Finite Group Databasebeta

← #13166006

Subgroups — #13166006

Order 1728.

478 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 76 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×C4×D3×A4C211yesmaximal, characteristic
2C3⋊(C3⋊(A4⋊C8))C211yesmaximal, characteristic, complemented
3C3×C3⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4#10593643—31nomaximal
5#10593182—33nomaximal
6#10591154—31nomaximal
7C2×C3×D3×A4C411yescharacteristic
8C32×C4×A4C2211yescharacteristic
9C3×Dic3×A4C411yescharacteristic
10#88786—41nomaximal
11C22×C3×C4×D3D311yescharacteristic
12C2×C3×C4×A4—31no—
13C4×D3×A4D313yes—
14C2×#3711—31no—
15C2×#3711—31no—
16C3⋊(A4⋊C8)—31no—
17C3⋊(A4⋊C8)—33no—
18C3⋊(A4⋊C8)—61no—
19C3⋊(A4⋊C8)—63no—
20C3⋊(A4⋊C8)D311yescharacteristic, complemented
21C3×A4⋊C8—33no—
22#63657—31no—
23C3×#830—31no—
24(C2×D3)⋊(C3⋊C8)—31no—
25(C2×D3)⋊(C3⋊C8)—31no—
26C2×C32×A4C2×C411yescharacteristic
27C3×D3×A4—21no—
28C32×C4×D3—41no—
29C3⋊(C3⋊(C3⋊C8))—41no—
30C3×C3⋊(C3⋊C8)—41no—
31#5409—93no—
32#5207—91no—
33#4726—91no—
34C23×C3×D3Dic311yescharacteristic
35C22×C3×A4—31no—
36C2×D3×A4Dic313yes—
37C22×C32×C4C2×D311yescharacteristic
38C22×C3×Dic3Dic311yescharacteristic
39C2×C3×C4×D3—31no—
40C2×C3×C4×D3—31no—
41C2×C3×C4×D3—31no—
42C2×C3×C4×D3—31no—
43C3×C4×A4—21no—
44C3×C4×A4—23no—
45C3×C4×A4—31no—
46C3×C4×A4C2×D311yescharacteristic
47C3×C4×A4C2×D313yes—
48Dic3×A4Dic313yes—
49#3748—121no—
50C2×C3⋊(C3⋊C8)—31no—
51C2×C3⋊(C3⋊C8)—31no—
52C2×C3×C3⋊C8—31no—
53C2×C3×C3⋊C8—31no—
54#3711—61no—
55#3711—61no—
56#3711—61no—
57#3711—61no—
58#3711—123no—
59C2×C32×D3—41no—
60C32×A4M1611yescharacteristic, complemented
61C33×C4—41no—
62C32×Dic3—41no—
63C23×C3×C4—31no—
64C22×C4×D3C3⋊D311yescharacteristic
65C2×C4×A4—33no—
66C2×#208—91no—
67C2×#208—91no—
68C2×ΓL(1,𝔽25)—91no—
69C2×ΓL(1,𝔽25)—91no—
70A4⋊C8—33no—
71A4⋊C8—93no—
72A4⋊C8—183no—
73C3×C22⋊C8—91no—
74#830—31no—
75#830—91no—
76#830—91no—
77#830—91no—
78#830—91no—
79#830—181no—
80(C2×D3)⋊C8—91no—
81(C2×D3)⋊C8—91no—
82C23×C32C2×Dic311yescharacteristic
83C22×C3×D3—21no—
84C22×C3×D3—31no—
85C22×C3×D3—31no—
86C22×C3×D3—61no—
87C2×C3×A4—21no—
88C2×C3×A4—23no—
89C2×C3×A4—61no—
90C2×C3×A4C4×D311yescharacteristic
91C2×C3×A4C2×Dic313yes—
92D3×A4—23no—
93C2×C32×C4—31no—
94C2×C32×C4—31no—
95C2×C32×C4—121no—
96C2×C3×Dic3—31no—
97C2×C3×Dic3—31no—
98C3×C4×D3—31no—
99C3×C4×D3—31no—
100C3×C4×D3—31no—
101C3×C4×D3—43no—
102C3×C4×D3—61no—
103C3×C4×D3S411yescharacteristic
104C3⋊(C3⋊C8)—41no—
105C3⋊(C3⋊C8)—61no—
106C3⋊(C3⋊C8)—61no—
107C3⋊(C3⋊C8)—121no—
108C3⋊(C3⋊C8)—123no—
109C3⋊(C3⋊C8)—241no—
110C3⋊(C3⋊C8)—243no—
111C3×C3⋊C8—61no—
112C3×C3⋊C8—61no—
113C3×C3⋊C8—123no—
114#407—271noSylow
