Finite Group Databasebeta

← C3×#1500 #72215

Subgroups — C3×#1500 #72215

Order 384.

238 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 92 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×#529C211yesmaximal, characteristic
2C3×#431C211yesmaximal, characteristic, complemented
3C3×#376C211yesmaximal, characteristic, complemented
4C3×#328C212yesmaximal, complemented
5C3×#327C212yesmaximal
6#1500C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#126—21no—
8C3×#125C411yescharacteristic, complemented
9C3×#124C411yescharacteristic, complemented
10C3×#123—21no—
11C3×C22⋊Q8C2211yescharacteristic
12C3×#121C2211yescharacteristic
13C3×C4×Q8C411yescharacteristic, complemented
14C3×C4×D4—21no—
15C3×C4×D4C411yescharacteristic, complemented
16C3×#117—21no—
17C2×C3×C42—21no—
18C2×C3×C42C2211yescharacteristic
19C3×C4⋊C8—21no—
20C3×C22⋊C8C2212yes—
21C3×C4×C8—21no—
22C3×#95—21no—
23C3×#95C2212yes—
24#529C2×C311yescharacteristic
25#431C2×C311yescharacteristic, complemented
26#376C2×C311yescharacteristic, complemented
27#328C2×C312yescomplemented
28#327C2×C312yes—
29C2×C3×Q8C2×C411yescharacteristic
30C2×C3×D4—21no—
31C2×C3×D4C2×C411yescharacteristic
32C22×C3×C4—22no—
33C22×C3×C4—21no—
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C22×C3×C4C2311yescharacteristic
37C2×C3×C8—22no—
38C3×C4⋊C4—21no—
39C3×C4⋊C4—21no—
40C3×C4⋊C4—21no—
41C3×C4⋊C4—21no—
42C3×C4⋊C4C2×C411yescharacteristic
43C3×C4⋊C4C2×C411yescharacteristic
44C3×C22⋊C4—21no—
45C3×C22⋊C4—21no—
46C3×C22⋊C4—21no—
47C3×C22⋊C4—21no—
48C3×C22⋊C4—21no—
49C3×C42—24no—
50C3×C42—21no—
51C3×C42C2×C411yescharacteristic
52C3×C42C2×C411yescharacteristic
53C3×C42D411yescharacteristic
54C3×C42D411yescharacteristic
55#126—21no—
56#125C3×C411yescharacteristic, complemented
57#124C3×C411yescharacteristic, complemented
58#123—21no—
59C22⋊Q8C22×C311yescharacteristic
60#121C22×C311yescharacteristic
61C4×Q8C3×C411yescharacteristic, complemented
62C4×D4—21no—
63C4×D4C3×C411yescharacteristic, complemented
64#117—21no—
65C2×C42—21no—
66C2×C42C22×C311yescharacteristic
67C4⋊C8—21no—
68C22⋊C8C22×C312yes—
69C4×C8—21no—
70#95—21no—
71#95C22×C312yes—
72C23×C3—21no—
73C23×C3C2×D411yescharacteristic
74C3×Q8—41no—
75C3×D4—41no—
76C3×D4—42no—
77C2×C3×C4—22no—
78C2×C3×C4—22no—
79C2×C3×C4—24no—
80C2×C3×C4—21no—
81C2×C3×C4—21no—
82C2×C3×C4—21no—
83C2×C3×C4—41no—
84C2×C3×C4—42no—
85C2×C3×C4—42no—
86C2×C3×C4—42no—
87C2×C3×C4C22⋊C411yescharacteristic
88C2×C3×C4C22⋊C411yescharacteristic
89C2×C3×C4C22⋊C411yescharacteristic
90C2×C3×C4C22⋊C411yescharacteristic
91C2×C3×C4C22×C411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93C3×C8—42no—
94C2×Q8C2×C3×C411yescharacteristic
95C2×D4—21no—
96C2×D4C2×C3×C411yescharacteristic
97C22×C4—22no—
98C22×C4—21no—
99C22×C4C3×D411yescharacteristic
100C22×C4C3×D411yescharacteristic
101C22×C4C23×C311yescharacteristic
102C2×C8—22no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4C2×C3×C411yescharacteristic
108C4⋊C4C2×C3×C411yescharacteristic
109C22⋊C4—21no—
110C22⋊C4—21no—
111C22⋊C4—21no—
112C22⋊C4—21no—
113C22⋊C4—21no—
114C42—24no—
115C42—21no—
116C42C2×C3×C411yescharacteristic
117C42C2×C3×C411yescharacteristic
118C42C3×D411yescharacteristic
119C42C3×D411yescharacteristic
120C22×C3—21no—
121C22×C3—21no—
122C22×C3—41no—
123C22×C3—41no—
124C22×C3—41no—
125C22×C3C2×C22⋊C411yescharacteristic
126C22×C3#13512yes—
127C3×C4—21no—
128C3×C4—21no—
129C3×C4—21no—
130C3×C4—21no—
131C3×C4—21no—
132C3×C4—44no—
133C3×C4—41no—
134C3×C4—41no—
135C3×C4—41no—
136C3×C4#10412yes—
137C23—21no—
138C23C2×C3×D411yescharacteristic
139Q8—41no—
140D4—41no—
141D4—42no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4C3×C22⋊C411yescharacteristic
153C2×C4C3×C22⋊C411yescharacteristic
154C2×C4C3×C22⋊C411yescharacteristic
155C2×C4C3×C22⋊C411yescharacteristic
156C2×C4C22×C3×C411yescharacteristic
157C2×C4C2×C3×D411yescharacteristic
158C8—42no—
159C2×C3—22no—
160C2×C3—81no—
161C2×C3#41011yescharacteristic
162C2×C3#41611yescharacteristic
163C2×C3C2×#10411yescharacteristic
164C22—21no—
165C22—21no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22C2×C3×C22⋊C411yescharacteristic
170C22C3×#13512yes—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—44no—
177C4—41no—
178C4—41no—
179C4—41no—
180C4C3×#10412yes—
181C3#150011yescharacteristic, direct product factor, Sylow
182C2—22no—
183C2—81no—
184C2C3×#41011yescharacteristic
185C2C3×#41611yescharacteristic
186C2C2×C3×#10411yescharacteristic
subgroupG/H
centerC2×C3×C4C22⋊C4
commutatorC2×C4C22×C3×C4
Frattini, cosocleC22×C4C23×C3
socleC22×C3C2×C22⋊C4
prime radicalC3×#121C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#431C2semidirect
3. C3×#376C2semidirect
4. C3×#328C2semidirect
6. #1500C3direct (N × H)
8. C3×#125C4semidirect
9. C3×#124C4semidirect
13. C3×C4×Q8C4semidirect
15. C3×C4×D4C4semidirect
25. #431C2×C3semidirect
26. #376C2×C3semidirect
27. #328C2×C3semidirect
56. #125C3×C4semidirect
57. #124C3×C4semidirect
61. C4×Q8C3×C4semidirect
63. C4×D4C3×C4semidirect
181. C3#1500direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5292
C3×#4312
C3×#3762
C3×#3282
C3×#3272
#15003