Finite Group Databasebeta

← #13166002

Subgroups — #13166002

Order 1728.

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

normal subgroups only

This group has 520 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
1C4⋊(C3⋊(C3⋊S4))C211yesmaximal, characteristic
2C3×#64259C211yesmaximal, characteristic, complemented
3C3×A4×C3⋊C8C211yesmaximal, characteristic, complemented
4#10593635—31nomaximal
5#10593178—33nomaximal
6#10591177—31nomaximal
7C2×C3⋊(C3⋊S4)—21no—
8C32×C4×A4C2211yescharacteristic
9C3×C3⋊SL(2,ℤ/4)—21no—
10#88797—41nomaximal
11C4⋊(C3⋊S4)—31no—
12C4⋊(C3⋊S4)—33no—
13C4⋊(C3⋊S4)—61no—
14C4⋊(C3⋊S4)—63no—
15#64259D311yescharacteristic, complemented
16C3×A4⋊Q8—33no—
17#64139—31no—
18C3×#924—31no—
19C22×C3×C3⋊C8D311yescharacteristic, complemented
20C3×C8×A4—31no—
21C2×#3717—31no—
22A4×C3⋊C8D313yescomplemented
23#63573—31no—
24#63566—31no—
25#63564—31no—
26C2×C32×A4D411yescharacteristic
27C3⋊(C3⋊S4)—41no—
28C4⋊(C3⋊(C3⋊D3))—41no—
29C3×#634—41no—
30C32×C3⋊C8—41no—
31#5401—93no—
32#5114—91no—
33#5048—91no—
34C2×C3⋊S4—62no—
35C2×C3⋊S4—66no—
36C2×C3⋊S4—63no—
37C2×C3⋊S4—61no—
38C22×C32×C4C2×D311yescharacteristic
39C2×#638—61no—
40C2×C4⋊(C3⋊D3)—31no—
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#3752—61no—
49C4⋊(C3⋊Dic3)—31no—
50#3744—121no—
51C3×#217—61no—
52C3×C4⋊Dic3—31no—
53C3×Dic3⋊C4—61no—
54C2×C3×C3⋊C8—31no—
55C2×C3×C3⋊C8—31no—
56#3717—61no—
57#3717—61no—
58#3717—123no—
59C2×C3⋊(C3⋊D3)—81no—
60C32×A4SD1611yescharacteristic, complemented
61C33×C4—41no—
62C3×C3⋊Dic3—81no—
63C4⋊S4—93no—
64C4⋊S4—183no—
65A4⋊Q8—33no—
66C22×C3×C8—31no—
67C3×C22⋊Q8—91no—
68C2×#215—91no—
69#930—91no—
70#930—91no—
71#930—181no—
72#924—31no—
73C22×C3⋊C8C3⋊D311yescharacteristic, complemented
74C2×#204—91no—
75C8×A4—33no—
76#835—91no—
77#821—91no—
78#817—91no—
79#816—91no—
80#809—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—126no—
89C3⋊S4—122no—
90C3⋊S4—123no—
91C3⋊S4—121no—
92C2×C32×C4—31no—
93C2×C32×C4—31no—
94#638—121no—
95#638—121no—
96C2×C3⋊Dic3—61no—
97C4⋊(C3⋊D3)—61no—
98C4⋊(C3⋊D3)—61no—
99C4⋊(C3⋊D3)—121no—
100C4⋊(C3⋊D3)—123no—
101C4⋊(C3⋊D3)—241no—
102C4⋊(C3⋊D3)—243no—
103#634—41no—
104C2×C3×Dic3—61no—
105C2×C3×Dic3—61no—
106C3×Dic6—61no—
107C3×Dic6—61no—
108C3×Dic6—123no—
109C32×C8—121no—
110C3×C3⋊C8—31no—
111C3×C3⋊C8—43no—
112C3×C3⋊C8S411yescharacteristic, complemented
113#465—271noSylow
114C2×C33—41no—
115C3⋊(C3⋊D3)—161no—
116C2×S4—186no—
117C2×S4—183no—
118C2×C3×Q8—91no—
119C22×C3×C4—21no—
120C22×C3×C4D3211yescharacteristic
