Finite Group Databasebeta

← C3×A4×#217 #13173136

Subgroups — C3×A4×#217 #13173136

Order 1728.

1182 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C23×C32×A4C211yesmaximal, characteristic
2C2×C3×Dic3×A4C212yesmaximal, complemented
3C22×C3×#217C311yesmaximal, characteristic, complemented
4C3×A4×C22⋊C4—31nomaximal
5A4×#217C313yesmaximal, direct product factor
6C22×C32×A4—22no—
7C22×C32×A4C412yescomplemented
8C22×C32×A4C2211yescharacteristic
9C3×Dic3×A4—22no—
10C32×#217—41nomaximal
11C25×C32C2×C311yescharacteristic
12C23×C3×A4—23no—
13C23×C3×A4—21no—
14C23×C3×A4D311yescharacteristic
15C23×C3×A4C2×C313yes—
16C23×C3×Dic3C2×C312yescomplemented
17C2×C3×C4×A4—32no—
18C2×Dic3×A4C2×C316yescomplemented
19C2×C3×#217—31no—
20C2×C3×#217—31no—
21C2×C3×#217—32no—
22C23×C33—41no—
23C2×C32×A4—22no—
24C2×C32×A4C2×C411yescharacteristic
25C2×C32×A4D412yes—
26C2×C32×Dic3—42no—
27C22×C3×C22⋊C4—31no—
28C22×#217C3211yescharacteristic, complemented
29A4×C22⋊C4—33no—
30C24×C32—22no—
31C24×C32—31no—
32C24×C32—31no—
33C24×C32—32no—
34C24×C32—62no—
35C24×C32C3×C412yescomplemented
36C24×C32C22×C311yescharacteristic
37C22×C3×A4—24no—
38C22×C3×A4—212no—
39C22×C3×A4—22no—
40C22×C3×A4—21no—
41C22×C3×A4—26no—
42C22×C3×A4—23no—
43C22×C3×A4—22no—
44C22×C3×A4—26no—
45C22×C3×A4Dic312yescomplemented
46C22×C3×A4C3×C416yescomplemented
47C22×C3×A4C2×D311yescharacteristic
48C22×C3×A4C22×C313yes—
49C22×C3×Dic3—22no—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—32no—
52C22×C3×Dic3—62no—
53C3×C4×A4—62no—
54Dic3×A4—26no—
55C32×C22⋊C4—121no—
56C3×#217—32no—
57C3×#217—31no—
58C3×#217—32no—
59C3×#217—43no—
60C3×#217A411yescharacteristic, direct product factor
61C22×C33—41no—
62C22×C33—42no—
63C22×C33—82no—
64C32×A4C22⋊C411yescharacteristic, complemented
65C32×Dic3—82no—
66C25×C3—21no—
67C25×C3C3×D311yescharacteristic
68C25×C3C2×C3211yescharacteristic
69C23×A4—23no—
70C23×A4C3×D313yes—
71C23×C3×C4—32no—
72C23×Dic3C2×C3212yescomplemented
73C2×C4×A4—36no—
74C2×C3×C22⋊C4—91no—
75C2×C3×C22⋊C4—92no—
76C2×C3×C22⋊C4—91no—
77C2×#217—31no—
78C2×#217—31no—
79C2×#217—32no—
80C23×C32—22no—
81C23×C32—31no—
82C23×C32—32no—
83C23×C32—31no—
84C23×C32—32no—
85C23×C32—32no—
86C23×C32—32no—
87C23×C32—32no—
88C23×C32—31no—
89C23×C32—41no—
90C23×C32—43no—
91C23×C32—62no—
92C23×C32—62no—
93C23×C32—62no—
94C23×C32—64no—
95C23×C32—62no—
96C23×C32—62no—
97C23×C32—64no—
98C23×C32—83no—
99C23×C32—81no—
100C23×C32C2×C3×C411yescharacteristic
101C23×C32C3×D412yes—
102C23×C32C2×A411yescharacteristic
103C2×C3×A4—24no—
104C2×C3×A4—212no—
105C2×C3×A4—22no—
106C2×C3×A4—26no—
107C2×C3×A4—21no—
108C2×C3×A4—22no—
109C2×C3×A4—23no—
110C2×C3×A4—26no—
111C2×C3×A4C2×Dic311yescharacteristic
112C2×C3×A4#6712yes—
113C2×C3×A4C2×C3×C413yes—
114C2×C3×A4C3×D416yes—
