Finite Group Databasebeta

← C3×#1895 #72610

Subgroups — C3×#1895 #72610

Order 384.

412 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 128 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
1C22×C3×M16C211yesmaximal, characteristic, complemented
2C2×C3×#121C211yesmaximal, characteristic
3C2×C3×#102C212yesmaximal, complemented
4C3×#380C211yesmaximal, characteristic, complemented
5C3×#340C212yesmaximal, complemented
6#1895C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×D4—21no—
8C22×C3×D4C2211yescharacteristic
9C23×C3×C4C2211yescharacteristic
10C2×C3×M16—22no—
11C2×C3×M16—24no—
12C22×C3×C8C2212yes—
13C3×#121—22no—
14C3×#121C414yescomplemented
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4—21no—
18C2×C3×C22⋊C4—21no—
19C3×#102—24no—
20C3×#95—21no—
21C22×M16C2×C311yescharacteristic, complemented
22C2×#121C2×C311yescharacteristic
23C2×#102C2×C312yescomplemented
24#380C2×C311yescharacteristic, complemented
25#340C2×C312yescomplemented
26C24×C3—21no—
27C24×C3D411yescharacteristic
28C2×C3×D4—24no—
29C2×C3×D4—22no—
30C2×C3×D4—21no—
31C2×C3×D4—22no—
32C2×C3×D4—44no—
33C2×C3×D4C2×C412yes—
34C22×C3×C4—22no—
35C22×C3×C4—22no—
36C22×C3×C4—21no—
37C22×C3×C4—22no—
38C22×C3×C4C2×C412yes—
39C22×C3×C4D411yescharacteristic
40C22×C3×C4D412yes—
41C22×C3×C4C2311yescharacteristic
42C3×M16—28no—
43C2×C3×C8—22no—
44C2×C3×C8—22no—
45C2×C3×C8D414yescomplemented
46C3×C4⋊C4—21no—
47C3×C4⋊C4—22no—
48C3×C4⋊C4—22no—
49C3×C4⋊C4C2×C412yes—
50C3×C22⋊C4—24no—
51C3×C22⋊C4—42no—
52C22×D4—21no—
53C22×D4C22×C311yescharacteristic
54C23×C4C22×C311yescharacteristic
55C2×M16—22no—
56C2×M16—24no—
57C22×C8C22×C312yes—
58#121—22no—
59#121C3×C414yescomplemented
60C2×C4⋊C4C22×C312yes—
61C2×C4⋊C4C22×C311yescharacteristic
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64#102—24no—
65#95—21no—
66C23×C3—22no—
67C23×C3—21no—
68C23×C3—22no—
69C23×C3—21no—
70C23×C3—22no—
71C23×C3—42no—
72C23×C3—42no—
73C23×C3—41no—
74C23×C3—41no—
75C23×C3C22⋊C412yes—
76C23×C3C2×D411yescharacteristic
77C3×D4—24no—
78C3×D4—48no—
79C3×D4—41no—
80C3×D4—41no—
81C2×C3×C4—22no—
82C2×C3×C4—24no—
83C2×C3×C4—22no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—42no—
87C2×C3×C4—42no—
88C2×C3×C4—42no—
89C2×C3×C4—42no—
90C2×C3×C4C22⋊C412yes—
91C2×C3×C4C22×C411yescharacteristic
92C2×C3×C4C2×D412yes—
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4#4112yes—
95C3×C8—24no—
96C24—21no—
97C24C3×D411yescharacteristic
98C2×D4—24no—
99C2×D4—22no—
100C2×D4—21no—
101C2×D4—22no—
102C2×D4—44no—
103C2×D4C2×C3×C412yes—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4C2×C3×C412yes—
109C22×C4C3×D411yescharacteristic
110C22×C4C3×D412yes—
111C22×C4C23×C311yescharacteristic
112M16—28no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8C3×D414yescomplemented
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4C2×C3×C412yes—
120C22⋊C4—24no—
121C22⋊C4—42no—
122C22×C3—22no—
123C22×C3—22no—
124C22×C3—22no—
125C22×C3—42no—
126C22×C3—42no—
127C22×C3—42no—
128C22×C3—42no—
129C22×C3—42no—
130C22×C3—82no—
131C22×C3—82no—
132C22×C3C2×C22⋊C411yescharacteristic
133C22×C3#12112yes—
134C22×C3AGL(1,ℤ/8)11yescharacteristic
135C22×C3AGL(1,ℤ/8)12yes—
136C22×C3#13711yescharacteristic
137C3×C4—22no—
138C3×C4—42no—
139C3×C4—82no—
140C3×C4C4×D412yes—
141C3×C4#12011yescharacteristic
142C3×C4#12311yescharacteristic
143C23—22no—
144C23—21no—
145C23—22no—
146C23—21no—
147C23—22no—
148C23—42no—
149C23—42no—
150C23—41no—
151C23—41no—
152C23C3×C22⋊C412yes—
153C23C2×C3×D411yescharacteristic
154D4—24no—
155D4—48no—
156D4—41no—
157D4—41no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4C3×C22⋊C412yes—
168C2×C4C22×C3×C411yescharacteristic
169C2×C4C2×C3×D412yes—
170C2×C4C2×C3×D411yescharacteristic
171C2×C4C3×#4112yes—
172C8—24no—
173C2×C3—42no—
174C2×C3—82no—
175C2×C3#38611yescharacteristic
176C2×C3#41711yescharacteristic
177C2×C3#41811yescharacteristic
178C2×C3#46812yes—
179C2×C3#46912yes—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—82no—
189C22—82no—
190C22C2×C3×C22⋊C411yescharacteristic
191C22C3×#12112yes—
192C22C3×AGL(1,ℤ/8)11yescharacteristic
193C22C3×AGL(1,ℤ/8)12yes—
194C22C3×#13711yescharacteristic
195C4—22no—
196C4—42no—
197C4—82no—
198C4C3×C4×D412yes—
199C4C3×#12011yescharacteristic
200C4C3×#12311yescharacteristic
201C3#189511yescharacteristic, direct product factor, Sylow
202C2—42no—
203C2—82no—
204C2C3×#38611yescharacteristic
205C2C3×#41711yescharacteristic
206C2C3×#41811yescharacteristic
207C2C3×#46812yes—
208C2C3×#46912yes—
subgroupG/H
center, socleC23×C3C2×D4
commutatorC2×C4C22×C3×C4
Frattini, cosocleC22×C4C23×C3
prime radicalC2×C3×#121C2
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
1. C22×C3×M16C2semidirect
3. C2×C3×#102C2semidirect
4. C3×#380C2semidirect
5. C3×#340C2semidirect
6. #1895C3direct (N × H)
14. C3×#121C4semidirect
21. C22×M16C2×C3semidirect
23. C2×#102C2×C3semidirect
24. #380C2×C3semidirect
25. #340C2×C3semidirect
45. C2×C3×C8D4semidirect
59. #121C3×C4semidirect
115. C2×C8C3×D4semidirect
201. C3#1895direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×M162
C2×C3×#1212
C2×C3×#1022
C3×#3802
C3×#3402
#18953