Finite Group Databasebeta

← C3×#10591473 #13158485

Subgroups — C3×#10591473 #13158485

Order 1728.

794 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3753C211yesmaximal, characteristic
2C32×#936C211yesmaximal, characteristic, complemented
3C2×C32×#214C211yesmaximal, characteristic, complemented
4C3×#63864C211yesmaximal, characteristic, complemented
5C2×C3×#3718C211yesmaximal, characteristic, complemented
6C2×C3×#3717C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×#5168—31nomaximal
9C3×#5059—31nomaximal
10#10591473C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×#3753—22no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#217—22no—
18C32×#214—22no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×#3722—22no—
22C3×#3718—22no—
23C3×#3717—22no—
24C2×C32×SD16—31no—
25C32×#124—31no—
26C2×#3753C2×C311yescharacteristic
27C3×#947—31no—
28C2×C3×#216—31no—
29C2×C3×#216—31no—
30C2×C3×#215—31no—
31C3×#936—21no—
32C3×#936D311yescharacteristic, complemented
33C3×#936C2×C311yescharacteristic, complemented
34C2×C3×#214—21no—
35C2×C3×#214D311yescharacteristic, complemented
36C2×C3×#214C2×C311yescharacteristic, complemented
37C2×C3×#213—31no—
38C2×C3×#213—31no—
39C2×C3×#213—61no—
40C2×C3×#204—31no—
41C3×#875—31no—
42#63864C2×C311yescharacteristic, complemented
43C2×#3718C2×C311yescharacteristic, complemented
44C2×#3717C2×C311yescharacteristic, complemented
45C3×C8×Dic3—31no—
46C3×C4×C3⋊C8—31no—
47Dic3×C3⋊C8C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C33×D4—21no—
51C33×D4—21no—
52C2×C33×C4C2311yescharacteristic
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D412yes—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)D412yes—
63C32×C3⋊C8D412yes—
64C3×#495—91no—
65#5168—31no—
66#5059—31no—
67C2×C32×Q8—31no—
68C2×C32×Q8—31no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4—41no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C2×D311yescharacteristic
75C2×C32×D4C22×C311yescharacteristic
76C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—61no—
80C2×C3×Dic6—21no—
81C2×C3×Dic6—21no—
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C3×Dic6C2×D311yescharacteristic
84C2×C3×Dic6C22×C311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C32×SD16—62no—
87C2×C32×C8—31no—
88C32×C22⋊C4—62no—
89C32×C42—31no—
90#3753—22no—
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C3×#217—24no—
93C3×#217—22no—
94C3×#217—22no—
95C3×#216—62no—
96C3×#216—62no—
97C3×#215—62no—
98C3×#214—24no—
99C3×#214—22no—
100C3×#214—22no—
101C3×#213—62no—
102C3×#213—62no—
103C3×#213—64no—
104C3×(C2×D3)⋊C4—62no—
105C3×(C2×D3)⋊C4—62no—
106C3×C4×Dic3—21no—
107C3×C4×Dic3C2×D311yescharacteristic
108C3×C4×Dic3C22×C311yescharacteristic
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—61no—
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C2×C3×C3⋊C8C22×C311yescharacteristic
115C3×#204—62no—
116#3722—22no—
117#3718—22no—
118#3717—22no—
119C22×C33—41no—
120C22×C33—42no—
121C22×C33C2×D411yescharacteristic
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—42no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2×D411yescharacteristic
