Finite Group Databasebeta

← C3×#10590861 #13157786

Subgroups — C3×#10590861 #13157786

Order 1728.

618 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3746C211yesmaximal, characteristic, complemented
2C3×#63860C211yesmaximal, characteristic
3C3×#63844C211yesmaximal, characteristic, complemented
4C2×C3×#3716C211yesmaximal, characteristic
5C32×#821C211yesmaximal, characteristic
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C3×#63567C211yesmaximal, characteristic, complemented
8C3×#4845—31nomaximal
9C3×#4767—31nomaximal
10#10590861C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C33×C8C2211yescharacteristic
15C3×#3746—22no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C3×#3724—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×Dic32—21no—
23C3×#3716—22no—
24C2×#3746C2×C311yescharacteristic, complemented
25C3×#936—31no—
26C2×C3×#214—31no—
27C2×C3×Dic12—31no—
28C2×C3×Dic12—31no—
29C2×C3×Dic12—61no—
30C2×C3×#204—31no—
31C3×#887—31no—
32C3×#875—31no—
33C3×C4×Dic6—31no—
34#63860C2×C311yescharacteristic
35#63844C2×C311yescharacteristic, complemented
36C2×#3716C2×C311yescharacteristic
37C32×C4⋊C8—31no—
38C32×#102—31no—
39C3×#821—21no—
40C3×#821D311yescharacteristic
41C3×#821C2×C311yescharacteristic
42C3×#816—31no—
43C3×Dic3⋊C8—21no—
44C3×Dic3⋊C8D311yescharacteristic, complemented
45C3×Dic3⋊C8C2×C311yescharacteristic, complemented
46C3×#810—31no—
47#63567C2×C311yescharacteristic, complemented
48C22×C32×D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634—21no—
54C3×#634—21no—
55C3×#634D412yes—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×D12—21no—
60C32×D12—21no—
61C33×C8—21no—
62C32×C3⋊C8—21no—
63C3×#464—91no—
64#4845—31no—
65#4767—31no—
66C2×C32×D4—31no—
67C2×#634C22×C311yescharacteristic
68C2×#634C22×C311yescharacteristic
69C2×C3×D12—21no—
70C2×C3×D12C2×D311yescharacteristic
71C2×C3×D12C22×C311yescharacteristic
72C2×C3×Dic6—31no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—31no—
75C2×C3×Dic6—31no—
76C2×C3×Dic6—61no—
77C2×C3×Dic6—61no—
78C2×C32×C8—21no—
79C2×C32×C8—21no—
80C2×C32×C8—21no—
81C2×C32×C8—31no—
82C2×C32×C8—41no—
83C2×C32×C8C2×D311yescharacteristic
84C2×C32×C8C2×D311yescharacteristic
85C2×C32×C8C22×C311yescharacteristic
86C32×C4⋊C4—31no—
87C32×C42—31no—
88#3746—22no—
89C3×#217—61no—
90C3×#217—61no—
91C3×#214—62no—
92C3×(C2×D3)⋊C4—61no—
93C3×(C2×D3)⋊C4—61no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C22×C311yescharacteristic
97C3×Dic3⋊C4—61no—
98C3×Dic3⋊C4—61no—
99C3×C4×Dic3—21no—
100C3×C4×Dic3—61no—
101C3×C4×Dic3—61no—
102C3×C4×Dic3C2×D311yescharacteristic
103C3×C4×Dic3C22×C311yescharacteristic
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C22×C311yescharacteristic
107C3×Dic12—62no—
108C3×Dic12—62no—
109C3×Dic12—64no—
110C3×#204—62no—
111#3724—21no—
112(C2×D3)⋊Dic3—21no—
113(C2×D3)⋊Dic3—21no—
114Dic32—21no—
115#3716—22no—
116C22×C33C2×D411yescharacteristic
117C2×C32×D3—41no—
118C2×C32×D3—41no—
119C2×C32×D3—41no—
