Finite Group Databasebeta

← #13166001

Subgroups — #13166001

Order 1728.

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

normal subgroups only

This group has 482 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
1#11724463C211yesmaximal, characteristic, complemented
2C3×C4⋊(C3⋊S4)C211yesmaximal, characteristic
3C3×A4×C3⋊C8C211yesmaximal, characteristic, complemented
4#10593635—31nomaximal
5#10593177—33nomaximal
6#10591175—31nomaximal
7C2×C3×C3⋊S4—21no—
8C32×C4×A4C2211yescharacteristic
9C3⋊(C3⋊SL(2,ℤ/4))—21no—
10#88796—41nomaximal
11C4⋊(C3⋊S4)D311yescharacteristic
12#64259—31no—
13#64259—33no—
14#64259—61no—
15#64259—63no—
16C3×C4⋊S4—33no—
17#64133—31no—
18C3×#930—31no—
19C22×C3×C3⋊C8D311yescharacteristic, complemented
20C3×C8×A4—31no—
21C2×#3716—31no—
22A4×C3⋊C8D313yescomplemented
23#63573—31no—
24#63567—31no—
25#63563—31no—
26C2×C32×A4D411yescharacteristic
27C3×C3⋊S4—41no—
28#6308—41no—
29C3×C4⋊(C3⋊D3)—41no—
30C32×C3⋊C8—41no—
31#5401—93no—
32#5114—91no—
33#5041—91no—
34C2×C3⋊S4—21no—
35C2×C3×S4—63no—
36C22×C32×C4C2×D311yescharacteristic
37C2×#634—31no—
38C2×C3×#67—61no—
39C2×C3×D12—31no—
40C3×C4×A4—21no—
41C3×C4×A4—23no—
42C3×C4×A4C2×D313yes—
43C3×C4×A4C2×D311yescharacteristic
44C3⋊SL(2,ℤ/4)—62no—
45C3⋊SL(2,ℤ/4)—66no—
46C3⋊SL(2,ℤ/4)—63no—
47C3⋊SL(2,ℤ/4)—61no—
48#3757—61no—
49C4⋊(C3⋊Dic3)—31no—
50#3750—61no—
51#3744—121no—
52C3×(C2×D3)⋊C4—61no—
53C3×C4⋊Dic3—31no—
54C2×C3×C3⋊C8—31no—
55C2×C3×C3⋊C8—31no—
56#3716—61no—
57#3716—61no—
58#3716—123no—
59C2×C3×C3⋊D3—81no—
60C32×A4SD1611yescharacteristic, complemented
61C33×C4—41no—
62C3⋊(C3⋊Dic3)—81no—
63C4⋊S4—33no—
64A4⋊Q8—93no—
65A4⋊Q8—183no—
66C22×C3×C8—31no—
67C3×#121—91no—
68C2×#214—91no—
69#930—31no—
70#924—91no—
71#924—91no—
72#924—181no—
73C22×C3⋊C8C3⋊D311yescharacteristic, complemented
74C2×#204—91no—
75C8×A4—33no—
76#832—91no—
77#821—91no—
78#817—91no—
79#816—91no—
80#810—91no—
81#808—91no—
82C23×C32D1211yescharacteristic
83C22×C3×D3—61no—
84C2×C3×A4—21no—
85C2×C3×A4—23no—
86C2×C3×A4D1213yes—
87C2×C3×A4#6711yescharacteristic
88C3⋊S4—41no—
89C3×S4—123no—
90C2×C32×C4—31no—
91C2×C32×C4—31no—
92C2×C3⋊Dic3—61no—
93C2×C3⋊Dic3—61no—
94C4⋊(C3⋊D3)—41no—
95#634—61no—
96#634—61no—
97#634—121no—
98#634—123no—
99#634—241no—
100#634—243no—
101C3×#67—121no—
102C3×#67—121no—
103C2×C3×Dic3—61no—
104C3×D12—61no—
105C3×D12—61no—
106C3×D12—123no—
107C32×C8—121no—
108C3×C3⋊C8—31no—
109C3×C3⋊C8—43no—
110C3×C3⋊C8S411yescharacteristic, complemented
111#465—271noSylow
112C2×C33—41no—
