Finite Group Databasebeta

← C3×#10591488 #13158501

Subgroups — C3×#10591488 #13158501

Order 1728.

630 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#936C211yesmaximal, characteristic
2C3×#63918C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic
4C32×#832C211yesmaximal, characteristic
5C3×#63566C211yesmaximal, characteristic, complemented
6C3×#63565C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8C3×#5055—31nomaximal
9C3×#4765—31nomaximal
10#10591488C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×#627—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×#217—21no—
17C32×#217—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C32×#124—31no—
23C3×#936—21no—
24C3×#936D311yescharacteristic
25C3×#936C2×C311yescharacteristic
26C3×#888—31no—
27C3×#869—31no—
28#63918C2×C311yescharacteristic, complemented
29C32×#102—31no—
30#63659C2×C311yescharacteristic
31C3×#835—31no—
32C3×#835—31no—
33C3×#832—21no—
34C3×#832—31no—
35C3×#832—31no—
36C3×#832—61no—
37C3×#832D311yescharacteristic
38C3×#832C2×C311yescharacteristic
39C3×#816—31no—
40C3×#815—31no—
41C3×#810—31no—
42C3×#803—31no—
43#63566C2×C311yescharacteristic, complemented
44#63565C2×C311yescharacteristic, complemented
45#63554C2×C311yescharacteristic, complemented
46C23×C33—21no—
47C33×D4—21no—
48C33×D4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3×#627—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60C3×#489—91no—
61#5055—31no—
62#4765—31no—
63C2×C32×Q8—31no—
64C2×C32×D4—21no—
65C2×C32×D4—21no—
66C2×C32×D4—21no—
67C2×C32×D4—41no—
68C2×C32×D4C2×D311yescharacteristic
69C2×C32×D4C2×D311yescharacteristic
70C2×C32×D4C22×C311yescharacteristic
71C2×C3×Dic6—21no—
72C2×C3×Dic6—61no—
73C2×C3×Dic6—61no—
74C2×C3×Dic6C2×D311yescharacteristic
75C2×C3×Dic6C22×C311yescharacteristic
76C2×#627—21no—
77C2×C32×C8—31no—
78C32×C4⋊C4—31no—
79C32×C22⋊C4—61no—
80C32×C22⋊C4—61no—
81C32×C42—31no—
82C4⋊(C3⋊Dic3)C22×C311yescharacteristic
83C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
84C3×#217—22no—
85C3×#217—22no—
86C3×#217—21no—
87C3×#217—21no—
88C3×#217—21no—
89C3×#217—21no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C22×C311yescharacteristic
97C3×Dic3⋊C4—61no—
98C3×C4×Dic3—21no—
99C3×C4×Dic3C2×D311yescharacteristic
100C3×C4×Dic3C22×C311yescharacteristic
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C22×C311yescharacteristic
107Dic3⋊Dic3—21no—
108C22×C33—41no—
109C22×C33—41no—
110C22×C33—41no—
111C22×C33C2×D411yescharacteristic
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C3×C3⋊Dic3—41no—
115C32×Dic3—21no—
116C32×Dic3—21no—
117C32×Dic3—41no—
118C32×Dic3—41no—
119C3×C4⋊Q8—91no—
120C3×#124—31no—
121C3×#124—31no—
122C3×#124—61no—
123#936C3×D311yescharacteristic
124#888—31no—
125#869—31no—
126C3×#103—91no—
127C3×#103—91no—
128C3×#102—31no—
129C3×#102—31no—
130C3×#102—61no—
131C3×#102—91no—
132C3×#97—91no—
133#835—31no—
134#835—31no—
135#832—31no—
136#832—31no—
137#832—61no—
138#832C3×D311yescharacteristic
139#816—31no—
140#815—31no—
141#810—31no—
142#803—31no—
143C23×C32—21no—
144C23×C32—22no—
145C23×C32—21no—
146C23×C32—21no—
147C23×C32—41no—
148C23×C32—41no—
149C23×C32—42no—
150C32×Q8—61no—
151C32×Q8—61no—
152C32×D4—21no—
153C32×D4—21no—
154C32×D4—22no—
155C32×D4—22no—
156C32×D4—21no—
157C32×D4—21no—
158C32×D4—21no—
159C32×D4—21no—
160C32×D4—41no—
161C32×D4—41no—
162C32×D4—41no—
163C32×D4—41no—
164C32×D4—42no—
165C32×D4—42no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—31no—
170C2×C32×C4—31no—
171C2×C32×C4—41no—
172C2×C32×C4—61no—
173C2×C32×C4—61no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C23×C311yescharacteristic
177C2×C3⋊Dic3—21no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—41no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—121no—
189C2×C3×Dic3#6711yescharacteristic
190C2×C3×Dic3#6711yescharacteristic
