Finite Group Databasebeta

← C3×#10593380 #13166724

Subgroups — C3×#10593380 #13166724

Order 1728.

682 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 94 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×#64306C211yesmaximal, characteristic
2C3×#64283C211yesmaximal, characteristic, complemented
3C3×C4×#624C211yesmaximal, characteristic, complemented
4C3×#63741C214yesmaximal, complemented
5#10593380C311yesmaximal, characteristic, direct product factor
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C32×#235—21no—
8C3×#3811—21no—
9C2×C3×#624—21no—
10C3×#3805C2212yes—
11C3×#3802C411yescharacteristic, complemented
12C3×C4×D32—21no—
13C3×#3798C412yescomplemented
14C3×#3797C411yescharacteristic, complemented
15C3×#3796—21no—
16C3×#3792C2212yescomplemented
17C3×#3788—21no—
18C3×#3787—21no—
19C32×C4×Dic3—21no—
20C3×Dic32C2212yescomplemented
21C3×D3×#41—61no—
22#64306C2×C311yescharacteristic
23#64283C2×C311yescharacteristic, complemented
24C3×C42×D3—61no—
25C4×#624C2×C311yescharacteristic, complemented
26#63741C2×C314yescomplemented
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C3×D32—21no—
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3C2311yescharacteristic
31C3×C4×C3⋊D3C2×C411yescharacteristic
32C3×C4×C3⋊D3C2×C411yescharacteristic
33C3×C4×C3⋊D3D411yescharacteristic
34C3×C4×C3⋊D3D411yescharacteristic
35C32×#67—22no—
36C2×C32×Dic3—22no—
37C32×D12—21no—
38C32×C4×D3—22no—
39C32×Dic6—21no—
40C3×#627C2×C412yescomplemented
41C3×#626—22no—
42C3×#625C2×C412yescomplemented
43C3×#624—24no—
44C3×#624—21no—
45C3×D3×Dic3—22no—
46C3×#622—22no—
47C2×C3×#104—91nomaximal
48C32×#41—61no—
49C2×C4×C3⋊D3C22×C311yescharacteristic
50C3×#239—61no—
51C3×D3×Q8—61no—
52C3×#237—61no—
53C3×#237—62no—
54C3×D3×D4—61no—
55C3×D3×D4—62no—
56C3×#235—21no—
57C3×#235—21no—
58C3×#235—41no—
59C3×#235—62no—
60C2×C3×C4×D3—62no—
61C2×C3×C4×D3—62no—
62C2×C3×C4×D3—61no—
63C2×C3×C4×D3—61no—
64#3811—21no—
65C2×#624—21no—
66#3805C22×C312yes—
67#3802C3×C411yescharacteristic, complemented
68C4×D32—21no—
69#3798C3×C412yescomplemented
70#3797C3×C411yescharacteristic, complemented
71#3796—21no—
72#3792C22×C312yescomplemented
73#3788—21no—
74#3787—21no—
75C32×C42—61no—
76C3×C4×Dic3—21no—
77C3×C4×Dic3—21no—
78C3×C4×Dic3—41no—
79C3×C4×Dic3—62no—
80Dic32C22×C312yescomplemented
81C22×C33C2×D411yescharacteristic
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3C22⋊C411yescharacteristic
84C2×C3×C3⋊D3C22⋊C411yescharacteristic
85C2×C32×D3—22no—
86C3×D32—42no—
87C33×C4C22⋊C411yescharacteristic
88C33×C4C22⋊C411yescharacteristic
89C3×C3⋊Dic3C22×C411yescharacteristic
90C3×C3⋊Dic3C2×D411yescharacteristic
91C32×Dic3—24no—
92C32×Dic3—22no—
93C2×C3×#41—91no—
94D3×#41—61no—
95C2×C3×M16—91no—
96C2×C3×C42—91no—
97C42×D3—61no—
98C3×#104—94no—
99C22×C3⋊D3C3×D411yescharacteristic
100C22×C3×D3—61no—
101C22×C3×D3—61no—
102C22×C3×D3—62no—
103C2×D32—21no—
104C32×Q8—61no—
105C32×D4—61no—
106C32×D4—62no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4—62no—
112C2×C32×C4—62no—
113C2×C32×C4C3×D411yescharacteristic
114C2×C3⋊Dic3C23×C311yescharacteristic
115C4×C3⋊D3C2×C3×C411yescharacteristic
116C4×C3⋊D3C2×C3×C411yescharacteristic
117C4×C3⋊D3C3×D411yescharacteristic
118C4×C3⋊D3C3×D411yescharacteristic
119C3×#67—22no—
120C3×#67—22no—
121C3×#67—42no—
122C3×#67—62no—
