Finite Group Databasebeta

← C3×#1894 #72609

Subgroups — C3×#1894 #72609

Order 384.

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

normal subgroups only

This group has 360 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
2C2×C3×C22⋊Q8C211yesmaximal, characteristic
3C2×C3×#103C212yesmaximal, complemented
4C3×#380C211yesmaximal, characteristic
5C3×#340C212yesmaximal, complemented
6#1894C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×Q8C2211yescharacteristic
8C23×C3×C4C2211yescharacteristic
9C2×C3×M16—22no—
10C2×C3×M16—24no—
11C22×C3×C8C2212yes—
12C3×C22⋊Q8—22no—
13C3×C22⋊Q8C414yescomplemented
14C2×C3×C4⋊C4—21no—
15C2×C3×C4⋊C4C2212yes—
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4—21no—
18C2×C3×C22⋊C4—21no—
19C3×#103—24no—
20C3×#95—21no—
21C22×M16C2×C311yescharacteristic
22C2×C22⋊Q8C2×C311yescharacteristic
23C2×#103C2×C312yescomplemented
24#380C2×C311yescharacteristic
25#340C2×C312yescomplemented
26C24×C3D411yescharacteristic
27C2×C3×Q8—21no—
28C2×C3×Q8—22no—
29C2×C3×Q8—22no—
30C2×C3×Q8C2×C412yes—
31C22×C3×C4—22no—
32C22×C3×C4—22no—
33C22×C3×C4—21no—
34C22×C3×C4—21no—
35C22×C3×C4—22no—
36C22×C3×C4C2×C412yes—
37C22×C3×C4D411yescharacteristic
38C22×C3×C4D412yes—
39C22×C3×C4C2311yescharacteristic
40C3×M16—28no—
41C2×C3×C8—22no—
42C2×C3×C8—22no—
43C2×C3×C8D414yes—
44C3×C4⋊C4—24no—
45C3×C4⋊C4—21no—
46C3×C4⋊C4—22no—
47C3×C4⋊C4—22no—
48C3×C4⋊C4C2×C412yes—
49C3×C22⋊C4—24no—
50C3×C22⋊C4—42no—
51C22×Q8C22×C311yescharacteristic
52C23×C4C22×C311yescharacteristic
53C2×M16—22no—
54C2×M16—24no—
55C22×C8C22×C312yes—
56C22⋊Q8—22no—
57C22⋊Q8C3×C414yescomplemented
58C2×C4⋊C4—21no—
59C2×C4⋊C4C22×C312yes—
60C2×C4⋊C4C22×C311yescharacteristic
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63#103—24no—
64#95—21no—
65C23×C3—22no—
66C23×C3—21no—
67C23×C3—21no—
68C23×C3—22no—
69C23×C3C22⋊C412yes—
70C23×C3C2×D411yescharacteristic
71C3×Q8—24no—
72C3×Q8—41no—
73C3×Q8—41no—
74C2×C3×C4—22no—
75C2×C3×C4—24no—
76C2×C3×C4—22no—
77C2×C3×C4—22no—
78C2×C3×C4—22no—
79C2×C3×C4—22no—
80C2×C3×C4—42no—
81C2×C3×C4—41no—
82C2×C3×C4—41no—
83C2×C3×C4—42no—
84C2×C3×C4—42no—
85C2×C3×C4—42no—
86C2×C3×C4C22⋊C412yes—
87C2×C3×C4C22×C411yescharacteristic
88C2×C3×C4C2×D412yes—
89C2×C3×C4C2×D411yescharacteristic
90C2×C3×C4#4112yes—
91C3×C8—24no—
92C24C3×D411yescharacteristic
93C2×Q8—21no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8C2×C3×C412yes—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4C2×C3×C412yes—
103C22×C4C3×D411yescharacteristic
104C22×C4C3×D412yes—
105C22×C4C23×C311yescharacteristic
106M16—28no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8C3×D414yes—
110C4⋊C4—24no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4C2×C3×C412yes—
115C22⋊C4—24no—
116C22⋊C4—42no—
117C22×C3—22no—
118C22×C3—22no—
119C22×C3—22no—
120C22×C3—42no—
121C22×C3—42no—
122C22×C3C2×C22⋊C411yescharacteristic
123C22×C3#12112yes—
124C22×C3AGL(1,ℤ/8)11yescharacteristic
125C22×C3#13711yescharacteristic
126C22×C3#13712yes—
127C3×C4—22no—
128C3×C4—42no—
129C3×C4—42no—
130C3×C4—82no—
131C3×C4C4×D412yes—
132C3×C4#12011yescharacteristic
133C3×C4#12311yescharacteristic
134C23—22no—
135C23—21no—
136C23—21no—
137C23—22no—
138C23C3×C22⋊C412yes—
139C23C2×C3×D411yescharacteristic
140Q8—24no—
141Q8—41no—
142Q8—41no—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—42no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4C3×C22⋊C412yes—
156C2×C4C22×C3×C411yescharacteristic
157C2×C4C2×C3×D412yes—
158C2×C4C2×C3×D411yescharacteristic
159C2×C4C3×#4112yes—
160C8—24no—
161C2×C3—42no—
162C2×C3#38611yescharacteristic
163C2×C3#41711yescharacteristic
164C2×C3#41911yescharacteristic
165C2×C3#46812yes—
166C2×C3#47012yes—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—42no—
171C22—42no—
172C22C2×C3×C22⋊C411yescharacteristic
173C22C3×#12112yes—
174C22C3×AGL(1,ℤ/8)11yescharacteristic
175C22C3×#13711yescharacteristic
176C22C3×#13712yes—
177C4—22no—
178C4—42no—
179C4—42no—
180C4—82no—
181C4C3×C4×D412yes—
182C4C3×#12011yescharacteristic
183C4C3×#12311yescharacteristic
184C3#189411yescharacteristic, direct product factor, Sylow
185C2—42no—
186C2C3×#38611yescharacteristic
187C2C3×#41711yescharacteristic
188C2C3×#41911yescharacteristic
189C2C3×#46812yes—
190C2C3×#47012yes—
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. #1894C3direct (N × H)
13. C3×C22⋊Q8C4semidirect
23. C2×#103C2×C3semidirect
25. #340C2×C3semidirect
57. C22⋊Q8C3×C4semidirect
184. C3#1894direct (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×C22⋊Q82
C2×C3×#1032
C3×#3802
C3×#3402
#18943