112yes—
125C23C22⋊C4×C21112yes—
126C23C2×D4×C21111yescharacteristic
127Q8—22no—
128Q8—24no—
129Q8—41no—
130Q8—41no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4C22⋊C4×C21112yes—
140C2×C4C22×C4×C21111yescharacteristic
141C2×C4C2×D4×C21112yes—
142C2×C4C2×D4×C21111yescharacteristic
143C2×C4#41×C21112yes—
144C8—24no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×C22⋊C4×C21111yescharacteristic
149C22#121×C21112yes—
150C4—22no—
151C4—42no—
152C4—42no—
153C4—82no—
154C4C4×D4×C21112yes—
155C4#120×C21111yescharacteristic
156C4#123×C21111yescharacteristic
157C2—21no—
158C2—22no—
159C2—42no—
160C2#386×C21111yescharacteristic
subgroupG/H
center, socleC2×C211#386
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC2×#41×C211C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
3. #419×C211C2semidirect
5. H(3,ℤ/4)×C211C2semidirect
11. C4⋊Q8×C211C4semidirect
12. #124×C211C4semidirect
80. C211#1909direct (N × H)
81. #1909C211direct (N × H)
84. #419C2×C211semidirect
86. H(3,ℤ/4)C2×C211semidirect
92. C4⋊Q8C4×C211semidirect
93. #124C4×C211semidirect

Maximal subgroups

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

subgroupindex
#568×C2112
#536×C2112
#419×C2112
#410×C2112
H(3,ℤ/4)×C2112
#1909211