Finite Group Databasebeta

← C2×C3×AΣL(1,𝔽16) #72833

Subgroups — C2×C3×AΣL(1,𝔽16) #72833

Order 384.

438 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 88 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
2C2×C3×#100C211yesmaximal, characteristic, complemented
3C2×C3×#99C211yesmaximal, characteristic, complemented
4C3×AΣL(1,𝔽16)C214yesmaximal, direct product factor
5C2×AΣL(1,𝔽16)C311yesmaximal, characteristic, direct product factor, Sylow
6C25×C3C411yescharacteristic, complemented
7C22×C3×D4—21no—
8C22×C3×D4C2211yescharacteristic
9C2×C3×M16—21no—
10C3×#120—21no—
11C3×#120—21no—
12C3×#120C412yescomplemented
13C3×#120C2212yes—
14C2×C3×C22⋊C4—21no—
15C2×C3×C22⋊C4—21no—
16C2×C3×C22⋊C4C411yescharacteristic, complemented
17C3×#100—21no—
18C3×#100C2212yescomplemented
19C3×#99—21no—
20C3×#99C2212yescomplemented
21C2×#120C2×C311yescharacteristic
22C2×#100C2×C311yescharacteristic, complemented
23C2×#99C2×C311yescharacteristic, complemented
24AΣL(1,𝔽16)C2×C314yesdirect product factor
25C24×C3—21no—
26C24×C3—21no—
27C24×C3—44no—
28C24×C3—41no—
29C24×C3—41no—
30C24×C3C2×C411yescharacteristic
31C24×C3C2×C412yescomplemented
32C24×C3D411yescharacteristic
33C2×C3×D4—22no—
34C2×C3×D4—22no—
35C2×C3×D4—41no—
36C2×C3×D4—41no—
37C2×C3×D4—44no—
38C2×C3×D4C2×C411yescharacteristic
39C2×C3×D4D411yescharacteristic
40C2×C3×D4D411yescharacteristic
41C2×C3×D4C2311yescharacteristic
42C22×C3×C4—21no—
43C22×C3×C4—41no—
44C22×C3×C4D411yescharacteristic
45C3×M16—22no—
46C3×M16—41no—
47C2×C3×C8—41no—
48C3×C22⋊C4—21no—
49C3×C22⋊C4—22no—
50C3×C22⋊C4—22no—
51C3×C22⋊C4—22no—
52C3×C22⋊C4—41no—
53C3×C22⋊C4C2×C412yescomplemented
54C25C3×C411yescharacteristic, complemented
55C22×D4—21no—
56C22×D4C22×C311yescharacteristic
57C2×M16—21no—
58#120—21no—
59#120—21no—
60#120C3×C412yescomplemented
61#120C22×C312yes—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4C3×C411yescharacteristic, complemented
65#100—21no—
66#100C22×C312yescomplemented
67#99—21no—
68#99C22×C312yescomplemented
69C23×C3—24no—
70C23×C3—21no—
71C23×C3—21no—
72C23×C3—21no—
73C23×C3—22no—
74C23×C3—21no—
75C23×C3—42no—
76C23×C3—44no—
77C23×C3—42no—
78C23×C3—42no—
79C23×C3—44no—
80C23×C3—42no—
81C23×C3—42no—
82C23×C3—44no—
83C23×C3—48no—
84C23×C3—44no—
85C23×C3—41no—
86C23×C3—41no—
87C23×C3C22⋊C411yescharacteristic
88C23×C3C22⋊C411yescharacteristic
89C23×C3C22⋊C411yescharacteristic
90C23×C3C22×C411yescharacteristic
91C23×C3C2×D411yescharacteristic
92C3×D4—41no—
93C3×D4—41no—
94C3×D4—41no—
95C3×D4—41no—
96C3×D4—44no—
97C3×D4—44no—
98C2×C3×C4—21no—
99C2×C3×C4—21no—
100C2×C3×C4—22no—
101C2×C3×C4—42no—
102C2×C3×C4—42no—
103C2×C3×C4—82no—
104C2×C3×C4C22⋊C411yescharacteristic
105C2×C3×C4C2×D411yescharacteristic
106C3×C8—42no—
107C24—21no—
108C24—21no—
109C24—44no—
110C24—41no—
111C24—41no—
112C24C2×C3×C411yescharacteristic
113C24C2×C3×C412yescomplemented
114C24C3×D411yescharacteristic
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—41no—
118C2×D4—41no—
119C2×D4—44no—
120C2×D4C2×C3×C411yescharacteristic
121C2×D4C3×D411yescharacteristic
122C2×D4C3×D411yescharacteristic
