Finite Group Databasebeta

← C3×#2010 #72725

Subgroups — C3×#2010 #72725

Order 384.

394 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C3×#575C211yesmaximal, characteristic, complemented
2C3×#525C212yesmaximal, complemented
3C3×#421C212yesmaximal, complemented
4C3×#407C211yesmaximal, characteristic, complemented
5C3×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
6#2010C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#41C2212yes—
8C23×C3×C4C2211yescharacteristic
9C3×#137—22no—
10C3×AGL(1,ℤ/8)—22no—
11C3×#135—22no—
12C2×C3×M16C2211yescharacteristic
13C2×C3×M16C2211yescharacteristic
14C3×#123—22no—
15C3×C22⋊Q8—22no—
16C3×#121—22no—
17C3×#120—22no—
18C3×C4×D4—24no—
19C3×#117C2212yes—
20C3×#104—24no—
21C3×C22⋊C8—22no—
22#575C2×C311yescharacteristic, complemented
23#525C2×C312yescomplemented
24#421C2×C312yescomplemented
25#407C2×C311yescharacteristic, complemented
26H(3,ℤ/4)C2×C311yescharacteristic, complemented
27C24×C3D411yescharacteristic
28C3×#41—22no—
29C3×#41—22no—
30C3×#41—22no—
31C3×#41—22no—
32C2×C3×Q8D412yescomplemented
33C2×C3×D4—24no—
34C2×C3×D4—22no—
35C2×C3×D4D412yescomplemented
36C22×C3×C4—21no—
37C22×C3×C4—22no—
38C22×C3×C4—22no—
39C22×C3×C4—22no—
40C22×C3×C4—21no—
41C22×C3×C4D411yescharacteristic
42C22×C3×C4C2311yescharacteristic
43C3×Q16—22no—
44C3×SD16—24no—
45C3×D8—22no—
46C3×M16—22no—
47C3×M16—22no—
48C2×C3×C8—21no—
49C2×C3×C8D412yescomplemented
50C3×C4⋊C4—24no—
51C3×C4⋊C4—22no—
52C3×C22⋊C4—24no—
53C3×C22⋊C4—24no—
54C3×C22⋊C4—22no—
55C3×C42D414yescomplemented
56C2×#41C22×C312yes—
57C23×C4C22×C311yescharacteristic
58#137—22no—
59AGL(1,ℤ/8)—22no—
60#135—22no—
61C2×M16C22×C311yescharacteristic
62C2×M16C22×C311yescharacteristic
63#123—22no—
64C22⋊Q8—22no—
65#121—22no—
66#120—22no—
67C4×D4—24no—
68#117C22×C312yes—
69#104—24no—
70C22⋊C8—22no—
71C23×C3—22no—
72C23×C3—21no—
73C23×C3—22no—
74C23×C3—42no—
75C23×C3C2×D411yescharacteristic
76C3×Q8—22no—
77C3×Q8—22no—
78C3×D4—22no—
79C3×D4—22no—
80C3×D4—48no—
81C3×D4—42no—
82C3×D4—42no—
83C2×C3×C4—24no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—21no—
87C2×C3×C4—41no—
88C2×C3×C4—42no—
89C2×C3×C4—42no—
90C2×C3×C4—41no—
91C2×C3×C4—42no—
92C2×C3×C4C2×D412yes—
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4C2×D412yes—
95C2×C3×C4#4111yescharacteristic
96C3×C8—22no—
97C3×C8—41no—
98C24C3×D411yescharacteristic
99#41—22no—
100#41—22no—
101#41—22no—
102#41—22no—
103C2×Q8C3×D412yescomplemented
104C2×D4—24no—
105C2×D4—22no—
106C2×D4C3×D412yescomplemented
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4C3×D411yescharacteristic
113C22×C4C23×C311yescharacteristic
114Q16—22no—
115SD16—24no—
116D8—22no—
117M16—22no—
118M16—22no—
119C2×C8—21no—
120C2×C8C3×D412yescomplemented
121C4⋊C4—24no—
122C4⋊C4—22no—
123C22⋊C4—24no—
124C22⋊C4—24no—
125C22⋊C4—22no—
126C42C3×D414yescomplemented
127C22×C3—24no—
128C22×C3—22no—
129C22×C3—41no—
130C22×C3—42no—
131C22×C3—41no—
132C22×C3—44no—
133C22×C3—41no—
134C22×C3—42no—
135C22×C3#12012yes—
136C22×C3#12111yescharacteristic
137C3×C4—44no—
138C3×C4—41no—
139C3×C4—42no—
140C3×C4#12011yescharacteristic
141C3×C4#12112yes—
142C3×C4#12711yescharacteristic
143C23—22no—
144C23—21no—
145C23—22no—
146C23—42no—
147C23C2×C3×D411yescharacteristic
148Q8—22no—
149Q8—22no—
150D4—22no—
151D4—22no—
152D4—48no—
153D4—42no—
154D4—42no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4C2×C3×D412yes—
165C2×C4C2×C3×D411yescharacteristic
166C2×C4C2×C3×D412yes—
167C2×C4C3×#4111yescharacteristic
168C8—22no—
169C8—41no—
170C2×C3—22no—
171C2×C3—21no—
172C2×C3—42no—
173C2×C3—82no—
174C2×C3#39211yescharacteristic
175C22—24no—
176C22—22no—
177C22—41no—
178C22—42no—
179C22—41no—
180C22—44no—
181C22—41no—
182C22—42no—
183C22C3×#12012yes—
184C22C3×#12111yescharacteristic
185C4—44no—
186C4—41no—
187C4—42no—
188C4C3×#12011yescharacteristic
189C4C3×#12112yes—
190C4C3×#12711yescharacteristic
191C3#201011yescharacteristic, direct product factor, Sylow
192C2—22no—
193C2—21no—
194C2—42no—
195C2—82no—
196C2C3×#39211yescharacteristic
subgroupG/H
centerC3×C4#127
commutator, Frattini, cosocleC22×C4C23×C3
socleC2×C3#392
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#575C2semidirect
2. C3×#525C2semidirect
3. C3×#421C2semidirect
4. C3×#407C2semidirect
5. C3×H(3,ℤ/4)C2semidirect
6. #2010C3direct (N × H)
22. #575C2×C3semidirect
23. #525C2×C3semidirect
24. #421C2×C3semidirect
25. #407C2×C3semidirect
26. H(3,ℤ/4)C2×C3semidirect
32. C2×C3×Q8D4semidirect
35. C2×C3×D4D4semidirect
49. C2×C3×C8D4semidirect
55. C3×C42D4semidirect
103. C2×Q8C3×D4semidirect
106. C2×D4C3×D4semidirect
120. C2×C8C3×D4semidirect
126. C42C3×D4semidirect
191. C3#2010direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5752
C3×#5252
C3×#4212
C3×#4072
C3×H(3,ℤ/4)2
#20103