Finite Group Databasebeta

← C3×#1898 #72613

Subgroups — C3×#1898 #72613

Order 384.

598 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 120 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
1C2×C3×#120C211yesmaximal, characteristic
2C22×C3×C22⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#99C212yesmaximal, complemented
4C3×#379C211yesmaximal, characteristic, complemented
5C3×#342C212yesmaximal, complemented
6#1898C311yesmaximal, characteristic, direct product factor, Sylow
7C25×C3C2211yescharacteristic
8C22×C3×D4C412yescomplemented
9C22×C3×D4C2211yescharacteristic
10C23×C3×C4—21no—
11C3×#120—21no—
12C3×#120—22no—
13C3×#120—21no—
14C2×C3×C22⋊C4—21no—
15C2×C3×C22⋊C4—24no—
16C2×C3×C22⋊C4—22no—
17C2×C3×C22⋊C4C412yescomplemented
18C2×C3×C22⋊C4C2212yes—
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C3×#99—24no—
22C2×#120C2×C311yescharacteristic
23C22×C22⋊C4C2×C311yescharacteristic, complemented
24C2×#99C2×C312yescomplemented
25#379C2×C311yescharacteristic, complemented
26#342C2×C312yescomplemented
27C24×C3—22no—
28C24×C3—22no—
29C24×C3—44no—
30C24×C3C2×C412yes—
31C24×C3C2×C411yescharacteristic, complemented
32C24×C3D411yescharacteristic
33C24×C3D412yes—
34C24×C3D411yescharacteristic
35C24×C3C2311yescharacteristic
36C2×C3×D4—22no—
37C2×C3×D4—22no—
38C2×C3×D4—22no—
39C2×C3×D4—21no—
40C2×C3×D4—21no—
41C2×C3×D4—22no—
42C2×C3×D4—44no—
43C22×C3×C4—24no—
44C22×C3×C4—22no—
45C22×C3×C4—22no—
46C22×C3×C4—44no—
47C22×C3×C4C2×C411yescharacteristic, complemented
48C22×C3×C4C2×C412yescomplemented
49C3×C22⋊C4—24no—
50C3×C22⋊C4—22no—
51C3×C22⋊C4—22no—
52C3×C22⋊C4—22no—
53C3×C22⋊C4—21no—
54C3×C22⋊C4—21no—
55C3×C22⋊C4—22no—
56C3×C22⋊C4—22no—
57C3×C22⋊C4—44no—
58C3×C22⋊C4D414yescomplemented
59C25C22×C311yescharacteristic
60C22×D4C3×C412yescomplemented
61C22×D4C22×C311yescharacteristic
62C23×C4—21no—
63#120—21no—
64#120—22no—
65#120—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4C3×C412yescomplemented
70C2×C22⋊C4C22×C312yes—
71C2×C22⋊C4C22×C311yescharacteristic
72C2×C22⋊C4C22×C312yes—
73#99—24no—
74C23×C3—24no—
75C23×C3—21no—
76C23×C3—21no—
77C23×C3—24no—
78C23×C3—24no—
79C23×C3—24no—
80C23×C3—21no—
81C23×C3—24no—
82C23×C3—21no—
83C23×C3—21no—
84C23×C3—22no—
85C23×C3—42no—
86C23×C3—42no—
87C23×C3—42no—
88C23×C3—42no—
89C23×C3—48no—
90C23×C3—48no—
91C23×C3—42no—
92C23×C3C22⋊C412yes—
93C23×C3C22⋊C412yes—
94C23×C3C22×C411yescharacteristic
95C23×C3C2×D411yescharacteristic
96C23×C3C2×D411yescharacteristic
97C23×C3C2×D412yes—
98C23×C3#4112yes—
99C3×D4—44no—
100C3×D4—44no—
101C3×D4—41no—
102C3×D4—42no—
103C3×D4—41no—
104C2×C3×C4—21no—
105C2×C3×C4—22no—
106C2×C3×C4—22no—
107C2×C3×C4—24no—
108C2×C3×C4—48no—
109C2×C3×C4—44no—
110C2×C3×C4—42no—
111C2×C3×C4—44no—
112C2×C3×C4—42no—
113C24—22no—
114C24—22no—
115C24—44no—
116C24C2×C3×C412yes—
117C24C2×C3×C411yescharacteristic, complemented
118C24C3×D411yescharacteristic
119C24C3×D412yes—
120C24C3×D411yescharacteristic
121C24C23×C311yescharacteristic
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—22no—
128C2×D4—44no—
