Finite Group Databasebeta

← #13166004

Subgroups — #13166004

Order 1728.

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

normal subgroups only

This group has 446 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
2C3×#64259C211yesmaximal, characteristic
3C3×A4×C3⋊C8C211yesmaximal, characteristic
4#10593632—31nomaximal
5#10593180—33nomaximal
6#10591187—31nomaximal
7C32×C4×A4C2211yescharacteristic
8C3⋊(C3⋊SL(2,ℤ/4))—21no—
9C3×C3⋊SL(2,ℤ/4)—21no—
10#88800—41nomaximal
11#64259—31no—
12#64259—33no—
13#64259—61no—
14#64259—63no—
15#64259D311yescharacteristic
16C3×A4⋊Q8—33no—
17#64133—31no—
18C3×#924—31no—
19C22×C3×C3⋊C8D311yescharacteristic
20C3×C8×A4—31no—
21C2×#3719—31no—
22A4×C3⋊C8D313yes—
23#63575—31no—
24#63567—31no—
25#63566—31no—
26C2×C32×A4D411yescharacteristic
27#6308—41no—
28C3×#634—41no—
29C32×C3⋊C8—41no—
30#5398—93no—
31#5116—91no—
32#5051—91no—
33C22×C32×C4C2×D311yescharacteristic
34C2×#634—31no—
35C2×C3×Dic6—31no—
36C3×C4×A4—21no—
37C3×C4×A4—23no—
38C3×C4×A4C2×D313yes—
39C3×C4×A4C2×D311yescharacteristic
40C3⋊SL(2,ℤ/4)—21no—
41C3⋊SL(2,ℤ/4)—62no—
42C3⋊SL(2,ℤ/4)—66no—
43C3⋊SL(2,ℤ/4)—63no—
44C3⋊SL(2,ℤ/4)—61no—
45C3×SL(2,ℤ/4)—63no—
46#3757—61no—
47C4⋊(C3⋊Dic3)—31no—
48#3750—61no—
49#3746—121no—
50C3×#217—61no—
51C3×C4⋊Dic3—31no—
52C3×Dic3⋊C4—61no—
53C2×C3×C3⋊C8—31no—
54C2×C3×C3⋊C8—31no—
55#3719—61no—
56#3719—61no—
57#3719—123no—
58C32×A4Q1611yescharacteristic, complemented
59C33×C4—41no—
60C3⋊(C3⋊Dic3)—81no—
61C3×C3⋊Dic3—81no—
62A4⋊Q8—33no—
63A4⋊Q8—93no—
64A4⋊Q8—183no—
65C22×C3×C8—31no—
66C3×C22⋊Q8—91no—
67C2×#216—91no—
68#924—31no—
69#924—91no—
70#924—91no—
71#924—181no—
72C22×C3⋊C8C3⋊D311yescharacteristic
73C2×Dic12—91no—
74C8×A4—33no—
75#835—91no—
76#818—91no—
77#816—91no—
78#816—91no—
79#810—91no—
80#807—91no—
81C23×C32D1211yescharacteristic
82C2×C3×A4—21no—
83C2×C3×A4—23no—
84C2×C3×A4D1213yes—
85C2×C3×A4#6711yescharacteristic
86C2×C32×C4—31no—
87C2×C32×C4—31no—
88C2×C3⋊Dic3—61no—
89C2×C3⋊Dic3—61no—
90#634—41no—
91#634—61no—
92#634—61no—
93#634—121no—
94#634—123no—
95#634—241no—
96#634—243no—
97C2×C3×Dic3—61no—
98C2×C3×Dic3—61no—
99C3×Dic6—61no—
100C3×Dic6—61no—
101C3×Dic6—123no—
102C32×C8—121no—
103C3×C3⋊C8—31no—
104C3×C3⋊C8—43no—
105C3×C3⋊C8S411yescharacteristic
106#467—271noSylow
107C2×C33—41no—
108C2×C3×Q8—91no—
109C22×C3×C4—21no—
110C22×C3×C4D3211yescharacteristic
111C22×C3×C4C2×C3⋊D311yescharacteristic
112C2×Dic6—31no—
113C2×Dic6—91no—
114C2×Dic6—91no—
115C2×Dic6—181no—
116C4×A4—23no—
117C4×A4D3213yes—
118SL(2,ℤ/4)—63no—
119SL(2,ℤ/4)—186no—
120SL(2,ℤ/4)—183no—
121C2×C3×C8—91no—
122C2×C3×C8—91no—
123C3×C4⋊C4—91no—
124C3×C4⋊C4—181no—
125C3×C22⋊C4—181no—
126#217—61no—
127#217—182no—
128#217—181no—
129#217—181no—
130#216—181no—
131#216—181no—
132C4⋊Dic3—31no—
133C4⋊Dic3—91no—
134C4⋊Dic3—91no—
135C4⋊Dic3—181no—
136Dic3⋊C4—61no—
137Dic3⋊C4—181no—
138Dic3⋊C4—181no—
139Dic3⋊C4—182no—
140C2×C3⋊C8—31no—
141C2×C3⋊C8—31no—
142Dic12—181no—
143Dic12—181no—
144Dic12—363no—
145C22×C32—31no—
146C22×C32—31no—
147C22×C32Dic1211yescharacteristic, complemented
148C3×A4—21no—
149C3×A4—23no—
150C3×A4Dic1213yescomplemented
151C3×A4#21611yescharacteristic, complemented
152C32×C4—31no—
153C32×C4—41no—
154C32×C4—43no—
155C32×C4—83no—
156C32×C4—81no—
157C32×C4C2×S411yescharacteristic
158C3⋊Dic3—81no—
159C3⋊Dic3—121no—
160C3⋊Dic3—121no—
161C3⋊Dic3—243no—
162C3⋊Dic3—241no—
163C3⋊Dic3—246no—