120C33×C4C2×D411yescharacteristic
121C33×C4#4111yescharacteristic
122C3×C3⋊Dic3—22no—
123C3×C3⋊Dic3—41no—
124C32×Dic3—21no—
125C32×Dic3—21no—
126C32×Dic3—41no—
127C2×C3×Q16—91no—
128C2×C3×SD16—91no—
129C3×#124—91no—
130C3×C4×Q8—91no—
131#936—31no—
132C2×#214—31no—
133C2×Dic12—31no—
134C2×Dic12—31no—
135C2×Dic12—61no—
136C2×#204—31no—
137#887—31no—
138#875—31no—
139C4×Dic6—31no—
140C3×C4⋊C8—31no—
141C3×C4⋊C8—31no—
142C3×C4⋊C8—61no—
143C3×#103—91no—
144C3×#102—31no—
145C3×#102—31no—
146C3×#102—61no—
147#821C3×D311yescharacteristic
148#816—31no—
149Dic3⋊C8C3×D311yescharacteristic, complemented
150#810—31no—
151C23×C32—61no—
152C22×C3×D3—21no—
153C22×C3×D3—21no—
154C22×C3×D3—41no—
155C32×D4—61no—
156C32×D4—61no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—31no—
161C2×C32×C4—31no—
162C2×C32×C4—41no—
163C2×C32×C4—61no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C23×C311yescharacteristic
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—21no—
169#634—21no—
170#634—21no—
171#634—21no—
172#634C3×D412yes—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—41no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—121no—
183C2×C3×Dic3—121no—
184C2×C3×Dic3D1211yescharacteristic
185C2×C3×Dic3D1211yescharacteristic
186C2×C3×Dic3C3×D411yescharacteristic
187C2×C3×Dic3C3×D411yescharacteristic
188C3×D12—22no—
189C3×D12—21no—
190C3×D12—21no—
191C3×D12—21no—
192C3×D12—21no—
193C3×D12—41no—
194C3×Dic6—32no—
195C3×Dic6—32no—
196C3×Dic6—62no—
197C3×Dic6—61no—
198C3×Dic6—61no—
199C3×Dic6—62no—
200C3×Dic6—61no—
201C3×Dic6—61no—
202C3×Dic6—61no—
203C3×Dic6—61no—
204C3×Dic6—121no—
205C3×Dic6—121no—
206C32×C8—21no—
207C32×C8—22no—
208C32×C8—21no—
209C32×C8—21no—
210C32×C8—41no—
211C32×C8—41no—
212C32×C8—42no—
213C32×C8—61no—
214C3×C3⋊C8—21no—
215C3×C3⋊C8—21no—
216C3×C3⋊C8—41no—
217#464—91noSylow
218C2×C33#12111yescharacteristic
219C2×C33#13511yescharacteristic
220C2×C33#13711yescharacteristic
221C32×D3—81no—
222C2×C3×Q8—91no—
223C2×C3×Q8—91no—
224C2×C3×D4—31no—
225C2×C3×D4—31no—
226C2×C3×D4—61no—
227C2×D12C2×C3×D311yescharacteristic
228C2×Dic6—31no—
229C2×Dic6—31no—
230C2×Dic6—31no—
231C2×Dic6—31no—
232C2×Dic6—61no—
233C2×Dic6—61no—
234C3×Q16—182no—
235C3×SD16—182no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—41no—
242C2×C3×C8—61no—
243C2×C3×C8D3211yescharacteristic
244C2×C3×C8C2×C3×D311yescharacteristic
245C2×C3×C8C2×C3×D311yescharacteristic
246C3×C4⋊C4—31no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—61no—
249C3×C4⋊C4—181no—
250C3×C22⋊C4—181no—
251C3×C22⋊C4—181no—
252C3×C42—31no—
253C3×C42—31no—
254C3×C42—61no—
255C3×C42—181no—
256#217—61no—
257#217—61no—
258#214—62no—
259(C2×D3)⋊C4—61no—
260(C2×D3)⋊C4—61no—
261C4⋊Dic3C2×C3×D311yescharacteristic
262Dic3⋊C4—61no—
263Dic3⋊C4—61no—
264C4×Dic3—61no—
265C4×Dic3—61no—
266C4×Dic3C2×C3×D311yescharacteristic
267C2×C3⋊C8C2×C3×D311yescharacteristic
268Dic12—62no—
269Dic12—62no—
270Dic12—64no—
271#204—62no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—21no—
