Finite Group Databasebeta

← C3×#10591561 #13158564

Subgroups — C3×#10591561 #13158564

Order 1728.

580 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64155C211yesmaximal, characteristic
2C3×#63849C211yesmaximal, characteristic
3C32×#835C211yesmaximal, characteristic, complemented
4C32×#835C211yesmaximal, characteristic, complemented
5C3×#63575C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63557C211yesmaximal, characteristic
8C3×#4815—31nomaximal
9C3×#4814—31nomaximal
10#10591561C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C3×#3752—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×#3750—21no—
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3722—21no—
21C3×Dic32—21no—
22#64155C2×C311yescharacteristic
23C3×#946—31no—
24C3×#946—31no—
25C3×#946—61no—
26C3×#892—31no—
27C3×#892—31no—
28#63849C2×C311yescharacteristic
29C32×#103—31no—
30C32×#103—31no—
31C3×#835—21no—
32C3×#835—21no—
33C3×#835D311yescharacteristic, complemented
34C3×#835D311yescharacteristic, complemented
35C3×#835C2×C311yescharacteristic, complemented
36C3×#835C2×C311yescharacteristic, complemented
37C3×(C2×D3)⋊C8—31no—
38C3×(C2×D3)⋊C8—31no—
39C3×#818—31no—
40C3×#817—31no—
41C3×#808—31no—
42C3×#807—31no—
43#63575C2×C311yescharacteristic, complemented
44#63573C2×C311yescharacteristic, complemented
45#63557C2×C311yescharacteristic
46C22×C3×C3⋊D3D411yescharacteristic
47C33×Q8—21no—
48C33×Q8—21no—
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4×C3⋊D3—21no—
54C3×C4×C3⋊D3—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C32×C3⋊C8—21no—
58C32×C3⋊C8—21no—
59C3×#485—91no—
60#4815—31no—
61#4814—31no—
62C2×C32×Q8—21no—
63C2×C32×Q8—21no—
64C2×C32×Q8—21no—
65C2×C32×Q8—41no—
66C2×C32×Q8C2×D311yescharacteristic
67C2×C32×Q8C2×D311yescharacteristic
68C2×C32×Q8C22×C311yescharacteristic
69C2×C4×C3⋊D3C22×C311yescharacteristic
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—61no—
73C2×C32×C8—31no—
74C2×C32×C8—31no—
75C32×C4⋊C4—31no—
76C32×C4⋊C4—31no—
77#3752—21no—
78C4⋊(C3⋊Dic3)C22×C311yescharacteristic
79#3750—21no—
80C3×(C2×D3)⋊C4—61no—
81C3×(C2×D3)⋊C4—61no—
82C3×(C2×D3)⋊C4—61no—
83C3×(C2×D3)⋊C4—61no—
84C3×(C2×D3)⋊C4—62no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C22×C311yescharacteristic
93C3×C4⋊Dic3C22×C311yescharacteristic
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C3×Dic3⋊C4—62no—
97C3×C4×Dic3—61no—
98C3×C4×Dic3—61no—
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C22×C311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105#3722—21no—
106Dic32—21no—
107C22×C33C2×D411yescharacteristic
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3—21no—
110C2×C3×C3⋊D3—21no—
111C33×C4—41no—
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—41no—
116C32×Dic3—41no—
117C32×Dic3—41no—
118C3×C22⋊Q8—91no—
119C3×#117—91no—
120#946—31no—
121#946—31no—
122#946—61no—
123#892—31no—
124#892—31no—
125C3×#107—91no—
126C3×#106—91no—
127C3×#103—31no—
128C3×#103—31no—
129C3×#103—31no—
130C3×#103—31no—
131C3×#103—61no—
132C3×#103—61no—
133C3×C22⋊C8—91no—
134#835C3×D311yescharacteristic, complemented
135#835C3×D311yescharacteristic, complemented
136(C2×D3)⋊C8—31no—
137(C2×D3)⋊C8—31no—
138#818—31no—
139#817—31no—
140#808—31no—
141#807—31no—
142C22×C3⋊D3C3×D411yescharacteristic
143C22×C3×D3—31no—
144C22×C3×D3—31no—
145C22×C3×D3—61no—
146C32×Q8—21no—
147C32×Q8—21no—
148C32×Q8—22no—
149C32×Q8—22no—
150C32×Q8—21no—
151C32×Q8—21no—
152C32×Q8—21no—
153C32×Q8—21no—
154C32×Q8—41no—
155C32×Q8—41no—
156C32×Q8—41no—
157C32×Q8—41no—
158C32×Q8—42no—
159C32×Q8—42no—
160C2×C32×C4—21no—
161C2×C32×C4—22no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—41no—
168C2×C32×C4—41no—
169C2×C32×C4—41no—
