Finite Group Databasebeta

← C3×#10591478 #13158490

Subgroups — C3×#10591478 #13158490

Order 1728.

614 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#936C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic, complemented
3C3×#63659C211yesmaximal, characteristic
4C32×#832C211yesmaximal, characteristic
5C3×#63566C211yesmaximal, characteristic, complemented
6C3×#63565C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×#5056—31nomaximal
9C3×#4768—31nomaximal
10#10591478C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×#217—21no—
16C32×#217—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22C32×#124—31no—
23C3×#936—21no—
24C3×#936D311yescharacteristic
25C3×#936C2×C311yescharacteristic
26C3×#890—31no—
27C3×#870—31no—
28#63869C2×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×C8×Dic3—31no—
41C3×#810—31no—
42C3×C4×C3⋊C8—31no—
43#63566C2×C311yescharacteristic, complemented
44#63565C2×C311yescharacteristic, complemented
45Dic3×C3⋊C8C2×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×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59C3×#488—91no—
60#5056—31no—
61#4768—31no—
62C2×C32×Q8—31no—
63C2×C32×D4—21no—
64C2×C32×D4—21no—
65C2×C32×D4—21no—
66C2×C32×D4—41no—
67C2×C32×D4C2×D311yescharacteristic
68C2×C32×D4C2×D311yescharacteristic
69C2×C32×D4C22×C311yescharacteristic
70C2×C3×Dic6—21no—
71C2×C3×Dic6C2×D311yescharacteristic
72C2×C3×Dic6C22×C311yescharacteristic
73C2×C32×C8—31no—
74C32×C4⋊C4—31no—
75C32×C22⋊C4—61no—
76C32×C22⋊C4—61no—
77C32×C42—31no—
78C4⋊(C3⋊Dic3)C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C3×#217—22no—
81C3×#217—22no—
82C3×#217—21no—
83C3×#217—21no—
84C3×#217—21no—
85C3×#217—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C22×C311yescharacteristic
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C3×C4×Dic3—21no—
97C3×C4×Dic3C2×D311yescharacteristic
98C3×C4×Dic3C22×C311yescharacteristic
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105#3724—21no—
106Dic3⋊Dic3—21no—
107C22×C33—41no—
108C22×C33—41no—
109C22×C33—41no—
110C22×C33C2×D411yescharacteristic
111C33×C4#4111yescharacteristic
112C33×C4#4111yescharacteristic
113C3×C3⋊Dic3—41no—
114C32×Dic3—21no—
115C32×Dic3—21no—
116C32×Dic3—41no—
117C32×Dic3—41no—
118C3×#125—91no—
119C3×#124—31no—
120C3×#124—31no—
121C3×#124—61no—
122#936C3×D311yescharacteristic
123#890—31no—
124#870—31no—
125C3×#103—91no—
126C3×#103—91no—
127C3×#102—31no—
128C3×#102—31no—
129C3×#102—61no—
130C3×#102—91no—
131C3×C4×C8—91no—
132#835—31no—
133#835—31no—
134#832—31no—
135#832—31no—
136#832—61no—
137#832C3×D311yescharacteristic
138#816—31no—
139C8×Dic3—31no—
140#810—31no—
141C4×C3⋊C8—31no—
142C23×C32—21no—
143C23×C32—22no—
144C23×C32—21no—
145C23×C32—21no—
146C23×C32—41no—
147C23×C32—41no—
148C23×C32—42no—
149C32×Q8—61no—
150C32×Q8—61no—
151C32×D4—21no—
152C32×D4—21no—
153C32×D4—22no—
154C32×D4—22no—
155C32×D4—21no—
156C32×D4—21no—
157C32×D4—21no—
158C32×D4—21no—
159C32×D4—41no—
160C32×D4—41no—
161C32×D4—41no—
162C32×D4—41no—
163C32×D4—42no—
164C32×D4—42no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—31no—
169C2×C32×C4—31no—
170C2×C32×C4—41no—
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×Dic3—22no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—41no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—121no—
188C2×C3×Dic3#6711yescharacteristic
189C2×C3×Dic3#6711yescharacteristic
190C2×C3×Dic3C3×D411yescharacteristic
191C2×C3×Dic3C3×D411yescharacteristic
192C3×Dic6—21no—
193C3×Dic6—21no—
194C3×Dic6—21no—
195C3×Dic6—21no—
196C3×Dic6—41no—
197C3×Dic6—41no—
198C32×C8—61no—
199C3⋊(C3⋊C8)—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—41no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—121no—
