Finite Group Databasebeta

← C3×#1852 #72567

Subgroups — C3×#1852 #72567

Order 384.

360 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 144 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×C4⋊C4C211yesmaximal, characteristic
2C2×C3×#107C212yesmaximal, complemented
3C2×C3×C22⋊C8C211yesmaximal, characteristic
4C3×#380C211yesmaximal, characteristic
5C3×#340C212yesmaximal, complemented
6#1852C311yesmaximal, characteristic, direct product factor, Sylow
7C23×C3×C4—21no—
8C23×C3×C4C2211yescharacteristic
9C22×C3×C8C2212yes—
10C2×C3×C4⋊C4—21no—
11C2×C3×C4⋊C4—24no—
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2212yes—
15C2×C3×C22⋊C4—21no—
16C3×#107—24no—
17C3×C22⋊C8C414yescomplemented
18C3×#95—21no—
19C22×C4⋊C4C2×C311yescharacteristic
20C2×#107C2×C312yescomplemented
21C2×C22⋊C8C2×C311yescharacteristic
22#380C2×C311yescharacteristic
23#340C2×C312yescomplemented
24C24×C3D411yescharacteristic
25C22×C3×C4—22no—
26C22×C3×C4—22no—
27C22×C3×C4—24no—
28C22×C3×C4—22no—
29C22×C3×C4—21no—
30C22×C3×C4—21no—
31C22×C3×C4—42no—
32C22×C3×C4—42no—
33C22×C3×C4C2×C412yes—
34C22×C3×C4D411yescharacteristic
35C22×C3×C4Q812yes—
36C22×C3×C4C2311yescharacteristic
37C2×C3×C8—22no—
38C2×C3×C8—22no—
39C2×C3×C8C2×C414yescomplemented
40C3×C4⋊C4—24no—
41C3×C4⋊C4—22no—
42C3×C4⋊C4—22no—
43C3×C4⋊C4—22no—
44C3×C4⋊C4D414yes—
45C3×C22⋊C4—42no—
46C23×C4—21no—
47C23×C4C22×C311yescharacteristic
48C22×C8C22×C312yes—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—24no—
51C2×C4⋊C4C22×C311yescharacteristic
52C2×C4⋊C4C22×C311yescharacteristic
53C2×C4⋊C4C22×C312yes—
54C2×C22⋊C4—21no—
55#107—24no—
56C22⋊C8C3×C414yescomplemented
57#95—21no—
58C23×C3—21no—
59C23×C3—21no—
60C23×C3—21no—
61C23×C3—21no—
62C23×C3C4⋊C412yes—
63C23×C3D812yes—
64C23×C3Q1612yes—
65C23×C3C2×D411yescharacteristic
66C2×C3×C4—21no—
67C2×C3×C4—21no—
68C2×C3×C4—24no—
69C2×C3×C4—24no—
70C2×C3×C4—24no—
71C2×C3×C4—48no—
72C2×C3×C4—42no—
73C2×C3×C4—42no—
74C2×C3×C4—41no—
75C2×C3×C4—41no—
76C2×C3×C4—41no—
77C2×C3×C4—41no—
78C2×C3×C4—41no—
79C2×C3×C4—41no—
80C2×C3×C4C4⋊C412yes—
81C2×C3×C4C22×C411yescharacteristic
82C2×C3×C4C2×D412yes—
83C2×C3×C4C2×Q811yescharacteristic
84C2×C3×C4#4112yes—
85C3×C8—24no—
86C24C3×D411yescharacteristic
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—24no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4C2×C3×C412yes—
96C22×C4C3×D411yescharacteristic
97C22×C4C3×Q812yes—
98C22×C4C23×C311yescharacteristic
99C2×C8—22no—
100C2×C8—22no—
101C2×C8C2×C3×C414yescomplemented
102C4⋊C4—24no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4C3×D414yes—
107C22⋊C4—42no—
108C22×C3—22no—
109C22×C3—22no—
110C22×C3—22no—
111C22×C3—22no—
112C22×C3—22no—
113C22×C3—22no—
114C22×C3#10714yescomplemented
115C22×C3C2×C4⋊C411yescharacteristic
116C22×C3#12011yescharacteristic
117C22×C3#12311yescharacteristic
118C22×C3C2×D811yescharacteristic
119C22×C3C2×Q1611yescharacteristic
120C22×C3AGL(1,ℤ/8)11yescharacteristic
121C22×C3#13711yescharacteristic
122C3×C4—22no—
123C3×C4—44no—
124C3×C4—81no—
125C3×C4—81no—
126C3×C4C4×D412yes—
127C3×C4C22⋊Q812yes—
128C23—21no—
129C23—21no—
130C23—21no—
131C23—21no—
132C23C3×C4⋊C412yes—
133C23C3×D812yes—
134C23C3×Q1612yes—
135C23C2×C3×D411yescharacteristic
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—24no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—48no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4C3×C4⋊C412yes—
151C2×C4C22×C3×C411yescharacteristic
152C2×C4C2×C3×D412yes—
153C2×C4C2×C3×Q811yescharacteristic
154C2×C4C3×#4112yes—
155C8—24no—
156C2×C3—24no—
157C2×C3#38511yescharacteristic
158C2×C3C2×#10711yescharacteristic
159C2×C3#42811yescharacteristic
160C2×C3#44711yescharacteristic
161C2×C3#45111yescharacteristic
162C2×C3#48011yescharacteristic
163C2×C3#48411yescharacteristic
164C22—22no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22C3×#10714yescomplemented
171C22C2×C3×C4⋊C411yescharacteristic
172C22C3×#12011yescharacteristic
173C22C3×#12311yescharacteristic
174C22C2×C3×D811yescharacteristic
175C22C2×C3×Q1611yescharacteristic
176C22C3×AGL(1,ℤ/8)11yescharacteristic
177C22C3×#13711yescharacteristic
178C4—22no—
179C4—44no—
180C4—81no—
181C4—81no—
182C4C3×C4×D412yes—
183C4C3×C22⋊Q812yes—
184C3#185211yescharacteristic, direct product factor, Sylow
185C2—24no—
186C2C3×#38511yescharacteristic
187C2C2×C3×#10711yescharacteristic
188C2C3×#42811yescharacteristic
189C2C3×#44711yescharacteristic
190C2C3×#45111yescharacteristic
191C2C3×#48011yescharacteristic
192C2C3×#48411yescharacteristic
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
5. C3×#340C2semidirect
6. #1852C3direct (N × H)
17. C3×C22⋊C8C4semidirect
20. C2×#107C2×C3semidirect
23. #340C2×C3semidirect
39. C2×C3×C8C2×C4semidirect
56. C22⋊C8C3×C4semidirect
101. C2×C8C2×C3×C4semidirect
114. C22×C3#107semidirect
170. C22C3×#107semidirect
184. C3#1852direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊C42
C2×C3×#1072
C2×C3×C22⋊C82
C3×#3802
C3×#3402
#18523