275C22×C32—41no—
276C22×C32—121no—
277C22×C32—121no—
278C22×C32—121no—
279C22×C32C2×D1211yescharacteristic
280C22×C32D3×D411yescharacteristic
281C22×C32C2×C3×D411yescharacteristic
282C2×C3×D3—42no—
283C2×C3×D3—41no—
284C2×C3×D3—41no—
285C2×C3×D3—41no—
286C2×C3×D3—41no—
287C2×C3×D3—41no—
288C2×C3×D3—41no—
289C2×C3×D3—81no—
290C2×C3×D3—81no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—41no—
298C32×C4—41no—
299C32×C4—61no—
300C32×C4—61no—
301C32×C4—121no—
302C32×C4#23511yescharacteristic
303C32×C4D3×D411yescharacteristic
304C32×C4D3×D411yescharacteristic
305C32×C4#23911yescharacteristic
306C32×C4C2×C3×D411yescharacteristic
307C32×C4C3×#4111yescharacteristic
308C3⋊Dic3—22no—
309C3⋊Dic3—41no—
310C3×Dic3—22no—
311C3×Dic3—21no—
312C3×Dic3—21no—
313C3×Dic3—21no—
314C3×Dic3—21no—
315C3×Dic3—41no—
316C3×Dic3—41no—
317C3×Dic3—41no—
318C3×Dic3—62no—
319C3×Dic3—62no—
320C3×Dic3—81no—
321C3×Dic3—122no—
322C3×Dic3—122no—
323C3×Dic3—121no—
324C3×Dic3—121no—
325C2×Q16—91no—
326C2×SD16—91no—
327#124—91no—
328C4×Q8—91no—
329C4⋊C8—31no—
330#103—91no—
331#102—31no—
332C33#46411yescharacteristic, complemented, Sylow
333C23×C3—61no—
334C23×C3—61no—
335C23×C3—121no—
336C22×D3—21no—
337C3×Q8—92no—
338C3×Q8—181no—
339C3×Q8—181no—
340C3×Q8—181no—
341C3×D4—62no—
342C3×D4—61no—
343C3×D4—61no—
344C3×D4—61no—
345C3×D4—61no—
346C3×D4—121no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—41no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—121no—
360C2×C3×C4—181no—
361C2×C3×C4—181no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C22×C3×D311yescharacteristic
364C2×C3×C4C22×C3×D311yescharacteristic
365C2×Dic3—21no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—121no—
371C2×Dic3—121no—
372C2×Dic3C3×D1211yescharacteristic
373C2×Dic3C3×D1211yescharacteristic
374D12—21no—
375D12—21no—
376Dic6—32no—
377Dic6—32no—
378Dic6—62no—
379Dic6—61no—
380Dic6—61no—
381Dic6—62no—
382Dic6—61no—
383Dic6—61no—
384Dic6—61no—
385Dic6—61no—
386Dic6—121no—
387Dic6—121no—
388C3×C8—22no—
389C3×C8—21no—
390C3×C8—21no—
391C3×C8—21no—
392C3×C8—41no—
393C3×C8—41no—
394C3×C8—42no—
395C3×C8—61no—
396C3×C8—61no—
397C3×C8—121no—
398C3⋊C8—21no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—41no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—241no—
412C2×C32#88411yescharacteristic
413C2×C32#89511yescharacteristic
414C2×C32#90411yescharacteristic
415C2×C32#90911yescharacteristic
416C2×C32#91211yescharacteristic
417C2×C32#91511yescharacteristic
418C2×C32C3×#12111yescharacteristic
419C2×C32C3×#13511yescharacteristic
420C2×C32C3×#13711yescharacteristic
421C3×D3—82no—
422C3×D3—81no—
423C3×D3—81no—
424C2×Q8—91no—
425C2×Q8—91no—
426C2×D4—31no—
427Q16—182no—
428SD16—182no—
429C2×C8—31no—
430C2×C8C3×D3211yescharacteristic
431C4⋊C4—31no—
432C4⋊C4—181no—
433C22⋊C4—181no—
434C22⋊C4—181no—
435C42—31no—
436C42—181no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3—122no—
442C22×C3—121no—
443C22×C3—121no—
444C22×C3—121no—