170C2×C32×C4—42no—
171C2×C32×C4—61no—
172C2×C32×C4—61no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C23×C311yescharacteristic
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3C3×D411yescharacteristic
178C4×C3⋊D3—21no—
179C4×C3⋊D3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3—41no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C3×C4×D3—61no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×C4×D3—121no—
197C3×C4×D3—121no—
198C32×C8—61no—
199C32×C8—61no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—41no—
205C3×C3⋊C8—41no—
206#485—91noSylow
207C2×C33#12311yescharacteristic
208C2×C33#13511yescharacteristic
209C2×C33#13711yescharacteristic
210C3×C3⋊D3—41no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—41no—
215C2×C3×Q8D3211yescharacteristic
216C2×C3×Q8C2×C3×D311yescharacteristic
217C2×C3×Q8C2×C3×D311yescharacteristic
218C22×C3×C4—91no—
219C2×C4×D3—31no—
220C2×C4×D3—31no—
221C2×C4×D3—61no—
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—61no—
227C2×C3×C8—61no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4—181no—
236C3×C22⋊C4—181no—
237C3×C22⋊C4—181no—
238C3×C42—181no—
239(C2×D3)⋊C4—61no—
240(C2×D3)⋊C4—61no—
241(C2×D3)⋊C4—61no—
242(C2×D3)⋊C4—61no—
243(C2×D3)⋊C4—62no—
244C4⋊Dic3—31no—
245C4⋊Dic3—31no—
246C4⋊Dic3—61no—
247C4⋊Dic3C2×C3×D311yescharacteristic
248C4⋊Dic3C2×C3×D311yescharacteristic
249Dic3⋊C4—61no—
250Dic3⋊C4—61no—
251Dic3⋊C4—62no—
252C4×Dic3—61no—
253C4×Dic3—61no—
254C2×C3⋊C8C2×C3×D311yescharacteristic
255C2×C3⋊C8C2×C3×D311yescharacteristic
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—41no—
260C22×C32D3×D411yescharacteristic
261C22×C32D3×D411yescharacteristic
262C22×C32C2×C3×D411yescharacteristic
263C2×C3⋊D3—21no—
264C2×C3⋊D3—21no—
265C2×C3⋊D3—21no—
266C2×C3×D3—62no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—41no—
282C32×C4—41no—
283C32×C4—42no—
284C32×C4—42no—
285C32×C4—42no—
286C32×C4—44no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4—121no—
291C32×C4—121no—
292C32×C4#23511yescharacteristic
293C32×C4#23511yescharacteristic
294C32×C4#23711yescharacteristic
295C32×C4#23711yescharacteristic
296C32×C4C3×#4111yescharacteristic
297C32×C4C3×#4111yescharacteristic
298C3⋊Dic3—21no—
299C3⋊Dic3—41no—
300C3×Dic3—41no—
301C3×Dic3—41no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—61no—
305C3×Dic3—61no—
306C3×Dic3—81no—
307C3×Dic3—81no—
308C3×Dic3—121no—
309C3×Dic3—122no—
310C3×Dic3—121no—
311C3×Dic3—121no—
312C22⋊Q8—91no—
313#117—91no—
314#107—91no—
315#106—91no—
316#103—31no—
317#103—31no—
318C22⋊C8—91no—
319C33#48511yescharacteristic, complemented, Sylow
320C23×C3—91no—
321C22×D3—31no—
322C22×D3—31no—
323C22×D3—61no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—21no—
327C3×Q8—21no—
328C3×Q8—21no—
329C3×Q8—21no—
330C3×Q8—21no—
331C3×Q8—21no—
332C3×Q8—41no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×Q8—42no—
337C3×Q8—42no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—91no—
354C2×C3×C4—121no—
355C2×C3×C4—121no—
356C2×C3×C4—181no—
357C2×C3×C4—181no—
358C2×C3×C4—181no—
359C2×C3×C4C2×D3211yescharacteristic
360C2×C3×C4C22×C3×D311yescharacteristic
361C2×C3×C4C22×C3×D311yescharacteristic
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—31no—
365C2×Dic3—31no—
366C2×Dic3—61no—
367C2×Dic3—62no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C4×D3—61no—
371C4×D3—61no—
372C4×D3—61no—
373C4×D3—61no—
374C4×D3—121no—
375C4×D3—121no—
376C3×C8—61no—
377C3×C8—61no—
378C3×C8—61no—
379C3×C8—61no—
380C3×C8—121no—
381C3×C8—121no—
382C3⋊C8—21no—
383C3⋊C8—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—41no—
394C2×C32—41no—
395C2×C32—41no—
396C2×C32#88311yescharacteristic
