Finite Group Databasebeta

← C3×#10591224 #13158072

Subgroups — C3×#10591224 #13158072

Order 1728.

962 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C211yesmaximal, characteristic
3C2×C32×C4×Dic3C211yesmaximal, characteristic
4C3×#63579C214yesmaximal
5C3×#4977—31nomaximal
6C3×#4936—31nomaximal
7#10591224C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C32×Dic3C2212yes—
11C22×C32×Dic3C2212yes—
12C3×C4⋊(C3⋊Dic3)—22no—
13C32×C4⋊Dic3—22no—
14C32×C4×Dic3C414yescomplemented
15C2×C32×C4⋊C4—31no—
16C2×C32×C42—31no—
17C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
18C2×C3×C4⋊Dic3—21no—
19C2×C3×C4⋊Dic3—31no—
20C2×C3×C4⋊Dic3—31no—
21C2×C3×C4⋊Dic3—61no—
22C2×C3×C4⋊Dic3D311yescharacteristic
23C2×C3×C4⋊Dic3C2×C311yescharacteristic
24C2×C3×C4×Dic3—21no—
25C2×C3×C4×Dic3D311yescharacteristic
26C2×C3×C4×Dic3C2×C311yescharacteristic
27C3×C3⋊#95—34no—
28C3×C3⋊#95—34no—
29#63579C2×C314yes—
30C23×C33C2311yescharacteristic
31C2×C33×C4C2×C412yes—
32C2×C33×C4D412yes—
33C2×C33×C4Q812yes—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3—24no—
36C2×C32×Dic3—22no—
37C2×C32×Dic3—22no—
38C2×C32×Dic3—22no—
39C2×C32×Dic3—24no—
40C2×C32×Dic3C2×C414yes—
41C3×#382—91no—
42#4977—31no—
43#4936—31no—
44C22×C32×C4—21no—
45C22×C32×C4—21no—
46C22×C32×C4—21no—
47C22×C32×C4—32no—
48C22×C32×C4—32no—
49C22×C32×C4—41no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C22×C311yescharacteristic
53C22×C3⋊Dic3C22×C312yes—
54C22×C3×Dic3—22no—
55C22×C3×Dic3—22no—
56C22×C3×Dic3—32no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—62no—
59C22×C3×Dic3C2×D312yes—
60C22×C3×Dic3C2×D312yes—
61C22×C3×Dic3C22×C312yes—
62C22×C3×Dic3C22×C312yes—
63C32×C4⋊C4—62no—
64C32×C42—34no—
65C4⋊(C3⋊Dic3)—22no—
66C3×C4⋊Dic3—24no—
67C3×C4⋊Dic3—22no—
68C3×C4⋊Dic3—22no—
69C3×C4⋊Dic3—64no—
70C3×C4⋊Dic3—62no—
71C3×C4⋊Dic3—62no—
72C3×C4×Dic3—24no—
73C3×C4×Dic3Dic314yescomplemented
74C3×C4×Dic3C3×C414yescomplemented
75C22×C33C22×C411yescharacteristic
76C22×C33C2×D411yescharacteristic
77C22×C33C2×Q811yescharacteristic
78C22×C33#4112yes—
79C22×C33#4112yes—
80C33×C4C4⋊C414yes—
81C3×C3⋊Dic3—42no—
82C32×Dic3—24no—
83C32×Dic3—42no—
84C2×C3×C4⋊C4—31no—
85C2×C3×C4⋊C4—31no—
86C2×C3×C4⋊C4—61no—
87C2×C3×C4⋊C4—91no—
88C2×C3×C42—31no—
89C2×C3×C42—31no—
90C2×C3×C42—61no—
91C2×C4⋊Dic3—31no—
92C2×C4⋊Dic3—31no—
93C2×C4⋊Dic3—61no—
94C2×C4⋊Dic3C3×D311yescharacteristic
95C2×C4×Dic3C3×D311yescharacteristic
96C3×#95—94no—
97C3⋊#95—34no—
98C3⋊#95—34no—
99C23×C32—21no—
100C23×C32—21no—
101C23×C32—21no—
102C23×C32—41no—
103C23×C32C22×D311yescharacteristic
104C23×C32C22×D311yescharacteristic
105C23×C32C23×C311yescharacteristic
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—34no—
116C2×C32×C4—42no—
117C2×C32×C4—42no—
118C2×C32×C4—42no—
119C2×C32×C4—62no—
120C2×C32×C4—62no—
121C2×C32×C4—62no—
122C2×C32×C4—64no—
123C2×C32×C4Dic612yes—
124C2×C32×C4Dic612yes—
125C2×C32×C4C4×D312yes—
126C2×C32×C4D1212yes—
127C2×C32×C4C2×Dic312yes—
128C2×C32×C4#6712yes—
129C2×C32×C4C2×C3×C412yes—
130C2×C32×C4C3×D412yes—
131C2×C32×C4C3×Q812yes—
132C2×C3⋊Dic3—22no—
133C2×C3⋊Dic3—24no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—24no—
141C2×C3×Dic3—24no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—24no—
145C2×C3×Dic3—24no—
146C2×C3×Dic3—44no—
147C2×C3×Dic3—64no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3—64no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3—124no—
153C2×C3×Dic3C2×Dic314yes—
154C2×C3×Dic3C2×C3×C414yes—
155#382—91noSylow