191C2×C3×Dic3C3×D411yescharacteristic
192C2×C3×Dic3C3×D411yescharacteristic
193C3×Dic6—21no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6—21no—
197C3×Dic6—41no—
198C3×Dic6—41no—
199C3×Dic6—122no—
200C3×Dic6—122no—
201#627—42no—
202C32×C8—61no—
203C3⋊(C3⋊C8)—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—21no—
206C3×C3⋊C8—41no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—121no—
210#489—91noSylow
211C2×C33—81no—
212C2×C33#12411yescharacteristic
213C2×C33AGL(1,ℤ/8)11yescharacteristic
214C2×C33#13711yescharacteristic
215C2×C3×Q8—31no—
216C2×C3×Q8—31no—
217C2×C3×Q8—61no—
218C2×C3×Q8—181no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—41no—
223C2×C3×D4D3211yescharacteristic
224C2×C3×D4C2×C3×D311yescharacteristic
225C2×C3×D4C2×C3×D311yescharacteristic
226C2×Dic6—61no—
227C2×Dic6—61no—
228C2×Dic6C2×C3×D311yescharacteristic
229C2×C3×C8—31no—
230C2×C3×C8—31no—
231C2×C3×C8—61no—
232C2×C3×C8—91no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—181no—
238C3×C22⋊C4—61no—
239C3×C22⋊C4—61no—
240C3×C22⋊C4—61no—
241C3×C22⋊C4—61no—
242C3×C22⋊C4—62no—
243C3×C22⋊C4—62no—
244C3×C42—31no—
245C3×C42—31no—
246C3×C42—61no—
247#217—21no—
248#217—21no—
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251C4⋊Dic3—61no—
252C4⋊Dic3—61no—
253C4⋊Dic3C2×C3×D311yescharacteristic
254Dic3⋊C4—61no—
255C4×Dic3C2×C3×D311yescharacteristic
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—61no—
259C2×C3⋊C8C2×C3×D311yescharacteristic
260C22×C32—21no—
261C22×C32—21no—
262C22×C32—21no—
263C22×C32—41no—
264C22×C32—42no—
265C22×C32—42no—
266C22×C32—42no—
267C22×C32—44no—
268C22×C32—41no—
269C22×C32—41no—
270C22×C32—41no—
271C22×C32—41no—
272C22×C32—41no—
273C22×C32—42no—
274C22×C32—42no—
275C22×C32—41no—
276C22×C32—41no—
277C22×C32—41no—
278C22×C32—41no—
279C22×C32—81no—
280C22×C32—81no—
281C22×C32—81no—
282C22×C32—81no—
283C22×C32—82no—
284C22×C32—82no—
285C22×C32D3×D411yescharacteristic
286C22×C32C2×#6711yescharacteristic
287C22×C32C2×C3×D411yescharacteristic
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—41no—
295C32×C4—41no—
296C32×C4—61no—
297C32×C4—61no—
298C32×C4—121no—
299C32×C4—121no—
300C32×C4#23511yescharacteristic
301C32×C4#23711yescharacteristic
302C32×C4#23711yescharacteristic
303C32×C4#23711yescharacteristic
304C32×C4C3×#4111yescharacteristic
305C32×C4C3×#4111yescharacteristic
306C3⋊Dic3—41no—
307C3×Dic3—21no—
308C3×Dic3—21no—
309C3×Dic3—21no—
310C3×Dic3—21no—
311C3×Dic3—41no—
312C3×Dic3—41no—
313C3×Dic3—42no—
314C3×Dic3—41no—
315C3×Dic3—41no—
316C3×Dic3—41no—
317C3×Dic3—41no—
318C3×Dic3—81no—
319C3×Dic3—121no—
320C3×Dic3—121no—
321C3×Dic3—241no—
322C4⋊Q8—91no—
323#124—31no—
324#103—91no—
325#103—91no—
326#102—31no—
327#102—91no—
328#97—91no—
329C33#48911yescharacteristic, complemented, Sylow
330C23×C3—22no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—42no—
337C3×Q8—61no—
338C3×Q8—61no—
339C3×Q8—61no—
340C3×Q8—61no—
341C3×Q8—121no—
342C3×Q8—121no—
343C3×Q8—362no—
344C3×D4—22no—
345C3×D4—22no—
346C3×D4—21no—
347C3×D4—21no—
348C3×D4—21no—
349C3×D4—21no—
350C3×D4—21no—
351C3×D4—21no—
352C3×D4—41no—
353C3×D4—41no—
354C3×D4—41no—
355C3×D4—41no—
356C3×D4—42no—
357C3×D4—42no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—41no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—62no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—61no—
373C2×C3×C4—121no—
374C2×C3×C4—181no—
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C22×C3×D311yescharacteristic
377C2×C3×C4C22×C3×D311yescharacteristic
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—121no—
383C2×Dic3C3×#6711yescharacteristic
384C2×Dic3C3×#6711yescharacteristic
385Dic6—21no—
386Dic6—21no—
387Dic6—122no—
388Dic6—122no—
389C3×C8—61no—
390C3×C8—61no—
391C3×C8—121no—
392C3×C8—181no—
393C3⋊C8—21no—
394C3⋊C8—61no—
395C3⋊C8—61no—