115C2×C32×C4—122no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—46no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3C2×A412yescomplemented
123C22×C22⋊C4—31noSylow
124C2×C33—41no—
125C2×C33—42no—
126C2×C33—82no—
127C24×C3—24no—
128C24×C3—22no—
129C24×C3—22no—
130C24×C3—22no—
131C24×C3—21no—
132C24×C3—32no—
133C24×C3—31no—
134C24×C3—32no—
135C24×C3—31no—
136C24×C3—31no—
137C24×C3—31no—
138C24×C3—61no—
139C24×C3—61no—
140C24×C3—62no—
141C24×C3—62no—
142C24×C3—62no—
143C24×C3—64no—
144C24×C3C3×Dic312yescomplemented
145C24×C3C32×C412yescomplemented
146C24×C3C2×C3×D311yescharacteristic
147C24×C3C22×C3211yescharacteristic
148C22×A4—212no—
149C22×A4—26no—
150C22×A4—26no—
151C22×A4—23no—
152C22×A4C3×Dic316yescomplemented
153C22×A4C2×C3×D313yes—
154C22×C3×C4—62no—
155C22×C3×C4—92no—
156C22×C3×C4—92no—
157C22×C3×C4—182no—
158C22×Dic3—22no—
159C22×Dic3—32no—
160C22×Dic3—32no—
161C22×Dic3—62no—
162C4×A4—66no—
163C3×C22⋊C4—31no—
164C3×C22⋊C4—92no—
165C3×C22⋊C4—92no—
166C3×C22⋊C4—91no—
167C3×C22⋊C4—123no—
168#217—32no—
169#217—31no—
170#217—32no—
171#217C3×A411yescharacteristic, direct product factor
172C22×C32—22no—
173C22×C32—31no—
174C22×C32—32no—
175C22×C32—32no—
176C22×C32—31no—
177C22×C32—31no—
178C22×C32—32no—
179C22×C32—32no—
180C22×C32—32no—
181C22×C32—41no—
182C22×C32—42no—
183C22×C32—43no—
184C22×C32—46no—
185C22×C32—62no—
186C22×C32—64no—
187C22×C32—62no—
188C22×C32—62no—
189C22×C32—62no—
190C22×C32—64no—
191C22×C32—62no—
192C22×C32—83no—
193C22×C32—86no—
194C22×C32—82no—
195C22×C32—86no—
196C22×C32—812no—
197C22×C32—84no—
198C22×C32—81no—
199C22×C32—82no—
200C22×C32C3×C22⋊C411yescharacteristic, complemented
201C22×C32C4×A412yescomplemented
202C22×C32C22×A411yescharacteristic
203C3×A4—21no—
204C3×A4—23no—
205C3×A4#21711yescharacteristic, direct product factor
206C3×A4C3×C22⋊C413yescomplemented
207C32×C4—242no—
208C3×Dic3—22no—
209C3×Dic3—62no—
210C3×Dic3—86no—
211C25C32×D311yescharacteristic
212C23×C4—32no—
213C2×C22⋊C4—91no—
214C2×C22⋊C4—92no—
215C2×C22⋊C4—91no—
216C33—41noSylow
217C23×C3—24no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—21no—
221C23×C3—22no—
222C23×C3—21no—
223C23×C3—32no—
224C23×C3—32no—
225C23×C3—31no—
226C23×C3—31no—
227C23×C3—31no—
228C23×C3—31no—
229C23×C3—32no—
230C23×C3—32no—
231C23×C3—32no—
232C23×C3—32no—
233C23×C3—31no—
234C23×C3—31no—
235C23×C3—32no—
236C23×C3—32no—
237C23×C3—32no—
238C23×C3—32no—
239C23×C3—43no—
240C23×C3—61no—
241C23×C3—62no—
242C23×C3—61no—
243C23×C3—62no—
244C23×C3—62no—
245C23×C3—61no—
246C23×C3—62no—
247C23×C3—62no—
248C23×C3—62no—
249C23×C3—62no—
250C23×C3—62no—
251C23×C3—62no—
252C23×C3—64no—
253C23×C3—64no—
254C23×C3—62no—
255C23×C3—62no—
256C23×C3—64no—
257C23×C3—64no—
258C23×C3—62no—
259C23×C3—62no—
260C23×C3—62no—
261C23×C3—62no—
