Finite Group Databasebeta

← #13166008

Subgroups — #13166008

Order 1728.

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

normal subgroups only

This group has 588 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×D12C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊(C3⋊S4))C211yesmaximal, characteristic, complemented
3C3×C3⋊(A4⋊C8)C211yesmaximal, characteristic, complemented
4#10593649—31nomaximal
5#10593184—33nomaximal
6#10591162—31nomaximal
7C2×C3×D3×A4—21no—
8C2×C3⋊(C3⋊S4)—21no—
9C32×C4×A4C2211yescharacteristic
10#88790—41nomaximal
11C22×C3×D12D311yescharacteristic, complemented
12C3×D4×A4—31no—
13A4×D12D313yescomplemented
14C4⋊(C3⋊S4)—31no—
15C4⋊(C3⋊S4)—33no—
16C4⋊(C3⋊S4)—61no—
17C4⋊(C3⋊S4)—63no—
18#64139—31no—
19C2×#3714—31no—
20C2×#3714—31no—
21C3⋊(A4⋊C8)D311yescharacteristic, complemented
22C3×A4⋊C8—33no—
23C3×#830—31no—
24#63563—31no—
25#63563—31no—
26C2×C32×A4D411yescharacteristic
27C3×D3×A4—41no—
28C3⋊(C3⋊S4)—41no—
29C4⋊(C3⋊(C3⋊D3))—41no—
30C32×D12—41no—
31C3×C3⋊(C3⋊C8)—41no—
32#5415—93no—
33#5217—91no—
34#4732—91no—
35C23×C3×D3—21no—
36C22×C3×A4—61no—
37C2×D3×A4—23no—
38C2×C3⋊S4—62no—
39C2×C3⋊S4—66no—
40C2×C3⋊S4—61no—
41C2×C3⋊S4—63no—
42C22×C32×C4C2×D311yescharacteristic
43C2×#638—61no—
44C2×C4⋊(C3⋊D3)—31no—
45C2×C3×D12—31no—
46C2×C3×D12—31no—
47C2×C3×D12—61no—
48C3×C4×A4—21no—
49C3×C4×A4—23no—
50C3×C4×A4C2×D311yescharacteristic
51C3×C4×A4C2×D313yes—
52#3753—121no—
53#3752—61no—
54C4⋊(C3⋊Dic3)—31no—
55C2×C3×C3⋊C8—31no—
56C2×C3×C3⋊C8—31no—
57#3714—62no—
58#3714—61no—
59#3714—61no—
60#3714—123no—
61C2×C3⋊(C3⋊D3)—81no—
62C2×C32×D3—81no—
63C32×A4D811yescharacteristic, complemented
64C33×C4—41no—
65C22×C3×D4—31no—
66C22×D12C3⋊D311yescharacteristic, complemented
67D4×A4—33no—
68C4⋊S4—93no—
69C4⋊S4—183no—
70C2×#213—91no—
71C2×#213—91no—
72#930—91no—
73#930—91no—
74#930—181no—
75C2×D24—91no—
76C2×D24—91no—
77A4⋊C8—33no—
78C3×C22⋊C8—91no—
79#832—91no—
80#832—91no—
81#830—31no—
82#821—91no—
83#821—91no—
84C23×C32#6711yescharacteristic
85C22×C3⋊D3—61no—
86C22×C3×D3—41no—
87C22×C3×D3—61no—
88C22×C3×D3—61no—
89C22×C3×D3—121no—
90C2×C3×A4—21no—
91C2×C3×A4—23no—
92C2×C3×A4—121no—
93C2×C3×A4D1211yescharacteristic
94C2×C3×A4#6713yes—
95D3×A4—43no—
96C3⋊S4—126no—
97C3⋊S4—122no—
98C3⋊S4—121no—
99C3⋊S4—123no—
100C32×D4—121no—
101C2×C32×C4—31no—
102C2×C32×C4—31no—
103#638—121no—
104#638—121no—
105C2×C3⋊Dic3—61no—
106C4⋊(C3⋊D3)—61no—
107C4⋊(C3⋊D3)—61no—
108C4⋊(C3⋊D3)—121no—
109C4⋊(C3⋊D3)—123no—
110C4⋊(C3⋊D3)—241no—
111C4⋊(C3⋊D3)—243no—
112C3×D12—32no—
113C3×D12—31no—
114C3×D12—43no—
115C3×D12—61no—
116C3×D12S411yescharacteristic, complemented
117C3⋊(C3⋊C8)—41no—
118C3×C3⋊C8—61no—
119C3×C3⋊C8—61no—
120C3×C3⋊C8—123no—
121#447—271noSylow
122C2×C33—41no—
123C3⋊(C3⋊D3)—161no—
124C32×D3—161no—
125C24×C3—61no—
126C23×D3—21no—
127C22×A4—63no—
128C2×S4—186no—
129C2×S4—183no—
