Finite Group Databasebeta

← C3×#10591247 #13158179

Subgroups — C3×#10591247 #13158179

Order 1728.

914 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3750C211yesmaximal, characteristic
2C2×C3×#3724C211yesmaximal, characteristic
3C2×C3×Dic32C211yesmaximal, characteristic
4C32×C3⋊#95C212yesmaximal
5C3×#63579C212yesmaximal
6C3×#4650—32nomaximal
7#10591247C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C3×#3750—22no—
14C3×#3724—22no—
15C3×Dic32C414yescomplemented
16C2×#3750C2×C311yescharacteristic
17C2×C3×Dic3⋊C4—32no—
18C2×C3×Dic3⋊C4—32no—
19C2×C3×Dic3⋊C4—61no—
20C2×C3×C4×Dic3—32no—
21C2×#3724C2×C311yescharacteristic
22C2×Dic32C2×C311yescharacteristic
23C32×#95—32no—
24C3×C3⋊#95—22no—
25C3×C3⋊#95—32no—
26C3×C3⋊#95—32no—
27C3×C3⋊#95D312yes—
28C3×C3⋊#95C2×C312yes—
29#63579C2×C312yes—
30C23×C33C2311yescharacteristic
31C2×C33×C4—21no—
32C2×C33×C4—22no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3—22no—
35C2×C3×C3⋊Dic3C2×C412yes—
36C2×C3×C3⋊Dic3D412yes—
37C2×C3×C3⋊Dic3Q812yes—
38C2×C32×Dic3—22no—
39C2×C32×Dic3—22no—
40C2×C32×Dic3—22no—
41C2×C32×Dic3—22no—
42C2×C32×Dic3—22no—
43C2×C32×Dic3C2×C414yes—
44C3×#382—91no—
45#4650—32no—
46C22×C32×C4—21no—
47C22×C32×C4—22no—
48C22×C32×C4—32no—
49C22×C32×C4—32no—
50C22×C32×C4—41no—
51C22×C32×C4C2×D312yes—
52C22×C32×C4C22×C311yescharacteristic
53C22×C3⋊Dic3C22×C311yescharacteristic
54C22×C3⋊Dic3C22×C311yescharacteristic
55C22×C3×Dic3—22no—
56C22×C3×Dic3—22no—
57C22×C3×Dic3—32no—
58C22×C3×Dic3—32no—
59C22×C3×Dic3—61no—
60C22×C3×Dic3—61no—
61C22×C3×Dic3C2×D312yes—
62C22×C3×Dic3C2×D312yes—
63C22×C3×Dic3C22×C312yes—
64C22×C3×Dic3C22×C312yes—
65#3750—22no—
66C3×Dic3⋊C4—64no—
67C3×Dic3⋊C4—64no—
68C3×Dic3⋊C4—64no—
69C3×C4×Dic3—38no—
70#3724—22no—
71Dic32C3×C414yescomplemented
72C22×C33C22×C411yescharacteristic
73C22×C33C2×D411yescharacteristic
74C22×C33C2×Q811yescharacteristic
75C22×C33#4112yes—
76C22×C33#4112yes—
77C33×C4—41no—
78C3×C3⋊Dic3—41no—
79C3×C3⋊Dic3C4⋊C414yes—
80C32×Dic3—24no—
81C32×Dic3—42no—
82C2×C3×C4⋊C4—91no—
83C2×C3×C4⋊C4—91no—
84C2×C3×C42—91no—
85C2×Dic3⋊C4—32no—
86C2×Dic3⋊C4—32no—
87C2×Dic3⋊C4—61no—
88C2×C4×Dic3—32no—
89C3×#95—32no—
90C3×#95—32no—
91C3×#95—62no—
92C3×#95—92no—
93C3⋊#95—32no—
94C3⋊#95—32no—
95C3⋊#95C3×D312yes—
96C23×C32—21no—
97C23×C32—22no—
98C23×C32—41no—
99C23×C32C22×D312yes—
100C23×C32C23×C311yescharacteristic
101C2×C32×C4—22no—
102C2×C32×C4—21no—
103C2×C32×C4—22no—
104C2×C32×C4—24no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—34no—
109C2×C32×C4—42no—
110C2×C32×C4—42no—
111C2×C32×C4—42no—
112C2×C32×C4—42no—
113C2×C32×C4—44no—
114C2×C32×C4—62no—
115C2×C32×C4—62no—
116C2×C32×C4—62no—
117C2×C32×C4—62no—
118C2×C32×C4—62no—
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3C2×C3×C412yes—
122C2×C3⋊Dic3C3×D412yes—
123C2×C3⋊Dic3C3×Q812yes—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—22no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—34no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—42no—
140C2×C3×Dic3—42no—
141C2×C3×Dic3—42no—
142C2×C3×Dic3—42no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—62no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—122no—
