Finite Group Databasebeta

← #13166009

Subgroups — #13166009

Order 1728.

474 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C3×A4×D12C211yesmaximal, characteristic
2C3×#64259C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊(A4⋊C8))C211yesmaximal, characteristic, complemented
4#10593648—31nomaximal
5#10593185—33nomaximal
6#10591169—31nomaximal
7C2×C3×D3×A4—21no—
8C32×C4×A4C2211yescharacteristic
9C3×C3⋊SL(2,ℤ/4)—21no—
10#88792—41nomaximal
11C22×C3×D12D311yescharacteristic
12C3×D4×A4—31no—
13A4×D12D313yes—
14#64259D311yescharacteristic, complemented
15C3×A4⋊Q8—33no—
16C3×#924—31no—
17C2×#3715—31no—
18C2×#3715—31no—
19C3⋊(A4⋊C8)—31no—
20C3⋊(A4⋊C8)—33no—
21C3⋊(A4⋊C8)—61no—
22C3⋊(A4⋊C8)—63no—
23#63657—31no—
24#63562—31no—
25#63562—31no—
26C2×C32×A4D411yescharacteristic
27C3×D3×A4—41no—
28C3×#634—41no—
29C32×D12—41no—
30C3⋊(C3⋊(C3⋊C8))—41no—
31#5414—93no—
32#5218—91no—
33#5046—91no—
34C23×C3×D3—21no—
35C22×C3×A4—61no—
36C2×D3×A4—23no—
37C22×C32×C4C2×D311yescharacteristic
38C2×C3×D12—31no—
39C2×C3×D12—31no—
40C2×C3×D12—61no—
41C2×C3×Dic6—31no—
42C3×C4×A4—21no—
43C3×C4×A4—23no—
44C3×C4×A4C2×D313yes—
45C3×C4×A4C2×D311yescharacteristic
46C3⋊SL(2,ℤ/4)—21no—
47C3×SL(2,ℤ/4)—63no—
48#3754—121no—
49C2×C3⋊(C3⋊C8)—31no—
50C2×C3⋊(C3⋊C8)—31no—
51C3×#217—61no—
52C3×C4⋊Dic3—31no—
53C3×Dic3⋊C4—61no—
54#3715—62no—
55#3715—61no—
56#3715—61no—
57#3715—123no—
58C2×C32×D3—81no—
59C32×A4SD1611yescharacteristic, complemented
60C33×C4—41no—
61C3×C3⋊Dic3—81no—
62C22×C3×D4—31no—
63C22×D12C3⋊D311yescharacteristic
64D4×A4—33no—
65A4⋊Q8—33no—
66C3×C22⋊Q8—91no—
67C2×#215—91no—
68C2×#215—91no—
69C2×#214—91no—
70C2×#214—91no—
71#924—31no—
72A4⋊C8—93no—
73A4⋊C8—183no—
74#832—91no—
75#832—91no—
76#830—91no—
77#830—91no—
78#830—181no—
79#809—91no—
80#809—91no—
81C23×C32#6711yescharacteristic
82C22×C3×D3—41no—
83C22×C3×D3—61no—
84C22×C3×D3—61no—
85C22×C3×D3—121no—
86C2×C3×A4—21no—
87C2×C3×A4—23no—
88C2×C3×A4—121no—
89C2×C3×A4#6713yes—
90C2×C3×A4#6711yescharacteristic
91D3×A4—43no—
92C32×D4—121no—
93C2×C32×C4—31no—
94C2×C32×C4—31no—
95#634—41no—
96C2×C3×Dic3—61no—
97C2×C3×Dic3—61no—
98C3×D12—32no—
99C3×D12—31no—
100C3×D12—43no—
101C3×D12—61no—
102C3×D12S411yescharacteristic
103C3×Dic6—61no—
104C3×Dic6—61no—
105C3×Dic6—123no—
106C3⋊(C3⋊C8)—61no—
107C3⋊(C3⋊C8)—61no—
108C3⋊(C3⋊C8)—121no—
109C3⋊(C3⋊C8)—123no—
110C3⋊(C3⋊C8)—241no—
111C3⋊(C3⋊C8)—243no—
112#450—271noSylow
113C2×C33—41no—
114C32×D3—161no—
115C24×C3—61no—
116C23×D3—21no—
117C22×A4—63no—
118C2×C3×Q8—91no—
119C2×C3×D4—91no—
