Finite Group Databasebeta

← C3×#1872 #72587

Subgroups — C3×#1872 #72587

Order 384.

404 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C2×C3×C22⋊Q8C211yesmaximal, characteristic
2C22×C3×C4⋊C4C211yesmaximal, characteristic
3C2×C3×#103C212yesmaximal, complemented
4C2×C3×C22⋊C8C211yesmaximal, characteristic
5C3×#340C212yesmaximal, complemented
6#1872C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×Q8C2211yescharacteristic
8C23×C3×C4—21no—
9C23×C3×C4C2211yescharacteristic
10C22×C3×C8C2212yes—
11C3×C22⋊Q8—22no—
12C3×C22⋊Q8C414yescomplemented
13C2×C3×C4⋊C4—21no—
14C2×C3×C4⋊C4—21no—
15C2×C3×C4⋊C4—24no—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18C2×C3×C22⋊C4—21no—
19C3×#103—24no—
20C3×C22⋊C8—22no—
21C2×C22⋊Q8C2×C311yescharacteristic
22C22×C4⋊C4C2×C311yescharacteristic
23C2×#103C2×C312yescomplemented
24C2×C22⋊C8C2×C311yescharacteristic
25#340C2×C312yescomplemented
26C24×C3D411yescharacteristic
27C2×C3×Q8—21no—
28C2×C3×Q8—22no—
29C2×C3×Q8—22no—
30C2×C3×Q8C2×C412yescomplemented
31C22×C3×C4—22no—
32C22×C3×C4—22no—
33C22×C3×C4—24no—
34C22×C3×C4—21no—
35C22×C3×C4—21no—
36C22×C3×C4—22no—
37C22×C3×C4—42no—
38C22×C3×C4—42no—
39C22×C3×C4C2×C412yes—
40C22×C3×C4D411yescharacteristic
41C22×C3×C4D412yes—
42C22×C3×C4C2311yescharacteristic
43C2×C3×C8—22no—
44C2×C3×C8—22no—
45C2×C3×C8—22no—
46C3×C4⋊C4—24no—
47C3×C4⋊C4—24no—
48C3×C4⋊C4—22no—
49C3×C4⋊C4—21no—
50C3×C4⋊C4C2×C412yescomplemented
51C3×C4⋊C4D414yes—
52C3×C22⋊C4—24no—
53C22×Q8C22×C311yescharacteristic
54C23×C4—21no—
55C23×C4C22×C311yescharacteristic
56C22×C8C22×C312yes—
57C22⋊Q8—22no—
58C22⋊Q8C3×C414yescomplemented
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—24no—
62C2×C4⋊C4C22×C311yescharacteristic
63C2×C4⋊C4C22×C312yes—
64C2×C22⋊C4—21no—
65#103—24no—
66C22⋊C8—22no—
67C23×C3—21no—
68C23×C3—21no—
69C23×C3—21no—
70C23×C3—21no—
71C23×C3C22⋊C412yes—
72C23×C3SD1612yes—
73C23×C3Q1612yes—
74C23×C3C2×D411yescharacteristic
75C3×Q8—24no—
76C3×Q8—42no—
77C2×C3×C4—21no—
78C2×C3×C4—21no—
79C2×C3×C4—24no—
80C2×C3×C4—24no—
81C2×C3×C4—22no—
82C2×C3×C4—22no—
83C2×C3×C4—24no—
84C2×C3×C4—48no—
85C2×C3×C4—41no—
86C2×C3×C4—41no—
87C2×C3×C4—42no—
88C2×C3×C4—41no—
89C2×C3×C4—42no—
90C2×C3×C4—41no—
91C2×C3×C4C22⋊C412yes—
92C2×C3×C4C22×C411yescharacteristic
93C2×C3×C4C2×D412yes—
94C2×C3×C4C2×D411yescharacteristic
95C2×C3×C4#4112yes—
96C3×C8—42no—
97C24C3×D411yescharacteristic
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8C2×C3×C412yescomplemented
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—24no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4C2×C3×C412yes—
111C22×C4C3×D411yescharacteristic
112C22×C4C3×D412yes—
113C22×C4C23×C311yescharacteristic
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—21no—
121C4⋊C4C2×C3×C412yescomplemented
122C4⋊C4C3×D414yes—
123C22⋊C4—24no—
124C22×C3—22no—
125C22×C3—22no—
126C22×C3—22no—
127C22×C3—22no—
128C22×C3—22no—
129C22×C3—22no—
130C22×C3#10314yescomplemented
131C22×C3C2×C22⋊C411yescharacteristic
132C22×C3#12011yescharacteristic
133C22×C3#12311yescharacteristic
134C22×C3C2×SD1611yescharacteristic
135C22×C3C2×Q1611yescharacteristic
136C22×C3AGL(1,ℤ/8)11yescharacteristic
137C22×C3#13711yescharacteristic
138C3×C4—22no—
139C3×C4—42no—
140C3×C4—42no—
141C3×C4—44no—
142C3×C4C4×D412yes—
143C3×C4#12112yes—
144C23—21no—
145C23—21no—
146C23—21no—
147C23—21no—
148C23C3×C22⋊C412yes—
149C23C3×SD1612yes—
150C23C3×Q1612yes—
151C23C2×C3×D411yescharacteristic
152Q8—24no—
153Q8—42no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—48no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4C3×C22⋊C412yes—
169C2×C4C22×C3×C411yescharacteristic
170C2×C4C2×C3×D412yes—
171C2×C4C2×C3×D411yescharacteristic
172C2×C4C3×#4112yes—
173C8—42no—
174C2×C3—24no—
175C2×C3#38611yescharacteristic
176C2×C3C2×#10311yescharacteristic
177C2×C3#41711yescharacteristic
178C2×C3#45011yescharacteristic
179C2×C3#45111yescharacteristic
180C2×C3#48311yescharacteristic
181C2×C3#48411yescharacteristic
182C22—22no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22C3×#10314yescomplemented
189C22C2×C3×C22⋊C411yescharacteristic
190C22C3×#12011yescharacteristic
191C22C3×#12311yescharacteristic
192C22C2×C3×SD1611yescharacteristic
193C22C2×C3×Q1611yescharacteristic
194C22C3×AGL(1,ℤ/8)11yescharacteristic
195C22C3×#13711yescharacteristic
196C4—22no—
197C4—42no—
198C4—42no—
199C4—44no—
200C4C3×C4×D412yes—
201C4C3×#12112yes—
202C3#187211yescharacteristic, direct product factor, Sylow
203C2—24no—
204C2C3×#38611yescharacteristic
205C2C2×C3×#10311yescharacteristic
206C2C3×#41711yescharacteristic
207C2C3×#45011yescharacteristic
208C2C3×#45111yescharacteristic
209C2C3×#48311yescharacteristic
210C2C3×#48411yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutatorC2×C4C22×C3×C4
Frattini, cosocleC22×C4C23×C3
prime radicalC24×C3D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C3×#103C2semidirect
5. C3×#340C2semidirect
6. #1872C3direct (N × H)
12. C3×C22⋊Q8C4semidirect
23. C2×#103C2×C3semidirect
25. #340C2×C3semidirect
30. C2×C3×Q8C2×C4semidirect
50. C3×C4⋊C4C2×C4semidirect
58. C22⋊Q8C3×C4semidirect
101. C2×Q8C2×C3×C4semidirect
121. C4⋊C4C2×C3×C4semidirect
130. C22×C3#103semidirect
188. C22C3×#103semidirect
202. C3#1872direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C22×C3×C4⋊C42
C2×C3×#1032
C2×C3×C22⋊C82
C3×#3402
#18723