123C3×#67—62no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—42no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—62no—
130C3×D12—21no—
131C3×D12—21no—
132C3×D12—41no—
133C3×D12—62no—
134C3×C4×D3—22no—
135C3×C4×D3—22no—
136C3×C4×D3—42no—
137C3×C4×D3—61no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×C4×D3—61no—
144C3×C4×D3—61no—
145C3×C4×D3—68no—
146C3×C4×D3—62no—
147C3×C4×D3—62no—
148C3×Dic6—21no—
149C3×Dic6—21no—
150C3×Dic6—41no—
151C3×Dic6—62no—
152#627C2×C3×C412yescomplemented
153#626—22no—
154#625C2×C3×C412yescomplemented
155#624—24no—
156#624—21no—
157D3×Dic3—22no—
158#622—22no—
159C2×#104—91noSylow
160C2×C33—21no—
161C2×C33C2×C22⋊C411yescharacteristic
162C3×C3⋊D3—21no—
163C3×C3⋊D3#10412yescomplemented
164C32×D3—42no—
165C3×#41—61no—
166C3×#41—61no—
167C3×#41—92no—
168C3×#41—92no—
169C3×#41—121no—
170C3×#41—181no—
171C2×C3×Q8—91no—
172C2×C3×D4—91no—
173C2×C3×D4—181no—
174C22×C3×C4—91no—
175C22×C3×C4—181no—
176C22×C3×C4—181no—
177#239—61no—
178D3×Q8—61no—
179#237—62no—
180#237—61no—
181D3×D4—61no—
182D3×D4—62no—
183#235—21no—
184#235—62no—
185C2×C4×D3—61no—
186C2×C4×D3—61no—
187C2×C4×D3—62no—
188C2×C4×D3—62no—
189C3×M16—92no—
190C3×M16—181no—
191C2×C3×C8—181no—
192C3×C42—61no—
193C3×C42—61no—
194C3×C42—92no—
195C3×C42—121no—
196C4×Dic3—21no—
197C4×Dic3—62no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—62no—
203C22×C32C2×C3×D411yescharacteristic
204C2×C3⋊D3—22no—
205C2×C3⋊D3C3×C22⋊C411yescharacteristic
206C2×C3⋊D3C3×C22⋊C411yescharacteristic
207C2×C3×D3—22no—
208C2×C3×D3—22no—
209C2×C3×D3—42no—
210C2×C3×D3—61no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—64no—
215C2×C3×D3—122no—
216C2×C3×D3—124no—
217D32—42no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—41no—
223C32×C4—41no—
224C32×C4—41no—
225C32×C4—41no—
226C32×C4—62no—
227C32×C4—64no—
228C32×C4C3×C22⋊C411yescharacteristic
229C32×C4C3×C22⋊C411yescharacteristic
230C3⋊Dic3C22×C3×C411yescharacteristic
231C3⋊Dic3C2×C3×D411yescharacteristic
232C3×Dic3—24no—
233C3×Dic3—22no—
234C3×Dic3—24no—
235C3×Dic3—22no—
236C3×Dic3—42no—
237C3×Dic3—44no—
238C3×Dic3—61no—
239C3×Dic3—61no—
240C3×Dic3—61no—
241C3×Dic3—61no—
242C2×#41—91no—
243C2×M16—91no—
244C2×C42—91no—
245#104—94no—
246C33C2×#10411yescharacteristic, complemented, Sylow
247C23×C3—91no—
248C23×C3—181no—
249C22×D3—61no—
250C22×D3—61no—
251C22×D3—62no—
252C3×Q8—61no—
253C3×Q8—61no—
254C3×Q8—92no—
255C3×Q8—121no—
256C3×D4—61no—
257C3×D4—62no—
258C3×D4—62no—
259C3×D4—61no—
260C3×D4—92no—
261C3×D4—121no—
262C3×D4—122no—
263C3×D4—182no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—62no—
269C2×C3×C4—62no—
270C2×C3×C4—62no—
271C2×C3×C4—62no—
272C2×C3×C4—91no—
273C2×C3×C4—91no—
274C2×C3×C4—91no—
275C2×C3×C4—91no—
276C2×C3×C4—91no—
277C2×C3×C4—122no—
278C2×C3×C4—122no—
279C2×C3×C4—181no—
280C2×C3×C4—184no—
281C2×C3×C4—182no—
282C2×C3×C4GO+(4,𝔽2)11yescharacteristic
283#67—22no—
284#67—62no—
285#67—62no—
286C2×Dic3—22no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—62no—
290D12—21no—
291D12—62no—
292C4×D3—22no—
293C4×D3—61no—
294C4×D3—61no—
295C4×D3—61no—
296C4×D3—61no—
297C4×D3—61no—
298C4×D3—61no—
299C4×D3—61no—
300C4×D3—61no—
301C4×D3—68no—
302C4×D3—62no—
303C4×D3—62no—
304Dic6—21no—
305Dic6—62no—
306C3×C8—182no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—41no—
312C2×C32—41no—
313C2×C32—41no—
314C2×C32—42no—
315C2×C32—81no—
316C2×C32—122no—
