Finite Group Databasebeta

← C3×#10591141 #13158169

Subgroups — C3×#10591141 #13158169

Order 1728.

666 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3748C211yesmaximal, characteristic
2C2×C32×#208C211yesmaximal, characteristic
3C2×C32×#208C211yesmaximal, characteristic
4C3×#63577C214yesmaximal, complemented
5C3×#5125—31nomaximal
6C3×#5124—31nomaximal
7#10591141C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C3×#3748—21no—
10C3×#3748C2212yes—
11C2×C3×C3⋊(C3⋊C8)—21no—
12C32×#208—21no—
13C32×#208C412yes—
14C32×#208C412yes—
15C32×#208C2212yes—
16C2×C32×C3⋊C8—21no—
17C2×C32×C3⋊C8C2212yes—
18C2×C32×M16—31no—
19C2×C32×M16—31no—
20C2×#3748C2×C311yescharacteristic
21C2×C3×#208—21no—
22C2×C3×#208—21no—
23C2×C3×#208—31no—
24C2×C3×#208—31no—
25C2×C3×#208—61no—
26C2×C3×#208D311yescharacteristic
27C2×C3×#208D311yescharacteristic
28C2×C3×#208C2×C311yescharacteristic
29C2×C3×#208C2×C311yescharacteristic
30C3×#822—34no—
31C3×#819—34no—
32#63577C2×C314yescomplemented
33C23×C33Q811yescharacteristic
34C2×C33×C4C2×C411yescharacteristic
35C2×C33×C4C2×C411yescharacteristic
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4Q811yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C3×C3⋊(C3⋊C8)—22no—
41C32×C3⋊C8—22no—
42C32×C3⋊C8C2×C414yes—
43C3×#430—91no—
44#5125—31no—
45#5124—31no—
46C22×C32×C4—21no—
47C22×C32×C4—21no—
48C22×C32×C4—21no—
49C22×C32×C4—41no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C22×C311yescharacteristic
53C32×M16—32no—
54C32×M16—32no—
55C32×M16—32no—
56C32×M16—61no—
57C2×C32×C8—32no—
58C2×C32×C8—61no—
59#3748—21no—
60#3748C22×C312yes—
61C2×C3⋊(C3⋊C8)—21no—
62C3×#208—22no—
63C3×#208—22no—
64C3×#208—22no—
65C3×#208—22no—
66C3×#208—21no—
67C3×#208—21no—
68C3×#208—32no—
69C3×#208—32no—
70C3×#208—62no—
71C3×#208—62no—
72C3×#208—61no—
73C3×#208—61no—
74C3×#208Dic312yes—
75C3×#208Dic312yes—
76C3×#208C3×C412yes—
77C3×#208C3×C412yes—
78C3×#208C2×D312yes—
79C3×#208C22×C312yes—
80C2×C3×C3⋊C8—22no—
81C2×C3×C3⋊C8—22no—
82C2×C3×C3⋊C8—21no—
83C2×C3×C3⋊C8—21no—
84C2×C3×C3⋊C8—62no—
85C2×C3×C3⋊C8—61no—
86C2×C3×C3⋊C8—61no—
87C2×C3×C3⋊C8C2×D312yes—
88C2×C3×C3⋊C8C22×C312yes—
89C22×C33—41no—
90C22×C33C4⋊C411yescharacteristic
91C22×C33C4⋊C411yescharacteristic
92C22×C33C2×Q811yescharacteristic
93C33×C4C4⋊C411yescharacteristic
94C33×C4C4⋊C411yescharacteristic
95C33×C4C22×C411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C2×C3×M16—31no—
98C2×C3×M16—31no—
99C2×C3×M16—31no—
100C2×C3×M16—31no—
101C2×C3×M16—61no—
102C2×C3×M16—61no—
103C2×C3×M16—91no—
104C2×#208—31no—
105C2×#208—31no—
106C2×#208—61no—
107C2×#208C3×D311yescharacteristic
108C2×#208C3×D311yescharacteristic
109C3×#108—94no—
110#822—34no—
111#819—34no—
112C23×C32—21no—
113C23×C32—21no—
114C23×C32—21no—
115C23×C32—41no—
116C23×C32Dic611yescharacteristic
117C23×C32Dic611yescharacteristic
118C23×C32C3×Q811yescharacteristic
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—41no—
138C2×C32×C4—41no—
139C2×C32×C4—41no—
140C2×C32×C4—41no—
141C2×C32×C4—41no—
142C2×C32×C4—41no—
143C2×C32×C4Dic611yescharacteristic
144C2×C32×C4Dic611yescharacteristic
145C2×C32×C4C4×D311yescharacteristic
146C2×C32×C4C4×D311yescharacteristic
147C2×C32×C4D1211yescharacteristic
148C2×C32×C4D1211yescharacteristic
149C2×C32×C4C2×Dic311yescharacteristic
150C2×C32×C4C2×Dic311yescharacteristic
151C2×C32×C4#6711yescharacteristic
152C2×C32×C4#6711yescharacteristic
153C2×C32×C4C2×C3×C411yescharacteristic
154C2×C32×C4C2×C3×C411yescharacteristic
155C2×C32×C4C3×D411yescharacteristic
156C2×C32×C4C3×D411yescharacteristic