130C2×C3×D4—91no—
131C2×C3×D4—91no—
132C2×C3×D4—181no—
133C22×C3×C4—21no—
134C22×C3×C4D3211yescharacteristic
135C22×C3×C4C2×C3⋊D311yescharacteristic
136C2×#67—182no—
137C2×#67—181no—
138C2×#67—181no—
139C2×D12—31no—
140C2×D12—31no—
141C2×D12—61no—
142C2×D12—91no—
143C2×D12—91no—
144C2×D12—181no—
145C4×A4—23no—
146C4×A4D3213yes—
147C2×C3×C8—91no—
148C2×C3×C8—91no—
149#213—181no—
150#213—182no—
151#213—181no—
152#213—363no—
153(C2×D3)⋊C4—181no—
154(C2×D3)⋊C4—181no—
155(C2×D3)⋊C4—182no—
156C4⋊Dic3—91no—
157C4⋊Dic3—91no—
158C4⋊Dic3—181no—
159C2×C3⋊C8—31no—
160C2×C3⋊C8—31no—
161D24—182no—
162D24—181no—
163D24—181no—
164C22×C32—31no—
165C22×C32—31no—
166C22×C32—241no—
167C22×C32#21311yescharacteristic, complemented
168C2×C3⋊D3—121no—
169C2×C3⋊D3—121no—
170C2×C3⋊D3—121no—
171C2×C3⋊D3—243no—
172C2×C3⋊D3—241no—
173C2×C3⋊D3—246no—
174C2×C3⋊D3—242no—
175C2×C3×D3—21no—
176C2×C3×D3—61no—
177C2×C3×D3—83no—
178C2×C3×D3—121no—
179C2×C3×D3—121no—
180C2×C3×D3—121no—
181C2×C3×D3—121no—
182C3×A4—21no—
183C3×A4—23no—
184C3×A4D2411yescharacteristic, complemented
185C3×A4#21313yescomplemented
186C32×C4—31no—
187C32×C4—41no—
188C32×C4—43no—
189C32×C4—83no—
190C32×C4—81no—
191C32×C4C2×S411yescharacteristic
192C3⋊Dic3—121no—
193C22×D4—31no—
194C2×D8—271no—
195C2×D8—271no—
196#121—271no—
197#102—271no—
198#102—271no—
199C22⋊C8—91no—
200C33—41noSylow
201C23×C3—21no—
202C23×C3—121no—
203C23×C3—181no—
204C23×C3—181no—
205C23×C3—361no—
206C23×C3#62611yescharacteristic
207C23×C3#63811yescharacteristic
208C22×D3—41no—
209C22×D3—61no—
210C22×D3—61no—
211C22×D3—121no—
212C22×D3—181no—
213C22×D3—181no—
214C22×D3—182no—
215C2×A4—23no—
216C2×A4—123no—
217C2×A4#62613yes—
218S4—366no—
219S4—363no—
220C3×D4—31no—
221C3×D4—92no—
222C3×D4—91no—
223C3×D4—123no—
224C3×D4—181no—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—61no—
230C2×C3×C4—61no—
231#67—362no—
232#67—361no—
233#67—361no—
234#67—362no—
235#67—361no—
236#67—361no—
237C2×Dic3—181no—
238C2×Dic3—181no—
239C2×Dic3—182no—
240D12—32no—
241D12—31no—
242D12—61no—
243D12—181no—
244D12—181no—
245D12—181no—
246D12—181no—
247D12—361no—
248D12—363no—
249D12—361no—
250D12—723no—
251D12C3⋊S411yescharacteristic, complemented
252C3×C8—181no—
253C3×C8—181no—
254C3⋊C8—61no—
255C3⋊C8—61no—
256C3⋊C8—123no—
257C2×C32—31no—
258C2×C32—31no—
259C2×C32—41no—
260C2×C32—43no—
261C2×C32—83no—
262C2×C32—81no—
263C2×C32—481no—
264C2×C32GL(2,ℤ/4)11yescharacteristic
265C3⋊D3—241no—
266C3⋊D3—483no—
267C3⋊D3—481no—
268C3⋊D3—486no—
269C3⋊D3—482no—
270C3×D3—41no—
271C3×D3—121no—
272C3×D3—163no—
273C24—61no—
274C2×D4—91no—
275C2×D4—91no—
276C2×D4—181no—
277C2×D4—271no—
278C2×D4—541no—
279C22×C4D3×C3⋊D311yescharacteristic
280D8—541no—
281D8—542no—
282D8—541no—
283C2×C8—91no—
284C2×C8—91no—
285C4⋊C4—271no—
286C22⋊C4—541no—
287C22×C3—21no—
288C22×C3—31no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—61no—
