Finite Group Databasebeta

← C3×#10590541 #13157517

Subgroups — C3×#10590541 #13157517

Order 1728.

586 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64097C211yesmaximal, characteristic, complemented
2C3×#63863C211yesmaximal, characteristic
3C32×#810C211yesmaximal, characteristic, complemented
4C32×#809C211yesmaximal, characteristic
5C3×#63566C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic
7C3×#63555C211yesmaximal, characteristic, complemented
8C3×#4794—31nomaximal
9C3×#4765—31nomaximal
10#10590541C311yesmaximal, characteristic, direct product factor
11C2×C3×#634—21no—
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×(C2×D3)⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22#64097C2×C311yescharacteristic, complemented
23C3×#947—31no—
24C3×#936—31no—
25C3×#888—31no—
26C3×#888—31no—
27C3×#888—61no—
28#63863C2×C311yescharacteristic
29C32×#103—31no—
30C32×#102—31no—
31C3×#835—31no—
32C3×#832—31no—
33C3×#821—31no—
34C3×#816—31no—
35C3×#815—31no—
36C3×#815—31no—
37C3×#810—21no—
38C3×#810D311yescharacteristic, complemented
39C3×#810C2×C311yescharacteristic, complemented
40C3×#809—21no—
41C3×#809D311yescharacteristic
42C3×#809C2×C311yescharacteristic
43#63566C2×C311yescharacteristic, complemented
44#63563C2×C311yescharacteristic
45#63555C2×C311yescharacteristic, complemented
46C22×C32×D3—21no—
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×#634—42no—
53C2×C32×Dic3—21no—
54C32×D12—21no—
55C32×D12—21no—
56C32×Dic6—21no—
57C32×Dic6—21no—
58C32×C3⋊C8—21no—
59C32×C3⋊C8—21no—
60C3×#489—91no—
61#4794—31no—
62#4765—31no—
63C2×C32×Q8—31no—
64C2×C32×D4—31no—
65C2×#634—21no—
66C2×C3×D12—21no—
67C2×C3×D12C2×D311yescharacteristic
68C2×C3×D12C22×C311yescharacteristic
69C2×C3×Dic6—21no—
70C2×C3×Dic6—61no—
71C2×C3×Dic6—61no—
72C2×C3×Dic6—62no—
73C2×C3×Dic6C2×D311yescharacteristic
74C2×C3×Dic6C22×C311yescharacteristic
75C2×C32×C8—31no—
76C2×C32×C8—31no—
77C32×C4⋊C4—21no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4—41no—
81C32×C4⋊C4C2×D311yescharacteristic
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C4⋊C4C22×C311yescharacteristic
84C4⋊(C3⋊Dic3)C22×C311yescharacteristic
85#3750—21no—
86C4×C3⋊Dic3C22×C311yescharacteristic
87C3×#217—61no—
88C3×#217—61no—
89C3×(C2×D3)⋊C4—61no—
90C3×(C2×D3)⋊C4—61no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C3×Dic3⋊C4—62no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—61no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105C2×C3×C3⋊C8C22×C311yescharacteristic
106(C2×D3)⋊Dic3—21no—
107(C2×D3)⋊Dic3—21no—
108C22×C33C2×D411yescharacteristic
109C2×C32×D3—41no—
110C2×C32×D3—41no—
111C2×C32×D3—41no—
112C33×C4—41no—
113C33×C4#4111yescharacteristic
114C33×C4#4111yescharacteristic
115C3×C3⋊Dic3—21no—
116C3×C3⋊Dic3—21no—
117C3×C3⋊Dic3—41no—
118C32×Dic3—41no—
119C3×C4⋊Q8—91no—
120C3×#124—91no—
121#947—31no—
122#936—31no—
123#888—31no—
124#888—31no—
125#888—61no—
126C3×#103—31no—
127C3×#103—31no—
128C3×#103—61no—
129C3×#103—91no—
130C3×#102—31no—
131C3×#102—31no—
132C3×#102—61no—
133C3×#102—91no—
134C3×#97—91no—
135#835—31no—
136#832—31no—
137#821—31no—
138#816—31no—
139#815—31no—
140#815—31no—
141#810C3×D311yescharacteristic, complemented
142#809C3×D311yescharacteristic
143C23×C32—61no—
144C22×C3×D3—21no—
145C22×C3×D3—21no—
146C22×C3×D3—41no—
147C32×Q8—61no—
148C32×Q8—61no—
149C32×D4—61no—
150C32×D4—61no—
151C2×C32×C4—21no—
152C2×C32×C4—22no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—41no—
159C2×C32×C4—41no—
160C2×C32×C4—41no—
161C2×C32×C4—42no—
162C2×C32×C4—61no—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C23×C311yescharacteristic
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3C3×D411yescharacteristic
168C2×C3⋊Dic3C3×D411yescharacteristic