123C2×D4C23×C311yescharacteristic
124C22×C4—21no—
125C22×C4—41no—
126C22×C4C3×D411yescharacteristic
127M16—22no—
128M16—41no—
129C2×C8—41no—
130C22⋊C4—21no—
131C22⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—22no—
134C22⋊C4—41no—
135C22⋊C4C2×C3×C412yescomplemented
136C22×C3—22no—
137C22×C3—24no—
138C22×C3—21no—
139C22×C3—21no—
140C22×C3—21no—
141C22×C3—21no—
142C22×C3—41no—
143C22×C3—41no—
144C22×C3—41no—
145C22×C3—41no—
146C22×C3—42no—
147C22×C3—44no—
148C22×C3—42no—
149C22×C3—44no—
150C22×C3—44no—
151C22×C3—48no—
152C22×C3—42no—
153C22×C3—44no—
154C22×C3—42no—
155C22×C3—42no—
156C22×C3—81no—
157C22×C3#9911yescharacteristic
158C22×C3#9911yescharacteristic
159C22×C3C2×C22⋊C411yescharacteristic
160C3×C4—21no—
161C3×C4—21no—
162C3×C4—42no—
163C3×C4—82no—
164C23—24no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—22no—
169C23—21no—
170C23—42no—
171C23—44no—
172C23—42no—
173C23—42no—
174C23—44no—
175C23—42no—
176C23—42no—
177C23—44no—
178C23—48no—
179C23—44no—
180C23—41no—
181C23—41no—
182C23C3×C22⋊C411yescharacteristic
183C23C3×C22⋊C411yescharacteristic
184C23C3×C22⋊C411yescharacteristic
185C23C22×C3×C411yescharacteristic
186C23C2×C3×D411yescharacteristic
187D4—41no—
188D4—41no—
189D4—41no—
190D4—41no—
191D4—44no—
192D4—44no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4C3×C22⋊C411yescharacteristic
200C2×C4C2×C3×D411yescharacteristic
201C8—42no—
202C2×C3—21no—
203C2×C3—21no—
204C2×C3—41no—
205C2×C3—44no—
206C2×C3—41no—
207C2×C3—41no—
208C2×C3—41no—
209C2×C3AΣL(1,𝔽16)12yesdirect product factor
210C2×C3C2×#9911yescharacteristic
211C22—22no—
212C22—24no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—44no—
223C22—42no—
224C22—44no—
225C22—44no—
226C22—48no—
227C22—42no—
228C22—42no—
229C22—44no—
230C22—42no—
231C22—81no—
232C22C3×#9911yescharacteristic
233C22C3×#9911yescharacteristic
234C22C2×C3×C22⋊C411yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—82no—
239C3C2×AΣL(1,𝔽16)11yescharacteristic, direct product factor, Sylow
240C2—21no—
241C2—21no—
242C2—41no—
243C2—44no—
244C2—41no—
245C2—41no—
246C2—41no—
247C2C3×AΣL(1,𝔽16)12yesdirect product factor
248C2C2×C3×#9911yescharacteristic
subgroupG/H
center, socleC22×C3#99
commutatorC23C22×C3×C4
Frattini, cosocleC2×D4C23×C3
prime radicalC2×C3×#120C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#100C2semidirect
3. C2×C3×#99C2semidirect
4. C3×AΣL(1,𝔽16)C2direct (N × H)
5. C2×AΣL(1,𝔽16)C3direct (N × H)
6. C25×C3C4semidirect
12. C3×#120C4semidirect
16. C2×C3×C22⋊C4C4semidirect
18. C3×#100C22semidirect
20. C3×#99C22semidirect
22. C2×#100C2×C3semidirect
23. C2×#99C2×C3semidirect
24. AΣL(1,𝔽16)C2×C3direct (N × H)
31. C24×C3C2×C4semidirect
53. C3×C22⋊C4C2×C4semidirect
54. C25C3×C4semidirect
60. #120C3×C4semidirect
64. C2×C22⋊C4C3×C4semidirect
66. #100C22×C3semidirect
68. #99C22×C3semidirect
113. C24C2×C3×C4semidirect
135. C22⋊C4C2×C3×C4semidirect
209. C2×C3AΣL(1,𝔽16)direct (N × H)
239. C3C2×AΣL(1,𝔽16)direct (N × H)
247. C2C3×AΣL(1,𝔽16)direct (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
C2×C3×#1002
C2×C3×#992
C3×AΣL(1,𝔽16)2
C2×AΣL(1,𝔽16)3