Finite Group Databasebeta

← M16×#1094 #13152215

Subgroups — M16×#1094 #13152215

Order 1728.

418 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 90 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×C4×#1094C211yesmaximal, characteristic
2C8×#1094C212yesmaximal, complemented
3C22×C32×M16C311yesmaximal, characteristic
4M16×C22⋊C9C313yesmaximal, complemented
5C22×#1094C411yescharacteristic
6C4×#1094C411yescharacteristic
7C4×#1094C2211yescharacteristic
8M16×3−1+2—41nomaximal
9C23×C32×C4C2×C311yescharacteristic
10C2×C32×M16—31no—
11C2×C32×M16—32no—
12C2×C32×M16—31no—
13C22×C32×C8C2×C312yes—
14C2×C4×C22⋊C9C2×C313yes—
15C8×C22⋊C9C2×C316yescomplemented
16C2×#1094—21no—
17C2×#1094C2×C411yescharacteristic
18C2×C4×3−1+2—41no—
19C8×3−1+2—42no—
20C22×C3×M16—31no—
21C22×C3×M16C3211yescharacteristic
22C24×C32C3×C411yescharacteristic
23C22×C32×C4—31no—
24C22×C32×C4—31no—
25C22×C32×C4—31no—
26C22×C32×C4—31no—
27C22×C32×C4C3×C411yescharacteristic
28C22×C32×C4C22×C311yescharacteristic
29C22×C22⋊C9C3×C413yes—
30C32×M16—32no—
31C32×M16—31no—
32C32×M16—32no—
33C32×M16A411yescharacteristic
34C2×C32×C8—32no—
35C2×C32×C8—32no—
36C4×C22⋊C9C3×C413yes—
37C4×C22⋊C9C22×C313yes—
38C9×M16—43no—
39C22×3−1+2—41no—
40#1094M1611yescharacteristic, direct product factor
41C4×3−1+2—41no—
42C4×3−1+2—41no—
43C23×C3×C4—31no—
44C23×C3×C4C2×C3211yescharacteristic
45C2×C3×M16—33no—
46C2×C3×M16—36no—
47C2×C3×M16—33no—
48C2×C3×M16—31no—
49C2×C3×M16—32no—
50C2×C3×M16—31no—
51C22×C3×C8—32no—
52C22×C3×C8C2×C3212yes—
53C23×C32—21no—
54C23×C32—31no—
55C23×C32—31no—
56C23×C32—61no—
57C23×C32C2×C3×C411yescharacteristic
58C2×C32×C4—32no—
59C2×C32×C4—31no—
60C2×C32×C4—31no—
61C2×C32×C4—31no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C32×C4—31no—
65C2×C32×C4—31no—
66C2×C32×C4C2×A411yescharacteristic
67C2×C22⋊C9—23no—
68C2×C22⋊C9C2×C3×C413yes—
69C32×C8—32no—
70C32×C8C2×A412yes—
71C2×C4×C9—43no—
72C8×C9—46no—
73C22×M163−1+211yescharacteristic, complemented, Sylow
74C2×3−1+2—41no—
75C2×3−1+2—81no—
76C24×C3—31no—
77C24×C3C32×C411yescharacteristic
78C22×C3×C4—33no—
79C22×C3×C4—33no—
80C22×C3×C4—33no—
81C22×C3×C4—33no—
82C22×C3×C4—31no—
83C22×C3×C4—31no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—31no—
87C22×C3×C4—31no—
88C22×C3×C4C32×C411yescharacteristic
89C22×C3×C4C22×C3211yescharacteristic
90C3×M16—36no—
91C3×M16—32no—
92C3×M16—33no—
93C3×M16—36no—
94C3×M16—31no—
95C3×M16—32no—
96C3×M16—31no—
97C3×M16C3×A411yescharacteristic
98C2×C3×C8—36no—
99C2×C3×C8—36no—
100C2×C3×C8—32no—
101C2×C3×C8—32no—
102C22×C32—31no—
103C22×C32—31no—
104C22×C32—31no—
105C22×C32—61no—
106C22×C32—61no—
107C22×C32—61no—
108C22×C32—61no—
109C22×C32C3×M1611yescharacteristic
110C22×C32C4×A411yescharacteristic
111C32×C4—31no—
112C32×C4—31no—
113C32×C4C4×A411yescharacteristic
114C32×C4C22×A411yescharacteristic
115C22×C9—43no—
116C22⋊C9C3×M1613yescomplemented
117C4×C9—43no—
118C4×C9—43no—
119C23×C4C2×3−1+211yescharacteristic
120C2×M16—31no—
121C2×M16—32no—
122C2×M16—31no—
123C22×C8C2×3−1+212yescomplemented
1243−1+2—41noSylow
125C23×C3—21no—
