Finite Group Databasebeta

← M16×#1113 #13178454

Subgroups — M16×#1113 #13178454

Order 1728.

1416 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 255 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×#1113C211yesmaximal, characteristic
2C3×M16×C3⋊D3C213yesmaximal, complemented
3#11723656C213yesmaximal, complemented
4#11723645C216yesmaximal, complemented
5C8×#1113C212yesmaximal, complemented
6D32×M16—33nomaximal
7C22×#1113C411yescharacteristic
8C2×C3×C4×C3⋊D3C2213yes—
9C2×#6294C413yes—
10C4×#1113C413yes—
11C4×#1113C411yescharacteristic
12C4×#1113C2211yescharacteristic
13C4×#1113C2213yes—
14C33×M16C2211yescharacteristic, complemented
15C3×#3748C2213yescomplemented
16C3×#3743C2216yescomplemented
17C3×C8×C3⋊D3C2216yescomplemented
18#88809C2216yescomplemented
19#88808C2216yescomplemented
20C2×C4×D32—33no—
21M16×C3⋊D3D313yescomplemented
22C3×D3×M16—36no—
23C3×D3×M16—63no—
24#63818—33no—
25D3×#208—36no—
26#63791—36no—
27D3×ΓL(1,𝔽25)—312no—
28D32×C8—36no—
29C22×C3×C3⋊D3C2×C413yes—
30C2×#1113—24no—
31C2×#1113C2×C413yes—
32C2×#1113C2×C411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3C2×C413yes—
35C3×C4×C3⋊D3C2×C413yes—
36C3×C4×C3⋊D3C2×C413yes—
37C3×C4×C3⋊D3C2313yes—
38C3×C4×C3⋊D3C2313yes—
39#6294C2×C416yes—
40#6294C2×C413yes—
41#6294C2×C413yes—
42C33×C8C2312yescomplemented
43C3×C3⋊(C3⋊C8)C2316yescomplemented
44C2×D3×M16—93no—
45C22×D32—33no—
46C2×C4×C3⋊D3C2×D313yes—
47C2×C3×C4×D3—36no—
48C2×C3×C4×D3—63no—
49C2×#624—33no—
50C2×D3×Dic3—36no—
51C4×D32—33no—
52C4×D32—36no—
53C4×D32—33no—
54C4×D32—33no—
55C4×D32—36no—
56C4×D32—33no—
57C32×M16—23no—
58C32×M16—41no—
59C32×M16C2×D313yescomplemented
60#3748C2×D313yescomplemented
61#3743C2×D316yescomplemented
62C8×C3⋊D3C2×D316yescomplemented
63C3×#208—36no—
64C3×#208—63no—
65C3×ΓL(1,𝔽25)—312no—
66C3×ΓL(1,𝔽25)—66no—
67C3×D3×C8—312no—
68C3×D3×C8—66no—
69#3712—36no—
70#3711—312no—
71#3710—36no—
72D3×C3⋊C8—312no—
73C22×C33C22×C411yescharacteristic
74C2×C3×C3⋊D3—26no—
75C2×C3×C3⋊D3C22×C413yes—
76C2×C3×C3⋊D3C22×C413yes—
77#1113—26no—
78#1113M1614yesdirect product factor
79C33×C4C22×C411yescharacteristic
80C33×C4C2411yescharacteristic
81C3×C3⋊Dic3C22×C413yes—
82C3×C3⋊Dic3C22×C413yes—
83C22×C4×D3—93no—
84C2×C3×M16—93no—
85C2×#208—93no—
86D3×M16—36no—
87D3×M16—63no—
88D3×M16—912no—
89D3×M16—96no—
90C2×ΓL(1,𝔽25)—96no—
91C2×D3×C8—96no—
92C22×C3⋊D3C4×D313yes—
93C22×C3×D3—36no—
94C22×C3×D3—63no—
95C2×D32—33no—
96C2×D32—36no—
97C2×D32—33no—
98C2×D32—612no—
99C2×C32×C4—23no—
100C2×C32×C4—41no—
101C2×C32×C4C22×D313yes—
102C2×C3⋊Dic3C4×D313yes—
103C4×C3⋊D3C4×D313yes—
104C4×C3⋊D3C4×D313yes—
105C4×C3⋊D3C22×D313yes—
106C4×C3⋊D3C22×D313yes—
107C2×C3×Dic3—36no—
108C2×C3×Dic3—63no—
109C3×C4×D3—36no—
110C3×C4×D3—36no—
111C3×C4×D3—36no—
112C3×C4×D3—36no—
113C3×C4×D3—63no—
114C3×C4×D3—63no—
115C3×C4×D3—63no—
116C3×C4×D3—63no—
117#624—36no—
118#624—33no—
119#624—33no—
120D3×Dic3—36no—
121D3×Dic3—36no—
122D3×Dic3—36no—
123D3×Dic3—36no—
124C32×C8—26no—
125C32×C8—42no—
126C32×C8C22×D316yescomplemented
127C3⋊(C3⋊C8)C22×D316yescomplemented
128C3×C3⋊C8—312no—
129C3×C3⋊C8—66no—
130C22×M16—271noSylow
131C2×C33—21no—
132C2×C33C23×C411yescharacteristic
133C3×C3⋊D3—23no—
134C3×C3⋊D3C2×M1616yescomplemented
135C23×D3—93no—
136C22×C3×C4—93no—
137C22×Dic3—93no—
138C2×C4×D3—36no—
139C2×C4×D3—63no—
140C2×C4×D3—93no—
141C2×C4×D3—93no—
142C2×C4×D3—96no—
143C2×C4×D3—93no—
144C2×C4×D3—96no—
145C2×C4×D3—93no—
146C2×C4×D3—96no—
147C3×M16—23no—
148C3×M16—41no—
149C3×M16—93no—
150C3×M16—96no—