126C32×Dic3—22no—
127C32×Dic3—41no—
128C32×Dic3—41no—
129C2×C3×Q16—91no—
130C2×C3×SD16—31no—
131C2×C3×SD16—31no—
132C2×C3×SD16—61no—
133C2×C3×SD16—91no—
134C2×C3×D8—91no—
135C3×#124—31no—
136C3×#124—31no—
137C3×#124—61no—
138C3×#124—91no—
139#947—31no—
140C2×#216—31no—
141C2×#216—31no—
142C2×#215—31no—
143#936C3×D311yescharacteristic, complemented
144C2×#214C3×D311yescharacteristic, complemented
145C2×#213—31no—
146C2×#213—31no—
147C2×#213—61no—
148C2×#204—31no—
149#875—31no—
150C3×C4×C8—91no—
151C8×Dic3—31no—
152C4×C3⋊C8—31no—
153C23×C32—21no—
154C23×C32—22no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—41no—
158C23×C32—41no—
159C23×C32—42no—
160C22×C3⋊D3—21no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—121no—
164C32×Q8—61no—
165C32×Q8—61no—
166C32×Q8—61no—
167C32×Q8—61no—
168C32×D4—21no—
169C32×D4—21no—
170C32×D4—22no—
171C32×D4—22no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—21no—
175C32×D4—21no—
176C32×D4—41no—
177C32×D4—41no—
178C32×D4—41no—
179C32×D4—41no—
180C32×D4—42no—
181C32×D4—42no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—32no—
186C2×C32×C4—41no—
187C2×C32×C4—61no—
188C2×C32×C4—61no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C23×C311yescharacteristic
192C4⋊(C3⋊D3)—21no—
193C4⋊(C3⋊D3)—21no—
194C2×C3×Dic3—22no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—21no—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—41no—
201C2×C3×Dic3#6712yes—
202C2×C3×Dic3C3×D412yes—
203C3×D12—62no—
204C3×D12—61no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—121no—
209C3×Dic6—22no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—21no—
214C3×Dic6—21no—
215C3×Dic6—21no—
216C3×Dic6—21no—
217C3×Dic6—21no—
218C3×Dic6—41no—
219C3×Dic6—41no—
220C3×Dic6—41no—
221C32×C8—32no—
222C3⋊(C3⋊C8)C3×D412yes—
223C3×C3⋊C8—22no—
224C3×C3⋊C8—32no—
225C3×C3⋊C8—32no—
226C3×C3⋊C8—62no—
227C3×C3⋊C8#6712yes—
228C3×C3⋊C8C3×D412yes—
229#495—91noSylow
230C2×C33—81no—
231C2×C33#12711yescharacteristic
232C2×C33#13512yes—
233C3×C3⋊D3—81no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—31no—
237C2×C3×Q8—31no—
238C2×C3×Q8—61no—
239C2×C3×Q8—61no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—41no—
244C2×C3×D4—91no—
245C2×C3×D4D3211yescharacteristic
246C2×C3×D4C2×C3×D311yescharacteristic
247C2×C3×D4C2×C3×D311yescharacteristic
248C2×D12—31no—
249C2×D12—31no—
250C2×D12—61no—
251C2×Dic6C2×C3×D311yescharacteristic
252C2×Dic6C2×C3×D311yescharacteristic
253C3×Q16—182no—
254C3×SD16—62no—
255C3×SD16—62no—
256C3×SD16—64no—
257C3×SD16—182no—
258C3×D8—182no—
259C2×C3×C8—31no—
260C2×C3×C8—31no—
261C2×C3×C8—61no—
262C2×C3×C8—91no—
263C3×C22⋊C4—62no—
264C3×C22⋊C4—62no—
265C3×C22⋊C4—64no—
266C3×C22⋊C4—182no—
267C3×C42—31no—
268C3×C42—31no—
269C3×C42—61no—
270#217—22no—
271#216—62no—
272#216—62no—
273#215—62no—
274#214—22no—
275#213—62no—
276#213—62no—
277#213—64no—
278(C2×D3)⋊C4—62no—
279(C2×D3)⋊C4—62no—
280C4×Dic3C2×C3×D311yescharacteristic
281C2×C3⋊C8—31no—
282C2×C3⋊C8—31no—
283C2×C3⋊C8—61no—
284C2×C3⋊C8C2×C3×D311yescharacteristic
285#204—62no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32—42no—