169#634—42no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—41no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—62no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C3×D12—22no—
183C3×D12—21no—
184C3×D12—21no—
185C3×D12—21no—
186C3×D12—21no—
187C3×D12—41no—
188C3×Dic6—22no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—21no—
192C3×Dic6—21no—
193C3×Dic6—41no—
194C3×Dic6—124no—
195C3×Dic6—122no—
196C3×Dic6—122no—
197C32×C8—61no—
198C32×C8—61no—
199C3×C3⋊C8—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—41no—
204C3×C3⋊C8—41no—
205#489—91noSylow
206C2×C33#12411yescharacteristic
207C2×C33AGL(1,ℤ/8)11yescharacteristic
208C2×C33#13711yescharacteristic
209C32×D3—81no—
210C2×C3×Q8—31no—
211C2×C3×Q8—31no—
212C2×C3×Q8—61no—
213C2×C3×Q8—181no—
214C2×C3×D4—31no—
215C2×C3×D4—31no—
216C2×C3×D4—61no—
217C2×D12C2×C3×D311yescharacteristic
218C2×Dic6—61no—
219C2×Dic6—61no—
220C2×Dic6—62no—
221C2×Dic6C2×C3×D311yescharacteristic
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—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—41no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—181no—
234C3×C4⋊C4D3211yescharacteristic
235C3×C4⋊C4C2×C3×D311yescharacteristic
236C3×C4⋊C4C2×C3×D311yescharacteristic
237C3×C22⋊C4—181no—
238C3×C22⋊C4—181no—
239C3×C42—91no—
240#217—61no—
241#217—61no—
242(C2×D3)⋊C4—61no—
243(C2×D3)⋊C4—61no—
244C4⋊Dic3—31no—
245C4⋊Dic3—31no—
246C4⋊Dic3—61no—
247Dic3⋊C4—61no—
248Dic3⋊C4—61no—
249Dic3⋊C4—62no—
250C4×Dic3—31no—
251C4×Dic3—31no—
252C4×Dic3—61no—
253C2×C3⋊C8C2×C3×D311yescharacteristic
254C2×C3⋊C8C2×C3×D311yescharacteristic
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—41no—
259C22×C32—121no—
260C22×C32—121no—
261C22×C32—121no—
262C22×C32D3×D411yescharacteristic
263C22×C32D3×D411yescharacteristic
264C22×C32C2×C3×D411yescharacteristic
265C2×C3×D3—42no—
266C2×C3×D3—41no—
267C2×C3×D3—41no—
268C2×C3×D3—41no—
269C2×C3×D3—41no—
270C2×C3×D3—41no—
271C2×C3×D3—41no—
272C2×C3×D3—81no—
273C2×C3×D3—81no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—41no—
281C32×C4—41no—
282C32×C4—41no—
283C32×C4—42no—
284C32×C4—41no—
285C32×C4—41no—
286C32×C4—81no—
287C32×C4—81no—
288C32×C4—82no—
289C32×C4—121no—
290C32×C4#23511yescharacteristic
291C32×C4#23511yescharacteristic
292C32×C4#23711yescharacteristic
293C32×C4#23711yescharacteristic
294C32×C4C3×#4111yescharacteristic
295C32×C4C3×#4111yescharacteristic
296C3⋊Dic3—21no—
297C3⋊Dic3—21no—
298C3⋊Dic3—41no—
299C3×Dic3—42no—
300C3×Dic3—41no—
301C3×Dic3—41no—
302C3×Dic3—62no—
303C3×Dic3—62no—
304C3×Dic3—61no—
305C3×Dic3—61no—
306C3×Dic3—61no—
307C3×Dic3—61no—
308C3×Dic3—122no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311C4⋊Q8—91no—
312#124—91no—
313#103—31no—
314#103—91no—
315#102—31no—
316#102—91no—
317#97—91no—
318C33#48911yescharacteristic, complemented, Sylow
319C23×C3—61no—
320C23×C3—61no—
321C23×C3—121no—
322C22×D3—21no—
323C3×Q8—62no—
324C3×Q8—61no—
325C3×Q8—61no—
326C3×Q8—61no—
327C3×Q8—61no—
328C3×Q8—121no—
329C3×Q8—362no—
330C3×D4—62no—
331C3×D4—61no—
332C3×D4—61no—
333C3×D4—61no—
334C3×D4—61no—
335C3×D4—121no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—41no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—91no—
350C2×C3×C4—91no—
351C2×C3×C4—121no—
352C2×C3×C4—181no—
353C2×C3×C4C2×D3211yescharacteristic
354C2×C3×C4C22×C3×D311yescharacteristic
355C2×C3×C4C22×C3×D311yescharacteristic
356C2×Dic3—21no—
357C2×Dic3—31no—
358C2×Dic3—31no—
359C2×Dic3—31no—
360C2×Dic3—31no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—62no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366D12—21no—
367D12—21no—
368Dic6—21no—
369Dic6—21no—
370Dic6—124no—
371Dic6—122no—
372Dic6—122no—
373C3×C8—61no—
374C3×C8—61no—
375C3×C8—61no—
376C3×C8—61no—
377C3×C8—121no—
378C3×C8—121no—
379C3⋊C8—21no—