151C3×M16D3213yescomplemented
152C2×C3×C8—96no—
153#208—36no—
154#208—63no—
155#208—96no—
156C2×C3⋊C8—96no—
157ΓL(1,𝔽25)—312no—
158ΓL(1,𝔽25)—66no—
159ΓL(1,𝔽25)—912no—
160D3×C8—312no—
161D3×C8—66no—
162D3×C8—912no—
163C22×C32—23no—
164C22×C32—41no—
165C22×C32C2×C4×D313yes—
166C2×C3⋊D3—26no—
167C2×C3⋊D3C2×C4×D313yes—
168C2×C3⋊D3C2×C4×D313yes—
169C2×C3×D3—36no—
170C2×C3×D3—36no—
171C2×C3×D3—63no—
172C2×C3×D3—63no—
173C2×C3×D3—612no—
174C2×C3×D3—126no—
175D32—312no—
176D32—612no—
177D32—66no—
178C32×C4—23no—
179C32×C4—23no—
180C32×C4—41no—
181C32×C4—41no—
182C32×C4C2×C4×D313yes—
183C32×C4C23×D313yes—
184C3⋊Dic3C2×C4×D313yes—
185C3⋊Dic3C2×C4×D313yes—
186C3×Dic3—36no—
187C3×Dic3—36no—
188C3×Dic3—63no—
189C3×Dic3—63no—
190C23×C4—271no—
191C2×M16—93no—
192C2×M16—276no—
193C2×M16—273no—
194C22×C8—272no—
195C33C22×M1611yescharacteristic, complemented, Sylow
196C23×C3—93no—
197C22×D3—36no—
198C22×D3—63no—
199C22×D3—93no—
200C22×D3—96no—
201C22×D3—93no—
202C22×D3—1812no—
203C2×C3×C4—23no—
204C2×C3×C4—41no—
205C2×C3×C4—93no—
206C2×C3×C4—93no—
207C2×C3×C4—93no—
208C2×C3×C4—93no—
209C2×C3×C4—93no—
210C2×C3×C4C2×D3213yes—
211C2×Dic3—36no—
212C2×Dic3—63no—
213C2×Dic3—93no—
214C2×Dic3—93no—
215C2×Dic3—96no—
216C4×D3—36no—
217C4×D3—36no—
218C4×D3—36no—
219C4×D3—36no—
220C4×D3—63no—
221C4×D3—63no—
222C4×D3—63no—
223C4×D3—63no—
224C4×D3—96no—
225C4×D3—96no—
226C4×D3—96no—
227C4×D3—96no—
228C3×C8—26no—
229C3×C8—42no—
230C3×C8—96no—
231C3×C8C2×D3216yescomplemented
232C3⋊C8—312no—
233C3⋊C8—66no—
234C2×C32—23no—
235C2×C32—23no—
236C2×C32—41no—
237C2×C32—43no—
238C2×C32—81no—
239C2×C32C22×C4×D313yes—
240C3⋊D3—23no—
241C3⋊D3D3×M1616yescomplemented
242C3×D3—312no—
243C3×D3—66no—
244C3×D3—66no—
245C3×D3—123no—
246C24—271no—
247C22×C4—93no—
248C22×C4—271no—
249C22×C4—273no—
250C22×C4—271no—
251C22×C4—273no—
252C22×C4—273no—
253M16—93no—
254M16—96no—
255M16—276no—
256M16#111311yescharacteristic, direct product factor
257C2×C8—96no—
258C2×C8—276no—
259C22×C3—23no—
260C22×C3—41no—
261C22×C3—93no—
262C22×C3—93no—
263C22×C3—186no—
264C22×C3C4×D3213yes—
265C2×D3—36no—
266C2×D3—36no—
267C2×D3—63no—
268C2×D3—63no—
269C2×D3—612no—
270C2×D3—912no—
271C2×D3—126no—
272C2×D3—1812no—
273C2×D3—186no—
274C3×C4—23no—
275C3×C4—23no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—93no—
279C3×C4—93no—
280C3×C4C4×D3213yes—
281C3×C4C22×D3213yes—
282Dic3—36no—
283Dic3—36no—
284Dic3—63no—
285Dic3—63no—
286C32—23no—
287C32—41no—
288C32C2×D3×M1613yescomplemented
289C23—93no—
290C23—271no—
291C23—273no—
292C23—544no—
293C2×C4—93no—
294C2×C4—93no—
295C2×C4—93no—
296C2×C4—93no—
297C2×C4—93no—
298C2×C4—273no—
299C2×C4—273no—
300C2×C4—276no—
301C2×C4C2×#111311yescharacteristic
302C8—96no—
303C8C2×#111312yescomplemented
304C2×C3—23no—
305C2×C3—23no—
306C2×C3—43no—
307C2×C3—41no—
308C2×C3—81no—
309C2×C3—96no—
310C2×C3—183no—
311C2×C3C2×C4×D3213yes—
312D3—312no—
313D3—66no—
314D3—66no—
315D3—123no—
316C22—93no—
317C22—93no—
318C22—186no—
319C22—274no—
320C22—546no—
321C22C4×#111311yescharacteristic
322C4—93no—
323C4—93no—
324C4C4×#111311yescharacteristic
325C4C22×#111311yescharacteristic
326C3—23no—
327C3—41no—
328C3D32×M1613yescomplemented
329C2—21no—
330C2—96no—
331C2—183no—
332C2C2×C4×#111311yescharacteristic
subgroupG/H
center, FrattiniC4C22×#1113
commutator, socleC2×C33C23×C4
FittingC33×M16C22
prime radicalC22×#1113C4
cosocleC33×C4C24
hypercenterM16#1113
hypocenterC33C22×M16
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ M16