151C2×C3×Dic3C4×D314yes—
152C2×C3×Dic3C2×C3×C414yes—
153#382—91noSylow
154C2×C33C2×C4⋊C411yescharacteristic
155C2×C33#11712yes—
156C2×C33#12411yescharacteristic
157C2×C33#12411yescharacteristic
158C2×C33#12512yes—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—32no—
162C22×C3×C4—32no—
163C22×C3×C4—32no—
164C22×C3×C4—32no—
165C22×C3×C4—41no—
166C22×C3×C4—62no—
167C22×C3×C4—62no—
168C22×C3×C4—91no—
169C22×C3×C4—91no—
170C22×C3×C4D3211yescharacteristic
171C22×C3×C4C2×C3×D312yes—
172C22×Dic3—32no—
173C22×Dic3—32no—
174C22×Dic3—61no—
175C22×Dic3—61no—
176C22×Dic3C2×C3×D312yes—
177C22×Dic3C2×C3×D312yes—
178C3×C4⋊C4—182no—
179C3×C4⋊C4—182no—
180C3×C42—94no—
181Dic3⋊C4—64no—
182Dic3⋊C4—64no—
183Dic3⋊C4—64no—
184C4×Dic3—38no—
185C22×C32—22no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—22no—
189C22×C32—21no—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—22no—
193C22×C32—22no—
194C22×C32—22no—
195C22×C32—22no—
196C22×C32—22no—
197C22×C32—41no—
198C22×C32—41no—
199C22×C32—41no—
200C22×C32—42no—
201C22×C32—42no—
202C22×C32C2×C4×D312yes—
203C22×C32#23512yes—
204C22×C32#23512yes—
205C22×C32D3×D412yes—
206C22×C32#23712yes—
207C22×C32#23712yes—
208C22×C32D3×Q812yes—
209C22×C32C22×C3×C411yescharacteristic
210C22×C32C2×C3×D411yescharacteristic
211C22×C32C2×C3×Q811yescharacteristic
212C22×C32C3×#4112yes—
213C22×C32C3×#4112yes—
214C32×C4—44no—
215C32×C4—42no—
216C32×C4—44no—
217C32×C4—42no—
218C32×C4—41no—
219C32×C4—64no—
220C32×C4—122no—
221C3⋊Dic3—41no—
222C3⋊Dic3C3×C4⋊C414yes—
223C3×Dic3—24no—
224C3×Dic3—24no—
225C3×Dic3—38no—
226C3×Dic3—44no—
227C3×Dic3—44no—
228C3×Dic3—42no—
229C3×Dic3—42no—
230C3×Dic3—64no—
231C3×Dic3—122no—
232C3×Dic3—122no—
233C2×C4⋊C4—91no—
234C2×C4⋊C4—91no—
235C2×C42—91no—
236#95—32no—
237#95—92no—
238C33#38211yescharacteristic, complemented, Sylow
239C23×C3—22no—
240C23×C3—21no—
241C23×C3—41no—
242C23×C3C2×D3211yescharacteristic
243C23×C3C22×C3×D312yes—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—21no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—34no—
252C2×C3×C4—34no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—44no—
257C2×C3×C4—42no—
258C2×C3×C4—64no—
259C2×C3×C4—62no—
260C2×C3×C4—62no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—64no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—62no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—62no—
270C2×C3×C4—92no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—122no—
274C2×C3×C4—122no—
275C2×C3×C4—122no—
276C2×C3×C4—122no—
277C2×C3×C4—182no—
278C2×C3×C4—181no—
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—34no—
285C2×Dic3—34no—
286C2×Dic3—34no—
287C2×Dic3—62no—
288C2×Dic3—62no—
289C2×Dic3—62no—
290C2×Dic3—62no—
291C2×Dic3—62no—
292C2×Dic3—62no—
293C2×Dic3—62no—
294C2×Dic3—122no—
295C2×Dic3C3×C4×D314yes—
296C2×C32—22no—
297C2×C32—22no—
298C2×C32—22no—
299C2×C32—22no—
300C2×C32—22no—
301C2×C32—22no—
302C2×C32—22no—
303C2×C32—22no—
304C2×C32—21no—
305C2×C32—21no—
306C2×C32—21no—
307C2×C32—22no—
308C2×C32—41no—
309C2×C32—42no—
310C2×C32—41no—
