Finite Group Databasebeta

← #1642×C211 #41009262

Subgroups — #1642×C211 #41009262

Order 27008.

250 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 72 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#550×C211C211yesmaximal, characteristic
2#486×C211C211yesmaximal, characteristic, complemented
3#475×C211C211yesmaximal, characteristic, complemented
4#461×C211C211yesmaximal, characteristic, complemented
5#331×C211C211yesmaximal, characteristic, complemented
6Q8⋊C8×C211C211yesmaximal, characteristic, complemented
7D4⋊C8×C211C211yesmaximal, characteristic, complemented
8C2×#41×C211—21no—
9C2×SD16×C211—21no—
10C4⋊Q8×C211C2211yescharacteristic
11#127×C211—21no—
12#127×C211C2211yescharacteristic
13#121×C211—22no—
14#121×C211—21no—
15C4×Q8×C211C2211yescharacteristic
16C4×D4×C211—21no—
17C4×D4×C211C2211yescharacteristic
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
25#41×C211—42no—
26#41×C211—42no—
27C2×Q8×C211—21no—
28C2×Q8×C211D411yescharacteristic
29C2×D4×C211—22no—
30C2×D4×C211—22no—
31C2×D4×C211—21no—
32C2×D4×C211—21no—
33C2×D4×C211D411yescharacteristic
34C2×D4×C211D412yes—
35C22×C4×C211—22no—
36C22×C4×C211—21no—
37SD16×C211—42no—
38C2×C8×C211—21no—
39C2×C8×C211—21no—
40C2×C8×C211—21no—
41C4⋊C4×C211—21no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211D411yescharacteristic
45C4⋊C4×C211D411yescharacteristic
46C22⋊C4×C211—22no—
47C22⋊C4×C211—21no—
48C42×C211—21no—
49C42×C211C2311yescharacteristic
50C23×C211—21no—
51C23×C211—22no—
52Q8×C211—21no—
53Q8×C211—41no—
54Q8×C211—41no—
55Q8×C211SD1612yescomplemented
56D4×C211—21no—
57D4×C211—21no—
58D4×C211—42no—
59D4×C211—44no—
60D4×C211—41no—
61D4×C211—41no—
62D4×C211—41no—
63D4×C211—41no—
64D4×C211—82no—
65C2×C4×C211—21no—
66C2×C4×C211—21no—
67C2×C4×C211—21no—
68C2×C4×C211—44no—
69C2×C4×C211—41no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C2×C4×C211C2×D411yescharacteristic
73C2×C4×C211C2×D411yescharacteristic
74C2×C4×C211C2×D411yescharacteristic
75C8×C211—41no—
76C8×C211—41no—
77C8×C211—41no—
78C22×C211—41no—
79C22×C211—42no—
80C22×C211—42no—
81C22×C211—41no—
82C22×C211—42no—
83C22×C211—41no—
84C22×C211#12011yescharacteristic
85C4×C211—22no—
86C4×C211—21no—
87C4×C211—41no—
88C4×C211—41no—
89C4×C211—81no—
90C4×C211C2×SD1611yescharacteristic
91C4×C211#13511yescharacteristic
92C4×C211AGL(1,ℤ/8)11yescharacteristic
93C4×C211AGL(1,ℤ/8)11yescharacteristic
94C2×C211—81no—
95C2×C211—82no—
96C2×C211#44911yescharacteristic
97C2×C211#45011yescharacteristic
98C2×C211#45511yescharacteristic
99C211#164211yescharacteristic, direct product factor, Sylow
100#1642C21111yesmaximal, characteristic, direct product factor, Sylow
101#550C2×C21111yescharacteristic
102#486C2×C21111yescharacteristic, complemented
103#475C2×C21111yescharacteristic, complemented
104#461C2×C21111yescharacteristic, complemented
105#331C2×C21111yescharacteristic, complemented
106Q8⋊C8C2×C21111yescharacteristic, complemented
107D4⋊C8C2×C21111yescharacteristic, complemented
108C2×#41—21no—
109C2×SD16—21no—
110C4⋊Q8C22×C21111yescharacteristic
111#127—21no—
112#127C22×C21111yescharacteristic
113#121—22no—
114#121—21no—
115C4×Q8C22×C21111yescharacteristic
116C4×D4—21no—
117C4×D4C22×C21111yescharacteristic
118#106—21no—
119C4⋊C8C22×C21111yescharacteristic
120C4⋊C8C22×C21111yescharacteristic
121#103—21no—
122#103—21no—
123#102—22no—
124C4×C8C22×C21111yescharacteristic
125#41—42no—
126#41—42no—
127C2×Q8—21no—
128C2×Q8D4×C21111yescharacteristic
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4D4×C21111yescharacteristic
134C2×D4D4×C21112yes—
135C22×C4—22no—
136C22×C4—21no—
137SD16—42no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4D4×C21111yescharacteristic
145C4⋊C4D4×C21111yescharacteristic
146C22⋊C4—22no—
147C22⋊C4—21no—
148C42—21no—
149C42C23×C21111yescharacteristic
150C23—21no—
151C23—22no—
152Q8—21no—
153Q8—41no—
154Q8—41no—
155Q8SD16×C21112yescomplemented
156D4—21no—
157D4—21no—
158D4—42no—
159D4—44no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—41no—
164D4—82no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—44no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4C2×D4×C21111yescharacteristic
173C2×C4C2×D4×C21111yescharacteristic
174C2×C4C2×D4×C21111yescharacteristic
175C8—41no—
176C8—41no—
177C8—41no—
178C22—41no—
179C22—42no—
180C22—42no—
181C22—41no—
182C22—42no—
183C22—41no—
184C22#120×C21111yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—41no—
189C4—81no—
190C4C2×SD16×C21111yescharacteristic
191C4#135×C21111yescharacteristic
192C4AGL(1,ℤ/8)×C21111yescharacteristic
193C4AGL(1,ℤ/8)×C21111yescharacteristic
194C2—81no—
195C2—82no—
196C2#449×C21111yescharacteristic
197C2#450×C21111yescharacteristic
198C2#455×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC42C23×C211
prime radical#550×C211C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #486×C211C2semidirect
3. #475×C211C2semidirect
4. #461×C211C2semidirect
5. #331×C211C2semidirect
6. Q8⋊C8×C211C2semidirect
7. D4⋊C8×C211C2semidirect
55. Q8×C211SD16semidirect
99. C211#1642direct (N × H)
100. #1642C211direct (N × H)
102. #486C2×C211semidirect
103. #475C2×C211semidirect
104. #461C2×C211semidirect
105. #331C2×C211semidirect
106. Q8⋊C8C2×C211semidirect
107. D4⋊C8C2×C211semidirect
155. Q8SD16×C211semidirect

Maximal subgroups

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

subgroupindex
#550×C2112
#486×C2112
#475×C2112
#461×C2112
#331×C2112
Q8⋊C8×C2112
D4⋊C8×C2112
#1642211