121C22×C3×C4C2×C3⋊D311yescharacteristic
122C2×#67—182no—
123C2×#67—181no—
124C2×#67—181no—
125C2×D12—91no—
126C2×D12—91no—
127C2×D12—181no—
128C2×Dic6—31no—
129C4×A4—23no—
130C4×A4D3213yes—
131SL(2,ℤ/4)—63no—
132C2×C3×C8—91no—
133C2×C3×C8—91no—
134C3×C4⋊C4—91no—
135C3×C4⋊C4—181no—
136C3×C22⋊C4—181no—
137#217—61no—
138#215—181no—
139#215—181no—
140(C2×D3)⋊C4—181no—
141(C2×D3)⋊C4—181no—
142(C2×D3)⋊C4—182no—
143C4⋊Dic3—31no—
144C4⋊Dic3—91no—
145C4⋊Dic3—91no—
146C4⋊Dic3—181no—
147Dic3⋊C4—61no—
148C2×C3⋊C8—31no—
149C2×C3⋊C8—31no—
150#204—181no—
151#204—181no—
152#204—363no—
153C22×C32—31no—
154C22×C32—31no—
155C22×C32#20411yescharacteristic, complemented
156C2×C3⋊D3—121no—
157C2×C3⋊D3—121no—
158C2×C3⋊D3—121no—
159C2×C3⋊D3—243no—
160C2×C3⋊D3—241no—
161C2×C3⋊D3—246no—
162C2×C3⋊D3—242no—
163C3×A4—21no—
164C3×A4—23no—
165C3×A4#20413yescomplemented
166C3×A4#21511yescharacteristic, complemented
167C32×C4—31no—
168C32×C4—41no—
169C32×C4—43no—
170C32×C4—83no—
171C32×C4—81no—
172C32×C4C2×S411yescharacteristic
173C3⋊Dic3—81no—
174C3⋊Dic3—121no—
175C3×Dic3—121no—
176C3×Dic3—121no—
177C3×Dic3—243no—
178C2×SD16—271no—
179C22×C8—31no—
180C22⋊Q8—91no—
181#121—271no—
182#106—271no—
183#103—271no—
184#102—271no—
185C33—41noSylow
186C23×C3—21no—
187C23×C3#62611yescharacteristic
188C23×C3C4⋊(C3⋊D3)11yescharacteristic
189C22×D3—181no—
190C22×D3—181no—
191C22×D3—182no—
192C2×A4—23no—
193C2×A4#62613yes—
194S4—366no—
195S4—363no—
196C3×Q8—181no—
197C3×Q8—181no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—181no—
205C2×C3×C4—181no—
206#67—362no—
207#67—361no—
208#67—361no—
209#67—362no—
210#67—361no—
211#67—361no—
212C2×Dic3—61no—
213C2×Dic3—61no—
214C2×Dic3—181no—
215C2×Dic3—181no—
216C2×Dic3—182no—
217D12—181no—
218D12—181no—
219D12—181no—
220D12—181no—
221D12—361no—
222D12—363no—
223D12—361no—
224D12—723no—
225Dic6—61no—
226Dic6—61no—
227Dic6—123no—
228C3×C8—31no—
229C3×C8—91no—
230C3×C8—123no—
231C3⋊C8—31no—
232C3⋊C8C3⋊S411yescharacteristic, complemented
233C2×C32—31no—
234C2×C32—31no—
235C2×C32—41no—
236C2×C32—43no—
237C2×C32—83no—
238C2×C32—81no—
239C2×C32C4⋊S411yescharacteristic
240C3⋊D3—241no—
241C3⋊D3—483no—
242C3⋊D3—481no—
243C3⋊D3—486no—
244C3⋊D3—482no—
245C2×Q8—91no—
246C2×D4—271no—
247C2×D4—541no—
248C22×C4D3×C3⋊D311yescharacteristic
249SD16—541no—
250SD16—541no—
251C2×C8—91no—
252C2×C8—91no—
253C4⋊C4—91no—
254C4⋊C4—181no—
255C4⋊C4—271no—
256C22⋊C4—181no—
257C22⋊C4—541no—
258C22×C3—21no—
259C22×C3—31no—
260C22×C3—31no—
261C22×C3—31no—
262C22×C3—31no—
263C22×C3—61no—