115C2×C33—41no—
116C32×D3—81no—
117C24×C3—31no—
118C23×D3C3⋊Dic311yescharacteristic
119C22×A4—33no—
120C22×C3×C4—21no—
121C22×C3×C4—31no—
122C22×C3×C4—91no—
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C22×C3×C4—91no—
126C22×C3×C4D3211yescharacteristic
127C22×C3×C4C2×C3⋊D311yescharacteristic
128C22×Dic3C3⋊Dic311yescharacteristic
129C2×C4×D3—31no—
130C2×C4×D3—31no—
131C2×C4×D3—31no—
132C2×C4×D3—31no—
133C4×A4—23no—
134C4×A4—33no—
135C4×A4D3213yes—
136C2×C3×C8—91no—
137C2×C3×C8—91no—
138#208—181no—
139#208—181no—
140#208—181no—
141#208—181no—
142#208—363no—
143C2×C3⋊C8—31no—
144C2×C3⋊C8—31no—
145C2×C3⋊C8—91no—
146C2×C3⋊C8—91no—
147C2×C3⋊C8—91no—
148C2×C3⋊C8—91no—
149C2×C3⋊C8—181no—
150C2×C3⋊C8—181no—
151ΓL(1,𝔽25)—181no—
152ΓL(1,𝔽25)—181no—
153ΓL(1,𝔽25)—181no—
154ΓL(1,𝔽25)—181no—
155C22×C32—31no—
156C22×C32—31no—
157C22×C32—121no—
158C22×C32#20811yescharacteristic, complemented
159C2×C3×D3—32no—
160C2×C3×D3—31no—
161C2×C3×D3—43no—
162C2×C3×D3—61no—
163C2×C3×D3—61no—
164C2×C3×D3—61no—
165C2×C3×D3SL(2,ℤ/4)11yescharacteristic
166C3×A4—21no—
167C3×A4—23no—
168C3×A4ΓL(1,𝔽25)11yescharacteristic, complemented
169C3×A4#20813yescomplemented
170C32×C4—31no—
171C32×C4—41no—
172C32×C4—43no—
173C32×C4—83no—
174C32×C4—81no—
175C32×C4—121no—
176C32×C4C2×S411yescharacteristic
177C3×Dic3—31no—
178C3×Dic3—43no—
179C3×Dic3SL(2,ℤ/4)11yescharacteristic
180C23×C4—31no—
181C2×M16—271no—
182C2×M16—271no—
183C22⋊C8—91no—
184C22⋊C8—271no—
185C22⋊C8—271no—
186C22⋊C8—271no—
187C33—41noSylow
188C23×C3—21no—
189C23×C3—61no—
190C23×C3—91no—
191C23×C3—91no—
192C23×C3—181no—
193C23×C3D3×Dic311yescharacteristic
194C23×C3C2×C3⋊Dic311yescharacteristic
195C22×D3—21no—
196C22×D3—31no—
197C22×D3—31no—
198C22×D3—61no—
199C2×A4—23no—
200C2×A4—63no—
201C2×A4D3×Dic313yes—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—91no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—123no—
215C2×C3×C4—181no—
216C2×Dic3—31no—
217C2×Dic3—31no—
218C4×D3—31no—
219C4×D3—31no—
220C4×D3—31no—
221C4×D3—61no—
222C4×D3C3⋊S411yescharacteristic
223C3×C8—181no—
224C3×C8—181no—
225C3⋊C8—61no—
226C3⋊C8—61no—
227C3⋊C8—123no—
228C3⋊C8—181no—
229C3⋊C8—181no—
230C3⋊C8—181no—
231C3⋊C8—181no—
232C3⋊C8—361no—
233C3⋊C8—363no—
234C3⋊C8—361no—
235C3⋊C8—723no—
236C2×C32—31no—
237C2×C32—31no—
238C2×C32—41no—
239C2×C32—43no—
240C2×C32—83no—
241C2×C32—81no—
242C2×C32—241no—
243C2×C32C2×SL(2,ℤ/4)11yescharacteristic
244C3×D3—21no—
245C3×D3—61no—
246C3×D3—83no—
247C24—31no—
248C22×C4—31no—
249C22×C4—91no—
250C22×C4—91no—
251C22×C4—91no—
252C22×C4—91no—
253C22×C4D3×C3⋊D311yescharacteristic
254M16—541no—
255M16—541no—
256M16—541no—
257M16—541no—
258C2×C8—91no—
259C2×C8—91no—
