Finite Group Databasebeta

← C3×#2022 #72737

Subgroups — C3×#2022 #72737

Order 384.

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

normal subgroups only

This group has 430 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×#534C211yesmaximal, characteristic
2C2×C3×#123C212yesmaximal, complemented
3C2×C3×#99C212yesmaximal, complemented
4C3×#381C211yesmaximal, characteristic, complemented
5C3×#342C211yesmaximal, characteristic, complemented
6#2022C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×D4C2212yes—
8C23×C3×C4C2211yescharacteristic
9C3×#123—22no—
10C3×#123—24no—
11C3×#123—24no—
12C3×#121—22no—
13C3×#120—22no—
14C2×C3×C4⋊C4—22no—
15C2×C3×C22⋊C4—22no—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2212yes—
19C3×#99—24no—
20C3×#95—22no—
21#534C2×C311yescharacteristic
22C2×#123C2×C312yescomplemented
23C2×#99C2×C312yescomplemented
24#381C2×C311yescharacteristic, complemented
25#342C2×C311yescharacteristic, complemented
26C24×C3D412yes—
27C24×C3C2311yescharacteristic
28C2×C3×D4—22no—
29C2×C3×D4—22no—
30C2×C3×D4—22no—
31C2×C3×D4—22no—
32C2×C3×D4—22no—
33C2×C3×D4—22no—
34C2×C3×D4—22no—
35C22×C3×C4—21no—
36C22×C3×C4—22no—
37C22×C3×C4—22no—
38C22×C3×C4—22no—
39C22×C3×C4—21no—
40C22×C3×C4—21no—
41C22×C3×C4D412yes—
42C22×C3×C4D412yes—
43C22×C3×C4D412yescomplemented
44C3×C4⋊C4—28no—
45C3×C4⋊C4—22no—
46C3×C22⋊C4—28no—
47C3×C22⋊C4—24no—
48C3×C22⋊C4—22no—
49C3×C22⋊C4—22no—
50C3×C22⋊C4—22no—
51C3×C22⋊C4D414yescomplemented
52C22×D4C22×C312yes—
53C23×C4C22×C311yescharacteristic
54#123—22no—
55#123—24no—
56#123—24no—
57#121—22no—
58#120—22no—
59C2×C4⋊C4—22no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4C22×C311yescharacteristic
62C2×C22⋊C4C22×C311yescharacteristic
63C2×C22⋊C4C22×C312yes—
64#99—24no—
65#95—22no—
66C23×C3—22no—
67C23×C3—22no—
68C23×C3—22no—
69C23×C3—42no—
70C23×C3—44no—
71C23×C3C2×D412yes—
72C23×C3C2×D411yescharacteristic
73C23×C3C2×D411yescharacteristic
74C23×C3C2×D412yes—
75C23×C3#4111yescharacteristic
76C3×D4—42no—
77C3×D4—42no—
78C3×D4—42no—
79C3×D4—42no—
80C3×D4—82no—
81C2×C3×C4—24no—
82C2×C3×C4—24no—
83C2×C3×C4—22no—
84C2×C3×C4—22no—
85C2×C3×C4—22no—
86C2×C3×C4—22no—
87C2×C3×C4—22no—
88C2×C3×C4—44no—
89C2×C3×C4—41no—
90C2×C3×C4—42no—
91C2×C3×C4—41no—
92C2×C3×C4—42no—
93C2×C3×C4—42no—
94C2×C3×C4—41no—
95C24C3×D412yes—
96C24C23×C311yescharacteristic
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4C3×D412yes—
111C22×C4C3×D412yes—
112C22×C4C3×D412yescomplemented
113C4⋊C4—28no—
114C4⋊C4—22no—
115C22⋊C4—28no—
116C22⋊C4—24no—
117C22⋊C4—22no—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C22⋊C4C3×D414yescomplemented
121C22×C3—22no—
122C22×C3—24no—
123C22×C3—41no—
124C22×C3—42no—
125C22×C3—41no—
126C22×C3—44no—
127C22×C3—42no—
128C22×C3—44no—
129C22×C3—82no—
130C22×C3#12012yes—
131C22×C3#12011yescharacteristic
132C22×C3#12111yescharacteristic
133C22×C3#12112yes—
134C22×C3#12711yescharacteristic
135C3×C4—44no—
136C3×C4—42no—
137C3×C4—42no—
138C3×C4—42no—
139C3×C4—81no—
140C23—22no—
141C23—22no—
142C23—22no—
143C23—42no—
144C23—44no—
145C23C2×C3×D412yes—
146C23C2×C3×D411yescharacteristic
147C23C2×C3×D411yescharacteristic
148C23C2×C3×D412yes—
149C23C3×#4111yescharacteristic
150D4—42no—
151D4—42no—
152D4—42no—
153D4—42no—
154D4—82no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—44no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C3—22no—
170C2×C3—21no—
171C2×C3—22no—
172C2×C3—21no—
173C2×C3—82no—
174C2×C3#39211yescharacteristic
175C2×C3#45812yes—
176C22—22no—
177C22—24no—
178C22—41no—
179C22—42no—
180C22—41no—
181C22—44no—
182C22—42no—
183C22—44no—
184C22—82no—
185C22C3×#12012yes—
186C22C3×#12011yescharacteristic
187C22C3×#12111yescharacteristic
188C22C3×#12112yes—
189C22C3×#12711yescharacteristic
190C4—44no—
191C4—42no—
192C4—42no—
193C4—42no—
194C4—81no—
195C3#202211yescharacteristic, direct product factor, Sylow
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—82no—
201C2C3×#39211yescharacteristic
202C2C3×#45812yes—
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC24C23×C3
prime radicalC3×#534C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#123C2semidirect
3. C2×C3×#99C2semidirect
4. C3×#381C2semidirect
5. C3×#342C2semidirect
6. #2022C3direct (N × H)
22. C2×#123C2×C3semidirect
23. C2×#99C2×C3semidirect
24. #381C2×C3semidirect
25. #342C2×C3semidirect
43. C22×C3×C4D4semidirect
51. C3×C22⋊C4D4semidirect
112. C22×C4C3×D4semidirect
120. C22⋊C4C3×D4semidirect
195. C3#2022direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#5342
C2×C3×#1232
C2×C3×#992
C3×#3812
C3×#3422
#20223