264C22×C3—61no—
265C22×C3#371711yescharacteristic, complemented
266C22×C3#374411yescharacteristic, complemented
267C2×D3—362no—
268C2×D3—361no—
269C2×D3—361no—
270C2×D3—361no—
271C2×D3—361no—
272C2×D3—362no—
273C2×D3—362no—
274C2×D3—361no—
275C2×D3—361no—
276C2×D3—723no—
277C2×D3—726no—
278A4—23no—
279A4#371713yescomplemented
280C3×C4—21no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—43no—
284C3×C4—61no—
285C3×C4—83no—
286C3×C4—361no—
287C3×C4—361no—
288C3×C4D3×S411yescharacteristic
289C3×C4C2×C3⋊S411yescharacteristic
290Dic3—121no—
291Dic3—121no—
292Dic3—243no—
293Dic3—361no—
294Dic3—361no—
295Dic3—362no—
296C32—41no—
297C32—43no—
298C32—83no—
299C32—81no—
300C32#540111yescharacteristic, complemented
301C23—541no—
302C23#629211yescharacteristic
303Q8—181no—
304Q8—181no—
305D4—541no—
306D4—541no—
307D4—1081no—
308D4—1081no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—181no—
312C2×C4—181no—
313C2×C4—541no—
314C8—31no—
315C8—91no—
316C2×C3—21no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—43no—
322C2×C3—61no—
323C2×C3—61no—
324C2×C3—83no—
325C2×C3#6420711yescharacteristic
326C2×C3C4⋊(C3⋊S4)11yescharacteristic
327D3—722no—
328D3—721no—
329D3—721no—
330D3—1443no—
331D3—1446no—
332C22—31no—
333C22—31no—
334C22—1081no—
335C22—1081no—
336C22—1081no—
337C22#8879711yescharacteristic, complemented
338C4—31no—
339C4—361no—
340C4—361no—
341C4—1081no—
342C4D3×C3⋊S411yescharacteristic
343C3—21no—
344C3—43no—
345C3—83no—
346C3#1059317811yescharacteristic, complemented
347C3#1059363511yescharacteristic, complemented
348C2—31no—
349C2—31no—
350C2—2161no—
351C2#1172420211yescharacteristic
subgroupG/H
centerC2#11724202
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4D3×C3⋊S4
FittingC22×C32×C4C2×D3
socleC23×C32D12
prime radicalC4⋊(C3⋊(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
2. C3×#64259C2semidirect
3. C3×A4×C3⋊C8C2semidirect
15. #64259D3semidirect
19. C22×C3×C3⋊C8D3semidirect
22. A4×C3⋊C8D3semidirect
60. C32×A4SD16semidirect
73. C22×C3⋊C8C3⋊D3semidirect
112. C3×C3⋊C8S4semidirect
155. C22×C32#204semidirect
165. C3×A4#204semidirect
166. C3×A4#215semidirect
232. C3⋊C8C3⋊S4semidirect
265. C22×C3#3717semidirect
266. C22×C3#3744semidirect
279. A4#3717semidirect
300. C32#5401semidirect
337. C22#88797semidirect
346. C3#10593178semidirect
347. C3#10593635semidirect

Maximal subgroups

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

subgroupindex
C4⋊(C3⋊(C3⋊S4))2
C3×#642592
C3×A4×C3⋊C82
#105936353
#105931783
#105911773
#887974