Finite Group Databasebeta

← #13166012

Subgroups — #13166012

Order 1728.

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

normal subgroups only

This group has 540 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×Dic6C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊(C3⋊S4))C211yesmaximal, characteristic
3C3×C3⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4#10593651—31nomaximal
5#10593188—33nomaximal
6#10591174—31nomaximal
7C2×C3⋊(C3⋊S4)—21no—
8C32×C4×A4C2211yescharacteristic
9C3×Dic3×A4—21no—
10#88795—41nomaximal
11C22×C3×Dic6D311yescharacteristic, complemented
12C3×Q8×A4—31no—
13A4×Dic6D313yescomplemented
14C4⋊(C3⋊S4)—31no—
15C4⋊(C3⋊S4)—33no—
16C4⋊(C3⋊S4)—61no—
17C4⋊(C3⋊S4)—63no—
18#64139—31no—
19C2×#3717—31no—
20C2×#3717—31no—
21C3⋊(A4⋊C8)D311yescharacteristic, complemented
22C3×A4⋊C8—33no—
23C3×#830—31no—
24#63566—31no—
25#63566—31no—
26C2×C32×A4D411yescharacteristic
27C3⋊(C3⋊S4)—41no—
28C4⋊(C3⋊(C3⋊D3))—41no—
29C32×Dic6—41no—
30C3×C3⋊(C3⋊C8)—41no—
31#5417—93no—
32#5227—91no—
33#4738—91no—
34C2×C3⋊S4—62no—
35C2×C3⋊S4—66no—
36C2×C3⋊S4—61no—
37C2×C3⋊S4—63no—
38C22×C32×C4C2×D311yescharacteristic
39C2×#638—61no—
40C2×C4⋊(C3⋊D3)—31no—
41C22×C3×Dic3—21no—
42C2×C3×Dic6—31no—
43C2×C3×Dic6—31no—
44C2×C3×Dic6—61no—
45C3×C4×A4—21no—
46C3×C4×A4—23no—
47C3×C4×A4—61no—
48C3×C4×A4C2×D311yescharacteristic
49C3×C4×A4C2×D313yes—
50Dic3×A4—23no—
51#3755—121no—
52#3752—61no—
53C4⋊(C3⋊Dic3)—31no—
54C2×C3×C3⋊C8—31no—
55C2×C3×C3⋊C8—31no—
56#3717—62no—
57#3717—61no—
58#3717—61no—
59#3717—123no—
60C2×C3⋊(C3⋊D3)—81no—
61C32×A4SD1611yescharacteristic, complemented
62C33×C4—41no—
63C32×Dic3—81no—
64C22×C3×Q8—31no—
65C22×Dic6C3⋊D311yescharacteristic, complemented
66Q8×A4—33no—
67C4⋊S4—93no—
68C4⋊S4—183no—
69C2×#215—91no—
70C2×#215—91no—
71#930—91no—
72#930—91no—
73#930—181no—
74C2×#204—91no—
75C2×#204—91no—
76A4⋊C8—33no—
77C3×C22⋊C8—91no—
78#835—91no—
79#835—91no—
80#830—31no—
81#816—91no—
82#816—91no—
83C23×C32#6711yescharacteristic
84C22×C3⋊D3—61no—
85C2×C3×A4—21no—
86C2×C3×A4—23no—
87C2×C3×A4D1211yescharacteristic
88C2×C3×A4#6713yes—
89C3⋊S4—126no—
90C3⋊S4—122no—
91C3⋊S4—121no—
92C3⋊S4—123no—
93C32×Q8—121no—
94C2×C32×C4—31no—
95C2×C32×C4—31no—
96#638—121no—
97#638—121no—
98C2×C3⋊Dic3—61no—
99C4⋊(C3⋊D3)—61no—
100C4⋊(C3⋊D3)—61no—
101C4⋊(C3⋊D3)—121no—
102C4⋊(C3⋊D3)—123no—
103C4⋊(C3⋊D3)—241no—
104C4⋊(C3⋊D3)—243no—
105C2×C3×Dic3—61no—
106C2×C3×Dic3—61no—
107C3×Dic6—32no—
108C3×Dic6—31no—
109C3×Dic6—43no—
110C3×Dic6—61no—
111C3×Dic6S411yescharacteristic, complemented
112C3⋊(C3⋊C8)—41no—
113C3×C3⋊C8—61no—
114C3×C3⋊C8—61no—
115C3×C3⋊C8—123no—
116#448—271noSylow
117C2×C33—41no—