206#488—91noSylow
207C2×C33—81no—
208C2×C33#12411yescharacteristic
209C2×C33#13511yescharacteristic
210C2×C33#13511yescharacteristic
211C2×C3×Q8—31no—
212C2×C3×Q8—31no—
213C2×C3×Q8—61no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—41no—
218C2×C3×D4D3211yescharacteristic
219C2×C3×D4C2×C3×D311yescharacteristic
220C2×C3×D4C2×C3×D311yescharacteristic
221C2×Dic6C2×C3×D311yescharacteristic
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—61no—
225C2×C3×C8—91no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—61no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—181no—
231C3×C4⋊C4—181no—
232C3×C22⋊C4—61no—
233C3×C22⋊C4—61no—
234C3×C22⋊C4—61no—
235C3×C22⋊C4—61no—
236C3×C22⋊C4—62no—
237C3×C22⋊C4—62no—
238C3×C42—31no—
239C3×C42—31no—
240C3×C42—61no—
241#217—21no—
242#217—21no—
243C4⋊Dic3—31no—
244C4⋊Dic3—31no—
245C4⋊Dic3—61no—
246C4⋊Dic3—61no—
247C4⋊Dic3C2×C3×D311yescharacteristic
248Dic3⋊C4—61no—
249Dic3⋊C4—61no—
250Dic3⋊C4—61no—
251C4×Dic3C2×C3×D311yescharacteristic
252C2×C3⋊C8—31no—
253C2×C3⋊C8—31no—
254C2×C3⋊C8—61no—
255C2×C3⋊C8C2×C3×D311yescharacteristic
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—41no—
260C22×C32—42no—
261C22×C32—42no—
262C22×C32—42no—
263C22×C32—44no—
264C22×C32—41no—
265C22×C32—41no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32—41no—
269C22×C32—42no—
270C22×C32—42no—
271C22×C32—41no—
272C22×C32—41no—
273C22×C32—41no—
274C22×C32—41no—
275C22×C32—81no—
276C22×C32—81no—
277C22×C32—81no—
278C22×C32—81no—
279C22×C32—82no—
280C22×C32—82no—
281C22×C32D3×D411yescharacteristic
282C22×C32C2×#6711yescharacteristic
283C22×C32C2×C3×D411yescharacteristic
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—41no—
291C32×C4—41no—
292C32×C4—61no—
293C32×C4—61no—
294C32×C4—121no—
295C32×C4—121no—
296C32×C4#23511yescharacteristic
297C32×C4#23711yescharacteristic
298C32×C4#23711yescharacteristic
299C32×C4#23711yescharacteristic
300C32×C4C3×#4111yescharacteristic
301C32×C4C3×#4111yescharacteristic
302C3⋊Dic3—41no—
303C3×Dic3—21no—
304C3×Dic3—21no—
305C3×Dic3—21no—
306C3×Dic3—21no—
307C3×Dic3—41no—
308C3×Dic3—41no—
309C3×Dic3—42no—
310C3×Dic3—41no—
311C3×Dic3—41no—
312C3×Dic3—41no—
313C3×Dic3—41no—
314C3×Dic3—81no—
315C3×Dic3—121no—
316C3×Dic3—121no—
317C3×Dic3—241no—
318#125—91no—
319#124—31no—
320#103—91no—
321#103—91no—
322#102—31no—
323#102—91no—
324C4×C8—91no—
325C33#48811yescharacteristic, complemented, Sylow
326C23×C3—22no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—42no—
333C3×Q8—61no—
334C3×Q8—61no—
335C3×Q8—61no—
336C3×Q8—61no—
337C3×Q8—121no—
338C3×Q8—121no—
339C3×D4—22no—
340C3×D4—22no—
341C3×D4—21no—
342C3×D4—21no—
343C3×D4—21no—
344C3×D4—21no—
345C3×D4—21no—
346C3×D4—21no—
347C3×D4—41no—
348C3×D4—41no—
349C3×D4—41no—
350C3×D4—41no—
351C3×D4—42no—
352C3×D4—42no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—41no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—62no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—121no—
369C2×C3×C4—181no—
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C22×C3×D311yescharacteristic
372C2×C3×C4C22×C3×D311yescharacteristic
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—121no—
378C2×Dic3C3×#6711yescharacteristic
379C2×Dic3C3×#6711yescharacteristic
380Dic6—21no—
381Dic6—21no—
382C3×C8—61no—
383C3×C8—61no—
384C3×C8—121no—
385C3×C8—181no—
386C3⋊C8—21no—
387C3⋊C8—61no—
388C3⋊C8—61no—
389C3⋊C8—121no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—41no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—82no—
403C2×C32—82no—
404C2×C32—82no—
405C2×C32—84no—
406C2×C32—81no—
407C2×C32—81no—
408C2×C32—81no—