380C3⋊C8—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32—241no—
394C2×C32#88511yescharacteristic
395C2×C32#88511yescharacteristic
396C2×C32#91111yescharacteristic
397C2×C32#91411yescharacteristic
398C2×C32#91511yescharacteristic
399C2×C32#91811yescharacteristic
400C2×C32C3×#12411yescharacteristic
401C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
402C2×C32C3×#13711yescharacteristic
403C3×D3—82no—
404C3×D3—81no—
405C3×D3—81no—
406C2×Q8—31no—
407C2×Q8—181no—
408C2×D4—31no—
409C2×C8—31no—
410C2×C8—31no—
411C4⋊C4—91no—
412C4⋊C4—181no—
413C4⋊C4C3×D3211yescharacteristic
414C22⋊C4—181no—
415C22⋊C4—181no—
416C42—91no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—41no—
421C22×C3—122no—
422C22×C3—121no—
423C22×C3—121no—
424C22×C3—121no—
425C22×C3—121no—
426C22×C3—121no—
427C22×C3—121no—
428C22×C3—241no—
429C22×C3—241no—
430C22×C3#381111yescharacteristic
431C22×C3C3×D3×D411yescharacteristic
432C22×C3C3×D3×D411yescharacteristic
433C2×D3—41no—
434C2×D3—41no—
435C2×D3—41no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—42no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—81no—
449C3×C4—81no—
450C3×C4—82no—
451C3×C4—122no—
452C3×C4—121no—
453C3×C4—121no—
454C3×C4—181no—
455C3×C4—181no—
456C3×C4—361no—
457C3×C4#379511yescharacteristic
458C3×C4#379511yescharacteristic
459C3×C4C3×#23511yescharacteristic
460C3×C4C3×#23511yescharacteristic
461C3×C4C3×#23711yescharacteristic
462C3×C4C3×#23711yescharacteristic
463Dic3—41no—
464Dic3—62no—
465Dic3—62no—
466Dic3—61no—
467Dic3—61no—
468Dic3—61no—
469Dic3—61no—
470Dic3—122no—
471Dic3—121no—
472Dic3—121no—
473C32—21no—
474C32—21no—
475C32—21no—
476C32—41no—
477C32#476511yescharacteristic, complemented
478C32#479411yescharacteristic, complemented
479C32C3×#48911yescharacteristic, complemented
480C23—61no—
481Q8—61no—
482Q8—61no—
483Q8—362no—
484D4—61no—
485D4—61no—
486C2×C4—21no—
487C2×C4—61no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—181no—
491C2×C4C2×C3×D3211yescharacteristic
492C8—61no—
493C8—61no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—41no—
504C2×C3—41no—
505C2×C3—41no—
506C2×C3—242no—
507C2×C3—241no—
508C2×C3—241no—
509C2×C3#6391311yescharacteristic
510C2×C3#6393711yescharacteristic
511C2×C3#6395111yescharacteristic
512C2×C3C3×#88511yescharacteristic
513C2×C3C3×#88511yescharacteristic
514C2×C3C3×#91111yescharacteristic
515C2×C3C3×#91411yescharacteristic
516C2×C3C3×#91511yescharacteristic
517C2×C3C3×#91811yescharacteristic
518D3—81no—
519C22—121no—
520C22—121no—
521C22—121no—
522C22C3×#381111yescharacteristic
523C4—41no—
524C4—121no—
525C4—181no—
526C4—181no—
527C4—361no—
528C4C3×#379511yescharacteristic
529C4C3×#379511yescharacteristic
530C3—21no—
531C3—21no—
532C3—21no—
533C3—41no—
534C3#1059054111yescharacteristic, direct product factor
535C3C3×#476511yescharacteristic, complemented
536C3C3×#479411yescharacteristic, complemented
537C2—241no—
538C2C3×#6391311yescharacteristic
539C2C3×#6393711yescharacteristic
540C2C3×#6395111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#489
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64097C2semidirect
3. C32×#810C2semidirect
5. C3×#63566C2semidirect
7. C3×#63555C2semidirect
10. #10590541C3direct (N × H)
22. #64097C2×C3semidirect
38. C3×#810D3semidirect
39. C3×#810C2×C3semidirect
43. #63566C2×C3semidirect
45. #63555C2×C3semidirect
141. #810C3×D3semidirect
318. C33#489semidirect
477. C32#4765semidirect
478. C32#4794semidirect
479. C32C3×#489semidirect
534. C3#10590541direct (N × H)
535. C3C3×#4765semidirect
536. C3C3×#4794semidirect

Maximal subgroups

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

subgroupindex
C3×#640972
C3×#638632
C32×#8102
C32×#8092
C3×#635662
C3×#635632
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