262C23×C3—64no—
263C23×C3—64no—
264C23×C3—64no—
265C23×C3—64no—
266C23×C3—68no—
267C23×C3—64no—
268C23×C3—68no—
269C23×C3—83no—
270C23×C3C2×C3×Dic311yescharacteristic
271C23×C3C3×#6712yes—
272C23×C3C2×C32×C411yescharacteristic
273C23×C3C32×D412yes—
274C23×C3D3×A411yescharacteristic
275C23×C3C2×C3×A411yescharacteristic
276C2×A4—212no—
277C2×A4—26no—
278C2×A4—23no—
279C2×A4—26no—
280C2×A4C2×C3×Dic313yes—
281C2×A4C3×#6716yes—
282C2×C3×C4—32no—
283C2×C3×C4—92no—
284C2×C3×C4—126no—
285C2×C3×C4—182no—
286C2×C3×C4—182no—
287C2×C3×C4—182no—
288C2×C3×C4—182no—
289C2×Dic3—32no—
290C2×Dic3—62no—
291C2×Dic3—62no—
292C2×Dic3—62no—
293C2×Dic3—62no—
294C2×Dic3C2×C3×A412yescomplemented
295C2×C32—22no—
296C2×C32—31no—
297C2×C32—32no—
298C2×C32—31no—
299C2×C32—41no—
300C2×C32—42no—
301C2×C32—43no—
302C2×C32—46no—
303C2×C32—62no—
304C2×C32—83no—
305C2×C32—86no—
306C2×C32—82no—
307C2×C32—86no—
308C2×C32—812no—
309C2×C32—84no—
310C2×C32—81no—
311C2×C32—82no—
312C2×C32C2×C4×A411yescharacteristic
313C2×C32D4×A412yes—
314C24—22no—
315C24—31no—
316C24—32no—
317C24—31no—
318C24—62no—
319C24C32×Dic312yescomplemented
320C24C2×C32×D311yescharacteristic
321C22×C4—62no—
322C22×C4—92no—
323C22×C4—92no—
324C22×C4—182no—
325C22⋊C4—31no—
326C22⋊C4—92no—
327C22⋊C4—92no—
328C22⋊C4—91no—
329C22×C3—24no—
330C22×C3—22no—
331C22×C3—22no—
332C22×C3—21no—
333C22×C3—21no—
334C22×C3—22no—
335C22×C3—31no—
336C22×C3—32no—
337C22×C3—31no—
338C22×C3—32no—
339C22×C3—32no—
340C22×C3—32no—
341C22×C3—31no—
342C22×C3—31no—
343C22×C3—32no—
344C22×C3—32no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—32no—
348C22×C3—32no—
349C22×C3—32no—
350C22×C3—32no—
351C22×C3—43no—
352C22×C3—46no—
353C22×C3—61no—
354C22×C3—62no—
355C22×C3—61no—
356C22×C3—62no—
357C22×C3—62no—
358C22×C3—61no—
359C22×C3—62no—
360C22×C3—62no—
361C22×C3—62no—
362C22×C3—62no—
363C22×C3—64no—
364C22×C3—62no—
365C22×C3—64no—
366C22×C3—62no—
367C22×C3—64no—
368C22×C3—64no—
369C22×C3—68no—
370C22×C3—64no—
371C22×C3—62no—
372C22×C3—62no—
373C22×C3—62no—
374C22×C3—62no—
375C22×C3—64no—
376C22×C3—64no—
377C22×C3—64no—
378C22×C3—64no—
379C22×C3—68no—
380C22×C3—62no—
381C22×C3—62no—
382C22×C3—86no—
383C22×C3—812no—
384C22×C3—83no—
385C22×C3—86no—
386C22×C3C3×#21711yescharacteristic, complemented
387C22×C3C32×C22⋊C411yescharacteristic, complemented
388C22×C3Dic3×A412yescomplemented
389C22×C3C3×C4×A412yescomplemented
390C22×C3C2×D3×A411yescharacteristic
391C22×C3C22×C3×A411yescharacteristic
392A4—23no—
393A4C3×#21713yesdirect product factor
394C3×C4—62no—
395C3×C4—182no—
396C3×C4—246no—
397Dic3—22no—
398Dic3—62no—
399C32—41no—
400C32—43no—
401C32—83no—
402C32—81no—
403C32A4×C22⋊C411yescharacteristic, complemented
404C23—22no—
405C23—32no—
406C23—31no—
407C23—31no—
408C23—32no—
409C23—32no—