317C2×C32C2×C3×C22⋊C411yescharacteristic
318C3⋊D3—21no—
319C3⋊D3C3×#10412yescomplemented
320C3×D3—42no—
321C3×D3—42no—
322C3×D3—62no—
323C3×D3—62no—
324C3×D3—62no—
325C3×D3—82no—
326C3×D3—121no—
327#41—61no—
328#41—92no—
329#41—92no—
330#41—181no—
331C2×Q8—91no—
332C2×D4—91no—
333C2×D4—181no—
334C22×C4—91no—
335C22×C4—181no—
336C22×C4—181no—
337M16—92no—
338M16—181no—
339C2×C8—181no—
340C42—61no—
341C42—92no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—62no—
347C22×C3—62no—
348C22×C3—91no—
349C22×C3—91no—
350C22×C3—122no—
351C22×C3—182no—
352C22×C3—362no—
353C22×C3C2×GO+(4,𝔽2)11yescharacteristic
354C2×D3—22no—
355C2×D3—61no—
356C2×D3—61no—
357C2×D3—61no—
358C2×D3—61no—
359C2×D3—64no—
360C2×D3—122no—
361C2×D3—124no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—21no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—41no—
369C3×C4—41no—
370C3×C4—62no—
371C3×C4—64no—
372C3×C4—64no—
373C3×C4—62no—
374C3×C4—91no—
375C3×C4—91no—
376C3×C4—122no—
377C3×C4—124no—
378C3×C4D32⋊C411yescharacteristic
379C3×C4D32⋊C411yescharacteristic
380Dic3—22no—
381Dic3—24no—
382Dic3—61no—
383Dic3—61no—
384Dic3—61no—
385Dic3—61no—
386C32—21no—
387C32—21no—
388C32—41no—
389C32—41no—
390C32C2×C3×#10411yescharacteristic, complemented
391C23—91no—
392C23—181no—
393Q8—61no—
394Q8—92no—
395D4—62no—
396D4—61no—
397D4—92no—
398D4—182no—
399C2×C4—62no—
400C2×C4—62no—
401C2×C4—91no—
402C2×C4—91no—
403C2×C4—91no—
404C2×C4—91no—
405C2×C4—91no—
406C2×C4—181no—
407C2×C4—184no—
408C2×C4—182no—
409C2×C4C3×GO+(4,𝔽2)11yescharacteristic
410C8—182no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—41no—
416C2×C3—42no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—81no—
420C2×C3—92no—
421C2×C3—122no—
422C2×C3—122no—
423C2×C3—181no—
424C2×C3—242no—
425C2×C3C2×D32⋊C411yescharacteristic
426D3—42no—
427D3—62no—
428D3—62no—
429D3—62no—
430D3—121no—
431C22—62no—
432C22—91no—
433C22—91no—
434C22—182no—
435C22—362no—
436C22C2×C3×GO+(4,𝔽2)11yescharacteristic
437C4—64no—
438C4—62no—
439C4—91no—
440C4—91no—
441C4C3×D32⋊C411yescharacteristic
442C4C3×D32⋊C411yescharacteristic
443C3—21no—
444C3—21no—
445C3—41no—
446C3—41no—
447C3#1059338011yescharacteristic, direct product factor
448C2—21no—
449C2—92no—
450C2—122no—
451C2—181no—
452C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC3×C4D32⋊C4
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C2×C22⋊C4
prime radicalC3×#64306C2
cosocleC2×C3⋊Dic3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×#104
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×#64283C2semidirect
3. C3×C4×#624C2semidirect
4. C3×#63741C2semidirect
5. #10593380C3direct (N × H)
11. C3×#3802C4semidirect
13. C3×#3798C4semidirect
14. C3×#3797C4semidirect
16. C3×#3792C22semidirect
20. C3×Dic32C22semidirect
23. #64283C2×C3semidirect
25. C4×#624C2×C3semidirect
26. #63741C2×C3semidirect
40. C3×#627C2×C4semidirect
42. C3×#625C2×C4semidirect
67. #3802C3×C4semidirect
69. #3798C3×C4semidirect
70. #3797C3×C4semidirect
72. #3792C22×C3semidirect
80. Dic32C22×C3semidirect
152. #627C2×C3×C4semidirect
154. #625C2×C3×C4semidirect
163. C3×C3⋊D3#104semidirect
246. C33C2×#104semidirect
319. C3⋊D3C3×#104semidirect
390. C32C2×C3×#104semidirect
447. C3#10593380direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643062
C3×#642832
C3×C4×#6242
C3×#637412
#105933803
C2×C3×#1049