129C22×C4—24no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—44no—
133C22×C4C2×C3×C411yescharacteristic, complemented
134C22×C4C2×C3×C412yescomplemented
135C22⋊C4—24no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—44no—
144C22⋊C4C3×D414yescomplemented
145C22×C3—24no—
146C22×C3—24no—
147C22×C3—22no—
148C22×C3—22no—
149C22×C3—22no—
150C22×C3—22no—
151C22×C3—22no—
152C22×C3—22no—
153C22×C3—22no—
154C22×C3—22no—
155C22×C3—42no—
156C22×C3—42no—
157C22×C3—48no—
158C22×C3—48no—
159C22×C3—42no—
160C22×C3—42no—
161C22×C3—41no—
162C22×C3—42no—
163C22×C3—82no—
164C22×C3#9914yescomplemented
165C22×C3C2×C22⋊C411yescharacteristic
166C22×C3C4×D412yes—
167C22×C3#12011yescharacteristic
168C22×C3#12112yes—
169C22×C3#12311yescharacteristic
170C3×C4—44no—
171C3×C4—41no—
172C3×C4—42no—
173C3×C4—82no—
174C23—24no—
175C23—21no—
176C23—21no—
177C23—24no—
178C23—24no—
179C23—24no—
180C23—21no—
181C23—24no—
182C23—21no—
183C23—21no—
184C23—22no—
185C23—42no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—48no—
190C23—48no—
191C23—42no—
192C23C3×C22⋊C412yes—
193C23C3×C22⋊C412yes—
194C23C22×C3×C411yescharacteristic
195C23C2×C3×D411yescharacteristic
196C23C2×C3×D411yescharacteristic
197C23C2×C3×D412yes—
198C23C3×#4112yes—
199D4—44no—
200D4—44no—
201D4—41no—
202D4—42no—
203D4—41no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—48no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C3—21no—
214C2×C3—22no—
215C2×C3—24no—
216C2×C3—21no—
217C2×C3—22no—
218C2×C3—42no—
219C2×C3—81no—
220C2×C3#38611yescharacteristic
221C2×C3C2×#9912yes—
222C22—24no—
223C22—24no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—42no—
233C22—42no—
234C22—48no—
235C22—48no—
236C22—42no—
237C22—42no—
238C22—41no—
239C22—42no—
240C22—82no—
241C22C3×#9914yescomplemented
242C22C2×C3×C22⋊C411yescharacteristic
243C22C3×C4×D412yes—
244C22C3×#12011yescharacteristic
245C22C3×#12112yes—
246C22C3×#12311yescharacteristic
247C4—44no—
248C4—41no—
249C4—42no—
250C4—82no—
251C3#189811yescharacteristic, direct product factor, Sylow
252C2—21no—
253C2—22no—
254C2—24no—
255C2—21no—
256C2—22no—
257C2—42no—
258C2—81no—
259C2C3×#38611yescharacteristic
260C2C2×C3×#9912yes—
subgroupG/H
center, socleC22×C3C2×C22⋊C4
commutatorC23C22×C3×C4
Frattini, cosocleC24C23×C3
prime radicalC2×C3×#120C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×C22⋊C4C2semidirect
3. C2×C3×#99C2semidirect
4. C3×#379C2semidirect
5. C3×#342C2semidirect
6. #1898C3direct (N × H)
8. C22×C3×D4C4semidirect
17. C2×C3×C22⋊C4C4semidirect
23. C22×C22⋊C4C2×C3semidirect
24. C2×#99C2×C3semidirect
25. #379C2×C3semidirect
26. #342C2×C3semidirect
31. C24×C3C2×C4semidirect
47. C22×C3×C4C2×C4semidirect
48. C22×C3×C4C2×C4semidirect
58. C3×C22⋊C4D4semidirect
60. C22×D4C3×C4semidirect
69. C2×C22⋊C4C3×C4semidirect
117. C24C2×C3×C4semidirect
133. C22×C4C2×C3×C4semidirect
134. C22×C4C2×C3×C4semidirect
144. C22⋊C4C3×D4semidirect
164. C22×C3#99semidirect
241. C22C3×#99semidirect
251. C3#1898direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1202
C22×C3×C22⋊C42
C2×C3×#992
C3×#3792
C3×#3422
#18983