396C3⋊C8—121no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—41no—
407C2×C32—41no—
408C2×C32—41no—
409C2×C32—82no—
410C2×C32—82no—
411C2×C32—82no—
412C2×C32—84no—
413C2×C32—81no—
414C2×C32—81no—
415C2×C32—81no—
416C2×C32#88511yescharacteristic
417C2×C32#91111yescharacteristic
418C2×C32#91511yescharacteristic
419C2×C32#93611yescharacteristic
420C2×C32#94911yescharacteristic
421C2×C32#95111yescharacteristic
422C2×C32C3×#12411yescharacteristic
423C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
424C2×C32C3×#13711yescharacteristic
425C2×Q8—31no—
426C2×Q8—181no—
427C2×D4C3×D3211yescharacteristic
428C2×C8—31no—
429C2×C8—91no—
430C4⋊C4—31no—
431C4⋊C4—91no—
432C4⋊C4—181no—
433C22⋊C4—61no—
434C22⋊C4—61no—
435C42—31no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—42no—
444C22×C3—44no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—41no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—81no—
456C22×C3—81no—
457C22×C3—81no—
458C22×C3—81no—
459C22×C3—82no—
460C22×C3—82no—
461C22×C3D3×#6711yescharacteristic
462C22×C3C3×D3×D411yescharacteristic
463C22×C3C2×C3×#6711yescharacteristic
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—61no—
473C3×C4—61no—
474C3×C4—61no—
475C3×C4—61no—
476C3×C4—121no—
477C3×C4—121no—
478C3×C4—122no—
479C3×C4—121no—
480C3×C4—121no—
481C3×C4—121no—
482C3×C4—121no—
483C3×C4—241no—
484C3×C4—361no—
485C3×C4#380411yescharacteristic
486C3×C4#380511yescharacteristic
487C3×C4C3×#23511yescharacteristic
488C3×C4C3×#23711yescharacteristic
489C3×C4C3×#23711yescharacteristic
490C3×C4C3×#23711yescharacteristic
491Dic3—21no—
492Dic3—21no—
493Dic3—41no—
494Dic3—41no—
495Dic3—121no—
496Dic3—121no—
497Dic3—241no—
498C32—21no—
499C32—21no—
500C32—21no—
501C32—41no—
502C32#476511yescharacteristic, complemented
503C32#505511yescharacteristic, complemented
504C32C3×#48911yescharacteristic, complemented
505C23—21no—
506Q8—61no—
507Q8—61no—
508Q8—362no—
509D4—21no—
510D4—21no—
511C2×C4—31no—
512C2×C4—31no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—181no—
516C2×C4C2×C3×D3211yescharacteristic
517C8—61no—
518C8—181no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—41no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—82no—
532C2×C3—82no—
533C2×C3—82no—
534C2×C3—84no—
535C2×C3—81no—
536C2×C3—81no—
537C2×C3—81no—
538C2×C3#6392511yescharacteristic
539C2×C3#6392911yescharacteristic
540C2×C3#6395911yescharacteristic
541C2×C3C3×#88511yescharacteristic
542C2×C3C3×#91111yescharacteristic
543C2×C3C3×#91511yescharacteristic
544C2×C3C3×#93611yescharacteristic
545C2×C3C3×#94911yescharacteristic
546C2×C3C3×#95111yescharacteristic
547C22—41no—
548C22—41no—
549C22—41no—
550C22C3×D3×#6711yescharacteristic
551C4—61no—
552C4—61no—
553C4—121no—
554C4—121no—
555C4—361no—
556C4C3×#380411yescharacteristic
557C4C3×#380511yescharacteristic
558C3—21no—
559C3—21no—
560C3—21no—
561C3—41no—
562C3#1059148811yescharacteristic, direct product factor
563C3C3×#476511yescharacteristic, complemented
564C3C3×#505511yescharacteristic, complemented
565C2—81no—
566C2C3×#6392511yescharacteristic
567C2C3×#6392911yescharacteristic
568C2C3×#6395911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#489
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C3×#63918C2semidirect
5. C3×#63566C2semidirect
6. C3×#63565C2semidirect
7. C3×#63554C2semidirect
10. #10591488C3direct (N × H)
28. #63918C2×C3semidirect
43. #63566C2×C3semidirect
44. #63565C2×C3semidirect
45. #63554C2×C3semidirect
329. C33#489semidirect
502. C32#4765semidirect
503. C32#5055semidirect
504. C32C3×#489semidirect
562. C3#10591488direct (N × H)
563. C3C3×#4765semidirect
564. C3C3×#5055semidirect

Maximal subgroups

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

subgroupindex
C32×#9362
C3×#639182
C3×#636592
C32×#8322
C3×#635662
C3×#635652
C3×#635542
C3×#50553
C3×#47653
#105914883