Finite Group Databasebeta

← C3×#10591443 #13158529

Subgroups — C3×#10591443 #13158529

Order 1728.

696 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64145C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic
3C3×#63843C211yesmaximal, characteristic, complemented
4C2×C3×#3719C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic
6C3×#63572C211yesmaximal, characteristic, complemented
7C3×#63567C211yesmaximal, characteristic, complemented
8C3×#5166—31nomaximal
9C3×#4767—31nomaximal
10#10591443C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C3×#3757—21no—
15C3×#3757—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×#214—22no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C3×#3719—22no—
24C2×C32×SD16—31no—
25#64145C2×C311yescharacteristic
26C3×Q8×Dic3—31no—
27C2×C3×#216—31no—
28C3×#936—31no—
29C3×#936—31no—
30C3×#936—61no—
31C2×C3×#214—21no—
32C2×C3×#214D311yescharacteristic
33C2×C3×#214C2×C311yescharacteristic
34C2×C3×Dic12—31no—
35C3×#887—31no—
36#63843C2×C311yescharacteristic, complemented
37C2×#3719C2×C311yescharacteristic, complemented
38C32×#102—31no—
39C3×#832—21no—
40C3×#832D311yescharacteristic
41C3×#832C2×C311yescharacteristic
42C3×#816—31no—
43C3×Dic3⋊C8—31no—
44C3×Dic3⋊C8—31no—
45C3×#810—31no—
46#63572C2×C311yescharacteristic, complemented
47#63567C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C33×D4—21no—
50C33×D4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×#634—21no—
56C3×#634—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C32×Dic6—21no—
60C32×Dic6D412yes—
61C32×C3⋊C8—21no—
62C32×C3⋊C8—21no—
63C3×#464—91no—
64#5166—31no—
65#4767—31no—
66C2×C32×Q8—31no—
67C2×C32×D4—21no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—41no—
71C2×C32×D4C2×D311yescharacteristic
72C2×C32×D4C2×D311yescharacteristic
73C2×C32×D4C22×C311yescharacteristic
74C2×#634C22×C311yescharacteristic
75C2×C3×Dic6—21no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—31no—
78C2×C3×Dic6—61no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C32×SD16—62no—
82C2×C32×C8—31no—
83C2×C32×C8—31no—
84C32×C4⋊C4—31no—
85#3757—21no—
86#3757—21no—
87C4×C3⋊Dic3C22×C311yescharacteristic
88C3×#217—61no—
89C3×#217—61no—
90C3×#217—61no—
91C3×#217—61no—
92C3×#217—62no—
93C3×#217—62no—
94C3×#216—62no—
95C3×#214—24no—
96C3×#214—22no—
97C3×#214—22no—
98C3×C4⋊Dic3—21no—
99C3×C4⋊Dic3—61no—
100C3×C4⋊Dic3C2×D311yescharacteristic
101C3×C4⋊Dic3C22×C311yescharacteristic
102C3×Dic3⋊C4—61no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3—31no—
105C3×C4×Dic3—61no—
106C3×C4×Dic3—61no—
107C3×C4×Dic3—61no—
108C2×C3×C3⋊C8—21no—
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113C2×C3×C3⋊C8C22×C311yescharacteristic
114C3×Dic12—62no—
115Dic3⋊Dic3—21no—
116Dic32—21no—
117#3719—22no—
118C22×C33—41no—
119C22×C33—41no—
120C22×C33—41no—
121C22×C33C2×D411yescharacteristic
122C33×C4C2×D411yescharacteristic
123C33×C4#4111yescharacteristic
124C3×C3⋊Dic3—21no—
125C3×C3⋊Dic3—21no—
126C3×C3⋊Dic3—41no—
127C32×Dic3—22no—
128C32×Dic3—41no—
129C2×C3×Q16—91no—
130C2×C3×SD16—31no—
131C2×C3×SD16—31no—
132C2×C3×SD16—61no—
133C3×#124—91no—
134C3×C4×Q8—91no—
135Q8×Dic3—31no—
136C2×#216—31no—
137#936—31no—
138#936—31no—
139#936—61no—
140C2×#214C3×D311yescharacteristic
141C2×Dic12—31no—
142#887—31no—
143C3×C4⋊C8—91no—
144C3×#103—91no—
145C3×#102—31no—
146C3×#102—31no—
147C3×#102—61no—
