Finite Group Databasebeta

← C3×#2021 #72736

Subgroups — C3×#2021 #72736

Order 384.

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

normal subgroups only

This group has 638 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, complemented
2C2×C3×#120C212yesmaximal, complemented
3C2×C3×#99C212yesmaximal, complemented
4C3×#379C211yesmaximal, characteristic, complemented
5C3×#342C211yesmaximal, characteristic, complemented
6#2021C311yesmaximal, characteristic, direct product factor, Sylow
7C25×C3C2211yescharacteristic
8C22×C3×D4—22no—
9C22×C3×D4C2212yes—
10C3×#123—22no—
11C3×#121—22no—
12C3×#120—24no—
13C3×#120—22no—
14C3×#120—24no—
15C2×C3×C22⋊C4—22no—
16C2×C3×C22⋊C4—22no—
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C3×#99—24no—
21#534C2×C311yescharacteristic, complemented
22C2×#120C2×C312yescomplemented
23C2×#99C2×C312yescomplemented
24#379C2×C311yescharacteristic, complemented
25#342C2×C311yescharacteristic, complemented
26C24×C3—21no—
27C24×C3—22no—
28C24×C3—22no—
29C24×C3—21no—
30C24×C3—44no—
31C24×C3D412yescomplemented
32C24×C3D412yescomplemented
33C24×C3C2311yescharacteristic
34C2×C3×D4—28no—
35C2×C3×D4—22no—
36C2×C3×D4—22no—
37C2×C3×D4—22no—
38C2×C3×D4—22no—
39C2×C3×D4—22no—
40C2×C3×D4—22no—
41C2×C3×D4—22no—
42C2×C3×D4—48no—
43C22×C3×C4—22no—
44C22×C3×C4—21no—
45C22×C3×C4D412yescomplemented
46C22×C3×C4D412yescomplemented
47C3×C4⋊C4—22no—
48C3×C22⋊C4—28no—
49C3×C22⋊C4—24no—
50C3×C22⋊C4—22no—
51C3×C22⋊C4—22no—
52C3×C22⋊C4—22no—
53C3×C22⋊C4—44no—
54C3×C22⋊C4D414yescomplemented
55C25C22×C311yescharacteristic
56C22×D4—22no—
57C22×D4C22×C312yes—
58#123—22no—
59#121—22no—
60#120—24no—
61#120—22no—
62#120—24no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4C22×C311yescharacteristic
66C2×C22⋊C4C22×C311yescharacteristic
67C2×C22⋊C4C22×C312yes—
68#99—24no—
69C23×C3—24no—
70C23×C3—28no—
71C23×C3—24no—
72C23×C3—22no—
73C23×C3—22no—
74C23×C3—22no—
75C23×C3—22no—
76C23×C3—44no—
77C23×C3—42no—
78C23×C3—42no—
79C23×C3—44no—
80C23×C3—48no—
81C23×C3—44no—
82C23×C3—44no—
83C23×C3—41no—
84C23×C3—42no—
85C23×C3—41no—
86C23×C3C2×D412yes—
87C23×C3C2×D411yescharacteristic
88C23×C3C2×D411yescharacteristic
89C23×C3C2×D412yes—
90C23×C3#4111yescharacteristic
91C3×D4—416no—
92C3×D4—42no—
93C3×D4—42no—
94C3×D4—42no—
95C3×D4—42no—
96C3×D4—82no—
97C2×C3×C4—22no—
98C2×C3×C4—24no—
99C2×C3×C4—22no—
100C2×C3×C4—22no—
101C2×C3×C4—22no—
102C2×C3×C4—44no—
103C2×C3×C4—42no—
104C2×C3×C4—42no—
105C2×C3×C4—41no—
106C24—21no—
107C24—22no—
108C24—22no—
109C24—21no—
110C24—44no—
111C24C3×D412yescomplemented
112C24C3×D412yescomplemented
113C24C23×C311yescharacteristic
114C2×D4—28no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—48no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4C3×D412yescomplemented