126C23×C3—33no—
127C23×C3—33no—
128C23×C3—31no—
129C23×C3—31no—
130C23×C3—31no—
131C23×C3—61no—
132C23×C3—61no—
133C23×C3—63no—
134C23×C3C2×C32×C411yescharacteristic
135C2×C3×C4—33no—
136C2×C3×C4—33no—
137C2×C3×C4—36no—
138C2×C3×C4—32no—
139C2×C3×C4—33no—
140C2×C3×C4—33no—
141C2×C3×C4—33no—
142C2×C3×C4—33no—
143C2×C3×C4—33no—
144C2×C3×C4—31no—
145C2×C3×C4—31no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—31no—
149C2×C3×C4—31no—
150C2×C3×C4—31no—
151C2×C3×C4—31no—
152C2×C3×C4C2×C3×A411yescharacteristic
153C3×C8—36no—
154C3×C8—32no—
155C3×C8—32no—
156C3×C8C2×C3×A412yes—
157C2×C32—21no—
158C2×C32—31no—
159C2×C32—31no—
160C2×C32—61no—
161C2×C32C2×C4×A411yescharacteristic
162C2×C9—43no—
163C2×C9—83no—
164C24C4×3−1+211yescharacteristic
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4C4×3−1+211yescharacteristic
170C22×C4C22×3−1+211yescharacteristic
171M16—32no—
172M16—31no—
173M16—32no—
174M16#109411yescharacteristic, direct product factor
175C2×C8—32no—
176C2×C8—32no—
177C22×C3—31no—
178C22×C3—33no—
179C22×C3—33no—
180C22×C3—31no—
181C22×C3—33no—
182C22×C3—31no—
183C22×C3—31no—
184C22×C3—31no—
185C22×C3—61no—
186C22×C3—61no—
187C22×C3—61no—
188C22×C3—61no—
189C22×C3—63no—
190C22×C3—63no—
191C22×C3—63no—
192C22×C3—63no—
193C22×C3C32×M1611yescharacteristic
194C22×C3C3×C4×A411yescharacteristic
195C3×C4—33no—
196C3×C4—33no—
197C3×C4—31no—
198C3×C4—31no—
199C3×C4—31no—
200C3×C4—31no—
201C3×C4C3×C4×A411yescharacteristic
202C3×C4C22×C3×A411yescharacteristic
203C32A4×M1611yescharacteristic
204C9—43no—
205C23—21no—
206C23—31no—
207C23—31no—
208C23—61no—
209C23C2×C4×3−1+211yescharacteristic
210C2×C4—32no—
211C2×C4—31no—
212C2×C4—31no—
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4C2×#109411yescharacteristic
219C8—32no—
220C8C2×#109412yescomplemented
221C2×C3—21no—
222C2×C3—33no—
223C2×C3—33no—
224C2×C3—31no—
225C2×C3—31no—
226C2×C3—31no—
227C2×C3—61no—
228C2×C3—63no—
229C2×C3—61no—
230C2×C3C2×C3×C4×A411yescharacteristic
231C22—31no—
232C22—31no—
233C22—31no—
234C22—61no—
235C22—61no—
236C22—61no—
237C22—61no—
238C22M16×3−1+211yescharacteristic, complemented
239C22C4×#109411yescharacteristic
240C4—31no—
241C4—31no—
242C4C4×#109411yescharacteristic
243C4C22×#109411yescharacteristic
244C3—31no—
245C3C3×A4×M1611yescharacteristic
246C2—21no—
247C2—31no—
248C2—31no—
249C2—61no—
250C2C2×C4×#109411yescharacteristic
subgroupG/H
center, FrattiniC3×C4C22×C3×A4
commutator, socleC23×C3C2×C32×C4
FittingC22×C32×M16C3
prime radicalC24×C32C3×C4
cosocleC22×C3×C4C22×C32
hypercenterC32×M16A4
hypocenterC22M16×3−1+2
LCSG ▷ C23×C3 ▷ C22
UCSC1 ◁ C3×C4 ◁ C32×M16

Complements

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

NH ≅ G/Ntype
2. C8×#1094C2semidirect
4. M16×C22⋊C9C3semidirect
15. C8×C22⋊C9C2×C3semidirect
40. #1094M16direct (N × H)
73. C22×M163−1+2semidirect
116. C22⋊C9C3×M16semidirect
123. C22×C8C2×3−1+2semidirect
174. M16#1094direct (N × H)
220. C8C2×#1094semidirect
238. C22M16×3−1+2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10942
C8×#10942
C22×C32×M163
M16×C22⋊C93
M16×3−1+24