Finite Group Databasebeta

← C3×A4×#239 #13181445

Subgroups — C3×A4×#239 #13181445

Order 1728.

1138 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1138 conjugacy classes of subgroups — too many to draw. Only its 184 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
1C32×Q8×A4C211yesmaximal, characteristic, complemented
2C3×A4×D12C213yesmaximal, complemented
3C3×C4×D3×A4C213yesmaximal, complemented
4C22×C3×#239C311yesmaximal, characteristic, complemented
5C3×A4×#41—31nomaximal
6A4×#239C313yesmaximal, direct product factor
7C2×C3×D3×A4C2213yes—
8C32×#239—41nomaximal
9C32×C4×A4C2213yes—
10C3×Dic3×A4C2211yescharacteristic
11C22×C32×Q8C2×C311yescharacteristic, complemented
12C2×C3×#239—31no—
13C2×C3×#239—33no—
14C2×C3×#239—33no—
15C2×C3×#239—31no—
16C22×C3×D12C2×C313yescomplemented
17C22×C3×C4×D3C2×C313yescomplemented
18C3×Q8×A4—21no—
19C3×Q8×A4—23no—
20C3×Q8×A4D311yescharacteristic, complemented
21C3×Q8×A4C2×C313yescomplemented
22C3×D4×A4—33no—
23C2×C3×C4×A4—33no—
24A4×D12C2×C319yescomplemented
25C4×D3×A4C2×C319yescomplemented
26C2×C32×A4C2311yescharacteristic
27C3×D3×A4—23no—
28C33×Q8—41no—
29C32×D12—43no—
30C32×C4×D3—43no—
31C22×C3×#41—31no—
32C22×#239C3211yescharacteristic, complemented
33A4×#41—33no—
34C23×C3×D3C22×C313yes—
35C22×C3×A4—33no—
36C2×D3×A4C22×C319yes—
37C32×#41—121no—
38C2×C32×Q8—31no—
39C2×C32×Q8—33no—
40C22×C32×C4C22×C313yes—
41C22×C3×Dic3C22×C311yescharacteristic
42C3×#239—36no—
43C3×#239—36no—
44C3×#239—32no—
45C3×#239—31no—
46C3×#239—33no—
47C3×#239—33no—
48C3×#239—43no—
49C3×#239A411yescharacteristic, direct product factor
50C2×C3×D12—33no—
51C2×C3×D12—36no—
52C2×C3×D12—33no—
53C2×C3×C4×D3—33no—
54C2×C3×C4×D3—33no—
55C2×C3×C4×D3—33no—
56C2×C3×C4×D3—33no—
57C3×C4×A4—23no—
58C3×C4×A4—29no—
59C3×C4×A4—31no—
60C3×C4×A4C2×D313yes—
61C3×C4×A4C22×C319yes—
62Dic3×A4C22×C313yes—
63C2×C32×D3—43no—
64C32×A4#4111yescharacteristic, complemented
65C33×C4—43no—
66C32×Dic3—41no—
67C2×C3×#41—91no—
68C2×C3×#41—93no—
69C2×C3×#41—93no—
70C2×C3×#41—91no—
71C22×C3×Q8—21no—
72C22×C3×Q8C3×D311yescharacteristic, complemented
73C22×C3×Q8C2×C3211yescharacteristic, complemented
74C22×C3×D4—33no—
75C23×C3×C4—33no—
76C2×#239—31no—
77C2×#239—33no—
78C2×#239—33no—
79C2×#239—31no—
80C22×D12C2×C3213yescomplemented
81C22×C4×D3C2×C3213yescomplemented
82Q8×A4—23no—
83Q8×A4C3×D313yescomplemented
84D4×A4—39no—
85C2×C4×A4—39no—
86C23×C32C23×C311yescharacteristic
87C22×C3×D3—23no—
88C22×C3×D3—33no—
89C22×C3×D3—33no—
90C22×C3×D3—63no—
91C2×C3×A4—21no—
92C2×C3×A4—23no—
93C2×C3×A4—63no—
94C2×C3×A4C22×D311yescharacteristic
95C2×C3×A4C23×C313yes—
96D3×A4—29no—
97C32×Q8—32no—
98C32×Q8—33no—
99C32×Q8—41no—
100C32×Q8—43no—
101C32×Q8—83no—
102C32×Q8—81no—
103C32×Q8C2×A411yescharacteristic, complemented
104C32×D4—123no—
105C2×C32×C4—33no—
106C2×C32×C4—33no—
107C2×C32×C4—123no—
108C2×C3×Dic3—31no—
109C2×C3×Dic3—31no—
110C3×D12—36no—
111C3×D12—36no—
112C3×D12—33no—
113C3×D12—49no—
114C3×D12C2×A413yescomplemented
115C3×C4×D3—36no—
116C3×C4×D3—33no—
117C3×C4×D3—33no—
118C3×C4×D3—33no—
119C3×C4×D3—49no—
120C3×C4×D3C2×A413yescomplemented
121C22×#41—31noSylow
122C2×C33—41no—
123C32×D3—83no—
124C24×C3—33no—
125C23×D3C22×C3213yes—
126C22×A4—39no—
127C3×#41—31no—
128C3×#41—93no—