156C2×C33C2×C4⋊C411yescharacteristic
157C2×C33#11712yes—
158C2×C33#12411yescharacteristic
159C2×C33#12411yescharacteristic
160C2×C33#12512yes—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—32no—
165C22×C3×C4—32no—
166C22×C3×C4—32no—
167C22×C3×C4—32no—
168C22×C3×C4—41no—
169C22×C3×C4—62no—
170C22×C3×C4—62no—
171C22×C3×C4—92no—
172C22×C3×C4D3211yescharacteristic
173C22×C3×C4C2×C3×D311yescharacteristic
174C22×C3×C4C2×C3×D311yescharacteristic
175C22×Dic3—32no—
176C22×Dic3—32no—
177C22×Dic3—62no—
178C22×Dic3C2×C3×D312yes—
179C22×Dic3C2×C3×D312yes—
180C3×C4⋊C4—64no—
181C3×C4⋊C4—62no—
182C3×C4⋊C4—62no—
183C3×C4⋊C4—182no—
184C3×C42—34no—
185C3×C42—34no—
186C3×C42—64no—
187C4⋊Dic3—22no—
188C4⋊Dic3—64no—
189C4⋊Dic3—62no—
190C4⋊Dic3—62no—
191C4×Dic3C3×Dic314yescomplemented
192C22×C32—22no—
193C22×C32—22no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—22no—
204C22×C32—22no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—42no—
211C22×C32—42no—
212C22×C32C2×Dic611yescharacteristic
213C22×C32C2×Dic611yescharacteristic
214C22×C32C2×C4×D311yescharacteristic
215C22×C32C2×D1211yescharacteristic
216C22×C32#23512yes—
217C22×C32#23512yes—
218C22×C32#23712yes—
219C22×C32#23912yes—
220C22×C32C22×Dic311yescharacteristic
221C22×C32C2×#6711yescharacteristic
222C22×C32C22×C3×C411yescharacteristic
223C22×C32C2×C3×D411yescharacteristic
224C22×C32C2×C3×Q811yescharacteristic
225C22×C32C3×#4112yes—
226C22×C32C3×#4112yes—
227C32×C4—24no—
228C32×C4—24no—
229C32×C4—24no—
230C32×C4—44no—
231C32×C4—64no—
232C32×C4—122no—
233C32×C4Dic3⋊C414yes—
234C32×C4C4⋊Dic314yes—
235C32×C4C3×C4⋊C414yes—
236C3⋊Dic3—42no—
237C3×Dic3—28no—
238C3×Dic3—24no—
239C3×Dic3—24no—
240C3×Dic3—44no—
241C3×Dic3—42no—
242C3×Dic3—42no—
243C3×Dic3—124no—
244C3×Dic3—122no—
245C3×Dic3—122no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—91no—
248C2×C42—31no—
249#95—94no—
250C33#38211yescharacteristic, complemented, Sylow
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—41no—
255C23×C3C2×D3211yescharacteristic
256C23×C3C22×C3×D311yescharacteristic
257C23×C3C22×C3×D311yescharacteristic
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—34no—
268C2×C3×C4—34no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—64no—
273C2×C3×C4—64no—
274C2×C3×C4—64no—
275C2×C3×C4—62no—
276C2×C3×C4—62no—
277C2×C3×C4—62no—
278C2×C3×C4—62no—
279C2×C3×C4—64no—
280C2×C3×C4—64no—
281C2×C3×C4—62no—
282C2×C3×C4—62no—
283C2×C3×C4—64no—
284C2×C3×C4—124no—
285C2×C3×C4—184no—
286C2×C3×C4—182no—
287C2×C3×C4D3×Dic312yes—
288C2×C3×C4#62612yes—
289C2×C3×C4#62712yes—
290C2×C3×C4C3×Dic612yes—
291C2×C3×C4C3×Dic612yes—
292C2×C3×C4C3×C4×D312yes—
293C2×C3×C4C3×D1212yes—
294C2×C3×C4C2×C3×Dic312yes—
295C2×C3×C4C3×#6712yes—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—24no—
299C2×Dic3—22no—
300C2×Dic3—64no—
301C2×Dic3—62no—
302C2×Dic3—64no—
303C2×Dic3—64no—
304C2×Dic3—62no—
305C2×Dic3—124no—
306C2×Dic3C2×C3×Dic314yes—
307C2×C32—22no—
308C2×C32—22no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—22no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—22no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—22no—
318C2×C32—21no—
319C2×C32—22no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—41no—
323C2×C32—42no—
324C2×C32—41no—
325C2×C32—41no—
326C2×C32—42no—
327C2×C32#87012yes—
328C2×C32#87511yescharacteristic
329C2×C32#87511yescharacteristic
330C2×C32#89012yes—
331C2×C32#89212yes—
332C2×C32C2×Dic3⋊C411yescharacteristic
333C2×C32C2×C4⋊Dic311yescharacteristic
