Finite Group Databasebeta

← C3×#10591626 #13158629

Subgroups — C3×#10591626 #13158629

Order 1728.

714 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3756C211yesmaximal, characteristic
2C32×#944C211yesmaximal, characteristic
3C2×C32×#216C211yesmaximal, characteristic
4C3×#63866C211yesmaximal, characteristic
5C2×C3×#3719C211yesmaximal, characteristic
6C2×C3×#3718C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic
8C3×#5183—31nomaximal
9C3×#5092—31nomaximal
10#10591626C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×#3756—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#216—22no—
18C32×Dic3⋊C4—22no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—22no—
22C3×#3719—22no—
23C3×#3718—22no—
24C2×C32×Q16—31no—
25C32×C4⋊Q8—31no—
26C2×#3756C2×C311yescharacteristic
27C3×#944—21no—
28C3×#944—31no—
29C3×#944D311yescharacteristic
30C3×#944C2×C311yescharacteristic
31C2×C3×#216—21no—
32C2×C3×#216—31no—
33C2×C3×#216—31no—
34C2×C3×#216—31no—
35C2×C3×#216—31no—
36C2×C3×#216—31no—
37C2×C3×#216—61no—
38C2×C3×#216D311yescharacteristic
39C2×C3×#216C2×C311yescharacteristic
40C2×C3×Dic12—31no—
41C3×#869—31no—
42#63866C2×C311yescharacteristic
43C2×#3719C2×C311yescharacteristic
44C2×#3718C2×C311yescharacteristic
45C3×C8×Dic3—31no—
46C3×C4×C3⋊C8—31no—
47Dic3×C3⋊C8C2×C311yescharacteristic
48C33×Q8—21no—
49C33×Q8—21no—
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D412yes—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)D412yes—
63C32×C3⋊C8D412yes—
64C3×#494—91no—
65#5183—31no—
66#5092—31no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—21no—
70C2×C32×Q8—31no—
71C2×C32×Q8—31no—
72C2×C32×Q8—41no—
73C2×C32×Q8C2×D311yescharacteristic
74C2×C32×Q8C2×D311yescharacteristic
75C2×C32×Q8C22×C311yescharacteristic
76C2×#634C22×C311yescharacteristic
77C2×C3×Dic6—21no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—61no—
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C3×Dic6C2×D311yescharacteristic
84C2×C3×Dic6C22×C311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C32×Q16—62no—
87C2×C32×C8—31no—
88C32×C4⋊C4—62no—
89C32×C42—31no—
90#3756—22no—
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#216—24no—
93C3×#216—22no—
94C3×#216—22no—
95C3×#216—62no—
96C3×#216—62no—
97C3×#216—62no—
98C3×#216—62no—
99C3×#216—64no—
100C3×#216—62no—
101C3×C4⋊Dic3—62no—
102C3×Dic3⋊C4—24no—
103C3×Dic3⋊C4—22no—
104C3×Dic3⋊C4—22no—
105C3×Dic3⋊C4—62no—
106C3×C4×Dic3—21no—
107C3×C4×Dic3C2×D311yescharacteristic
108C3×C4×Dic3C22×C311yescharacteristic
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—61no—
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C2×C3×C3⋊C8C22×C311yescharacteristic
115C3×Dic12—62no—
116Dic3⋊Dic3—22no—
117#3719—22no—
118#3718—22no—
119C22×C33C2×D411yescharacteristic
120C33×C4—41no—
121C33×C4C2×D411yescharacteristic
122C33×C4C2×D411yescharacteristic
123C3×C3⋊Dic3—41no—
124C32×Dic3—41no—
125C32×Dic3—41no—
126C32×Dic3Q1614yescomplemented
127C2×C3×Q16—31no—
128C2×C3×Q16—31no—
129C2×C3×Q16—61no—
130C2×C3×Q16—91no—
131C2×C3×Q16—91no—
132C2×C3×Q16—91no—
133C3×C4⋊Q8—31no—
134C3×C4⋊Q8—31no—
135C3×C4⋊Q8—61no—
136C3×C4⋊Q8—91no—
137#944—31no—
138#944C3×D311yescharacteristic
139C2×#216—31no—
140C2×#216—31no—
141C2×#216—31no—
142C2×#216—31no—
143C2×#216—31no—
144C2×#216—61no—
145C2×#216C3×D311yescharacteristic
146C2×Dic12—31no—
147#869—31no—
148C3×C4×C8—91no—
149C8×Dic3—31no—
150C4×C3⋊C8—31no—
151C32×Q8—21no—
152C32×Q8—21no—
153C32×Q8—22no—
154C32×Q8—22no—
155C32×Q8—21no—
156C32×Q8—21no—
157C32×Q8—21no—
158C32×Q8—21no—
159C32×Q8—41no—
160C32×Q8—41no—
161C32×Q8—41no—
162C32×Q8—41no—
163C32×Q8—42no—
164C32×Q8—42no—
165C32×Q8—61no—
166C32×Q8—61no—
167C32×Q8—61no—
168C32×Q8—61no—
169C2×C32×C4—21no—