410C23—31no—
411C23—32no—
412C23—32no—
413C23—62no—
414C23—62no—
415C23—62no—
416C23—64no—
417C23—62no—
418C23—62no—
419C23—64no—
420C23C2×C32×Dic311yescharacteristic
421C23C32×#6712yes—
422C23C3×D3×A411yescharacteristic
423C2×C4—32no—
424C2×C4—92no—
425C2×C4—182no—
426C2×C4—182no—
427C2×C4—182no—
428C2×C4—182no—
429C2×C3—24no—
430C2×C3—22no—
431C2×C3—22no—
432C2×C3—21no—
433C2×C3—22no—
434C2×C3—31no—
435C2×C3—31no—
436C2×C3—32no—
437C2×C3—31no—
438C2×C3—31no—
439C2×C3—32no—
440C2×C3—43no—
441C2×C3—46no—
442C2×C3—61no—
443C2×C3—61no—
444C2×C3—62no—
445C2×C3—64no—
446C2×C3—62no—
447C2×C3—62no—
448C2×C3—86no—
449C2×C3—812no—
450C2×C3—83no—
451C2×C3—86no—
452C2×C3C2×Dic3×A411yescharacteristic
453C2×C3A4×#6712yes—
454C2×C3C2×C3×C4×A411yescharacteristic
455C2×C3C3×D4×A412yes—
456C22—22no—
457C22—31no—
458C22—32no—
459C22—32no—
460C22—31no—
461C22—31no—
462C22—32no—
463C22—32no—
464C22—32no—
465C22—62no—
466C22—62no—
467C22—64no—
468C22—62no—
469C22—62no—
470C22—64no—
471C22—62no—
472C22C32×#21711yescharacteristic, complemented
473C22C3×Dic3×A412yescomplemented
474C22C2×C3×D3×A411yescharacteristic
475C4—62no—
476C4—182no—
477C3—21no—
478C3—43no—
479C3—83no—
480C3A4×#21711yescharacteristic, direct product factor
481C3C3×A4×C22⋊C411yescharacteristic, complemented
482C2—22no—
483C2—31no—
484C2—31no—
485C2—32no—
486C2—62no—
487C2C2×C3×Dic3×A411yescharacteristic
488C2C3×A4×#6712yes—
subgroupG/H
centerC22×C3C2×D3×A4
commutatorC23×C3C2×C32×C4
FrattiniC22C2×C3×D3×A4
FittingC25×C32C2×C3
socleC24×C32C22×C3
prime radicalC23×C32×A4C2
cosocleC24×C3C22×C32
hypercenterC23×C3D3×A4
hypocenterC22×C3C32×C22⋊C4
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic3×A4C2semidirect
3. C22×C3×#217C3semidirect
5. A4×#217C3direct (N × H)
7. C22×C32×A4C4semidirect
16. C23×C3×Dic3C2×C3semidirect
18. C2×Dic3×A4C2×C3semidirect
28. C22×#217C32semidirect
35. C24×C32C3×C4semidirect
45. C22×C3×A4Dic3semidirect
46. C22×C3×A4C3×C4semidirect
60. C3×#217A4direct (N × H)
64. C32×A4C22⋊C4semidirect
72. C23×Dic3C2×C32semidirect
122. C2×C3×Dic3C2×A4semidirect
144. C24×C3C3×Dic3semidirect
145. C24×C3C32×C4semidirect
152. C22×A4C3×Dic3semidirect
171. #217C3×A4direct (N × H)
200. C22×C32C3×C22⋊C4semidirect
201. C22×C32C4×A4semidirect
205. C3×A4#217direct (N × H)
206. C3×A4C3×C22⋊C4semidirect
294. C2×Dic3C2×C3×A4semidirect
319. C24C32×Dic3semidirect
386. C22×C3C3×#217semidirect
387. C22×C3C32×C22⋊C4semidirect
388. C22×C3Dic3×A4semidirect
389. C22×C3C3×C4×A4semidirect
393. A4C3×#217direct (N × H)
403. C32A4×C22⋊C4semidirect
472. C22C32×#217semidirect
473. C22C3×Dic3×A4semidirect
480. C3A4×#217direct (N × H)
481. C3C3×A4×C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×C32×A42
C2×C3×Dic3×A42
C22×C3×#2173
C3×A4×C22⋊C43
A4×#2173
C32×#2174