291C22×C32—42no—
292C22×C32—42no—
293C22×C32—44no—
294C22×C32—42no—
295C22×C32—41no—
296C22×C32—41no—
297C22×C32—41no—
298C22×C32—44no—
299C22×C32—42no—
300C22×C32—42no—
301C22×C32—82no—
302C22×C32—82no—
303C22×C32—84no—
304C22×C32D3×D411yescharacteristic
305C22×C32C2×#6711yescharacteristic
306C22×C32C2×C3×D411yescharacteristic
307C2×C3⋊D3—41no—
308C2×C3⋊D3—42no—
309C2×C3×D3—122no—
310C2×C3×D3—122no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C2×C3×D3—122no—
314C2×C3×D3—242no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4—62no—
324C32×C4—121no—
325C32×C4—121no—
326C32×C4D3×D411yescharacteristic
327C32×C4D3×D411yescharacteristic
328C32×C4D3×D411yescharacteristic
329C32×C4C2×#6711yescharacteristic
330C32×C4C2×C3×D411yescharacteristic
331C32×C4C2×C3×D411yescharacteristic
332C3×Dic3—22no—
333C3×Dic3—22no—
334C3×Dic3—42no—
335C3×Dic3—42no—
336C3×Dic3—42no—
337C3×Dic3—41no—
338C3×Dic3—41no—
339C3×Dic3—41no—
340C3×Dic3—41no—
341C2×Q16—91no—
342C2×SD16—31no—
343C2×SD16—91no—
344C2×D8—91no—
345#124—31no—
346#124—91no—
347C4×C8—91no—
348C33#49511yescharacteristic, complemented, Sylow
349C23×C3—22no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—42no—
356C23×C3—181no—
357C22×D3—61no—
358C22×D3—61no—
359C22×D3—121no—
360C3×Q8—62no—
361C3×Q8—61no—
362C3×Q8—61no—
363C3×Q8—61no—
364C3×Q8—61no—
365C3×Q8—61no—
366C3×Q8—61no—
367C3×Q8—61no—
368C3×Q8—61no—
369C3×Q8—121no—
370C3×Q8—121no—
371C3×Q8—121no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4—21no—
375C3×D4—21no—
376C3×D4—21no—
377C3×D4—21no—
378C3×D4—21no—
379C3×D4—21no—
380C3×D4—41no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—42no—
385C3×D4—42no—
386C3×D4—181no—
387C3×D4—181no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—32no—
392C2×C3×C4—32no—
393C2×C3×C4—41no—
394C2×C3×C4—62no—
395C2×C3×C4—62no—
396C2×C3×C4—61no—
397C2×C3×C4—61no—
398C2×C3×C4—61no—
399C2×C3×C4—61no—
400C2×C3×C4—121no—
401C2×C3×C4C2×D3211yescharacteristic
402C2×C3×C4C22×C3×D311yescharacteristic
403C2×C3×C4C22×C3×D311yescharacteristic
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3C3×#6712yes—
407D12—62no—
408D12—61no—
409D12—61no—
410D12—61no—
411D12—61no—
412D12—121no—
413Dic6—21no—
414Dic6—21no—
415Dic6—21no—
416Dic6—21no—
417C3×C8—32no—
418C3×C8—32no—
419C3×C8—62no—
420C3×C8—92no—
421C3⋊C8—32no—
422C3⋊C8—32no—
423C3⋊C8—62no—
424C3⋊C8C3×#6712yes—
425C2×C32—22no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—22no—
429C2×C32—22no—
430C2×C32—21no—
431C2×C32—41no—
432C2×C32—42no—
433C2×C32—82no—
434C2×C32—82no—
435C2×C32—82no—
436C2×C32—84no—
437C2×C32—81no—
438C2×C32—81no—
439C2×C32—81no—
440C2×C32#91612yes—
441C2×C32#94011yescharacteristic
442C2×C32#94011yescharacteristic
443C2×C32#95012yes—
444C2×C32C3×#12711yescharacteristic
445C2×C32C3×#13512yes—
446C3⋊D3—81no—
447C3×D3—242no—
448C3×D3—241no—
449C3×D3—241no—
450C2×Q8—31no—
451C2×Q8—31no—
452C2×D4—91no—
453C2×D4C3×D3211yescharacteristic
454Q16—182no—
455SD16—62no—
456SD16—182no—
457D8—182no—
458C2×C8—31no—
459C2×C8—91no—
460C22⋊C4—62no—
461C22⋊C4—182no—