170C2×C32×C4—22no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—32no—
177C2×C32×C4—41no—
178C2×C32×C4—41no—
179C2×C32×C4—41no—
180C2×C32×C4—42no—
181C2×C32×C4—61no—
182C2×C32×C4—61no—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C23×C311yescharacteristic
186C2×C3⋊Dic3—21no—
187#634—21no—
188#634—21no—
189C2×C3×Dic3—22no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—22no—
195C2×C3×Dic3—41no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—121no—
199C2×C3×Dic3#6712yes—
200C2×C3×Dic3C3×D412yes—
201C3×Dic6—22no—
202C3×Dic6—21no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—21no—
209C3×Dic6—21no—
210C3×Dic6—41no—
211C3×Dic6—41no—
212C3×Dic6—41no—
213C3×Dic6—62no—
214C3×Dic6—61no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—121no—
219C32×C8—32no—
220C3⋊(C3⋊C8)C3×D412yes—
221C3×C3⋊C8—22no—
222C3×C3⋊C8—32no—
223C3×C3⋊C8—32no—
224C3×C3⋊C8—62no—
225C3×C3⋊C8#6712yes—
226C3×C3⋊C8C3×D412yes—
227#494—91noSylow
228C2×C33#12711yescharacteristic
229C2×C33C2×Q1612yes—
230C2×C3×Q8—21no—
231C2×C3×Q8—21no—
232C2×C3×Q8—21no—
233C2×C3×Q8—31no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—31no—
237C2×C3×Q8—41no—
238C2×C3×Q8—61no—
239C2×C3×Q8—61no—
240C2×C3×Q8—91no—
241C2×C3×Q8D3211yescharacteristic
242C2×C3×Q8C2×C3×D311yescharacteristic
243C2×C3×Q8C2×C3×D311yescharacteristic
244C2×Dic6—31no—
245C2×Dic6—31no—
246C2×Dic6—61no—
247C2×Dic6C2×C3×D311yescharacteristic
248C2×Dic6C2×C3×D311yescharacteristic
249C3×Q16—62no—
250C3×Q16—62no—
251C3×Q16—64no—
252C3×Q16—182no—
253C3×Q16—182no—
254C3×Q16—182no—
255C2×C3×C8—31no—
256C2×C3×C8—31no—
257C2×C3×C8—61no—
258C2×C3×C8—91no—
259C3×C4⋊C4—62no—
260C3×C4⋊C4—62no—
261C3×C4⋊C4—64no—
262C3×C4⋊C4—182no—
263C3×C42—31no—
264C3×C42—31no—
265C3×C42—61no—
266#216—22no—
267#216—62no—
268#216—62no—
269#216—62no—
270#216—62no—
271#216—64no—
272#216—62no—
273C4⋊Dic3—62no—
274Dic3⋊C4—22no—
275Dic3⋊C4—62no—
276C4×Dic3C2×C3×D311yescharacteristic
277C2×C3⋊C8—31no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—61no—
280C2×C3⋊C8C2×C3×D311yescharacteristic
281Dic12—62no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—41no—
286C22×C32D3×D411yescharacteristic
287C22×C32C2×#6711yescharacteristic
288C22×C32C2×C3×D411yescharacteristic
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—34no—
296C32×C4—41no—
297C32×C4—41no—
298C32×C4—42no—
299C32×C4—42no—
300C32×C4—42no—
301C32×C4—44no—
302C32×C4—41no—
303C32×C4—41no—
304C32×C4—41no—
305C32×C4—121no—
306C32×C4—121no—
307C32×C4D3×D411yescharacteristic
308C32×C4D3×D411yescharacteristic
309C32×C4D3×D411yescharacteristic
310C32×C4C2×#6711yescharacteristic
311C32×C4C2×C3×D411yescharacteristic
312C32×C4C2×C3×D411yescharacteristic
313C3⋊Dic3—41no—
314C3×Dic3—24no—
315C3×Dic3—42no—
316C3×Dic3—42no—
317C3×Dic3—41no—
318C3×Dic3—41no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—122no—
322C3×Dic3—121no—
323C3×Dic3—121no—
324C3×Dic3#21614yescomplemented
325C3×Dic3C3×Q1614yescomplemented
326C2×Q16—31no—
327C2×Q16—91no—
328C2×Q16—91no—
329C2×Q16—91no—
330C4⋊Q8—31no—
331C4⋊Q8—91no—
332C4×C8—91no—
333C33#49411yescharacteristic, complemented, Sylow
334C3×Q8—22no—
335C3×Q8—22no—
336C3×Q8—21no—
337C3×Q8—21no—
338C3×Q8—21no—
339C3×Q8—21no—
340C3×Q8—21no—
341C3×Q8—21no—
342C3×Q8—41no—
343C3×Q8—41no—
344C3×Q8—41no—
345C3×Q8—41no—
346C3×Q8—42no—
347C3×Q8—42no—
348C3×Q8—62no—
349C3×Q8—61no—
350C3×Q8—61no—
351C3×Q8—61no—
352C3×Q8—61no—
353C3×Q8—61no—
354C3×Q8—61no—
355C3×Q8—61no—
356C3×Q8—61no—
357C3×Q8—121no—
358C3×Q8—121no—
359C3×Q8—121no—
360C3×Q8—181no—
361C3×Q8—181no—
362C2×C3×C4—22no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—32no—
370C2×C3×C4—32no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—41no—
