Finite Group Databasebeta

← 3−1+2×AΣL(1,𝔽16) #13136056

Subgroups — 3−1+2×AΣL(1,𝔽16) #13136056

Order 1728.

376 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 91 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
13−1+2×#120C211yesmaximal, characteristic
23−1+2×#100C211yesmaximal, characteristic, complemented
33−1+2×#99C211yesmaximal, characteristic, complemented
4C32×AΣL(1,𝔽16)C311yesmaximal, characteristic
5C9×AΣL(1,𝔽16)C313yesmaximal, complemented
6C24×3−1+2C411yescharacteristic, complemented
7C2×D4×3−1+2—21no—
8C2×D4×3−1+2C2211yescharacteristic
9M16×3−1+2—21no—
10C22⋊C4×3−1+2—21no—
11C22⋊C4×3−1+2—21no—
12C22⋊C4×3−1+2C411yescharacteristic, complemented
13C32×#120C2×C311yescharacteristic
14C32×#100C2×C311yescharacteristic
15C32×#99C2×C311yescharacteristic
16C9×#120C2×C313yes—
17C9×#100C2×C313yescomplemented
18C9×#99C2×C313yescomplemented
19C23×3−1+2—21no—
20C23×3−1+2—42no—
21C23×3−1+2—41no—
22C23×3−1+2C2×C411yescharacteristic
23C23×3−1+2D411yescharacteristic
24D4×3−1+2—41no—
25D4×3−1+2—42no—
26C2×C4×3−1+2—21no—
27C2×C4×3−1+2—41no—
28C2×C4×3−1+2D411yescharacteristic
29C8×3−1+2—41no—
30C3×AΣL(1,𝔽16)—31no—
31C3×AΣL(1,𝔽16)C3211yescharacteristic
32C24×C32C3×C411yescharacteristic
33C2×C32×D4—21no—
34C2×C32×D4C22×C311yescharacteristic
35C24×C9C3×C413yescomplemented
36C32×M16—21no—
37C32×C22⋊C4—21no—
38C32×C22⋊C4—21no—
39C32×C22⋊C4C3×C411yescharacteristic
40C2×D4×C9—23no—
41C2×D4×C9C22×C313yes—
42C9×M16—23no—
43C9×C22⋊C4—23no—
44C9×C22⋊C4—23no—
45C9×C22⋊C4C3×C413yescomplemented
46C22×3−1+2—22no—
47C22×3−1+2—21no—
48C22×3−1+2—21no—
49C22×3−1+2—41no—
50C22×3−1+2—41no—
51C22×3−1+2—41no—
52C22×3−1+2—42no—
53C22×3−1+2—41no—
54C22×3−1+2—42no—
55C22×3−1+2C22⋊C411yescharacteristic
56C4×3−1+2—21no—
57C4×3−1+2—41no—
58C4×3−1+2—81no—
59C3×#120—31no—
60C3×#120C2×C3211yescharacteristic
61C3×#100—31no—
62C3×#100C2×C3211yescharacteristic
63C3×#99—31no—
64C3×#99C2×C3211yescharacteristic
65C23×C32—21no—
66C23×C32—42no—
67C23×C32—41no—
68C23×C32C2×C3×C411yescharacteristic
69C23×C32C3×D411yescharacteristic
70C32×D4—41no—
71C32×D4—42no—
72C2×C32×C4—21no—
73C2×C32×C4—41no—
74C2×C32×C4C3×D411yescharacteristic
75C23×C9—23no—
76C23×C9—46no—
77C23×C9—43no—
78C23×C9C2×C3×C413yes—
79C23×C9C3×D413yes—
80C32×C8—41no—
81D4×C9—43no—
82D4×C9—46no—
83C2×C4×C9—23no—
84C2×C4×C9—43no—
85C2×C4×C9C3×D413yes—
86C8×C9—43no—
87AΣL(1,𝔽16)3−1+211yescharacteristic, direct product factor, Sylow
88C2×3−1+2—21no—
89C2×3−1+2—41no—
90C2×3−1+2—42no—
91C2×3−1+2—41no—
92C2×3−1+2#9911yescharacteristic
93C24×C3—31no—
94C24×C3C32×C411yescharacteristic
95C2×C3×D4—21no—
96C2×C3×D4—31no—
97C2×C3×D4—61no—
98C2×C3×D4C22×C3211yescharacteristic
99C3×M16—21no—
100C3×M16—61no—
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4—31no—
104C3×C22⋊C4—61no—
105C3×C22⋊C4—61no—
106C3×C22⋊C4C32×C411yescharacteristic
107C22×C32—22no—
108C22×C32—21no—
109C22×C32—21no—
110C22×C32—41no—
111C22×C32—41no—
112C22×C32—41no—
113C22×C32—42no—
114C22×C32—42no—
115C22×C32—41no—
116C22×C32C3×C22⋊C411yescharacteristic
117C32×C4—21no—
118C32×C4—41no—
119C32×C4—81no—
120C22×C9—26no—
121C22×C9—23no—
122C22×C9—23no—
123C22×C9—43no—
124C22×C9—43no—
125C22×C9—43no—
126C22×C9—46no—
127C22×C9—46no—
