Finite Group Databasebeta

← #10592825⋊C3 #13143210

Subgroups — #10592825⋊C3 #13143210

Order 1728.

733 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×M16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4#63647⋊C3C214yesmaximal, complemented
5#10592825C311yesmaximal, characteristic, complemented
6C3×#5126—31nomaximal
7C23×(C3⋊D3)⋊C3C411yescharacteristic
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3C411yescharacteristic
10C2×C4×(C3⋊D3)⋊C3—22no—
11C2×C4×(C3⋊D3)⋊C3—22no—
12C2×C4×(C3⋊D3)⋊C3C412yes—
13C2×C4×(C3⋊D3)⋊C3C2212yes—
14M16×H(3,𝔽3)—22no—
15C2×C8×H(3,𝔽3)C2212yescomplemented
16#3748⋊C3—22no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18C22×C4×C3⋊D3C2×C311yescharacteristic
19C22×C3×C4×D3—31no—
20C2×C32×M16—21no—
21C2×C32×M16D311yescharacteristic, complemented
22C2×C32×M16C2×C311yescharacteristic, complemented
23C2×#3748C2×C311yescharacteristic, complemented
24C2×C3×#208—31no—
25#63647C2×C314yescomplemented
26C3×(C2×D3)⋊C8—34no—
27C23×H(3,𝔽3)C2×C411yescharacteristic
28C22×(C3⋊D3)⋊C3—22no—
29C22×(C3⋊D3)⋊C3—24no—
30C22×(C3⋊D3)⋊C3C2×C412yes—
31C2×C4×H(3,𝔽3)C2×C411yescharacteristic
32C2×C4×H(3,𝔽3)D412yes—
33C2×C4×H(3,𝔽3)D412yes—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×(C3⋊Dic3)⋊C3—22no—
36C2×(C3⋊Dic3)⋊C3C2×C412yes—
37C4×(C3⋊D3)⋊C3—24no—
38C4×(C3⋊D3)⋊C3—22no—
39C4×(C3⋊D3)⋊C3—22no—
40C8×H(3,𝔽3)—22no—
41(C3⋊(C3⋊C8))⋊C3—22no—
42C3×#407—91no—
43#5126—31no—
44#5126—91no—
45C23×C3⋊D3C3×C411yescharacteristic
46C23×C3×D3—31no—
47C22×C32×C4—21no—
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C22×C311yescharacteristic
50C22×C3⋊Dic3C3×C411yescharacteristic
51C2×C4×C3⋊D3—22no—
52C2×C4×C3⋊D3—22no—
53C2×C4×C3⋊D3C3×C412yes—
54C2×C4×C3⋊D3C22×C312yes—
55C22×C3×Dic3—31no—
56C2×C3×C4×D3—32no—
57C2×C3×C4×D3—32no—
58C2×C3×C4×D3—62no—
59C2×C3×C4×D3—62no—
60C32×M16—24no—
61C32×M16—22no—
62C32×M16—22no—
63C2×C32×C8—22no—
64C2×C32×C8C2×D312yescomplemented
65C2×C32×C8C22×C312yescomplemented
66#3748—22no—
67C2×C3⋊(C3⋊C8)C22×C312yescomplemented
68C3×#208—62no—
69C2×C3×C3⋊C8—32no—
70C22×H(3,𝔽3)—22no—
71C22×H(3,𝔽3)C22⋊C412yes—
72C22×H(3,𝔽3)C22×C411yescharacteristic
73C2×(C3⋊D3)⋊C3—28no—
74C2×(C3⋊D3)⋊C3—22no—
75C2×(C3⋊D3)⋊C3—22no—
76C2×(C3⋊D3)⋊C3M1614yescomplemented
77C4×H(3,𝔽3)C22⋊C412yes—
78C4×H(3,𝔽3)C2×D411yescharacteristic
79C4×H(3,𝔽3)C2×D411yescharacteristic
80(C3⋊Dic3)⋊C3—22no—
81C23×C3×C4—91no—
82C22×C4×D3—31no—
83C22×C4×D3—91no—
84C2×C3×M16—31no—
85C2×C3×M16—31no—
86C2×C3×M16—61no—
87C2×C3×M16—91no—
88C2×C3×M16C3×D311yescharacteristic
89C2×#208—31no—
90C2×#208—91no—
91C3×C22⋊C8—94no—
92(C2×D3)⋊C8—34no—
93(C2×D3)⋊C8—94no—
94C23×C32—21no—
95C23×C32C4×D311yescharacteristic
96C23×C32C2×C3×C411yescharacteristic
97C22×C3⋊D3—22no—
98C22×C3⋊D3—24no—
99C22×C3⋊D3C2×C3×C412yes—
100C22×C3×D3—32no—
101C22×C3×D3—64no—
102C22×C3×D3—62no—
103C2×C32×C4—21no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—21no—
107C2×C32×C4C4×D311yescharacteristic
108C2×C32×C4D1212yes—
109C2×C32×C4#6712yes—
110C2×C32×C4C2×C3×C411yescharacteristic
111C2×C32×C4C3×D412yes—
112C2×C32×C4C3×D412yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C23×C311yescharacteristic