397C2×C32#88311yescharacteristic
398C2×C32#91511yescharacteristic
399C2×C32#91611yescharacteristic
400C2×C32#91811yescharacteristic
401C2×C32#91911yescharacteristic
402C2×C32C3×#12311yescharacteristic
403C2×C32C3×#13511yescharacteristic
404C2×C32C3×#13711yescharacteristic
405C3⋊D3—41no—
406C3×D3—122no—
407C3×D3—121no—
408C3×D3—121no—
409C2×Q8C3×D3211yescharacteristic
410C22×C4—91no—
411C2×C8—31no—
412C2×C8—31no—
413C4⋊C4—31no—
414C4⋊C4—31no—
415C4⋊C4—91no—
416C4⋊C4—181no—
417C22⋊C4—181no—
418C22⋊C4—181no—
419C42—181no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—41no—
424C22×C3—181no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3#381111yescharacteristic
428C22×C3C3×D3×D411yescharacteristic
429C22×C3C3×D3×D411yescharacteristic
430C2×D3—62no—
431C2×D3—61no—
432C2×D3—61no—
433C2×D3—61no—
434C2×D3—61no—
435C2×D3—61no—
436C2×D3—61no—
437C2×D3—121no—
438C2×D3—121no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—44no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—121no—
455C3×C4—121no—
456C3×C4—121no—
457C3×C4—121no—
458C3×C4—181no—
459C3×C4—241no—
460C3×C4—241no—
461C3×C4—361no—
462C3×C4#380411yescharacteristic
463C3×C4#380411yescharacteristic
464C3×C4C3×#23511yescharacteristic
465C3×C4C3×#23511yescharacteristic
466C3×C4C3×#23711yescharacteristic
467C3×C4C3×#23711yescharacteristic
468Dic3—41no—
469Dic3—41no—
470Dic3—61no—
471Dic3—61no—
472Dic3—121no—
473Dic3—122no—
474Dic3—121no—
475Dic3—121no—
476C32—21no—
477C32—21no—
478C32—21no—
479C32—41no—
480C32#481411yescharacteristic, complemented
481C32#481511yescharacteristic, complemented
482C32C3×#48511yescharacteristic, complemented
483C23—91no—
484Q8—21no—
485Q8—21no—
486C2×C4—21no—
487C2×C4—61no—
488C2×C4—61no—
489C2×C4—91no—
490C2×C4—181no—
491C2×C4—181no—
492C2×C4—181no—
493C2×C4C2×C3×D3211yescharacteristic
494C8—61no—
495C8—61no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—41no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—361no—
509C2×C3#6395011yescharacteristic
510C2×C3#6395111yescharacteristic
511C2×C3#6397511yescharacteristic
512C2×C3C3×#88311yescharacteristic
513C2×C3C3×#88311yescharacteristic
514C2×C3C3×#91511yescharacteristic
515C2×C3C3×#91611yescharacteristic
516C2×C3C3×#91811yescharacteristic
517C2×C3C3×#91911yescharacteristic
518D3—122no—
519D3—121no—
520D3—121no—
521C22—181no—
522C22—181no—
523C22—181no—
524C22C3×#381111yescharacteristic
525C4—41no—
526C4—121no—
527C4—121no—
528C4—181no—
529C4—361no—
530C4C3×#380411yescharacteristic
531C4C3×#380411yescharacteristic
532C3—21no—
533C3—21no—
534C3—21no—
535C3—41no—
536C3#1059156111yescharacteristic, direct product factor
537C3C3×#481411yescharacteristic, complemented
538C3C3×#481511yescharacteristic, complemented
539C2—361no—
540C2C3×#6395011yescharacteristic
541C2C3×#6395111yescharacteristic
542C2C3×#6397511yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×C3×Q8D32
hypocenterC32C3×#485
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
3. C32×#835C2semidirect
4. C32×#835C2semidirect
5. C3×#63575C2semidirect
6. C3×#63573C2semidirect
10. #10591561C3direct (N × H)
33. C3×#835D3semidirect
34. C3×#835D3semidirect
35. C3×#835C2×C3semidirect
36. C3×#835C2×C3semidirect
43. #63575C2×C3semidirect
44. #63573C2×C3semidirect
134. #835C3×D3semidirect
135. #835C3×D3semidirect
319. C33#485semidirect
480. C32#4814semidirect
481. C32#4815semidirect
482. C32C3×#485semidirect
536. C3#10591561direct (N × H)
537. C3C3×#4814semidirect
538. C3C3×#4815semidirect

Maximal subgroups

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

subgroupindex
C3×#641552
C3×#638492
C32×#8352
C3×#635752
C3×#635732
C3×#635572
C3×#48153
C3×#48143
#105915613