462C42—31no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—42no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—42no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—82no—
479C22×C3—82no—
480C22×C3—84no—
481C22×C3—362no—
482C22×C3—361no—
483C22×C3D3×#6711yescharacteristic
484C22×C3C3×D3×D411yescharacteristic
485C22×C3C2×C3×#6711yescharacteristic
486C2×D3—122no—
487C2×D3—122no—
488C2×D3—121no—
489C2×D3—121no—
490C2×D3—122no—
491C2×D3—242no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—62no—
501C3×C4—62no—
502C3×C4—122no—
503C3×C4—122no—
504C3×C4—122no—
505C3×C4—121no—
506C3×C4—121no—
507C3×C4—121no—
508C3×C4—121no—
509C3×C4D3×#6711yescharacteristic
510C3×C4#381111yescharacteristic
511C3×C4C3×D3×D411yescharacteristic
512C3×C4C3×D3×D411yescharacteristic
513C3×C4C3×D3×D411yescharacteristic
514C3×C4C2×C3×#6711yescharacteristic
515Dic3—22no—
516Dic3—41no—
517Dic3—41no—
518C32—21no—
519C32—21no—
520C32—21no—
521C32—41no—
522C32#505911yescharacteristic, complemented
523C32#516811yescharacteristic, complemented
524C32C3×#49511yescharacteristic, complemented
525C23—21no—
526C23—181no—
527Q8—61no—
528Q8—61no—
529Q8—61no—
530Q8—61no—
531D4—21no—
532D4—21no—
533D4—181no—
534D4—181no—
535C2×C4—32no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4C2×C3×D3211yescharacteristic
539C8—32no—
540C8—92no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—22no—
547C2×C3—41no—
548C2×C3—42no—
549C2×C3—82no—
550C2×C3—82no—
551C2×C3—82no—
552C2×C3—84no—
553C2×C3—81no—
554C2×C3—81no—
555C2×C3—81no—
556C2×C3—721no—
557C2×C3#6393512yes—
558C2×C3#6397911yescharacteristic
559C2×C3C3×#91612yes—
560C2×C3C3×#94011yescharacteristic
561C2×C3C3×#94011yescharacteristic
562C2×C3C3×#95012yes—
563D3—242no—
564D3—241no—
565D3—241no—
566C22—41no—
567C22—42no—
568C22—361no—
569C22—362no—
570C22C3×D3×#6711yescharacteristic
571C4—62no—
572C4—121no—
573C4—121no—
574C4C3×D3×#6711yescharacteristic
575C4C3×#381111yescharacteristic
576C3—21no—
577C3—21no—
578C3—21no—
579C3—41no—
580C3#1059147311yescharacteristic, direct product factor
581C3C3×#505911yescharacteristic, complemented
582C3C3×#516811yescharacteristic, complemented
583C2—81no—
584C2—721no—
585C2C3×#6393512yes—
586C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#495
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. C32×#936C2semidirect
3. C2×C32×#214C2semidirect
4. C3×#63864C2semidirect
5. C2×C3×#3718C2semidirect
6. C2×C3×#3717C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10591473C3direct (N × H)
32. C3×#936D3semidirect
33. C3×#936C2×C3semidirect
35. C2×C3×#214D3semidirect
36. C2×C3×#214C2×C3semidirect
42. #63864C2×C3semidirect
43. C2×#3718C2×C3semidirect
44. C2×#3717C2×C3semidirect
47. Dic3×C3⋊C8C2×C3semidirect
143. #936C3×D3semidirect
144. C2×#214C3×D3semidirect
348. C33#495semidirect
522. C32#5059semidirect
523. C32#5168semidirect
524. C32C3×#495semidirect
580. C3#10591473direct (N × H)
581. C3C3×#5059semidirect
582. C3C3×#5168semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37532
C32×#9362
C2×C32×#2142
C3×#638642
C2×C3×#37182
C2×C3×#37172
C3×Dic3×C3⋊C82
C3×#51683
C3×#50593
#105914733