Finite Group Databasebeta

← #1912×C211 #41009532

Subgroups — #1912×C211 #41009532

Order 27008.

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

normal subgroups only

This group has 314 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#536×C211C211yesmaximal, characteristic
2#519×C211C211yesmaximal, characteristic, complemented
3#421×C211C212yesmaximal
4C2×#101×C211C211yesmaximal, characteristic, complemented
5H(3,ℤ/4)×C211C212yesmaximal, complemented
6C2×#41×C211C2211yescharacteristic
7C22×Q8×C211C2211yescharacteristic
8C2×M16×C211C2212yes—
9C4⋊Q8×C211C412yescomplemented
10#124×C211C412yescomplemented
11#123×C211—22no—
12C22⋊Q8×C211—22no—
13C4×Q8×C211—24no—
14#117×C211—22no—
15#117×C211C2211yescharacteristic
16#117×C211C2212yes—
17#104×C211—24no—
18#101×C211—22no—
19#41×C211—22no—
20#41×C211—22no—
21C2×Q8×C211—22no—
22C2×Q8×C211—22no—
23C2×Q8×C211C2×C411yescharacteristic
24C2×Q8×C211C2×C412yes—
25C2×Q8×C211D412yes—
26C2×D4×C211—21no—
27C2×D4×C211C2×C411yescharacteristic
28C22×C4×C211—21no—
29C22×C4×C211D412yes—
30C22×C4×C211C2311yescharacteristic
31M16×C211—22no—
32M16×C211—22no—
33C2×C8×C211—22no—
34C4⋊C4×C211—24no—
35C4⋊C4×C211—24no—
36C4⋊C4×C211—22no—
37C4⋊C4×C211—21no—
38C22⋊C4×C211—22no—
39C22⋊C4×C211—21no—
40C22⋊C4×C211—24no—
41C42×C211—24no—
42C42×C211C2×C412yes—
43C42×C211D414yes—
44C23×C211—21no—
45C23×C211C2×D411yescharacteristic
46Q8×C211—24no—
47Q8×C211—24no—
48Q8×C211—22no—
49D4×C211—22no—
50D4×C211—41no—
51D4×C211—41no—
52C2×C4×C211—22no—
53C2×C4×C211—22no—
54C2×C4×C211—22no—
55C2×C4×C211—21no—
56C2×C4×C211—24no—
57C2×C4×C211—42no—
58C2×C4×C211C22⋊C414yes—
59C2×C4×C211C22×C411yescharacteristic
60C2×C4×C211C2×D412yes—
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211#4112yes—
63C8×C211—42no—
64C22×C211—41no—
65C22×C211—42no—
66C22×C211—41no—
67C22×C211C2×C22⋊C411yescharacteristic
68C22×C211#12011yescharacteristic
69C22×C211#12311yescharacteristic
70C4×C211—24no—
71C4×C211—44no—
72C4×C211—42no—
73C4×C211—41no—
74C4×C211C4×D412yes—
75C4×C211#12112yes—
76C2×C211—21no—
77C2×C211—21no—
78C2×C211—21no—
79C2×C211—81no—
80C2×C211#38611yescharacteristic
81C211#191211yescharacteristic, direct product factor, Sylow
82#1912C21111yesmaximal, characteristic, direct product factor, Sylow
83#536C2×C21111yescharacteristic
84#519C2×C21111yescharacteristic, complemented
85#421C2×C21112yes—
86C2×#101C2×C21111yescharacteristic, complemented
87H(3,ℤ/4)C2×C21112yescomplemented
88C2×#41C22×C21111yescharacteristic
89C22×Q8C22×C21111yescharacteristic
90C2×M16C22×C21112yes—
91C4⋊Q8C4×C21112yescomplemented
92#124C4×C21112yescomplemented
93#123—22no—
94C22⋊Q8—22no—
95C4×Q8—24no—
96#117—22no—
97#117C22×C21111yescharacteristic
98#117C22×C21112yes—
99#104—24no—
100#101—22no—
101#41—22no—
102#41—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×C4×C21111yescharacteristic
106C2×Q8C2×C4×C21112yes—
107C2×Q8D4×C21112yes—
108C2×D4—21no—
109C2×D4C2×C4×C21111yescharacteristic
110C22×C4—21no—
111C22×C4D4×C21112yes—
112C22×C4C23×C21111yescharacteristic
113M16—22no—
114M16—22no—
115C2×C8—22no—
116C4⋊C4—24no—
117C4⋊C4—24no—
118C4⋊C4—22no—
119C4⋊C4—21no—
120C22⋊C4—22no—
121C22⋊C4—21no—
122C22⋊C4—24no—
123C42—24no—
124C42C2×C4×C21112yes—
125C42D4×C21114yes—
126C23—21no—
127C23C2×D4×C21111yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—22no—
131D4—22no—
132D4—41no—
133D4—41no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—24no—
139C2×C4—42no—
140C2×C4C22⋊C4×C21114yes—
141C2×C4C22×C4×C21111yescharacteristic
142C2×C4C2×D4×C21112yes—
143C2×C4C2×D4×C21111yescharacteristic
144C2×C4#41×C21112yes—
145C8—42no—
146C22—41no—
147C22—42no—
148C22—41no—
149C22C2×C22⋊C4×C21111yescharacteristic
150C22#120×C21111yescharacteristic
151C22#123×C21111yescharacteristic
152C4—24no—
153C4—44no—
154C4—42no—
155C4—41no—
156C4C4×D4×C21112yes—
157C4#121×C21112yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—81no—
162C2#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
2. #519×C211C2semidirect
4. C2×#101×C211C2semidirect
5. H(3,ℤ/4)×C211C2semidirect
9. C4⋊Q8×C211C4semidirect
10. #124×C211C4semidirect
81. C211#1912direct (N × H)
82. #1912C211direct (N × H)
84. #519C2×C211semidirect
86. C2×#101C2×C211semidirect
87. H(3,ℤ/4)C2×C211semidirect
91. C4⋊Q8C4×C211semidirect
92. #124C4×C211semidirect

Maximal subgroups

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

subgroupindex
#536×C2112
#519×C2112
#421×C2112
C2×#101×C2112
H(3,ℤ/4)×C2112
#1912211