Finite Group Databasebeta

← #1640×C211 #41009260

Subgroups — #1640×C211 #41009260

Order 27008.

212 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 68 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#559×C211C211yesmaximal, characteristic
2#486×C211C211yesmaximal, characteristic
3#475×C211C211yesmaximal, characteristic, complemented
4#462×C211C211yesmaximal, characteristic, complemented
5#331×C211C211yesmaximal, characteristic
6Q8⋊C8×C211C211yesmaximal, characteristic, complemented
7Q8⋊C8×C211C211yesmaximal, characteristic, complemented
8C2×Q16×C211—21no—
9C4⋊Q8×C211C2211yescharacteristic
10#127×C211C2211yescharacteristic
11#124×C211—21no—
12#124×C211—21no—
13#123×C211—22no—
14C4×Q8×C211C2211yescharacteristic
15C4×Q8×C211C2211yescharacteristic
16C4×D4×C211—21no—
17C4×D4×C211—21no—
18#106×C211—21no—
19C4⋊C8×C211C2211yescharacteristic
20C4⋊C8×C211C2211yescharacteristic
21#103×C211—21no—
22#103×C211—21no—
23#102×C211—22no—
24C4×C8×C211C2211yescharacteristic
25C2×Q8×C211—21no—
26C2×Q8×C211D411yescharacteristic
27C2×Q8×C211D411yescharacteristic
28C2×D4×C211—21no—
29C2×D4×C211—21no—
30C2×D4×C211D412yes—
31C22×C4×C211—22no—
32Q16×C211—42no—
33C2×C8×C211—21no—
34C2×C8×C211—21no—
35C2×C8×C211—21no—
36C4⋊C4×C211—21no—
37C4⋊C4×C211—21no—
38C4⋊C4×C211—21no—
39C4⋊C4×C211—21no—
40C4⋊C4×C211D411yescharacteristic
41C4⋊C4×C211D411yescharacteristic
42C22⋊C4×C211—22no—
43C22⋊C4×C211—22no—
44C22⋊C4×C211—22no—
45C42×C211—21no—
46C42×C211—21no—
47C42×C211C2311yescharacteristic
48C23×C211—22no—
49Q8×C211—21no—
50Q8×C211—21no—
51Q8×C211—21no—
52Q8×C211—21no—
53Q8×C211—41no—
54Q8×C211—41no—
55D4×C211—42no—
56D4×C211—41no—
57D4×C211—41no—
58D4×C211—41no—
59D4×C211—41no—
60C2×C4×C211—21no—
61C2×C4×C211—21no—
62C2×C4×C211—21no—
63C2×C4×C211—21no—
64C2×C4×C211—41no—
65C2×C4×C211—82no—
66C2×C4×C211C2×D411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C8×C211—41no—
70C8×C211—41no—
71C8×C211—41no—
72C22×C211—42no—
73C22×C211—42no—
74C22×C211—42no—
75C22×C211#12011yescharacteristic
76C4×C211—21no—
77C4×C211—41no—
78C4×C211—41no—
79C4×C211—41no—
80C4×C211—41no—
81C4×C211—81no—
82C4×C211#13511yescharacteristic
83C4×C211#13511yescharacteristic
84C4×C211AGL(1,ℤ/8)11yescharacteristic
85C4×C211#13711yescharacteristic
86C2×C211—82no—
87C2×C211#44911yescharacteristic
88C2×C211#45211yescharacteristic
89C2×C211#45511yescharacteristic
90C211#164011yescharacteristic, direct product factor, Sylow
91#1640C21111yesmaximal, characteristic, direct product factor, Sylow
92#559C2×C21111yescharacteristic
93#486C2×C21111yescharacteristic
94#475C2×C21111yescharacteristic, complemented
95#462C2×C21111yescharacteristic, complemented
96#331C2×C21111yescharacteristic
97Q8⋊C8C2×C21111yescharacteristic, complemented
98Q8⋊C8C2×C21111yescharacteristic, complemented
99C2×Q16—21no—
100C4⋊Q8C22×C21111yescharacteristic
101#127C22×C21111yescharacteristic
102#124—21no—
103#124—21no—
104#123—22no—
105C4×Q8C22×C21111yescharacteristic
106C4×Q8C22×C21111yescharacteristic
107C4×D4—21no—
108C4×D4—21no—
109#106—21no—
110C4⋊C8C22×C21111yescharacteristic
111C4⋊C8C22×C21111yescharacteristic
112#103—21no—
113#103—21no—
114#102—22no—
115C4×C8C22×C21111yescharacteristic
116C2×Q8—21no—
117C2×Q8D4×C21111yescharacteristic
118C2×Q8D4×C21111yescharacteristic
119C2×D4—21no—
120C2×D4—21no—
121C2×D4D4×C21112yes—
122C22×C4—22no—
123Q16—42no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4D4×C21111yescharacteristic
132C4⋊C4D4×C21111yescharacteristic
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C42—21no—
137C42—21no—
138C42C23×C21111yescharacteristic
139C23—22no—
140Q8—21no—
141Q8—21no—
142Q8—21no—
143Q8—21no—
144Q8—41no—
145Q8—41no—
146D4—42no—
147D4—41no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—82no—
157C2×C4C2×D4×C21111yescharacteristic
158C2×C4C2×D4×C21111yescharacteristic
159C2×C4C2×D4×C21111yescharacteristic
160C8—41no—
161C8—41no—
162C8—41no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22#120×C21111yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—81no—
173C4#135×C21111yescharacteristic
174C4#135×C21111yescharacteristic
175C4AGL(1,ℤ/8)×C21111yescharacteristic
176C4#137×C21111yescharacteristic
177C2—82no—
178C2#449×C21111yescharacteristic
179C2#452×C21111yescharacteristic
180C2#455×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC42C23×C211
prime radical#127×C211C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
3. #475×C211C2semidirect
4. #462×C211C2semidirect
6. Q8⋊C8×C211C2semidirect
7. Q8⋊C8×C211C2semidirect
90. C211#1640direct (N × H)
91. #1640C211direct (N × H)
94. #475C2×C211semidirect
95. #462C2×C211semidirect
97. Q8⋊C8C2×C211semidirect
98. Q8⋊C8C2×C211semidirect

Maximal subgroups

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

subgroupindex
#559×C2112
#486×C2112
#475×C2112
#462×C2112
#331×C2112
Q8⋊C8×C2112
#1640211