118C3⋊(C3⋊D3)—161no—
119C2×S4—186no—
120C2×S4—183no—
121C2×C3×Q8—91no—
122C2×C3×Q8—91no—
123C2×C3×Q8—181no—
124C22×C3×C4—21no—
125C22×C3×C4—61no—
126C22×C3×C4D3211yescharacteristic
127C22×C3×C4C2×C3⋊D311yescharacteristic
128C2×#67—182no—
129C2×#67—181no—
130C2×#67—181no—
131C22×Dic3—21no—
132C2×D12—91no—
133C2×D12—91no—
134C2×D12—181no—
135C2×Dic6—31no—
136C2×Dic6—31no—
137C2×Dic6—61no—
138C4×A4—23no—
139C4×A4—63no—
140C4×A4D3213yes—
141C2×C3×C8—91no—
142C2×C3×C8—91no—
143#215—181no—
144#215—182no—
145#215—181no—
146#215—363no—
147(C2×D3)⋊C4—181no—
148(C2×D3)⋊C4—181no—
149(C2×D3)⋊C4—182no—
150C4⋊Dic3—91no—
151C4⋊Dic3—91no—
152C4⋊Dic3—181no—
153C2×C3⋊C8—31no—
154C2×C3⋊C8—31no—
155#204—182no—
156#204—181no—
157#204—181no—
158C22×C32—31no—
159C22×C32—31no—
160C22×C32#21511yescharacteristic, complemented
161C2×C3⋊D3—121no—
162C2×C3⋊D3—121no—
163C2×C3⋊D3—121no—
164C2×C3⋊D3—243no—
165C2×C3⋊D3—241no—
166C2×C3⋊D3—246no—
167C2×C3⋊D3—242no—
168C3×A4—21no—
169C3×A4—23no—
170C3×A4#20411yescharacteristic, complemented
171C3×A4#21513yescomplemented
172C32×C4—31no—
173C32×C4—41no—
174C32×C4—43no—
175C32×C4—83no—
176C32×C4—81no—
177C32×C4—241no—
178C32×C4C2×S411yescharacteristic
179C3⋊Dic3—121no—
180C3×Dic3—21no—
181C3×Dic3—61no—
182C3×Dic3—83no—
183C22×Q8—31no—
184C2×SD16—271no—
185C2×SD16—271no—
186#121—271no—
187#103—271no—
188#103—271no—
189C22⋊C8—91no—
190C33—41noSylow
191C23×C3—21no—
192C23×C3#62611yescharacteristic
193C23×C3#63811yescharacteristic
194C22×D3—181no—
195C22×D3—181no—
196C22×D3—182no—
197C2×A4—23no—
198C2×A4#62613yes—
199S4—366no—
200S4—363no—
201C3×Q8—31no—
202C3×Q8—92no—
203C3×Q8—91no—
204C3×Q8—123no—
205C3×Q8—181no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—61no—
211C2×C3×C4—61no—
212C2×C3×C4—181no—
213C2×C3×C4—181no—
214#67—362no—
215#67—361no—
216#67—361no—
217#67—362no—
218#67—361no—
219#67—361no—
220C2×Dic3—61no—
221C2×Dic3—61no—
222C2×Dic3—181no—
223C2×Dic3—181no—
224C2×Dic3—182no—
225D12—181no—
226D12—181no—
227D12—181no—
228D12—181no—
229D12—361no—
230D12—363no—
231D12—361no—
232D12—723no—
233Dic6—32no—
234Dic6—31no—
235Dic6—61no—
236Dic6C3⋊S411yescharacteristic, complemented
237C3×C8—181no—
238C3×C8—181no—
239C3⋊C8—61no—
240C3⋊C8—61no—
241C3⋊C8—123no—
242C2×C32—31no—
243C2×C32—31no—
244C2×C32—41no—
245C2×C32—43no—
246C2×C32—83no—
247C2×C32—81no—
248C2×C32GL(2,ℤ/4)11yescharacteristic
249C3⋊D3—241no—
250C3⋊D3—483no—
251C3⋊D3—481no—
252C3⋊D3—486no—
253C3⋊D3—482no—
254C2×Q8—91no—
255C2×Q8—91no—
256C2×Q8—181no—
257C2×D4—271no—
258C2×D4—541no—
259C22×C4—61no—
260C22×C4D3×C3⋊D311yescharacteristic
261SD16—541no—
262SD16—542no—
263SD16—541no—
264C2×C8—91no—
265C2×C8—91no—