120C2×C3×D4—91no—
121C2×C3×D4—181no—
122C22×C3×C4—21no—
123C22×C3×C4D3211yescharacteristic
124C22×C3×C4C2×C3⋊D311yescharacteristic
125C2×D12—31no—
126C2×D12—31no—
127C2×D12—61no—
128C2×Dic6—31no—
129C4×A4—23no—
130C4×A4D3213yes—
131SL(2,ℤ/4)—63no—
132C3×C4⋊C4—91no—
133C3×C4⋊C4—181no—
134C3×C22⋊C4—181no—
135#217—61no—
136#215—182no—
137#215—181no—
138#215—181no—
139#214—181no—
140#214—182no—
141#214—181no—
142#214—363no—
143C4⋊Dic3—31no—
144Dic3⋊C4—61no—
145C2×C3⋊C8—91no—
146C2×C3⋊C8—91no—
147C2×C3⋊C8—91no—
148C2×C3⋊C8—91no—
149C2×C3⋊C8—181no—
150C2×C3⋊C8—181no—
151C22×C32—31no—
152C22×C32—31no—
153C22×C32—241no—
154C22×C32#21411yescharacteristic, complemented
155C2×C3×D3—21no—
156C2×C3×D3—61no—
157C2×C3×D3—83no—
158C2×C3×D3—121no—
159C2×C3×D3—121no—
160C2×C3×D3—121no—
161C2×C3×D3—121no—
162C3×A4—21no—
163C3×A4—23no—
164C3×A4#21413yescomplemented
165C3×A4#21511yescharacteristic, complemented
166C32×C4—31no—
167C32×C4—41no—
168C32×C4—43no—
169C32×C4—83no—
170C32×C4—81no—
171C32×C4C2×S411yescharacteristic
172C3⋊Dic3—81no—
173C3×Dic3—121no—
174C3×Dic3—121no—
175C3×Dic3—243no—
176C22×D4—31no—
177C2×SD16—271no—
178C2×SD16—271no—
179C22⋊Q8—91no—
180#102—271no—
181#102—271no—
182C22⋊C8—271no—
183C33—41noSylow
184C23×C3—21no—
185C23×C3—121no—
186C23×C3—181no—
187C23×C3—181no—
188C23×C3—361no—
189C23×C3#62511yescharacteristic
190C23×C3#63811yescharacteristic
191C22×D3—41no—
192C22×D3—61no—
193C22×D3—61no—
194C22×D3—121no—
195C2×A4—23no—
196C2×A4—123no—
197C2×A4#62513yes—
198C3×Q8—181no—
199C3×Q8—181no—
200C3×D4—31no—
201C3×D4—92no—
202C3×D4—91no—
203C3×D4—123no—
204C3×D4—181no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—61no—
210C2×C3×C4—61no—
211C2×C3×C4—181no—
212C2×C3×C4—181no—
213C2×Dic3—61no—
214C2×Dic3—61no—
215D12—32no—
216D12—31no—
217D12—61no—
218D12C3⋊S411yescharacteristic
219Dic6—61no—
220Dic6—61no—
221Dic6—123no—
222C3⋊C8—181no—
223C3⋊C8—181no—
224C3⋊C8—181no—
225C3⋊C8—181no—
226C3⋊C8—361no—
227C3⋊C8—363no—
228C3⋊C8—361no—
229C3⋊C8—723no—
230C2×C32—31no—
231C2×C32—31no—
232C2×C32—41no—
233C2×C32—43no—
234C2×C32—83no—
235C2×C32—81no—
236C2×C32—481no—
237C2×C32GL(2,ℤ/4)11yescharacteristic
238C3×D3—41no—
239C3×D3—121no—
240C3×D3—163no—
241C24—61no—
242C2×Q8—91no—
243C2×D4—91no—
244C2×D4—91no—
245C2×D4—181no—
246C22×C4D3×C3⋊D311yescharacteristic
247SD16—541no—
248SD16—542no—
249SD16—541no—
250C2×C8—271no—
251C2×C8—271no—
252C4⋊C4—91no—
253C4⋊C4—181no—
254C22⋊C4—181no—
255C22×C3—21no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—31no—
259C22×C3—31no—