375C2×C3×C4—62no—
376C2×C3×C4—62no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—61no—
381C2×C3×C4—121no—
382C2×C3×C4—181no—
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C22×C3×D311yescharacteristic
385C2×C3×C4C22×C3×D311yescharacteristic
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—121no—
391C2×Dic3C3×#6712yes—
392Dic6—21no—
393Dic6—21no—
394Dic6—21no—
395Dic6—21no—
396Dic6—62no—
397Dic6—61no—
398Dic6—61no—
399Dic6—61no—
400Dic6—61no—
401Dic6—121no—
402C3×C8—32no—
403C3×C8—32no—
404C3×C8—62no—
405C3×C8—92no—
406C3⋊C8—32no—
407C3⋊C8—32no—
408C3⋊C8—62no—
409C3⋊C8C3×#6712yes—
410C2×C32—22no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—22no—
414C2×C32—22no—
415C2×C32—21no—
416C2×C32—41no—
417C2×C32—42no—
418C2×C32D3×Q1612yes—
419C2×C32#94011yescharacteristic
420C2×C32#94011yescharacteristic
421C2×C32C2×#21612yes—
422C2×C32C3×#12711yescharacteristic
423C2×C32C2×C3×Q1612yes—
424C2×Q8—31no—
425C2×Q8—31no—
426C2×Q8—91no—
427C2×Q8C3×D3211yescharacteristic
428Q16—62no—
429Q16—182no—
430Q16—182no—
431Q16—182no—
432C2×C8—31no—
433C2×C8—91no—
434C4⋊C4—62no—
435C4⋊C4—182no—
436C42—31no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3D3×#6711yescharacteristic
442C22×C3C3×D3×D411yescharacteristic
443C22×C3C2×C3×#6711yescharacteristic
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—34no—
451C3×C4—34no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—44no—
456C3×C4—41no—
457C3×C4—41no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—41no—
461C3×C4—64no—
462C3×C4—122no—
463C3×C4—122no—
464C3×C4—121no—
465C3×C4—121no—
466C3×C4—121no—
467C3×C4—121no—
468C3×C4—361no—
469C3×C4D3×#6711yescharacteristic
470C3×C4#381111yescharacteristic
471C3×C4C3×D3×D411yescharacteristic
472C3×C4C3×D3×D411yescharacteristic
473C3×C4C3×D3×D411yescharacteristic
474C3×C4C2×C3×#6711yescharacteristic
475Dic3—41no—
476Dic3—41no—
477Dic3—122no—
478Dic3—121no—
479Dic3—121no—
480Dic3C3×#21614yescomplemented
481C32—21no—
482C32—21no—
483C32—21no—
484C32—41no—
485C32#509211yescharacteristic, complemented
486C32#518311yescharacteristic, complemented
487C32C3×#49411yescharacteristic, complemented
488Q8—21no—
489Q8—21no—
490Q8—61no—
491Q8—61no—
492Q8—61no—
493Q8—61no—
494Q8—181no—
495Q8—181no—
496C2×C4—21no—
497C2×C4—32no—
498C2×C4—61no—
499C2×C4—61no—
500C2×C4—181no—
501C2×C4C2×C3×D3211yescharacteristic
502C8—32no—
503C8—92no—
504C2×C3—22no—
505C2×C3—22no—
506C2×C3—22no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—42no—
511C2×C3—41no—
512C2×C3D3×#21612yes—
513C2×C3#6397911yescharacteristic
514C2×C3C3×D3×Q1612yes—
515C2×C3C3×#94011yescharacteristic
516C2×C3C3×#94011yescharacteristic
517C2×C3C2×C3×#21612yes—
518C22C3×D3×#6711yescharacteristic
519C4—34no—
520C4—41no—
521C4—121no—
522C4—121no—
523C4—361no—
524C4C3×D3×#6711yescharacteristic
525C4C3×#381111yescharacteristic
526C3—21no—
527C3—21no—
528C3—21no—
529C3—41no—
530C3#1059162611yescharacteristic, direct product factor
531C3C3×#509211yescharacteristic, complemented
532C3C3×#518311yescharacteristic, complemented
533C2C3×D3×#21612yes—
534C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×Q8D32
hypocenterC32C3×#494
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
10. #10591626C3direct (N × H)
126. C32×Dic3Q16semidirect
324. C3×Dic3#216semidirect
325. C3×Dic3C3×Q16semidirect
333. C33#494semidirect
480. Dic3C3×#216semidirect
485. C32#5092semidirect
486. C32#5183semidirect
487. C32C3×#494semidirect
530. C3#10591626direct (N × H)
531. C3C3×#5092semidirect
532. C3C3×#5183semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37562
C32×#9442
C2×C32×#2162
C3×#638662
C2×C3×#37192
C2×C3×#37182
C3×Dic3×C3⋊C82
C3×#51833
C3×#50923
#105916263