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3C2×C3×C412yes—
117C4×C3⋊D3—24no—
118C4×C3⋊D3—22no—
119C4×C3⋊D3—22no—
120C2×C3×Dic3—32no—
121C2×C3×Dic3—62no—
122C3×C4×D3—64no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125C32×C8—24no—
126C32×C8—22no—
127C32×C8—22no—
128C3⋊(C3⋊C8)—22no—
129C3×C3⋊C8—62no—
130#407—91noSylow
131C2×H(3,𝔽3)—22no—
132C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
133C2×H(3,𝔽3)C2×M1612yes—
134(C3⋊D3)⋊C3—24no—
135C24×C3—91no—
136C23×D3—31no—
137C23×D3—91no—
138C22×C3×C4—31no—
139C22×C3×C4—31no—
140C22×C3×C4—61no—
141C22×C3×C4—92no—
142C22×C3×C4—92no—
143C22×C3×C4—91no—
144C22×C3×C4—182no—
145C22×C3×C4—182no—
146C22×C3×C4C2×C3×D311yescharacteristic
147C22×Dic3—31no—
148C22×Dic3—91no—
149C2×C4×D3—32no—
150C2×C4×D3—32no—
151C2×C4×D3—62no—
152C2×C4×D3—62no—
153C2×C4×D3—92no—
154C2×C4×D3—92no—
155C2×C4×D3—182no—
156C2×C4×D3—182no—
157C3×M16—22no—
158C3×M16—62no—
159C3×M16—62no—
160C3×M16—64no—
161C3×M16—182no—
162C2×C3×C8—32no—
163C2×C3×C8—32no—
164C2×C3×C8—62no—
165C2×C3×C8—92no—
166C2×C3×C8C2×C3×D312yes—
167#208—62no—
168#208—182no—
169C2×C3⋊C8—32no—
170C2×C3⋊C8—92no—
171C22×C32—22no—
172C22×C32—22no—
173C22×C32—22no—
174C22×C32—21no—
175C22×C32—42no—
176C22×C32(C2×D3)⋊C412yes—
177C22×C32C3×C22⋊C412yes—
178C22×C32C2×C4×D311yescharacteristic
179C22×C32C22×C3×C411yescharacteristic
180C2×C3⋊D3—28no—
181C2×C3⋊D3—22no—
182C2×C3⋊D3—22no—
183C2×C3⋊D3C3×M1614yescomplemented
184C2×C3×D3—34no—
185C2×C3×D3—68no—
186C2×C3×D3—62no—
187C2×C3×D3—62no—
188C32×C4—22no—
189C32×C4—21no—
190C32×C4—21no—
191C32×C4(C2×D3)⋊C412yes—
192C32×C4C3×C22⋊C412yes—
193C32×C4C2×D1211yescharacteristic
194C32×C4C2×#6711yescharacteristic
195C32×C4C2×C3×D411yescharacteristic
196C32×C4C2×C3×D411yescharacteristic
197C3⋊Dic3—22no—
198C3×Dic3—62no—
199C23×C4—91no—
200C2×M16—91no—
201C2×M16(C3⋊D3)⋊C311yescharacteristic, complemented
202C22⋊C8—94no—
203H(3,𝔽3)#40711yescharacteristic, complemented, Sylow
204C23×C3—31no—
205C23×C3—31no—
206C23×C3—61no—
207C23×C3—92no—
208C23×C3—182no—
209C23×C3—184no—
210C23×C3C3×C4×D311yescharacteristic
211C22×D3—32no—
212C22×D3—64no—
213C22×D3—62no—
214C22×D3—92no—
215C22×D3—184no—
216C22×D3—182no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—32no—
220C2×C3×C4—32no—
221C2×C3×C4—32no—
222C2×C3×C4—32no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—61no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—61no—
229C2×C3×C4—92no—
230C2×C3×C4—182no—
231C2×C3×C4—182no—
232C2×C3×C4—184no—
233C2×C3×C4—182no—
234C2×C3×C4C3×C4×D311yescharacteristic
235C2×C3×C4C3×D1212yes—
236C2×C3×C4C3×#6712yes—
237C2×C3×C4C22×C3×D311yescharacteristic
238C2×Dic3—32no—
239C2×Dic3—62no—
240C2×Dic3—92no—
241C2×Dic3—182no—
242C4×D3—64no—
243C4×D3—62no—
244C4×D3—62no—
245C4×D3—182no—
246C4×D3—182no—
247C4×D3—184no—
248C3×C8—22no—
249C3×C8—62no—
250C3×C8—62no—
251C3×C8—64no—
252C3×C8—182no—
253C3⋊C8—62no—
254C3⋊C8—182no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—42no—
260C2×C32D3×M1612yes—
261C2×C32C2×(C2×D3)⋊C411yescharacteristic
262C2×C32C2×C3×C22⋊C411yescharacteristic
263C2×C32C2×C3×M1612yes—