266C4⋊C4—271no—
267C22⋊C4—541no—
268C22×C3—21no—
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—31no—
272C22×C3—31no—
273C22×C3—61no—
274C22×C3—61no—
275C22×C3#371711yescharacteristic, complemented
276C22×C3#375511yescharacteristic, complemented
277C2×D3—362no—
278C2×D3—361no—
279C2×D3—361no—
280C2×D3—361no—
281C2×D3—361no—
282C2×D3—362no—
283C2×D3—362no—
284C2×D3—361no—
285C2×D3—361no—
286C2×D3—723no—
287C2×D3—726no—
288A4—23no—
289A4#371713yescomplemented
290C3×C4—21no—
291C3×C4—31no—
292C3×C4—31no—
293C3×C4—43no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—83no—
297C3×C4—181no—
298C3×C4—243no—
299C3×C4D3×S411yescharacteristic
300C3×C4C2×C3⋊S411yescharacteristic
301Dic3—21no—
302Dic3—61no—
303Dic3—361no—
304Dic3—361no—
305Dic3—362no—
306C32—41no—
307C32—43no—
308C32—83no—
309C32—81no—
310C32#541711yescharacteristic, complemented
311C23—541no—
312C23#629111yescharacteristic
313Q8—31no—
314Q8—92no—
315Q8—91no—
316Q8—181no—
317D4—541no—
318D4—541no—
319D4—1081no—
320D4—1081no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—181no—
324C2×C4—181no—
325C2×C4—541no—
326C8—181no—
327C8—181no—
328C2×C3—21no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—43no—
334C2×C3—61no—
335C2×C3—61no—
336C2×C3—83no—
337C2×C3#6421011yescharacteristic
338C2×C3#6426911yescharacteristic
339D3—722no—
340D3—721no—
341D3—721no—
342D3—1443no—
343D3—1446no—
344C22—31no—
345C22—31no—
346C22—1081no—
347C22—1081no—
348C22—1081no—
349C22#8879511yescharacteristic, complemented
350C4—31no—
351C4—61no—
352C4—181no—
353C4—1081no—
354C4D3×C3⋊S411yescharacteristic
355C3—21no—
356C3—43no—
357C3—83no—
358C3#1059318811yescharacteristic, complemented
359C3#1059365111yescharacteristic, complemented
360C2—31no—
361C2—31no—
362C2—2161no—
363C2#1172420511yescharacteristic
subgroupG/H
centerC2#11724205
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4D3×C3⋊S4
FittingC22×C32×C4C2×D3
socleC23×C32#67
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
1. C3×A4×Dic6C2semidirect
3. C3×C3⋊(A4⋊C8)C2semidirect
11. C22×C3×Dic6D3semidirect
13. A4×Dic6D3semidirect
21. C3⋊(A4⋊C8)D3semidirect
61. C32×A4SD16semidirect
65. C22×Dic6C3⋊D3semidirect
111. C3×Dic6S4semidirect
160. C22×C32#215semidirect
170. C3×A4#204semidirect
171. C3×A4#215semidirect
236. Dic6C3⋊S4semidirect
275. C22×C3#3717semidirect
276. C22×C3#3755semidirect
289. A4#3717semidirect
310. C32#5417semidirect
349. C22#88795semidirect
358. C3#10593188semidirect
359. C3#10593651semidirect

Maximal subgroups

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

subgroupindex
C3×A4×Dic62
C4⋊(C3⋊(C3⋊S4))2
C3×C3⋊(A4⋊C8)2
#105936513
#105931883
#105911743
#887954