Finite Group Databasebeta

← C3×#1524 #72239

Subgroups — C3×#1524 #72239

Order 384.

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

normal subgroups only

This group has 234 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×C4×M16C211yesmaximal, characteristic, complemented
3C2×C3×#97C211yesmaximal, characteristic, complemented
4C3×#330C212yesmaximal, complemented
5C3×#329C212yesmaximal, complemented
6#1524C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×M16—21no—
8C22×C3×C8—21no—
9C3×#126—21no—
10C3×#125C2211yescharacteristic
11C3×#124C2211yescharacteristic
12C3×#123—21no—
13C3×C22⋊Q8C411yescharacteristic
14C3×#121C411yescharacteristic
15C3×C4×Q8C411yescharacteristic
16C3×C4×D4—21no—
17C3×C4×D4C411yescharacteristic
18C3×#117—21no—
19C2×C3×C42C2211yescharacteristic
20C3×#97—21no—
21C3×#97C2212yes—
22C3×#97C2212yes—
23C3×C4×C8—21no—
24#529C2×C311yescharacteristic
25C4×M16C2×C311yescharacteristic, complemented
26C2×#97C2×C311yescharacteristic, complemented
27#330C2×C312yescomplemented
28#329C2×C312yescomplemented
29C2×C3×Q8C2×C411yescharacteristic
30C2×C3×D4—21no—
31C2×C3×D4C2×C411yescharacteristic
32C22×C3×C4—21no—
33C22×C3×C4C2×C411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4D411yescharacteristic
36C3×M16—24no—
37C2×C3×C8—24no—
38C2×C3×C8—22no—
39C2×C3×C8—22no—
40C3×C4⋊C4—21no—
41C3×C4⋊C4—21no—
42C3×C4⋊C4—21no—
43C3×C4⋊C4—21no—
44C3×C4⋊C4C2×C411yescharacteristic
45C3×C4⋊C4C2×C411yescharacteristic
46C3×C22⋊C4—21no—
47C3×C22⋊C4—21no—
48C3×C22⋊C4—21no—
49C3×C22⋊C4—21no—
50C3×C22⋊C4—21no—
51C3×C42—21no—
52C3×C42C2×C411yescharacteristic
53C3×C42D411yescharacteristic
54C3×C42D411yescharacteristic
55C3×C42C2311yescharacteristic
56C2×M16—21no—
57C22×C8—21no—
58#126—21no—
59#125C22×C311yescharacteristic
60#124C22×C311yescharacteristic
61#123—21no—
62C22⋊Q8C3×C411yescharacteristic
63#121C3×C411yescharacteristic
64C4×Q8C3×C411yescharacteristic
65C4×D4—21no—
66C4×D4C3×C411yescharacteristic
67#117—21no—
68C2×C42C22×C311yescharacteristic
69#97—21no—
70#97C22×C312yes—
71#97C22×C312yes—
72C4×C8—21no—
73C23×C3—21no—
74C23×C3C22⋊C411yescharacteristic
75C3×Q8—41no—
76C3×D4—41no—
77C3×D4—42no—
78C2×C3×C4—22no—
79C2×C3×C4—22no—
80C2×C3×C4—21no—
81C2×C3×C4—21no—
82C2×C3×C4—21no—
83C2×C3×C4—41no—
84C2×C3×C4—42no—
85C2×C3×C4C22⋊C411yescharacteristic
86C2×C3×C4C22⋊C411yescharacteristic
87C2×C3×C4C22⋊C411yescharacteristic
88C2×C3×C4C22×C411yescharacteristic
89C2×C3×C4C2×D411yescharacteristic
90C2×C3×C4C2×D411yescharacteristic
91C3×C8—24no—
92C3×C8—24no—
93C2×Q8C2×C3×C411yescharacteristic
94C2×D4—21no—
95C2×D4C2×C3×C411yescharacteristic
96C22×C4—21no—
97C22×C4C2×C3×C411yescharacteristic
98C22×C4C3×D411yescharacteristic
99C22×C4C3×D411yescharacteristic
100M16—24no—
101C2×C8—24no—
102C2×C8—22no—
103C2×C8—22no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4C2×C3×C411yescharacteristic
109C4⋊C4C2×C3×C411yescharacteristic
110C22⋊C4—21no—
111C22⋊C4—21no—
112C22⋊C4—21no—
113C22⋊C4—21no—
114C22⋊C4—21no—
115C42—21no—
116C42C2×C3×C411yescharacteristic
117C42C3×D411yescharacteristic
118C42C3×D411yescharacteristic
119C42C23×C311yescharacteristic
120C22×C3—21no—
121C22×C3—21no—
122C22×C3—41no—
123C22×C3—41no—
124C22×C3—41no—
125C22×C3#10412yes—
126C22×C3C2×C22⋊C411yescharacteristic
127C3×C4—21no—
128C3×C4—21no—
129C3×C4—21no—
130C3×C4—21no—
131C3×C4—21no—
132C3×C4—41no—
133C3×C4—41no—
134C3×C4—41no—
135C3×C4#10412yes—
136C23—21no—
137C23C3×C22⋊C411yescharacteristic
138Q8—41no—
139D4—41no—
140D4—42no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4C3×C22⋊C411yescharacteristic
149C2×C4C3×C22⋊C411yescharacteristic
150C2×C4C3×C22⋊C411yescharacteristic
151C2×C4C22×C3×C411yescharacteristic
152C2×C4C2×C3×D411yescharacteristic
153C2×C4C2×C3×D411yescharacteristic
154C8—24no—
155C8—24no—
156C2×C3—22no—
157C2×C3—81no—
158C2×C3#41311yescharacteristic
159C2×C3C2×#10411yescharacteristic
160C2×C3C2×#10411yescharacteristic
161C22—21no—
162C22—21no—
163C22—41no—
164C22—41no—
165C22—41no—
166C22C3×#10412yes—
167C22C2×C3×C22⋊C411yescharacteristic
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—41no—
174C4—41no—
175C4—41no—
176C4C3×#10412yes—
177C3#152411yescharacteristic, direct product factor, Sylow
178C2—22no—
179C2—81no—
180C2C3×#41311yescharacteristic
181C2C2×C3×#10411yescharacteristic
182C2C2×C3×#10411yescharacteristic
subgroupG/H
centerC2×C3×C4C22⋊C4
commutatorC2×C4C22×C3×C4
Frattini, cosocleC42C23×C3
socleC22×C3C2×C22⋊C4
prime radicalC3×#121C4
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×C4×M16C2semidirect
3. C2×C3×#97C2semidirect
4. C3×#330C2semidirect
5. C3×#329C2semidirect
6. #1524C3direct (N × H)
25. C4×M16C2×C3semidirect
26. C2×#97C2×C3semidirect
27. #330C2×C3semidirect
28. #329C2×C3semidirect
177. C3#1524direct (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×C4×M162
C2×C3×#972
C3×#3302
C3×#3292
#15243