Finite Group Databasebeta

← #1601×C211 #41009221

Subgroups — #1601×C211 #41009221

Order 27008.

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

normal subgroups only

This group has 308 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#534×C211C211yesmaximal, characteristic
2#448×C211C212yesmaximal, complemented
3#393×C211C211yesmaximal, characteristic, complemented
4#327×C211C212yesmaximal, complemented
5#323×C211C211yesmaximal, characteristic, complemented
6C22×Q8×C211C2211yescharacteristic, complemented
7C22×D4×C211—21no—
8C2×SD16×C211—22no—
9#123×C211—22no—
10#121×C211—21no—
11#121×C211C2212yes—
12#120×C211—22no—
13C2×C22⋊C4×C211—21no—
14C2×C22⋊C4×C211C2211yescharacteristic
15#103×C211—22no—
16C22⋊C8×C211C2212yescomplemented
17#95×C211—21no—
18#95×C211C2211yescharacteristic, complemented
19C24×C211—21no—
20C24×C211D411yescharacteristic
21C2×Q8×C211—22no—
22C2×Q8×C211—42no—
23C2×D4×C211—22no—
24C2×D4×C211—22no—
25C2×D4×C211—21no—
26C2×D4×C211—42no—
27C2×D4×C211—42no—
28C2×D4×C211D412yes—
29C22×C4×C211—21no—
30C22×C4×C211—21no—
31C22×C4×C211—21no—
32C22×C4×C211D411yescharacteristic
33C22×C4×C211C2311yescharacteristic
34SD16×C211—44no—
35C2×C8×C211—22no—
36C4⋊C4×C211—21no—
37C4⋊C4×C211D412yes—
38C22⋊C4×C211—22no—
39C22⋊C4×C211—22no—
40C22⋊C4×C211—22no—
41C22⋊C4×C211—42no—
42C23×C211—21no—
43C23×C211—21no—
44C23×C211—22no—
45C23×C211—41no—
46C23×C211—41no—
47C23×C211—42no—
48C23×C211—81no—
49C23×C211—81no—
50C23×C211SD1612yescomplemented
51C23×C211C2×D411yescharacteristic
52Q8×C211—24no—
53Q8×C211—42no—
54D4×C211—42no—
55D4×C211—44no—
56D4×C211—42no—
57D4×C211—42no—
58D4×C211—81no—
59C2×C4×C211—22no—
60C2×C4×C211—21no—
61C2×C4×C211—22no—
62C2×C4×C211—41no—
63C2×C4×C211—42no—
64C2×C4×C211—41no—
65C2×C4×C211—41no—
66C2×C4×C211—41no—
67C2×C4×C211—41no—
68C2×C4×C211—41no—
69C2×C4×C211—41no—
70C2×C4×C211C2×D412yes—
71C8×C211—42no—
72C22×C211—21no—
73C22×C211—21no—
74C22×C211—22no—
75C22×C211—42no—
76C22×C211—41no—
77C22×C211—41no—
78C22×C211—42no—
79C22×C211—42no—
80C22×C211—42no—
81C22×C211—81no—
82C22×C211—84no—
83C22×C211#12011yescharacteristic
84C22×C211C2×SD1611yescharacteristic
85C22×C211AGL(1,ℤ/8)11yescharacteristic
86C4×C211—22no—
87C4×C211—42no—
88C4×C211—41no—
89C4×C211—42no—
90C4×C211—81no—
91C2×C211—21no—
92C2×C211—21no—
93C2×C211—42no—
94C2×C211—82no—
95C2×C211#45011yescharacteristic
96C2×C211#45511yescharacteristic
97C2×C211B(4,𝔽2)11yescharacteristic
98C211#160111yescharacteristic, direct product factor, Sylow
99#1601C21111yesmaximal, characteristic, direct product factor, Sylow
100#534C2×C21111yescharacteristic
101#448C2×C21112yescomplemented
102#393C2×C21111yescharacteristic, complemented
103#327C2×C21112yescomplemented
104#323C2×C21111yescharacteristic, complemented
105C22×Q8C22×C21111yescharacteristic, complemented
106C22×D4—21no—
107C2×SD16—22no—
108#123—22no—
109#121—21no—
110#121C22×C21112yes—
111#120—22no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4C22×C21111yescharacteristic
114#103—22no—
115C22⋊C8C22×C21112yescomplemented
116#95—21no—
117#95C22×C21111yescharacteristic, complemented
118C24—21no—
119C24D4×C21111yescharacteristic
120C2×Q8—22no—
121C2×Q8—42no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4D4×C21112yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4D4×C21111yescharacteristic
132C22×C4C23×C21111yescharacteristic
133SD16—44no—
134C2×C8—22no—
135C4⋊C4—21no—
136C4⋊C4D4×C21112yes—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—42no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—41no—
145C23—41no—
146C23—42no—
147C23—81no—
148C23—81no—
149C23SD16×C21112yescomplemented
150C23C2×D4×C21111yescharacteristic
151Q8—24no—
152Q8—42no—
153D4—42no—
154D4—44no—
155D4—42no—
156D4—42no—
157D4—81no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4C2×D4×C21112yes—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—81no—
181C22—84no—
182C22#120×C21111yescharacteristic
183C22C2×SD16×C21111yescharacteristic
184C22AGL(1,ℤ/8)×C21111yescharacteristic
185C4—22no—
186C4—42no—
187C4—41no—
188C4—42no—
189C4—81no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—82no—
194C2#450×C21111yescharacteristic
195C2#455×C21111yescharacteristic
196C2B(4,𝔽2)×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC22×C4C23×C211
prime radical#534×C211C2
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. #448×C211C2semidirect
3. #393×C211C2semidirect
4. #327×C211C2semidirect
5. #323×C211C2semidirect
6. C22×Q8×C211C22semidirect
16. C22⋊C8×C211C22semidirect
18. #95×C211C22semidirect
50. C23×C211SD16semidirect
98. C211#1601direct (N × H)
99. #1601C211direct (N × H)
101. #448C2×C211semidirect
102. #393C2×C211semidirect
103. #327C2×C211semidirect
104. #323C2×C211semidirect
105. C22×Q8C22×C211semidirect
115. C22⋊C8C22×C211semidirect
117. #95C22×C211semidirect
149. C23SD16×C211semidirect

Maximal subgroups

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

subgroupindex
#534×C2112
#448×C2112
#393×C2112
#327×C2112
#323×C2112
#1601211