Finite Group Databasebeta

← C3×#1809 #72524

Subgroups — C3×#1809 #72524

Order 384.

316 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 136 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×#107C211yesmaximal, characteristic, complemented
3C2×C3×#106C211yesmaximal, characteristic, complemented
4C3×#380C212yesmaximal
5C3×#340C212yesmaximal, complemented
6#1809C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×C4C2211yescharacteristic
8C2×C3×M16—22no—
9C2×C3×M16—22no—
10C2×C3×M16C414yescomplemented
11C22×C3×C8C2211yescharacteristic
12C22×C3×C8C2211yescharacteristic
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C4⋊C4C2212yes—
15C2×C3×C22⋊C4—22no—
16C3×#107—22no—
17C3×#106—22no—
18C3×#95—22no—
19C22×M16C2×C311yescharacteristic
20C2×#107C2×C311yescharacteristic, complemented
21C2×#106C2×C311yescharacteristic, complemented
22#380C2×C312yes—
23#340C2×C312yescomplemented
24C24×C3D411yescharacteristic
25C22×C3×C4—21no—
26C22×C3×C4—21no—
27C22×C3×C4—21no—
28C22×C3×C4—21no—
29C22×C3×C4—22no—
30C22×C3×C4—22no—
31C22×C3×C4C2×C412yes—
32C22×C3×C4D411yescharacteristic
33C22×C3×C4Q812yes—
34C22×C3×C4C2311yescharacteristic
35C3×M16—28no—
36C2×C3×C8—21no—
37C2×C3×C8—21no—
38C2×C3×C8C2×C412yes—
39C2×C3×C8C2×C412yes—
40C2×C3×C8D412yes—
41C2×C3×C8D412yes—
42C3×C4⋊C4—22no—
43C3×C4⋊C4—22no—
44C3×C4⋊C4—22no—
45C3×C4⋊C4—22no—
46C3×C22⋊C4—44no—
47C23×C4C22×C311yescharacteristic
48C2×M16—22no—
49C2×M16—22no—
50C2×M16C3×C414yescomplemented
51C22×C8C22×C311yescharacteristic
52C22×C8C22×C311yescharacteristic
53C2×C4⋊C4C22×C312yes—
54C2×C4⋊C4C22×C312yes—
55C2×C22⋊C4—22no—
56#107—22no—
57#106—22no—
58#95—22no—
59C23×C3—21no—
60C23×C3—21no—
61C23×C3—22no—
62C23×C3—22no—
63C23×C3C4⋊C412yes—
64C23×C3C2×D411yescharacteristic
65C2×C3×C4—21no—
66C2×C3×C4—21no—
67C2×C3×C4—22no—
68C2×C3×C4—22no—
69C2×C3×C4—22no—
70C2×C3×C4—22no—
71C2×C3×C4—42no—
72C2×C3×C4—42no—
73C2×C3×C4—42no—
74C2×C3×C4—42no—
75C2×C3×C4—42no—
76C2×C3×C4—42no—
77C2×C3×C4C4⋊C412yes—
78C2×C3×C4C22×C411yescharacteristic
79C2×C3×C4C2×D411yescharacteristic
80C2×C3×C4C2×D411yescharacteristic
81C2×C3×C4C2×Q811yescharacteristic
82C2×C3×C4#4111yescharacteristic
83C2×C3×C4#4111yescharacteristic
84C3×C8—22no—
85C3×C8C22⋊C414yescomplemented
86C24C3×D411yescharacteristic
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4C2×C3×C412yes—
94C22×C4C3×D411yescharacteristic
95C22×C4C3×Q812yes—
96C22×C4C23×C311yescharacteristic
97M16—28no—
98C2×C8—21no—
99C2×C8—21no—
100C2×C8C2×C3×C412yes—
101C2×C8C2×C3×C412yes—
102C2×C8C3×D412yes—
103C2×C8C3×D412yes—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C22⋊C4—44no—
109C22×C3—22no—
110C22×C3—24no—
111C22×C3—42no—
112C22×C3—41no—
113C22×C3—41no—
114C22×C3C2×C4⋊C411yescharacteristic
115C22×C3#12111yescharacteristic
116C22×C3#12111yescharacteristic
117C22×C3AGL(1,ℤ/8)12yes—
118C22×C3#13712yes—
119C3×C4—22no—
120C3×C4—82no—
121C3×C4—82no—
122C3×C4C2×C22⋊C411yescharacteristic
123C3×C4#11711yescharacteristic
124C3×C4C22⋊Q811yescharacteristic
125C3×C4C22⋊Q811yescharacteristic
126C23—21no—
127C23—21no—
128C23—22no—
129C23—22no—
130C23C3×C4⋊C412yes—
131C23C2×C3×D411yescharacteristic
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4C3×C4⋊C412yes—
145C2×C4C22×C3×C411yescharacteristic
146C2×C4C2×C3×D411yescharacteristic
147C2×C4C2×C3×D411yescharacteristic
148C2×C4C2×C3×Q811yescharacteristic
149C2×C4C3×#4111yescharacteristic
150C2×C4C3×#4111yescharacteristic
151C8—22no—
152C8C3×C22⋊C414yescomplemented
153C2×C3—42no—
154C2×C3#38011yescharacteristic
155C2×C3#42812yes—
156C2×C3#46812yes—
157C2×C3#46911yescharacteristic
158C2×C3#47011yescharacteristic
159C22—22no—
160C22—24no—
161C22—42no—
162C22—41no—
163C22—41no—
164C22C2×C3×C4⋊C411yescharacteristic
165C22C3×#12111yescharacteristic
166C22C3×#12111yescharacteristic
167C22C3×AGL(1,ℤ/8)12yes—
168C22C3×#13712yes—
169C4—22no—
170C4—82no—
171C4—82no—
172C4C2×C3×C22⋊C411yescharacteristic
173C4C3×#11711yescharacteristic
174C4C3×C22⋊Q811yescharacteristic
175C4C3×C22⋊Q811yescharacteristic
176C3#180911yescharacteristic, direct product factor, Sylow
177C2—42no—
178C2C3×#38011yescharacteristic
179C2C3×#42812yes—
180C2C3×#46812yes—
181C2C3×#46911yescharacteristic
182C2C3×#47011yescharacteristic
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
2. C2×C3×#107C2semidirect
3. C2×C3×#106C2semidirect
5. C3×#340C2semidirect
6. #1809C3direct (N × H)
10. C2×C3×M16C4semidirect
20. C2×#107C2×C3semidirect
21. C2×#106C2×C3semidirect
23. #340C2×C3semidirect
50. C2×M16C3×C4semidirect
85. C3×C8C22⋊C4semidirect
152. C8C3×C22⋊C4semidirect
176. C3#1809direct (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×#1072
C2×C3×#1062
C3×#3802
C3×#3402
#18093