Complements

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

NH ≅ G/Ntype
2. C3×M16×C3⋊D3C2semidirect
3. #11723656C2semidirect
4. #11723645C2semidirect
5. C8×#1113C2semidirect
14. C33×M16C22semidirect
15. C3×#3748C22semidirect
16. C3×#3743C22semidirect
17. C3×C8×C3⋊D3C22semidirect
18. #88809C22semidirect
19. #88808C22semidirect
21. M16×C3⋊D3D3semidirect
42. C33×C8C23semidirect
43. C3×C3⋊(C3⋊C8)C23semidirect
59. C32×M16C2×D3semidirect
60. #3748C2×D3semidirect
61. #3743C2×D3semidirect
62. C8×C3⋊D3C2×D3semidirect
78. #1113M16direct (N × H)
126. C32×C8C22×D3semidirect
127. C3⋊(C3⋊C8)C22×D3semidirect
134. C3×C3⋊D3C2×M16semidirect
151. C3×M16D32semidirect
195. C33C22×M16semidirect
231. C3×C8C2×D32semidirect
241. C3⋊D3D3×M16semidirect
256. M16#1113direct (N × H)
288. C32C2×D3×M16semidirect
303. C8C2×#1113semidirect
328. C3D32×M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11132
C3×M16×C3⋊D32
#117236562
#117236452
C8×#11132
D32×M163