157C2×C32×C4C3×Q811yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C23×C311yescharacteristic
161C32×C8—34no—
162C32×C8—62no—
163C3⋊(C3⋊C8)—22no—
164C3×C3⋊C8—24no—
165C3×C3⋊C8—22no—
166C3×C3⋊C8—22no—
167C3×C3⋊C8—24no—
168C3×C3⋊C8—64no—
169C3×C3⋊C8—62no—
170C3×C3⋊C8—62no—
171C3×C3⋊C8C2×Dic314yes—
172C3×C3⋊C8C2×C3×C414yes—
173#430—91noSylow
174C2×C33—21no—
175C2×C33—21no—
176C2×C33—21no—
177C2×C33C2×C4⋊C411yescharacteristic
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—41no—
182C22×C3×C4D3211yescharacteristic
183C22×C3×C4C2×C3×D311yescharacteristic
184C22×C3×C4C2×C3×D311yescharacteristic
185C3×M16—32no—
186C3×M16—32no—
187C3×M16—32no—
188C3×M16—32no—
189C3×M16—32no—
190C3×M16—32no—
191C3×M16—62no—
192C3×M16—62no—
193C3×M16—62no—
194C3×M16—62no—
195C3×M16—61no—
196C3×M16—61no—
197C3×M16—92no—
198C3×M16—181no—
199C2×C3×C8—32no—
200C2×C3×C8—32no—
201C2×C3×C8—62no—
202C2×C3×C8—62no—
203C2×C3×C8—61no—
204C2×C3×C8—61no—
205C2×C3×C8—181no—
206#208—21no—
207#208—32no—
208#208—32no—
209#208—62no—
210#208—62no—
211#208—61no—
212#208—61no—
213#208C3×Dic312yes—
214#208C3×Dic312yes—
215#208C2×C3×D312yes—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—62no—
218C2×C3⋊C8—61no—
219C2×C3⋊C8—61no—
220C2×C3⋊C8C2×C3×D312yes—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32—41no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—42no—
237C22×C32—42no—
238C22×C32—42no—
239C22×C32—44no—
240C22×C32Dic3⋊C411yescharacteristic
241C22×C32Dic3⋊C411yescharacteristic
242C22×C32C4⋊Dic311yescharacteristic
243C22×C32C4⋊Dic311yescharacteristic
244C22×C32C3×C4⋊C411yescharacteristic
245C22×C32C3×C4⋊C411yescharacteristic
246C22×C32C2×Dic611yescharacteristic
247C22×C32C2×Dic611yescharacteristic
248C22×C32C2×C3×Q811yescharacteristic
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—41no—
262C32×C4—41no—
263C32×C4—41no—
264C32×C4—41no—
265C32×C4Dic3⋊C411yescharacteristic
266C32×C4Dic3⋊C411yescharacteristic
267C32×C4C4⋊Dic311yescharacteristic
268C32×C4C4⋊Dic311yescharacteristic
269C32×C4C3×C4⋊C411yescharacteristic
270C32×C4C3×C4⋊C411yescharacteristic
271C32×C4C2×C4×D311yescharacteristic
272C32×C4C2×D1211yescharacteristic
273C32×C4C22×Dic311yescharacteristic
274C32×C4C2×#6711yescharacteristic
275C32×C4C22×C3×C411yescharacteristic
276C32×C4C2×C3×D411yescharacteristic
277C2×M16—31no—
278C2×M16—31no—
279C2×M16—91no—
280#108—94no—
281C33#43011yescharacteristic, complemented, Sylow
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—21no—
285C23×C3—41no—
286C23×C3#62711yescharacteristic
287C23×C3C3×Dic611yescharacteristic
288C23×C3C3×Dic611yescharacteristic
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4D3×Dic311yescharacteristic
314C2×C3×C4D3×Dic311yescharacteristic
315C2×C3×C4#62611yescharacteristic
316C2×C3×C4#62611yescharacteristic
317C2×C3×C4#62711yescharacteristic
318C2×C3×C4C3×Dic611yescharacteristic
319C2×C3×C4C3×Dic611yescharacteristic
320C2×C3×C4C3×C4×D311yescharacteristic
321C2×C3×C4C3×C4×D311yescharacteristic
322C2×C3×C4C3×D1211yescharacteristic
323C2×C3×C4C3×D1211yescharacteristic
324C2×C3×C4C2×C3×Dic311yescharacteristic
325C2×C3×C4C2×C3×Dic311yescharacteristic
326C2×C3×C4C3×#6711yescharacteristic
327C2×C3×C4C3×#6711yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×C3×C4C22×C3×D311yescharacteristic
330C2×C3×C4C22×C3×D311yescharacteristic
331C3×C8—34no—
332C3×C8—34no—
333C3×C8—64no—
334C3×C8—64no—
335C3×C8—62no—
336C3×C8—62no—
337C3×C8—182no—
338C3⋊C8—22no—
339C3⋊C8—64no—
340C3⋊C8—62no—
341C3⋊C8—62no—