260C2×C8—271no—
261C2×C8—271no—
262C22×C3—21no—
263C22×C3—31no—
264C22×C3—31no—
265C22×C3—31no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—61no—
269C22×C3—61no—
270C22×C3—92no—
271C22×C3—91no—
272C22×C3—123no—
273C22×C3—181no—
274C22×C3—181no—
275C22×C3—181no—
276C22×C3#371111yescharacteristic, complemented
277C22×C3#374811yescharacteristic, complemented
278C2×D3—32no—
279C2×D3—31no—
280C2×D3—61no—
281C2×D3—61no—
282C2×D3—61no—
283C2×D3C3⋊SL(2,ℤ/4)11yescharacteristic
284A4—23no—
285A4#371113yescomplemented
286C3×C4—21no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—43no—
291C3×C4—61no—
292C3×C4—83no—
293C3×C4—91no—
294C3×C4—123no—
295C3×C4D3×S411yescharacteristic
296C3×C4C2×C3⋊S411yescharacteristic
297Dic3—31no—
298Dic3C3⋊SL(2,ℤ/4)11yescharacteristic
299C32—41no—
300C32—43no—
301C32—83no—
302C32—81no—
303C32#540911yescharacteristic, complemented
304C23—61no—
305C23—91no—
306C23—91no—
307C23—181no—
308C23D3×C3⋊Dic311yescharacteristic
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—91no—
313C2×C4—91no—
314C2×C4—91no—
315C2×C4—91no—
316C2×C4—91no—
317C2×C4—181no—
318C8—181no—
319C8—181no—
320C8—541no—
321C8—541no—
322C2×C3—21no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—43no—
328C2×C3—61no—
329C2×C3—61no—
330C2×C3—61no—
331C2×C3—83no—
332C2×C3—181no—
333C2×C3—243no—
334C2×C3D3×SL(2,ℤ/4)11yescharacteristic
335C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
336D3—21no—
337D3—61no—
338C22—31no—
339C22—31no—
340C22—31no—
341C22—92no—
342C22—91no—
343C22—181no—
344C22—181no—
345C22—181no—
346C22#8878611yescharacteristic, complemented
347C4—31no—
348C4—31no—
349C4—91no—
350C4D3×C3⋊S411yescharacteristic
351C3—21no—
352C3—43no—
353C3—83no—
354C3#1059318211yescharacteristic, complemented
355C3#1059364311yescharacteristic, complemented
356C2—31no—
357C2—31no—
358C2—61no—
359C2—181no—
360C2D3×C3⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D3×C3⋊S4
commutatorC2×C32×A4C2×C4
FittingC22×C32×C4C2×D3
socleC23×C32C2×Dic3
prime radicalC2×C3×D3×A4C4
cosocleC32×C4×A4C22
hypocenterC32×A4M16
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊(A4⋊C8))C2semidirect
3. C3×C3⋊(A4⋊C8)C2semidirect
20. C3⋊(A4⋊C8)D3semidirect
60. C32×A4M16semidirect
158. C22×C32#208semidirect
168. C3×A4ΓL(1,𝔽25)semidirect
169. C3×A4#208semidirect
276. C22×C3#3711semidirect
277. C22×C3#3748semidirect
285. A4#3711semidirect
303. C32#5409semidirect
346. C22#88786semidirect
354. C3#10593182semidirect
355. C3#10593643semidirect

Maximal subgroups

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

subgroupindex
C3×C4×D3×A42
C3⋊(C3⋊(A4⋊C8))2
C3×C3⋊(A4⋊C8)2
#105936433
#105931823
#105911543
#887864