293C22×C3—61no—
294C22×C3—61no—
295C22×C3—181no—
296C22×C3—243no—
297C22×C3—361no—
298C22×C3—361no—
299C22×C3—361no—
300C22×C3—361no—
301C22×C3#371411yescharacteristic, complemented
302C22×C3#375311yescharacteristic, complemented
303C2×D3—21no—
304C2×D3—61no—
305C2×D3—121no—
306C2×D3—121no—
307C2×D3—121no—
308C2×D3—121no—
309C2×D3—362no—
310C2×D3—361no—
311C2×D3—361no—
312C2×D3—361no—
313C2×D3—361no—
314C2×D3—362no—
315C2×D3—362no—
316C2×D3—361no—
317C2×D3—361no—
318C2×D3—723no—
319C2×D3—726no—
320A4—23no—
321A4#371413yescomplemented
322C3×C4—21no—
323C3×C4—31no—
324C3×C4—31no—
325C3×C4—43no—
326C3×C4—61no—
327C3×C4—83no—
328C3×C4D3×S411yescharacteristic
329C3×C4C2×C3⋊S411yescharacteristic
330Dic3—361no—
331Dic3—361no—
332Dic3—362no—
333C32—41no—
334C32—43no—
335C32—83no—
336C32—81no—
337C32#541511yescharacteristic, complemented
338C23—121no—
339C23—181no—
340C23—181no—
341C23—361no—
342C23—541no—
343C23#629111yescharacteristic
344D4—31no—
345D4—92no—
346D4—91no—
347D4—181no—
348D4—541no—
349D4—541no—
350D4—1081no—
351D4—1081no—
352C2×C4—31no—
353C2×C4—31no—
354C2×C4—541no—
355C8—181no—
356C8—181no—
357C2×C3—21no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—43no—
363C2×C3—61no—
364C2×C3—61no—
365C2×C3—83no—
366C2×C3—121no—
367C2×C3—361no—
368C2×C3—483no—
369C2×C3#6421011yescharacteristic
370C2×C3#6426911yescharacteristic
371D3—41no—
372D3—121no—
373D3—722no—
374D3—721no—
375D3—721no—
376D3—1443no—
377D3—1446no—
378C22—31no—
379C22—31no—
380C22—61no—
381C22—181no—
382C22—361no—
383C22—361no—
384C22—361no—
385C22—361no—
386C22—1081no—
387C22—1081no—
388C22—1081no—
389C22#8879011yescharacteristic, complemented
390C4—31no—
391C4—1081no—
392C4D3×C3⋊S411yescharacteristic
393C3—21no—
394C3—43no—
395C3—83no—
396C3#1059318411yescharacteristic, complemented
397C3#1059364911yescharacteristic, complemented
398C2—31no—
399C2—31no—
400C2—121no—
401C2—361no—
402C2—2161no—
403C2#1172420511yescharacteristic
subgroupG/H
centerC2#11724205
commutator, cosocleC32×C4×A4C22
Frattini, hypercenterC4D3×C3⋊S4
FittingC22×C32×C4C2×D3
socleC23×C32#67
hypocenterC32×A4D8
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×D12C2semidirect
2. C4⋊(C3⋊(C3⋊S4))C2semidirect
3. C3×C3⋊(A4⋊C8)C2semidirect
11. C22×C3×D12D3semidirect
13. A4×D12D3semidirect
21. C3⋊(A4⋊C8)D3semidirect
63. C32×A4D8semidirect
66. C22×D12C3⋊D3semidirect
116. C3×D12S4semidirect
167. C22×C32#213semidirect
184. C3×A4D24semidirect
185. C3×A4#213semidirect
251. D12C3⋊S4semidirect
301. C22×C3#3714semidirect
302. C22×C3#3753semidirect
321. A4#3714semidirect
337. C32#5415semidirect
389. C22#88790semidirect
396. C3#10593184semidirect
397. C3#10593649semidirect

Maximal subgroups

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

subgroupindex
C3×A4×D122
C4⋊(C3⋊(C3⋊S4))2
C3×C3⋊(A4⋊C8)2
#105936493
#105931843
#105911623
#887904