113C3×C3⋊D3—161no—
114C2×S4—63no—
115C2×C3×D4—91no—
116C2×C3×D4—181no—
117C22×C3×C4—21no—
118C22×C3×C4D3211yescharacteristic
119C22×C3×C4C2×C3⋊D311yescharacteristic
120C2×#67—61no—
121C2×D12—31no—
122C2×Dic6—91no—
123C2×Dic6—91no—
124C2×Dic6—181no—
125C4×A4—23no—
126C4×A4D3213yes—
127SL(2,ℤ/4)—186no—
128SL(2,ℤ/4)—183no—
129C2×C3×C8—91no—
130C2×C3×C8—91no—
131C3×C4⋊C4—91no—
132C3×C22⋊C4—181no—
133#217—182no—
134#217—181no—
135#217—181no—
136#214—181no—
137#214—181no—
138(C2×D3)⋊C4—61no—
139C4⋊Dic3—31no—
140C4⋊Dic3—91no—
141C4⋊Dic3—91no—
142C4⋊Dic3—181no—
143Dic3⋊C4—181no—
144Dic3⋊C4—181no—
145Dic3⋊C4—182no—
146C2×C3⋊C8—31no—
147C2×C3⋊C8—31no—
148#204—181no—
149#204—181no—
150#204—363no—
151C22×C32—31no—
152C22×C32—31no—
153C22×C32#20411yescharacteristic, complemented
154C2×C3⋊D3—81no—
155C2×C3×D3—121no—
156C2×C3×D3—121no—
157C2×C3×D3—121no—
158C2×C3×D3—243no—
159C3×A4—21no—
160C3×A4—23no—
161C3×A4#20413yescomplemented
162C3×A4#21411yescharacteristic, complemented
163C32×C4—31no—
164C32×C4—41no—
165C32×C4—43no—
166C32×C4—83no—
167C32×C4—81no—
168C32×C4C2×S411yescharacteristic
169C3⋊Dic3—121no—
170C3⋊Dic3—121no—
171C3⋊Dic3—243no—
172C3⋊Dic3—241no—
173C3⋊Dic3—246no—
174C3⋊Dic3—242no—
175C3×Dic3—121no—
176C2×SD16—271no—
177C22×C8—31no—
178C22⋊Q8—271no—
179#121—91no—
180#106—271no—
181#103—271no—
182#102—271no—
183C33—41noSylow
184C23×C3—21no—
185C23×C3—181no—
186C23×C3#62611yescharacteristic
187C23×C3C4⋊(C3⋊D3)11yescharacteristic
188C22×D3—61no—
189C2×A4—23no—
190C2×A4#62613yes—
191S4—123no—
192C3×D4—181no—
193C3×D4—181no—
194C3×D4—361no—
195C3×D4—361no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4—181no—
203#67—121no—
204#67—121no—
205C2×Dic3—61no—
206C2×Dic3—181no—
207C2×Dic3—181no—
208C2×Dic3—181no—
209C2×Dic3—181no—
210C2×Dic3—182no—
211C2×Dic3—182no—
212D12—61no—
213D12—61no—
214D12—123no—
215Dic6—181no—
216Dic6—181no—
217Dic6—181no—
218Dic6—181no—
219Dic6—361no—
220Dic6—361no—
221Dic6—363no—
222Dic6—723no—
223C3×C8—31no—
224C3×C8—91no—
225C3×C8—123no—
226C3⋊C8—31no—
227C3⋊C8C3⋊S411yescharacteristic, complemented
228C2×C32—31no—
229C2×C32—31no—
230C2×C32—41no—
231C2×C32—43no—
232C2×C32—83no—
233C2×C32—81no—
234C2×C32C4⋊S411yescharacteristic
235C3⋊D3—161no—
236C3×D3—241no—
237C3×D3—483no—
238C2×Q8—271no—
239C2×D4—91no—
240C2×D4—181no—
241C22×C4D3×C3⋊D311yescharacteristic
242SD16—541no—
243SD16—541no—
244C2×C8—91no—
245C2×C8—91no—
246C4⋊C4—91no—
247C4⋊C4—271no—
248C4⋊C4—541no—
249C22⋊C4—181no—
250C22⋊C4—541no—
251C22×C3—21no—
252C22×C3—31no—