164C3⋊Dic3—242no—
165C3×Dic3—121no—
166C3×Dic3—121no—
167C3×Dic3—243no—
168C2×Q16—271no—
169C22×C8—31no—
170C22⋊Q8—91no—
171C22⋊Q8—271no—
172#107—271no—
173#103—271no—
174#103—271no—
175C33—41noSylow
176C23×C3—21no—
177C23×C3#62611yescharacteristic
178C23×C3C4⋊(C3⋊D3)11yescharacteristic
179C2×A4—23no—
180C2×A4#62613yes—
181C3×Q8—181no—
182C3×Q8—181no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—181no—
190C2×C3×C4—181no—
191C2×Dic3—61no—
192C2×Dic3—61no—
193C2×Dic3—181no—
194C2×Dic3—181no—
195C2×Dic3—181no—
196C2×Dic3—181no—
197C2×Dic3—182no—
198C2×Dic3—182no—
199Dic6—61no—
200Dic6—61no—
201Dic6—123no—
202Dic6—181no—
203Dic6—181no—
204Dic6—181no—
205Dic6—181no—
206Dic6—361no—
207Dic6—361no—
208Dic6—363no—
209Dic6—723no—
210C3×C8—31no—
211C3×C8—91no—
212C3×C8—123no—
213C3⋊C8—31no—
214C3⋊C8C3⋊S411yescharacteristic
215C2×C32—31no—
216C2×C32—31no—
217C2×C32—41no—
218C2×C32—43no—
219C2×C32—83no—
220C2×C32—81no—
221C2×C32C4⋊S411yescharacteristic
222C2×Q8—91no—
223C2×Q8—271no—
224C22×C4D3×C3⋊D311yescharacteristic
225Q16—541no—
226Q16—541no—
227C2×C8—91no—
228C2×C8—91no—
229C4⋊C4—91no—
230C4⋊C4—181no—
231C4⋊C4—271no—
232C4⋊C4—541no—
233C22⋊C4—181no—
234C22⋊C4—541no—
235C22×C3—21no—
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—61no—
241C22×C3—61no—
242C22×C3#371911yescharacteristic, complemented
243C22×C3#374611yescharacteristic, complemented
244A4—23no—
245A4#371913yescomplemented
246C3×C4—21no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—43no—
250C3×C4—61no—
251C3×C4—83no—
252C3×C4—361no—
253C3×C4—361no—
254C3×C4D3×S411yescharacteristic
255C3×C4C2×C3⋊S411yescharacteristic
256Dic3—121no—
257Dic3—121no—
258Dic3—243no—
259Dic3—362no—
260Dic3—361no—
261Dic3—361no—
262Dic3—362no—
263Dic3—361no—
264Dic3—361no—
265Dic3—723no—
266Dic3—726no—
267C32—41no—
268C32—43no—
269C32—83no—
270C32—81no—
271C32#539811yescharacteristic, complemented
272C23#629211yescharacteristic
273Q8—181no—
274Q8—181no—
275Q8—541no—
276Q8—541no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—181no—
280C2×C4—181no—
281C2×C4—541no—
282C2×C4—541no—
283C8—31no—
284C8—91no—
285C2×C3—21no—
286C2×C3—31no—
287C2×C3—31no—
288C2×C3—31no—
289C2×C3—31no—
290C2×C3—43no—
291C2×C3—61no—
292C2×C3—61no—
293C2×C3—83no—
294C2×C3#6420711yescharacteristic
295C2×C3C4⋊(C3⋊S4)11yescharacteristic
296C22—31no—
297C22—31no—
298C22#8880011yescharacteristic, complemented
299C4—31no—
300C4—361no—
301C4—361no—
302C4—1081no—
303C4—1081no—
304C4D3×C3⋊S411yescharacteristic
305C3—21no—
306C3—43no—
307C3—83no—
308C3#1059318011yescharacteristic, complemented
309C3#1059363211yescharacteristic, complemented
310C2—31no—
311C2—31no—
312C2#1172420211yescharacteristic
subgroupG/H
centerC2#11724202
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4D3×C3⋊S4
FittingC22×C32×C4C2×D3
socleC23×C32D12
prime radicalC2×C32×A4D4
hypocenterC32×A4Q16
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
58. C32×A4Q16semidirect
147. C22×C32Dic12semidirect
150. C3×A4Dic12semidirect
151. C3×A4#216semidirect
242. C22×C3#3719semidirect
243. C22×C3#3746semidirect
245. A4#3719semidirect
271. C32#5398semidirect
298. C22#88800semidirect
308. C3#10593180semidirect
309. C3#10593632semidirect

Maximal subgroups

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

subgroupindex
#117244632
C3×#642592
C3×A4×C3⋊C82
#105936323
#105931803
#105911873
#888004