311C2×C32—41no—
312C2×C32—42no—
313C2×C32#87212yes—
314C2×C32#87712yes—
315C2×C32#88512yes—
316C2×C32#88512yes—
317C2×C32#88912yes—
318C2×C32#88912yes—
319C2×C32D3×C4⋊C412yes—
320C2×C32C2×C3×C4⋊C411yescharacteristic
321C2×C32C3×#11712yes—
322C2×C32C3×#12411yescharacteristic
323C2×C32C3×#12411yescharacteristic
324C2×C32C3×#12512yes—
325C22×C4—32no—
326C22×C4—32no—
327C22×C4—91no—
328C22×C4—91no—
329C22×C4C3×D3211yescharacteristic
330C4⋊C4—182no—
331C4⋊C4—182no—
332C42—94no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3—21no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—21no—
344C22×C3—22no—
345C22×C3—42no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—42no—
349C22×C3—41no—
350C22×C3#379512yes—
351C22×C3#379711yescharacteristic
352C22×C3C4×D3211yescharacteristic
353C22×C3#380412yes—
354C22×C3#381111yescharacteristic
355C22×C3C2×C3×C4×D312yes—
356C22×C3C3×#23512yes—
357C22×C3C3×#23512yes—
358C22×C3C3×D3×D412yes—
359C22×C3C3×#23712yes—
360C22×C3C3×#23712yes—
361C22×C3C3×D3×Q812yes—
362C3×C4—42no—
363C3×C4—44no—
364C3×C4—44no—
365C3×C4—42no—
366C3×C4—41no—
367C3×C4—64no—
368C3×C4—64no—
369C3×C4—94no—
370C3×C4—124no—
371C3×C4—124no—
372C3×C4—122no—
373C3×C4—122no—
374C3×C4—361no—
375Dic3—24no—
376Dic3—38no—
377Dic3—42no—
378Dic3—64no—
379Dic3—122no—
380Dic3—122no—
381C32—22no—
382C32—21no—
383C32—41no—
384C32#465012yescomplemented
385C32C3×#38211yescharacteristic, complemented
386C23C2×C3×D3211yescharacteristic
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—34no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—62no—
394C2×C4—62no—
395C2×C4—92no—
396C2×C4—92no—
397C2×C4—92no—
398C2×C4—182no—
399C2×C4—181no—
400C2×C3—21no—
401C2×C3—22no—
402C2×C3—22no—
403C2×C3—22no—
404C2×C3—22no—
405C2×C3—22no—
406C2×C3—22no—
407C2×C3—22no—
408C2×C3—21no—
409C2×C3—22no—
410C2×C3—22no—
411C2×C3—21no—
412C2×C3—41no—
413C2×C3—42no—
414C2×C3—42no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3#6384712yes—
418C2×C3#6387712yes—
419C2×C3#6388311yescharacteristic
420C2×C3#6391311yescharacteristic
421C2×C3#6395311yescharacteristic
422C2×C3C3×#87212yes—
423C2×C3C3×#87712yes—
424C2×C3C3×#88512yes—
425C2×C3C3×#88512yes—
426C2×C3C3×#88912yes—
427C2×C3C3×#88912yes—
428C2×C3C3×D3×C4⋊C412yes—
429C22C3×#379512yes—
430C22C3×#379711yescharacteristic
431C22C3×C4×D3211yescharacteristic
432C22C3×#380412yes—
433C22C3×#381111yescharacteristic
434C4—41no—
435C4—64no—
436C4—94no—
437C4—122no—
438C4—361no—
439C3—22no—
440C3—21no—
441C3—41no—
442C3#1059124711yescharacteristic, direct product factor
443C3C3×#465012yescomplemented
444C2C3×#6384712yes—
445C2C3×#6387712yes—
446C2C3×#6388311yescharacteristic
447C2C3×#6391311yescharacteristic
448C2C3×#6395311yescharacteristic
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. #10591247C3direct (N × H)
15. C3×Dic32C4semidirect
71. Dic32C3×C4semidirect
238. C33#382semidirect
384. C32#4650semidirect
385. C32C3×#382semidirect
442. C3#10591247direct (N × H)
443. C3C3×#4650semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37502
C2×C3×#37242
C2×C3×Dic322
C32×C3⋊#952
C3×#635792
C3×#46503
#105912473