253C22×C3—31no—
254C22×C3—31no—
255C22×C3—31no—
256C22×C3—61no—
257C22×C3—61no—
258C22×C3—361no—
259C22×C3—361no—
260C22×C3—361no—
261C22×C3#371611yescharacteristic, complemented
262C22×C3#374411yescharacteristic, complemented
263C2×D3—121no—
264C2×D3—121no—
265C2×D3—121no—
266C2×D3—243no—
267A4—23no—
268A4#371613yescomplemented
269C3×C4—21no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—43no—
273C3×C4—61no—
274C3×C4—83no—
275C3×C4—361no—
276C3×C4D3×S411yescharacteristic
277C3×C4C2×C3⋊S411yescharacteristic
278Dic3—121no—
279Dic3—362no—
280Dic3—361no—
281Dic3—361no—
282Dic3—362no—
283Dic3—361no—
284Dic3—361no—
285Dic3—723no—
286Dic3—726no—
287C32—41no—
288C32—43no—
289C32—83no—
290C32—81no—
291C32#540111yescharacteristic, complemented
292C23—181no—
293C23#629211yescharacteristic
294Q8—541no—
295Q8—541no—
296D4—181no—
297D4—181no—
298D4—361no—
299D4—361no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—181no—
303C2×C4—541no—
304C2×C4—541no—
305C8—31no—
306C8—91no—
307C2×C3—21no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—43no—
313C2×C3—61no—
314C2×C3—61no—
315C2×C3—83no—
316C2×C3—721no—
317C2×C3#6420711yescharacteristic
318C2×C3C4⋊(C3⋊S4)11yescharacteristic
319D3—241no—
320D3—483no—
321C22—31no—
322C22—31no—
323C22—361no—
324C22—361no—
325C22—361no—
326C22#8879611yescharacteristic, complemented
327C4—31no—
328C4—361no—
329C4—1081no—
330C4—1081no—
331C4D3×C3⋊S411yescharacteristic
332C3—21no—
333C3—43no—
334C3—83no—
335C3#1059317711yescharacteristic, complemented
336C3#1059363511yescharacteristic, complemented
337C2—31no—
338C2—31no—
339C2—721no—
340C2#1172420211yescharacteristic
subgroupG/H
centerC2#11724202
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4D3×C3⋊S4
FittingC22×C32×C4C2×D3
socleC23×C32D12
prime radicalC3×C4⋊(C3⋊S4)C2
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
1. #11724463C2semidirect
3. C3×A4×C3⋊C8C2semidirect
19. C22×C3×C3⋊C8D3semidirect
22. A4×C3⋊C8D3semidirect
60. C32×A4SD16semidirect
73. C22×C3⋊C8C3⋊D3semidirect
110. C3×C3⋊C8S4semidirect
153. C22×C32#204semidirect
161. C3×A4#204semidirect
162. C3×A4#214semidirect
227. C3⋊C8C3⋊S4semidirect
261. C22×C3#3716semidirect
262. C22×C3#3744semidirect
268. A4#3716semidirect
291. C32#5401semidirect
326. C22#88796semidirect
335. C3#10593177semidirect
336. C3#10593635semidirect

Maximal subgroups

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

subgroupindex
#117244632
C3×C4⋊(C3⋊S4)2
C3×A4×C3⋊C82
#105936353
#105931773
#105911753
#887964