342C3⋊C8C2×C3×Dic314yes—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—22no—
347C2×C32—22no—
348C2×C32—22no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—41no—
359C2×C32—41no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32—41no—
363C2×C32—41no—
364C2×C32—41no—
365C2×C32—42no—
366C2×C32—42no—
367C2×C32—42no—
368C2×C32C2×Dic3⋊C411yescharacteristic
369C2×C32C2×C4⋊Dic311yescharacteristic
370C2×C32C2×C3×C4⋊C411yescharacteristic
371C22×C4C3×D3211yescharacteristic
372M16—32no—
373M16—32no—
374M16—32no—
375M16—61no—
376M16—92no—
377M16—181no—
378C2×C8—32no—
379C2×C8—61no—
380C2×C8—181no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—44no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3Dic3⋊Dic311yescharacteristic
401C22×C3Dic3⋊Dic311yescharacteristic
402C22×C3C3×Dic3⋊C411yescharacteristic
403C22×C3C3×Dic3⋊C411yescharacteristic
404C22×C3C3×C4⋊Dic311yescharacteristic
405C22×C3C3×C4⋊Dic311yescharacteristic
406C22×C3C2×#62711yescharacteristic
407C22×C3C2×C3×Dic611yescharacteristic
408C22×C3C2×C3×Dic611yescharacteristic
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—41no—
422C3×C4—41no—
423C3×C4—41no—
424C3×C4—41no—
425C3×C4Dic3⋊Dic311yescharacteristic
426C3×C4Dic3⋊Dic311yescharacteristic
427C3×C4C3×Dic3⋊C411yescharacteristic
428C3×C4C3×Dic3⋊C411yescharacteristic
429C3×C4C3×C4⋊Dic311yescharacteristic
430C3×C4C3×C4⋊Dic311yescharacteristic
431C3×C4C2×D3×Dic311yescharacteristic
432C3×C4C2×#62611yescharacteristic
433C3×C4C2×C3×C4×D311yescharacteristic
434C3×C4C2×C3×D1211yescharacteristic
435C3×C4C22×C3×Dic311yescharacteristic
436C3×C4C2×C3×#6711yescharacteristic
437C32—21no—
438C32—21no—
439C32—21no—
440C32—41no—
441C32#512411yescharacteristic, complemented
442C32#512511yescharacteristic, complemented
443C32C3×#43011yescharacteristic, complemented
444C23C3×#62711yescharacteristic
445C2×C4C3×D3×Dic311yescharacteristic
446C2×C4C3×D3×Dic311yescharacteristic
447C2×C4C3×#62611yescharacteristic
448C2×C4C3×#62611yescharacteristic
449C2×C4C3×#62711yescharacteristic
450C2×C4C2×C3×D3211yescharacteristic
451C8—34no—
452C8—62no—
453C8—182no—
454C2×C3—22no—
455C2×C3—22no—
456C2×C3—22no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—42no—
476C2×C3—42no—
477C2×C3—42no—
478C2×C3—41no—
479C2×C3C2×Dic3⋊Dic311yescharacteristic
480C2×C3C2×C3×Dic3⋊C411yescharacteristic
481C2×C3C2×C3×C4⋊Dic311yescharacteristic
482C22—41no—
483C22C3×Dic3⋊Dic311yescharacteristic
484C22C3×Dic3⋊Dic311yescharacteristic
485C22C2×C3×#62711yescharacteristic
486C4C3×Dic3⋊Dic311yescharacteristic
487C4C3×Dic3⋊Dic311yescharacteristic
488C4C2×C3×D3×Dic311yescharacteristic
489C4C2×C3×#62611yescharacteristic
490C3—21no—
491C3—21no—
492C3—21no—
493C3—41no—
494C3#1059114111yescharacteristic, direct product factor
495C3C3×#512411yescharacteristic, complemented
496C3C3×#512511yescharacteristic, complemented
497C2—21no—
498C2—21no—
499C2—21no—
500C2C2×C3×Dic3⋊Dic311yescharacteristic
subgroupG/H
centerC3×C4C2×#626
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC2×C33C2×C4⋊C4
prime radicalC23×C33Q8
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#430
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63577C2semidirect
7. #10591141C3direct (N × H)
32. #63577C2×C3semidirect
281. C33#430semidirect
441. C32#5124semidirect
442. C32#5125semidirect
443. C32C3×#430semidirect
494. C3#10591141direct (N × H)
495. C3C3×#5124semidirect
496. C3C3×#5125semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37482
C2×C32×#2082
C3×#635772
C3×#51253
C3×#51243
#105911413