Finite Group Databasebeta

← #1645×C211 #41009265

Subgroups — #1645×C211 #41009265

Order 27008.

200 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 76 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
1Q82×C211C211yesmaximal, characteristic
2#487×C211C211yesmaximal, characteristic
3#475×C211C211yesmaximal, characteristic
4#462×C211C211yesmaximal, characteristic
5#333×C211C211yesmaximal, characteristic
6Q8⋊C8×C211C211yesmaximal, characteristic
7Q8⋊C8×C211C211yesmaximal, characteristic
8C2×Q16×C211—21no—
9C4⋊Q8×C211—22no—
10C4⋊Q8×C211—21no—
11C4⋊Q8×C211—21no—
12C4⋊Q8×C211C2211yescharacteristic
13C4⋊Q8×C211C2211yescharacteristic
14C4×Q8×C211—21no—
15C4×Q8×C211—21no—
16C4×Q8×C211C2211yescharacteristic
17C4×Q8×C211C2211yescharacteristic
18#106×C211—21no—
19C4⋊C8×C211C2211yescharacteristic
20C4⋊C8×C211C2211yescharacteristic
21#103×C211—21no—
22#103×C211—22no—
23#103×C211—21no—
24C4×C8×C211C2211yescharacteristic
25C2×Q8×C211—22no—
26C2×Q8×C211—21no—
27C2×Q8×C211D411yescharacteristic
28C2×Q8×C211D411yescharacteristic
29C2×Q8×C211D412yes—
30Q16×C211—42no—
31C2×C8×C211—21no—
32C2×C8×C211—21no—
33C2×C8×C211—21no—
34C4⋊C4×C211—22no—
35C4⋊C4×C211—22no—
36C4⋊C4×C211—21no—
37C4⋊C4×C211—21no—
38C4⋊C4×C211—21no—
39C4⋊C4×C211—21no—
40C4⋊C4×C211—21no—
41C4⋊C4×C211—21no—
42C4⋊C4×C211D411yescharacteristic
43C4⋊C4×C211D411yescharacteristic
44C42×C211—22no—
45C42×C211—21no—
46C42×C211—21no—
47C42×C211C2311yescharacteristic
48Q8×C211—21no—
49Q8×C211—21no—
50Q8×C211—42no—
51Q8×C211—41no—
52Q8×C211—41no—
53Q8×C211—82no—
54Q8×C211SD1612yes—
55Q8×C211Q1612yescomplemented
56C2×C4×C211—21no—
57C2×C4×C211—21no—
58C2×C4×C211—22no—
59C2×C4×C211—21no—
60C2×C4×C211—21no—
61C2×C4×C211—41no—
62C2×C4×C211C2×D411yescharacteristic
63C2×C4×C211C2×D411yescharacteristic
64C2×C4×C211C2×D411yescharacteristic
65C8×C211—41no—
66C8×C211—41no—
67C8×C211—41no—
68C22×C211#12011yescharacteristic
69C4×C211—22no—
70C4×C211—22no—
71C4×C211—21no—
72C4×C211—42no—
73C4×C211—41no—
74C4×C211—41no—
75C4×C211—81no—
76C4×C211C2×SD1611yescharacteristic
77C4×C211C2×Q1611yescharacteristic
78C4×C211#13711yescharacteristic
79C4×C211#13711yescharacteristic
80C2×C211#44811yescharacteristic
81C2×C211#45111yescharacteristic
82C2×C211#45511yescharacteristic
83C211#164511yescharacteristic, direct product factor, Sylow
84#1645C21111yesmaximal, characteristic, direct product factor, Sylow
85Q82C2×C21111yescharacteristic
86#487C2×C21111yescharacteristic
87#475C2×C21111yescharacteristic
88#462C2×C21111yescharacteristic
89#333C2×C21111yescharacteristic
90Q8⋊C8C2×C21111yescharacteristic
91Q8⋊C8C2×C21111yescharacteristic
92C2×Q16—21no—
93C4⋊Q8—22no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8C22×C21111yescharacteristic
97C4⋊Q8C22×C21111yescharacteristic
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8C22×C21111yescharacteristic
101C4×Q8C22×C21111yescharacteristic
102#106—21no—
103C4⋊C8C22×C21111yescharacteristic
104C4⋊C8C22×C21111yescharacteristic
105#103—21no—
106#103—22no—
107#103—21no—
108C4×C8C22×C21111yescharacteristic
109C2×Q8—22no—
110C2×Q8—21no—
111C2×Q8D4×C21111yescharacteristic
112C2×Q8D4×C21111yescharacteristic
113C2×Q8D4×C21112yes—
114Q16—42no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4D4×C21111yescharacteristic
127C4⋊C4D4×C21111yescharacteristic
128C42—22no—
129C42—21no—
130C42—21no—
131C42C23×C21111yescharacteristic
132Q8—21no—
133Q8—21no—
134Q8—42no—
135Q8—41no—
136Q8—41no—
137Q8—82no—
138Q8SD16×C21112yes—
139Q8Q16×C21112yescomplemented
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4C2×D4×C21111yescharacteristic
147C2×C4C2×D4×C21111yescharacteristic
148C2×C4C2×D4×C21111yescharacteristic
149C8—41no—
150C8—41no—
151C8—41no—
152C22#120×C21111yescharacteristic
153C4—22no—
154C4—22no—
155C4—21no—
156C4—42no—
157C4—41no—
158C4—41no—
159C4—81no—
160C4C2×SD16×C21111yescharacteristic
161C4C2×Q16×C21111yescharacteristic
162C4#137×C21111yescharacteristic
163C4#137×C21111yescharacteristic
164C2#448×C21111yescharacteristic
165C2#451×C21111yescharacteristic
166C2#455×C21111yescharacteristic
subgroupG/H
center, socle, prime radicalC22×C211#120
commutator, Frattini, cosocleC42C23×C211
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
55. Q8×C211Q16semidirect
83. C211#1645direct (N × H)
84. #1645C211direct (N × H)
139. Q8Q16×C211semidirect

Maximal subgroups

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

subgroupindex
Q82×C2112
#487×C2112
#475×C2112
#462×C2112
#333×C2112
Q8⋊C8×C2112
#1645211