445C22×C3—121no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—241no—
449C22×C3—241no—
450C22×C3D3×D1211yescharacteristic
451C22×C3C2×C3×D1211yescharacteristic
452C22×C3C3×D3×D411yescharacteristic
453C2×D3—41no—
454C2×D3—41no—
455C2×D3—41no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—41no—
463C3×C4—41no—
464C3×C4—62no—
465C3×C4—61no—
466C3×C4—61no—
467C3×C4—61no—
468C3×C4—61no—
469C3×C4—121no—
470C3×C4—121no—
471C3×C4—121no—
472C3×C4—182no—
473C3×C4—241no—
474C3×C4—361no—
475C3×C4#379911yescharacteristic
476C3×C4#381111yescharacteristic
477C3×C4C3×#23511yescharacteristic
478C3×C4C3×D3×D411yescharacteristic
479C3×C4C3×D3×D411yescharacteristic
480C3×C4C3×#23911yescharacteristic
481Dic3—21no—
482Dic3—21no—
483Dic3—41no—
484Dic3—62no—
485Dic3—62no—
486Dic3—122no—
487Dic3—122no—
488Dic3—121no—
489Dic3—121no—
490C32—21no—
491C32—21no—
492C32—21no—
493C32—41no—
494C32#476711yescharacteristic, complemented
495C32#484511yescharacteristic, complemented
496C32C3×#46411yescharacteristic, complemented
497C23—61no—
498Q8—92no—
499Q8—181no—
500Q8—181no—
501Q8—181no—
502D4—61no—
503D4—61no—
504C2×C4—31no—
505C2×C4—31no—
506C2×C4—61no—
507C2×C4—181no—
508C2×C4—181no—
509C2×C4C2×C3×D3211yescharacteristic
510C8—21no—
511C8—61no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—41no—
522C2×C3—41no—
523C2×C3—41no—
524C2×C3—242no—
525C2×C3—241no—
526C2×C3—241no—
527C2×C3#6380011yescharacteristic
528C2×C3#6380711yescharacteristic
529C2×C3#6391011yescharacteristic
530C2×C3C3×#88411yescharacteristic
531C2×C3C3×#89511yescharacteristic
532C2×C3C3×#90411yescharacteristic
533C2×C3C3×#90911yescharacteristic
534C2×C3C3×#91211yescharacteristic
535C2×C3C3×#91511yescharacteristic
536D3—81no—
537C22—121no—
538C22—121no—
539C22—121no—
540C22C3×D3×D1211yescharacteristic
541C4—61no—
542C4—61no—
543C4—121no—
544C4—182no—
545C4—361no—
546C4C3×#379911yescharacteristic
547C4C3×#381111yescharacteristic
548C3—21no—
549C3—21no—
550C3—21no—
551C3—41no—
552C3#1059086111yescharacteristic, direct product factor
553C3C3×#476711yescharacteristic, complemented
554C3C3×#484511yescharacteristic, complemented
555C2—241no—
556C2C3×#6380011yescharacteristic
557C2C3×#6380711yescharacteristic
558C2C3×#6391011yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×C3×C8D32
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3746C2semidirect
3. C3×#63844C2semidirect
6. C32×Dic3⋊C8C2semidirect
7. C3×#63567C2semidirect
10. #10590861C3direct (N × H)
24. C2×#3746C2×C3semidirect
35. #63844C2×C3semidirect
44. C3×Dic3⋊C8D3semidirect
45. C3×Dic3⋊C8C2×C3semidirect
47. #63567C2×C3semidirect
149. Dic3⋊C8C3×D3semidirect
332. C33#464semidirect
494. C32#4767semidirect
495. C32#4845semidirect
496. C32C3×#464semidirect
552. C3#10590861direct (N × H)
553. C3C3×#4767semidirect
554. C3C3×#4845semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37462
C3×#638602
C3×#638442
C2×C3×#37162
C32×#8212
C32×Dic3⋊C82
C3×#635672
C3×#48453
C3×#47673
#105908613