409C2×C32#88511yescharacteristic
410C2×C32#91211yescharacteristic
411C2×C32#91611yescharacteristic
412C2×C32#93611yescharacteristic
413C2×C32#95011yescharacteristic
414C2×C32#95011yescharacteristic
415C2×C32C3×#12411yescharacteristic
416C2×C32C3×#13511yescharacteristic
417C2×C32C3×#13511yescharacteristic
418C2×Q8—31no—
419C2×D4C3×D3211yescharacteristic
420C2×C8—31no—
421C2×C8—91no—
422C4⋊C4—31no—
423C4⋊C4—91no—
424C4⋊C4—181no—
425C4⋊C4—181no—
426C22⋊C4—61no—
427C22⋊C4—61no—
428C42—31no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—42no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—44no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—41no—
444C22×C3—41no—
445C22×C3—41no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—81no—
449C22×C3—81no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—82no—
453C22×C3—82no—
454C22×C3D3×#6711yescharacteristic
455C22×C3C3×D3×D411yescharacteristic
456C22×C3C2×C3×#6711yescharacteristic
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—61no—
466C3×C4—61no—
467C3×C4—61no—
468C3×C4—61no—
469C3×C4—121no—
470C3×C4—121no—
471C3×C4—122no—
472C3×C4—121no—
473C3×C4—121no—
474C3×C4—121no—
475C3×C4—121no—
476C3×C4—241no—
477C3×C4—361no—
478C3×C4#380411yescharacteristic
479C3×C4#380511yescharacteristic
480C3×C4C3×#23511yescharacteristic
481C3×C4C3×#23711yescharacteristic
482C3×C4C3×#23711yescharacteristic
483C3×C4C3×#23711yescharacteristic
484Dic3—21no—
485Dic3—21no—
486Dic3—41no—
487Dic3—41no—
488Dic3—121no—
489Dic3—121no—
490Dic3—241no—
491C32—21no—
492C32—21no—
493C32—21no—
494C32—41no—
495C32#476811yescharacteristic, complemented
496C32#505611yescharacteristic, complemented
497C32C3×#48811yescharacteristic, complemented
498C23—21no—
499Q8—61no—
500Q8—61no—
501D4—21no—
502D4—21no—
503C2×C4—31no—
504C2×C4—31no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—181no—
508C2×C4C2×C3×D3211yescharacteristic
509C8—61no—
510C8—181no—
511C2×C3—21no—
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—41no—
521C2×C3—41no—
522C2×C3—41no—
523C2×C3—82no—
524C2×C3—82no—
525C2×C3—82no—
526C2×C3—84no—
527C2×C3—81no—
528C2×C3—81no—
529C2×C3—81no—
530C2×C3#6393311yescharacteristic
531C2×C3#6393511yescharacteristic
532C2×C3#6395911yescharacteristic
533C2×C3C3×#88511yescharacteristic
534C2×C3C3×#91211yescharacteristic
535C2×C3C3×#91611yescharacteristic
536C2×C3C3×#93611yescharacteristic
537C2×C3C3×#95011yescharacteristic
538C2×C3C3×#95011yescharacteristic
539C22—41no—
540C22—41no—
541C22—41no—
542C22C3×D3×#6711yescharacteristic
543C4—61no—
544C4—61no—
545C4—121no—
546C4—121no—
547C4—361no—
548C4C3×#380411yescharacteristic
549C4C3×#380511yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059147811yescharacteristic, direct product factor
555C3C3×#476811yescharacteristic, complemented
556C3C3×#505611yescharacteristic, complemented
557C2—81no—
558C2C3×#6393311yescharacteristic
559C2C3×#6393511yescharacteristic
560C2C3×#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×#488
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×#63869C2semidirect
5. C3×#63566C2semidirect
6. C3×#63565C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10591478C3direct (N × H)
28. #63869C2×C3semidirect
43. #63566C2×C3semidirect
44. #63565C2×C3semidirect
45. Dic3×C3⋊C8C2×C3semidirect
325. C33#488semidirect
495. C32#4768semidirect
496. C32#5056semidirect
497. C32C3×#488semidirect
554. C3#10591478direct (N × H)
555. C3C3×#4768semidirect
556. C3C3×#5056semidirect

Maximal subgroups

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

subgroupindex
C32×#9362
C3×#638692
C3×#636592
C32×#8322
C3×#635662
C3×#635652
C3×Dic3×C3⋊C82
C3×#50563
C3×#47683
#105914783