264C3⋊D3—24no—
265C3×D3—64no—
266C24—91no—
267C22×C4—91no—
268C22×C4—92no—
269C22×C4—92no—
270C22×C4—182no—
271C22×C4—182no—
272C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
273M16—22no—
274M16—182no—
275C2×C8—92no—
276C2×C8C2×(C3⋊D3)⋊C312yescomplemented
277C22×C3—22no—
278C22×C3—31no—
279C22×C3—31no—
280C22×C3—32no—
281C22×C3—32no—
282C22×C3—62no—
283C22×C3—62no—
284C22×C3—61no—
285C22×C3—62no—
286C22×C3—94no—
287C22×C3—122no—
288C22×C3—182no—
289C22×C3—182no—
290C22×C3—188no—
291C22×C3C3×(C2×D3)⋊C412yes—
292C22×C3C2×C3×C4×D311yescharacteristic
293C2×D3—34no—
294C2×D3—68no—
295C2×D3—62no—
296C2×D3—62no—
297C2×D3—94no—
298C2×D3—182no—
299C2×D3—182no—
300C2×D3—188no—
301C3×C4—31no—
302C3×C4—31no—
303C3×C4—31no—
304C3×C4—31no—
305C3×C4—32no—
306C3×C4—32no—
307C3×C4—61no—
308C3×C4—61no—
309C3×C4—62no—
310C3×C4—182no—
311C3×C4C3×(C2×D3)⋊C412yes—
312C3×C4C2×C3×D1211yescharacteristic
313C3×C4C2×C3×#6711yescharacteristic
314Dic3—62no—
315Dic3—182no—
316C32—21no—
317C32#512611yescharacteristic, complemented
318C32C3×#40711yescharacteristic, complemented
319C23—92no—
320C23—184no—
321C23—182no—
322C23C4×(C3⋊D3)⋊C311yescharacteristic
323C2×C4—92no—
324C2×C4—182no—
325C2×C4—182no—
326C2×C4—182no—
327C2×C4—184no—
328C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
329C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
330C2×C4#638⋊C312yes—
331C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
332C8—22no—
333C8—182no—
334C2×C3—22no—
335C2×C3—32no—
336C2×C3—32no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—62no—
340C2×C3—62no—
341C2×C3—62no—
342C2×C3—61no—
343C2×C3—122no—
344C2×C3—184no—
345C2×C3C3×D3×M1612yes—
346C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
347D3—64no—
348D3—184no—
349C22—22no—
350C22—94no—
351C22—188no—
352C22—182no—
353C22—182no—
354C22#3752⋊C312yes—
355C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
356C4—182no—
357C4#3752⋊C312yes—
358C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
359C4C2×#638⋊C311yescharacteristic
360C3—31no—
361C3—31no—
362C3—61no—
363C3C3×#512611yescharacteristic
364C2—22no—
365C2—184no—
366C2M16×(C3⋊D3)⋊C312yes—
367C2C2×#3752⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×H(3,𝔽3)C2
socleC22×C3C2×C3×C4×D3
prime radicalC23×(C3⋊D3)⋊C3C4
cosocleC2×C32×C4C23×C3
hypercenterC2×M16(C3⋊D3)⋊C3
hypocenterC32C3×#407
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×M16×H(3,𝔽3)C2semidirect
3. C2×#3748⋊C3C2semidirect
4. #63647⋊C3C2semidirect
5. #10592825C3semidirect
15. C2×C8×H(3,𝔽3)C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
21. C2×C32×M16D3semidirect
22. C2×C32×M16C2×C3semidirect
23. C2×#3748C2×C3semidirect
25. #63647C2×C3semidirect
64. C2×C32×C8C2×D3semidirect
65. C2×C32×C8C22×C3semidirect
67. C2×C3⋊(C3⋊C8)C22×C3semidirect
76. C2×(C3⋊D3)⋊C3M16semidirect
183. C2×C3⋊D3C3×M16semidirect
201. C2×M16(C3⋊D3)⋊C3semidirect
203. H(3,𝔽3)#407semidirect
276. C2×C8C2×(C3⋊D3)⋊C3semidirect
317. C32#5126semidirect
318. C32C3×#407semidirect

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C2×M16×H(3,𝔽3)2
C2×#3748⋊C32
#63647⋊C32
#105928253
C3×#51263