128C22×C9—43no—
129C22×C9C3×C22⋊C413yes—
130C4×C9—23no—
131C4×C9—43no—
132C4×C9—83no—
133#120C2×3−1+211yescharacteristic
134#100C2×3−1+211yescharacteristic, complemented
135#99C2×3−1+211yescharacteristic, complemented
1363−1+2AΣL(1,𝔽16)11yescharacteristic, direct product factor, Sylow
137C23×C3—21no—
138C23×C3—31no—
139C23×C3—31no—
140C23×C3—42no—
141C23×C3—41no—
142C23×C3—61no—
143C23×C3—121no—
144C23×C3—122no—
145C23×C3C2×C32×C411yescharacteristic
146C23×C3C32×D411yescharacteristic
147C3×D4—41no—
148C3×D4—42no—
149C3×D4—121no—
150C3×D4—122no—
151C2×C3×C4—21no—
152C2×C3×C4—31no—
153C2×C3×C4—41no—
154C2×C3×C4—61no—
155C2×C3×C4—121no—
156C2×C3×C4C32×D411yescharacteristic
157C3×C8—41no—
158C3×C8—121no—
159C2×C32—21no—
160C2×C32—41no—
161C2×C32—42no—
162C2×C32—41no—
163C2×C32C3×#9911yescharacteristic
164C2×C9—23no—
165C2×C9—43no—
166C2×C9—46no—
167C2×C9—43no—
168C2×C9C3×#9913yes—
169C24C4×3−1+211yescharacteristic, complemented
170C2×D4—21no—
171C2×D4C22×3−1+211yescharacteristic
172M16—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4C4×3−1+211yescharacteristic, complemented
176C22×C3—22no—
177C22×C3—21no—
178C22×C3—21no—
179C22×C3—31no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—42no—
184C22×C3—42no—
185C22×C3—41no—
186C22×C3—61no—
187C22×C3—61no—
188C22×C3—62no—
189C22×C3—121no—
190C22×C3—121no—
191C22×C3—121no—
192C22×C3—121no—
193C22×C3—122no—
194C22×C3—122no—
195C22×C3C32×C22⋊C411yescharacteristic
196C3×C4—21no—
197C3×C4—41no—
198C3×C4—61no—
199C3×C4—81no—
200C3×C4—121no—
201C3×C4—241no—
202C32C3×AΣL(1,𝔽16)11yescharacteristic
203C9C3×AΣL(1,𝔽16)13yescomplemented
204C23—21no—
205C23—42no—
206C23—41no—
207C23C2×C4×3−1+211yescharacteristic
208C23D4×3−1+211yescharacteristic
209D4—41no—
210D4—42no—
211C2×C4—21no—
212C2×C4—41no—
213C2×C4D4×3−1+211yescharacteristic
214C8—41no—
215C2×C3—21no—
216C2×C3—31no—
217C2×C3—41no—
218C2×C3—42no—
219C2×C3—41no—
220C2×C3—61no—
221C2×C3—121no—
222C2×C3—121no—
223C2×C3—122no—
224C2×C3C32×#9911yescharacteristic
225C22—22no—
226C22—21no—
227C22—21no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—42no—
232C22—42no—
233C22—41no—
234C22C22⋊C4×3−1+211yescharacteristic
235C4—21no—
236C4—41no—
237C4—81no—
238C3—31no—
239C3C32×AΣL(1,𝔽16)11yescharacteristic
240C2—21no—
241C2—41no—
242C2—42no—
243C2—41no—
244C23−1+2×#9911yescharacteristic
subgroupG/H
center, socleC2×C3C32×#99
commutatorC23×C3C2×C32×C4
Frattini, cosocleC2×C3×D4C22×C32
prime radicalC32×#120C2×C3
LCSG ▷ C23×C3 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×3−1+2 ◁ C2×D4×3−1+2 ◁ G

Complements

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

NH ≅ G/Ntype
2. 3−1+2×#100C2semidirect
3. 3−1+2×#99C2semidirect
5. C9×AΣL(1,𝔽16)C3semidirect
6. C24×3−1+2C4semidirect
12. C22⋊C4×3−1+2C4semidirect
17. C9×#100C2×C3semidirect
18. C9×#99C2×C3semidirect
35. C24×C9C3×C4semidirect
45. C9×C22⋊C4C3×C4semidirect
87. AΣL(1,𝔽16)3−1+2direct (N × H)
134. #100C2×3−1+2semidirect
135. #99C2×3−1+2semidirect
136. 3−1+2AΣL(1,𝔽16)direct (N × H)
169. C24C4×3−1+2semidirect
175. C22⋊C4C4×3−1+2semidirect
203. C9C3×AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
3−1+2×#1202
3−1+2×#1002
3−1+2×#992
C32×AΣL(1,𝔽16)3
C9×AΣL(1,𝔽16)3