129C3×#41—93no—
130C3×#41—92no—
131C3×#41—96no—
132C3×#41—96no—
133C3×#41—91no—
134C3×#41—123no—
135C2×C3×Q8—33no—
136C2×C3×Q8—33no—
137C2×C3×Q8—31no—
138C2×C3×Q8—31no—
139C2×C3×Q8—61no—
140C2×C3×Q8—63no—
141C2×C3×D4—93no—
142C2×C3×D4—93no—
143C2×C3×D4—96no—
144C22×C3×C4—23no—
145C22×C3×C4—31no—
146C22×C3×C4—93no—
147C22×C3×C4—93no—
148C22×C3×C4—93no—
149C22×C3×C4—93no—
150C22×C3×C4C2×C3×D313yes—
151C22×C3×C4C22×C3213yes—
152C22×Dic3C22×C3211yescharacteristic
153#239—36no—
154#239—36no—
155#239—32no—
156#239—31no—
157#239—33no—
158#239—33no—
159#239C3×A411yescharacteristic, direct product factor
160C2×D12—33no—
161C2×D12—36no—
162C2×D12—33no—
163C2×C4×D3—33no—
164C2×C4×D3—33no—
165C2×C4×D3—33no—
166C2×C4×D3—33no—
167C4×A4—29no—
168C4×A4—33no—
169C4×A4C2×C3×D319yes—
170C22×C32—31no—
171C22×C32—31no—
172C22×C32—123no—
173C22×C32C3×#4111yescharacteristic, complemented
174C2×C3×D3—33no—
175C2×C3×D3—49no—
176C2×C3×D3—63no—
177C2×C3×D3—63no—
178C2×C3×D3—63no—
179C2×C3×D3—63no—
180C2×C3×D3C22×A413yes—
181C3×A4—21no—
182C3×A4—23no—
183C3×A4#23911yescharacteristic, direct product factor
184C3×A4C3×#4113yescomplemented
185C32×C4—33no—
186C32×C4—43no—
187C32×C4—49no—
188C32×C4—89no—
189C32×C4—83no—
190C32×C4—121no—
191C32×C4C22×A413yes—
192C3×Dic3—31no—
193C3×Dic3—43no—
194C3×Dic3C22×A411yescharacteristic
195C2×#41—91no—
196C2×#41—93no—
197C2×#41—93no—
198C2×#41—91no—
199C22×Q8C32×D311yescharacteristic, complemented
200C22×D4—33no—
201C23×C4—33no—
202C33—41noSylow
203C23×C3—21no—
204C23×C3—63no—
205C23×C3—93no—
206C23×C3—93no—
207C23×C3—183no—
208C23×C3C22×C3×D311yescharacteristic
209C23×C3C23×C3211yescharacteristic
210C22×D3—23no—
211C22×D3—33no—
212C22×D3—33no—
213C22×D3—63no—
214C2×A4—23no—
215C2×A4—69no—
216C2×A4C22×C3×D313yes—
217C3×Q8—21no—
218C3×Q8—32no—
219C3×Q8—32no—
220C3×Q8—33no—
221C3×Q8—33no—
222C3×Q8—43no—
223C3×Q8—63no—
224C3×Q8—62no—
225C3×Q8—83no—
226C3×Q8D3×A411yescharacteristic, complemented
227C3×Q8C2×C3×A411yescharacteristic, complemented
228C3×D4—33no—
229C3×D4—96no—
230C3×D4—96no—
231C3×D4—93no—
232C3×D4—129no—
233C2×C3×C4—33no—
234C2×C3×C4—33no—
235C2×C3×C4—33no—
236C2×C3×C4—33no—
237C2×C3×C4—33no—
238C2×C3×C4—63no—
239C2×C3×C4—63no—
240C2×C3×C4—93no—
241C2×C3×C4—93no—
242C2×C3×C4—91no—
243C2×C3×C4—96no—
244C2×C3×C4—93no—
245C2×C3×C4—91no—
246C2×C3×C4—129no—
247C2×Dic3—31no—
248C2×Dic3—31no—
249D12—36no—
250D12—36no—
251D12—33no—
252D12C2×C3×A413yescomplemented
253C4×D3—36no—
254C4×D3—33no—
255C4×D3—33no—
256C4×D3—33no—
257C4×D3C2×C3×A413yescomplemented
258C2×C32—31no—
259C2×C32—31no—
260C2×C32—41no—
261C2×C32—43no—
262C2×C32—83no—
263C2×C32—81no—
264C2×C32—243no—
265C2×C32C23×A411yescharacteristic
266C3×D3—23no—
267C3×D3—63no—
268C3×D3—89no—
269C24—33no—
270#41—31no—
271#41—93no—
272#41—93no—
273#41—92no—
274#41—96no—
275#41—96no—
276#41—91no—
277C2×Q8—33no—
278C2×Q8—31no—
279C2×D4—93no—
280C2×D4—93no—
281C2×D4—96no—
282C22×C4—31no—
283C22×C4—93no—
284C22×C4—93no—
285C22×C4—93no—
286C22×C4—93no—
287C22×C4C2×C32×D313yes—
288C22×C3—21no—
289C22×C3—31no—
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—31no—
293C22×C3—33no—
294C22×C3—61no—
295C22×C3—61no—
296C22×C3—93no—
297C22×C3—129no—
298C22×C3—183no—
299C22×C3—183no—
300C22×C3—183no—
