Finite Group Databasebeta

← #1602×C211 #41009222

Subgroups — #1602×C211 #41009222

Order 27008.

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

normal subgroups only

This group has 256 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#543×C211C211yesmaximal, characteristic
2#451×C211C212yesmaximal, complemented
3#393×C211C211yesmaximal, characteristic
4#328×C211C212yesmaximal, complemented
5#323×C211C211yesmaximal, characteristic
6C22×Q8×C211C2211yescharacteristic
7C2×Q16×C211—22no—
8C22⋊Q8×C211—22no—
9C22⋊Q8×C211—21no—
10C22⋊Q8×C211—22no—
11C22⋊Q8×C211C2212yes—
12C2×C22⋊C4×C211—21no—
13C2×C22⋊C4×C211—21no—
14C2×C22⋊C4×C211C2211yescharacteristic
15#103×C211—22no—
16C22⋊C8×C211C2212yes—
17#95×C211—21no—
18#95×C211C2211yescharacteristic
19C24×C211D411yescharacteristic
20C2×Q8×C211—21no—
21C2×Q8×C211—22no—
22C2×Q8×C211—42no—
23C2×Q8×C211D412yes—
24C22×C4×C211—21no—
25C22×C4×C211—21no—
26C22×C4×C211—21no—
27C22×C4×C211—21no—
28C22×C4×C211D411yescharacteristic
29C22×C4×C211C2311yescharacteristic
30Q16×C211—44no—
31C2×C8×C211—22no—
32C4⋊C4×C211—21no—
33C4⋊C4×C211—22no—
34C4⋊C4×C211—22no—
35C4⋊C4×C211D412yes—
36C22⋊C4×C211—22no—
37C22⋊C4×C211—22no—
38C22⋊C4×C211—22no—
39C22⋊C4×C211—42no—
40C23×C211—21no—
41C23×C211—21no—
42C23×C211—41no—
43C23×C211—41no—
44C23×C211Q1612yescomplemented
45C23×C211C2×D411yescharacteristic
46Q8×C211—24no—
47Q8×C211—42no—
48Q8×C211—42no—
49Q8×C211—81no—
50C2×C4×C211—22no—
51C2×C4×C211—22no—
52C2×C4×C211—21no—
53C2×C4×C211—22no—
54C2×C4×C211—41no—
55C2×C4×C211—41no—
56C2×C4×C211—41no—
57C2×C4×C211—41no—
58C2×C4×C211—41no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211—81no—
62C2×C4×C211—81no—
63C2×C4×C211C2×D412yes—
64C8×C211—42no—
65C22×C211—21no—
66C22×C211—21no—
67C22×C211—22no—
68C22×C211—42no—
69C22×C211—41no—
70C22×C211—41no—
71C22×C211—81no—
72C22×C211#12011yescharacteristic
73C22×C211C2×Q1611yescharacteristic
74C22×C211#13711yescharacteristic
75C4×C211—22no—
76C4×C211—41no—
77C4×C211—42no—
78C4×C211—42no—
79C4×C211—42no—
80C4×C211—81no—
81C2×C211—21no—
82C2×C211—21no—
83C2×C211—42no—
84C2×C211#45111yescharacteristic
85C2×C211#45511yescharacteristic
86C2×C211B(4,𝔽2)11yescharacteristic
87C211#160211yescharacteristic, direct product factor, Sylow
88#1602C21111yesmaximal, characteristic, direct product factor, Sylow
89#543C2×C21111yescharacteristic
90#451C2×C21112yescomplemented
91#393C2×C21111yescharacteristic
92#328C2×C21112yescomplemented
93#323C2×C21111yescharacteristic
94C22×Q8C22×C21111yescharacteristic
95C2×Q16—22no—
96C22⋊Q8—22no—
97C22⋊Q8—21no—
98C22⋊Q8—22no—
99C22⋊Q8C22×C21112yes—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4C22×C21111yescharacteristic
103#103—22no—
104C22⋊C8C22×C21112yes—
105#95—21no—
106#95C22×C21111yescharacteristic
107C24D4×C21111yescharacteristic
108C2×Q8—21no—
109C2×Q8—22no—
110C2×Q8—42no—
111C2×Q8D4×C21112yes—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4D4×C21111yescharacteristic
117C22×C4C23×C21111yescharacteristic
118Q16—44no—
119C2×C8—22no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4D4×C21112yes—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C22⋊C4—42no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Q16×C21112yescomplemented
133C23C2×D4×C21111yescharacteristic
134Q8—24no—
135Q8—42no—
136Q8—42no—
137Q8—81no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—81no—
150C2×C4—81no—
151C2×C4C2×D4×C21112yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#120×C21111yescharacteristic
161C22C2×Q16×C21111yescharacteristic
162C22#137×C21111yescharacteristic
163C4—22no—
164C4—41no—
165C4—42no—
166C4—42no—
167C4—42no—
168C4—81no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#451×C21111yescharacteristic
173C2#455×C21111yescharacteristic
174C2B(4,𝔽2)×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC22×C4C23×C211
prime radicalC24×C211D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #451×C211C2semidirect
4. #328×C211C2semidirect
44. C23×C211Q16semidirect
87. C211#1602direct (N × H)
88. #1602C211direct (N × H)
90. #451C2×C211semidirect
92. #328C2×C211semidirect
132. C23Q16×C211semidirect

Maximal subgroups

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

subgroupindex
#543×C2112
#451×C2112
#393×C2112
#328×C2112
#323×C2112
#1602211