334C2×C32#92612yes—
335C2×C32#93611yescharacteristic
336C2×C32#94711yescharacteristic
337C2×C32C2×C3×C4⋊C411yescharacteristic
338C2×C32C3×#11712yes—
339C2×C32C3×#12411yescharacteristic
340C2×C32C3×#12411yescharacteristic
341C2×C32C3×#12512yes—
342C22×C4—32no—
343C22×C4—32no—
344C22×C4—92no—
345C22×C4C3×D3211yescharacteristic
346C4⋊C4—62no—
347C4⋊C4—182no—
348C42—34no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3#379512yes—
370C22×C3#379912yes—
371C22×C3C2×D3×Dic311yescharacteristic
372C22×C3C2×#62611yescharacteristic
373C22×C3C2×#62711yescharacteristic
374C22×C3C2×C3×Dic611yescharacteristic
375C22×C3C2×C3×Dic611yescharacteristic
376C22×C3C2×C3×C4×D311yescharacteristic
377C22×C3C2×C3×D1211yescharacteristic
378C22×C3C3×#23512yes—
379C22×C3C3×#23512yes—
380C22×C3C3×#23712yes—
381C22×C3C3×#23912yes—
382C22×C3C22×C3×Dic311yescharacteristic
383C22×C3C2×C3×#6711yescharacteristic
384C3×C4—24no—
385C3×C4—24no—
386C3×C4—24no—
387C3×C4—44no—
388C3×C4—68no—
389C3×C4—64no—
390C3×C4—64no—
391C3×C4—124no—
392C3×C4—122no—
393C3×C4—122no—
394C3×C4—362no—
395C3×C4Dic3⋊Dic314yes—
396C3×C4C3×Dic3⋊C414yes—
397C3×C4C3×C4⋊Dic314yes—
398Dic3—24no—
399Dic3—42no—
400Dic3—124no—
401Dic3—122no—
402Dic3—122no—
403C32—21no—
404C32—21no—
405C32—21no—
406C32—41no—
407C32#493611yescharacteristic, complemented
408C32#497711yescharacteristic, complemented
409C32C3×#38211yescharacteristic, complemented
410C23C2×C3×D3211yescharacteristic
411C2×C4—34no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—64no—
415C2×C4—62no—
416C2×C4—184no—
417C2×C4—182no—
418C2×C4C3×D3×Dic312yes—
419C2×C4C3×#62612yes—
420C2×C4C3×#62712yes—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—22no—
425C2×C3—22no—
426C2×C3—22no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—22no—
434C2×C3—22no—
435C2×C3—22no—
436C2×C3—41no—
437C2×C3—42no—
438C2×C3—41no—
439C2×C3—41no—
440C2×C3—42no—
441C2×C3#6384112yes—
442C2×C3#6386011yescharacteristic
443C2×C3#6386411yescharacteristic
444C2×C3#6386912yes—
445C2×C3C2×Dic3⋊Dic311yescharacteristic
446C2×C3C3×#87012yes—
447C2×C3C3×#87511yescharacteristic
448C2×C3C3×#87511yescharacteristic
449C2×C3C3×#89012yes—
450C2×C3C3×#89212yes—
451C2×C3C2×C3×Dic3⋊C411yescharacteristic
452C2×C3C2×C3×C4⋊Dic311yescharacteristic
453C2×C3C3×#92612yes—
454C2×C3C3×#93611yescharacteristic
455C2×C3C3×#94711yescharacteristic
456C22C3×#379512yes—
457C22C3×#379912yes—
458C22C2×C3×D3×Dic311yescharacteristic
459C22C2×C3×#62611yescharacteristic
460C22C2×C3×#62711yescharacteristic
461C4—64no—
462C4—122no—
463C4—362no—
464C4C3×Dic3⋊Dic314yes—
465C3—21no—
466C3—21no—
467C3—21no—
468C3—41no—
469C3#1059122411yescharacteristic, direct product factor
470C3C3×#493611yescharacteristic, complemented
471C3C3×#497711yescharacteristic, complemented
472C2C3×#6384112yes—
473C2C3×#6386011yescharacteristic
474C2C3×#6386411yescharacteristic
475C2C3×#6386912yes—
476C2C2×C3×Dic3⋊Dic311yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#382
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
7. #10591224C3direct (N × H)
14. C32×C4×Dic3C4semidirect
73. C3×C4×Dic3Dic3semidirect
74. C3×C4×Dic3C3×C4semidirect
191. C4×Dic3C3×Dic3semidirect
250. C33#382semidirect
407. C32#4936semidirect
408. C32#4977semidirect
409. C32C3×#382semidirect
469. C3#10591224direct (N × H)
470. C3C3×#4936semidirect
471. C3C3×#4977semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊Dic3)2
C2×C32×C4⋊Dic32
C2×C32×C4×Dic32
C3×#635792
C3×#49773
C3×#49363
#105912243