301C22×C3—183no—
302C22×C3C3×#23911yescharacteristic, complemented
303C22×C3C32×#4111yescharacteristic, complemented
304C2×D3—33no—
305C2×D3—63no—
306C2×D3—63no—
307C2×D3—63no—
308C2×D3—63no—
309C2×D3C22×C3×A413yes—
310A4—23no—
311A4C3×#23913yesdirect product factor
312C3×C4—23no—
313C3×C4—31no—
314C3×C4—33no—
315C3×C4—33no—
316C3×C4—49no—
317C3×C4—63no—
318C3×C4—89no—
319C3×C4—91no—
320C3×C4—123no—
321C3×C4C2×D3×A413yes—
322C3×C4C22×C3×A413yes—
323Dic3—31no—
324Dic3C22×C3×A411yescharacteristic
325C32—41no—
326C32—43no—
327C32—83no—
328C32—81no—
329C32A4×#4111yescharacteristic, complemented
330C23—63no—
331C23—93no—
332C23—93no—
333C23—183no—
334C23C22×C32×D311yescharacteristic
335Q8—32no—
336Q8—33no—
337Q8C3×D3×A411yescharacteristic, complemented
338D4—33no—
339D4—96no—
340D4—96no—
341D4—93no—
342C2×C4—33no—
343C2×C4—33no—
344C2×C4—33no—
345C2×C4—93no—
346C2×C4—91no—
347C2×C4—93no—
348C2×C4—96no—
349C2×C4—93no—
350C2×C4—91no—
351C2×C3—21no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—43no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—63no—
360C2×C3—83no—
361C2×C3—183no—
362C2×C3—249no—
363C2×C3C22×D3×A411yescharacteristic
364C2×C3C23×C3×A411yescharacteristic
365D3—23no—
366D3—63no—
367C22—31no—
368C22—31no—
369C22—33no—
370C22—93no—
371C22—183no—
372C22—183no—
373C22—183no—
374C22—183no—
375C22C32×#23911yescharacteristic, complemented
376C4—31no—
377C4—33no—
378C4—91no—
379C4C2×C3×D3×A413yes—
380C3—21no—
381C3—43no—
382C3—83no—
383C3A4×#23911yescharacteristic, direct product factor
384C3C3×A4×#4111yescharacteristic, complemented
385C2—31no—
386C2—31no—
387C2—63no—
388C2—183no—
389C2C22×C3×D3×A411yescharacteristic
subgroupG/H
centerC2×C3C22×D3×A4
commutator, cosocleC23×C3C23×C32
FrattiniC2C22×C3×D3×A4
FittingC22×C32×Q8C2×C3
socleC23×C32C23×C3
hypercenterC3×Q8D3×A4
hypocenterC22×C3C32×#41
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C2×C3 ◁ C3×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C32×Q8×A4C2semidirect
2. C3×A4×D12C2semidirect
3. C3×C4×D3×A4C2semidirect
4. C22×C3×#239C3semidirect
6. A4×#239C3direct (N × H)
11. C22×C32×Q8C2×C3semidirect
16. C22×C3×D12C2×C3semidirect
17. C22×C3×C4×D3C2×C3semidirect
20. C3×Q8×A4D3semidirect
21. C3×Q8×A4C2×C3semidirect
24. A4×D12C2×C3semidirect
25. C4×D3×A4C2×C3semidirect
32. C22×#239C32semidirect
49. C3×#239A4direct (N × H)
64. C32×A4#41semidirect
72. C22×C3×Q8C3×D3semidirect
73. C22×C3×Q8C2×C32semidirect
80. C22×D12C2×C32semidirect
81. C22×C4×D3C2×C32semidirect
83. Q8×A4C3×D3semidirect
103. C32×Q8C2×A4semidirect
114. C3×D12C2×A4semidirect
120. C3×C4×D3C2×A4semidirect
159. #239C3×A4direct (N × H)
173. C22×C32C3×#41semidirect
183. C3×A4#239direct (N × H)
184. C3×A4C3×#41semidirect
199. C22×Q8C32×D3semidirect
226. C3×Q8D3×A4semidirect
227. C3×Q8C2×C3×A4semidirect
252. D12C2×C3×A4semidirect
257. C4×D3C2×C3×A4semidirect
302. C22×C3C3×#239semidirect
303. C22×C3C32×#41semidirect
311. A4C3×#239direct (N × H)
329. C32A4×#41semidirect
337. Q8C3×D3×A4semidirect
375. C22C32×#239semidirect
383. C3A4×#239direct (N × H)
384. C3C3×A4×#41semidirect

Maximal subgroups

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

subgroupindex
C32×Q8×A42
C3×A4×D122
C3×C4×D3×A42
C22×C3×#2393
C3×A4×#413
A4×#2393
C32×#2394