126C22×C4C3×D412yescomplemented
127C4⋊C4—22no—
128C22⋊C4—28no—
129C22⋊C4—24no—
130C22⋊C4—22no—
131C22⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—44no—
134C22⋊C4C3×D414yescomplemented
135C22×C3—22no—
136C22×C3—24no—
137C22×C3—28no—
138C22×C3—24no—
139C22×C3—41no—
140C22×C3—42no—
141C22×C3—41no—
142C22×C3—44no—
143C22×C3—44no—
144C22×C3—42no—
145C22×C3—48no—
146C22×C3—44no—
147C22×C3—42no—
148C22×C3—44no—
149C22×C3—42no—
150C22×C3—44no—
151C22×C3—82no—
152C22×C3#12012yes—
153C22×C3#12011yescharacteristic
154C22×C3#12112yes—
155C22×C3#12111yescharacteristic
156C22×C3#12711yescharacteristic
157C3×C4—44no—
158C3×C4—42no—
159C3×C4—42no—
160C3×C4—81no—
161C23—24no—
162C23—28no—
163C23—24no—
164C23—22no—
165C23—22no—
166C23—22no—
167C23—22no—
168C23—44no—
169C23—42no—
170C23—42no—
171C23—44no—
172C23—48no—
173C23—44no—
174C23—44no—
175C23—41no—
176C23—42no—
177C23—41no—
178C23C2×C3×D412yes—
179C23C2×C3×D411yescharacteristic
180C23C2×C3×D411yescharacteristic
181C23C2×C3×D412yes—
182C23C3×#4111yescharacteristic
183D4—416no—
184D4—42no—
185D4—42no—
186D4—42no—
187D4—42no—
188D4—82no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C3—22no—
199C2×C3—21no—
200C2×C3—21no—
201C2×C3—22no—
202C2×C3—44no—
203C2×C3—82no—
204C2×C3#39211yescharacteristic
205C2×C3B(4,𝔽2)12yes—
206C22—22no—
207C22—24no—
208C22—28no—
209C22—24no—
210C22—41no—
211C22—42no—
212C22—41no—
213C22—44no—
214C22—44no—
215C22—42no—
216C22—48no—
217C22—44no—
218C22—42no—
219C22—44no—
220C22—42no—
221C22—44no—
222C22—82no—
223C22C3×#12012yes—
224C22C3×#12011yescharacteristic
225C22C3×#12112yes—
226C22C3×#12111yescharacteristic
227C22C3×#12711yescharacteristic
228C4—44no—
229C4—42no—
230C4—42no—
231C4—81no—
232C3#202111yescharacteristic, direct product factor, Sylow
233C2—22no—
234C2—21no—
235C2—21no—
236C2—22no—
237C2—44no—
238C2—82no—
239C2C3×#39211yescharacteristic
240C2C3×B(4,𝔽2)12yes—
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC24C23×C3
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#534C2semidirect
2. C2×C3×#120C2semidirect
3. C2×C3×#99C2semidirect
4. C3×#379C2semidirect
5. C3×#342C2semidirect
6. #2021C3direct (N × H)
21. #534C2×C3semidirect
22. C2×#120C2×C3semidirect
23. C2×#99C2×C3semidirect
24. #379C2×C3semidirect
25. #342C2×C3semidirect
31. C24×C3D4semidirect
32. C24×C3D4semidirect
45. C22×C3×C4D4semidirect
46. C22×C3×C4D4semidirect
54. C3×C22⋊C4D4semidirect
111. C24C3×D4semidirect
112. C24C3×D4semidirect
125. C22×C4C3×D4semidirect
126. C22×C4C3×D4semidirect
134. C22⋊C4C3×D4semidirect
232. C3#2021direct (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×#1202
C2×C3×#992
C3×#3792
C3×#3422
#20213