260C22×C3—61no—
261C22×C3—61no—
262C22×C3—61no—
263C22×C3—181no—
264C22×C3—243no—
265C22×C3—361no—
266C22×C3—361no—
267C22×C3—361no—
268C22×C3—361no—
269C22×C3#371511yescharacteristic, complemented
270C22×C3#375411yescharacteristic, complemented
271C2×D3—21no—
272C2×D3—61no—
273C2×D3—121no—
274C2×D3—121no—
275C2×D3—121no—
276C2×D3—121no—
277A4—23no—
278A4#371513yescomplemented
279C3×C4—21no—
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—43no—
283C3×C4—61no—
284C3×C4—83no—
285C3×C4—361no—
286C3×C4—361no—
287C3×C4D3×S411yescharacteristic
288C3×C4C2×C3⋊S411yescharacteristic
289Dic3—121no—
290Dic3—121no—
291Dic3—243no—
292C32—41no—
293C32—43no—
294C32—83no—
295C32—81no—
296C32#541411yescharacteristic, complemented
297C23—121no—
298C23—181no—
299C23—181no—
300C23—361no—
301C23#629011yescharacteristic
302Q8—181no—
303Q8—181no—
304D4—31no—
305D4—92no—
306D4—91no—
307D4—181no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—181no—
311C2×C4—181no—
312C8—541no—
313C8—541no—
314C2×C3—21no—
315C2×C3—31no—
316C2×C3—31no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—43no—
320C2×C3—61no—
321C2×C3—61no—
322C2×C3—83no—
323C2×C3—121no—
324C2×C3—361no—
325C2×C3—483no—
326C2×C3#6420911yescharacteristic
327C2×C3#6426911yescharacteristic
328D3—41no—
329D3—121no—
330C22—31no—
331C22—31no—
332C22—61no—
333C22—181no—
334C22—361no—
335C22—361no—
336C22—361no—
337C22—361no—
338C22#8879211yescharacteristic, complemented
339C4—31no—
340C4—361no—
341C4—361no—
342C4D3×C3⋊S411yescharacteristic
343C3—21no—
344C3—43no—
345C3—83no—
346C3#1059318511yescharacteristic, complemented
347C3#1059364811yescharacteristic, complemented
348C2—31no—
349C2—31no—
350C2—121no—
351C2—361no—
352C2#1172420411yescharacteristic
subgroupG/H
centerC2#11724204
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4D3×C3⋊S4
FittingC22×C32×C4C2×D3
socleC23×C32#67
prime radicalC3×A4×D12C2
hypocenterC32×A4SD16
LCSG ▷ C32×C4×A4 ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3×#64259C2semidirect
3. C3⋊(C3⋊(A4⋊C8))C2semidirect
14. #64259D3semidirect
59. C32×A4SD16semidirect
154. C22×C32#214semidirect
164. C3×A4#214semidirect
165. C3×A4#215semidirect
269. C22×C3#3715semidirect
270. C22×C3#3754semidirect
278. A4#3715semidirect
296. C32#5414semidirect
338. C22#88792semidirect
346. C3#10593185semidirect
347. C3#10593648semidirect

Maximal subgroups

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

subgroupindex
C3×A4×D122
C3×#642592
C3⋊(C3⋊(A4⋊C8))2
#105936483
#105931853
#105911693
#887924