Finite Group Databasebeta

← #1599×C211 #41009219

Subgroups — #1599×C211 #41009219

Order 27008.

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

normal subgroups only

This group has 266 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#544×C211C211yesmaximal, characteristic, complemented
2#450×C211C212yesmaximal
3#394×C211C211yesmaximal, characteristic
4#328×C211C212yesmaximal, complemented
5#324×C211C211yesmaximal, characteristic, complemented
6C22×D4×C211C2211yescharacteristic
7C2×SD16×C211—22no—
8C4⋊Q8×C211—22no—
9#125×C211—22no—
10C22⋊Q8×C211—21no—
11C22⋊Q8×C211C2212yes—
12#117×C211—21no—
13C2×C4⋊C4×C211C2211yescharacteristic
14C2×C22⋊C4×C211—21no—
15C2×C22⋊C4×C211—21no—
16#102×C211—22no—
17C22⋊C8×C211C2212yes—
18#95×C211C2211yescharacteristic
19C24×C211—21no—
20C2×Q8×C211—21no—
21C2×Q8×C211D412yes—
22C2×D4×C211—22no—
23C2×D4×C211—42no—
24C22×C4×C211—21no—
25C22×C4×C211D411yescharacteristic
26C22×C4×C211D411yescharacteristic
27C22×C4×C211C2311yescharacteristic
28SD16×C211—44no—
29C2×C8×C211—22no—
30C4⋊C4×C211—21no—
31C4⋊C4×C211—21no—
32C4⋊C4×C211—21no—
33C4⋊C4×C211—22no—
34C4⋊C4×C211—22no—
35C4⋊C4×C211—22no—
36C4⋊C4×C211D412yes—
37C22⋊C4×C211—22no—
38C22⋊C4×C211—42no—
39C22⋊C4×C211—42no—
40C42×C211—22no—
41C23×C211—22no—
42C23×C211—41no—
43C23×C211—41no—
44C23×C211—41no—
45C23×C211—41no—
46C23×C211—41no—
47C23×C211—41no—
48C23×C211C2×D411yescharacteristic
49Q8×C211—42no—
50Q8×C211—81no—
51D4×C211—24no—
52D4×C211—42no—
53C2×C4×C211—21no—
54C2×C4×C211—21no—
55C2×C4×C211—22no—
56C2×C4×C211—22no—
57C2×C4×C211—21no—
58C2×C4×C211—41no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211SD1612yescomplemented
64C2×C4×C211C2×D412yes—
65C8×C211—42no—
66C22×C211—21no—
67C22×C211—21no—
68C22×C211—42no—
69C22×C211—42no—
70C22×C211—42no—
71C22×C211—81no—
72C22×C211—81no—
73C22×C211—81no—
74C22×C211—81no—
75C22×C211#12011yescharacteristic
76C22×C211C2×SD1611yescharacteristic
77C22×C211#13711yescharacteristic
78C4×C211—22no—
79C4×C211—22no—
80C4×C211—42no—
81C4×C211—41no—
82C4×C211—42no—
83C4×C211—81no—
84C2×C211—21no—
85C2×C211—21no—
86C2×C211—82no—
87C2×C211#44811yescharacteristic
88C2×C211#45411yescharacteristic
89C2×C211B(4,𝔽2)11yescharacteristic
90C211#159911yescharacteristic, direct product factor, Sylow
91#1599C21111yesmaximal, characteristic, direct product factor, Sylow
92#544C2×C21111yescharacteristic, complemented
93#450C2×C21112yes—
94#394C2×C21111yescharacteristic
95#328C2×C21112yescomplemented
96#324C2×C21111yescharacteristic, complemented
97C22×D4C22×C21111yescharacteristic
98C2×SD16—22no—
99C4⋊Q8—22no—
100#125—22no—
101C22⋊Q8—21no—
102C22⋊Q8C22×C21112yes—
103#117—21no—
104C2×C4⋊C4C22×C21111yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107#102—22no—
108C22⋊C8C22×C21112yes—
109#95C22×C21111yescharacteristic
110C24—21no—
111C2×Q8—21no—
112C2×Q8D4×C21112yes—
113C2×D4—22no—
114C2×D4—42no—
115C22×C4—21no—
116C22×C4D4×C21111yescharacteristic
117C22×C4D4×C21111yescharacteristic
118C22×C4C23×C21111yescharacteristic
119SD16—44no—
120C2×C8—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4D4×C21112yes—
128C22⋊C4—22no—
129C22⋊C4—42no—
130C22⋊C4—42no—
131C42—22no—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23C2×D4×C21111yescharacteristic
140Q8—42no—
141Q8—81no—
142D4—24no—
143D4—42no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4SD16×C21112yescomplemented
155C2×C4C2×D4×C21112yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22—81no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22#120×C21111yescharacteristic
167C22C2×SD16×C21111yescharacteristic
168C22#137×C21111yescharacteristic
169C4—22no—
170C4—22no—
171C4—42no—
172C4—41no—
173C4—42no—
174C4—81no—
175C2—21no—
176C2—21no—
177C2—82no—
178C2#448×C21111yescharacteristic
179C2#454×C21111yescharacteristic
180C2B(4,𝔽2)×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC22×C4C23×C211
prime radicalC22×D4×C211C22
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
1. #544×C211C2semidirect
4. #328×C211C2semidirect
5. #324×C211C2semidirect
63. C2×C4×C211SD16semidirect
90. C211#1599direct (N × H)
91. #1599C211direct (N × H)
92. #544C2×C211semidirect
95. #328C2×C211semidirect
96. #324C2×C211semidirect
154. C2×C4SD16×C211semidirect

Maximal subgroups

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

subgroupindex
#544×C2112
#450×C2112
#394×C2112
#328×C2112
#324×C2112
#1599211