148#832C3×D311yescharacteristic
149#816—31no—
150Dic3⋊C8—31no—
151Dic3⋊C8—31no—
152#810—31no—
153C23×C32—21no—
154C23×C32—22no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—41no—
158C23×C32—41no—
159C23×C32—42no—
160C32×Q8—32no—
161C32×Q8—61no—
162C32×D4—21no—
163C32×D4—21no—
164C32×D4—22no—
165C32×D4—22no—
166C32×D4—21no—
167C32×D4—21no—
168C32×D4—21no—
169C32×D4—21no—
170C32×D4—41no—
171C32×D4—41no—
172C32×D4—41no—
173C32×D4—41no—
174C32×D4—42no—
175C32×D4—42no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—41no—
180C2×C32×C4—61no—
181C2×C32×C4—61no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C23×C311yescharacteristic
185C2×C3⋊Dic3—21no—
186C2×C3⋊Dic3C3×D411yescharacteristic
187C2×C3⋊Dic3C3×D411yescharacteristic
188#634—21no—
189#634—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—41no—
199C2×C3×Dic3—41no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—62no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—61no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—22no—
208C3×Dic6—41no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—121no—
214C3×Dic6—121no—
215C3×Dic6#6712yes—
216C3×Dic6C3×D412yes—
217C32×C8—61no—
218C32×C8—61no—
219C3×C3⋊C8—22no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—21no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—41no—
225#464—91noSylow
226C2×C33—81no—
227C2×C33#12111yescharacteristic
228C2×C33#13511yescharacteristic
229C2×C33#13711yescharacteristic
230C2×C3×Q8—31no—
231C2×C3×Q8—31no—
232C2×C3×Q8—61no—
233C2×C3×Q8—91no—
234C2×C3×D4—21no—
235C2×C3×D4—21no—
236C2×C3×D4—21no—
237C2×C3×D4—41no—
238C2×C3×D4D3211yescharacteristic
239C2×C3×D4C2×C3×D311yescharacteristic
240C2×C3×D4C2×C3×D311yescharacteristic
241C2×Dic6—31no—
242C2×Dic6—31no—
243C2×Dic6—61no—
244C2×Dic6C2×C3×D311yescharacteristic
245C3×Q16—182no—
246C3×SD16—62no—
247C3×SD16—62no—
248C3×SD16—64no—
249C2×C3×C8—31no—
250C2×C3×C8—31no—
251C2×C3×C8—31no—
252C2×C3×C8—31no—
253C2×C3×C8—61no—
254C2×C3×C8—61no—
255C3×C4⋊C4—31no—
256C3×C4⋊C4—31no—
257C3×C4⋊C4—61no—
258C3×C4⋊C4—181no—
259C3×C22⋊C4—181no—
260C3×C22⋊C4—181no—
261C3×C42—91no—
262C3×C42—181no—
263#217—61no—
264#217—61no—
265#217—61no—
266#217—61no—
267#217—62no—
268#217—62no—
269#216—62no—
270#214—22no—
271C4⋊Dic3—61no—
272C4⋊Dic3C2×C3×D311yescharacteristic
273Dic3⋊C4—61no—
274C4×Dic3—31no—
275C4×Dic3—31no—
276C4×Dic3—61no—
277C4×Dic3—61no—
278C4×Dic3—61no—
279C2×C3⋊C8C2×C3×D311yescharacteristic
280C2×C3⋊C8C2×C3×D311yescharacteristic
281Dic12—62no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—41no—
286C22×C32—42no—
287C22×C32—42no—
288C22×C32—42no—
289C22×C32—44no—
290C22×C32—41no—
291C22×C32—41no—
292C22×C32—41no—
293C22×C32—41no—
294C22×C32—41no—
295C22×C32—42no—
296C22×C32—42no—
297C22×C32—41no—
298C22×C32—41no—
299C22×C32—41no—
300C22×C32—41no—
301C22×C32—81no—
302C22×C32—81no—
303C22×C32—81no—
304C22×C32—81no—
305C22×C32—82no—
306C22×C32—82no—
307C22×C32D3×D411yescharacteristic
308C22×C32D3×D411yescharacteristic
309C22×C32C2×C3×D411yescharacteristic
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—41no—
317C32×C4—41no—
318C32×C4—62no—
319C32×C4—121no—
320C32×C4#23511yescharacteristic
321C32×C4D3×D411yescharacteristic
322C32×C4#23711yescharacteristic
323C32×C4C2×#6711yescharacteristic
324C32×C4C2×C3×D411yescharacteristic
325C32×C4C3×#4111yescharacteristic
326C3⋊Dic3—21no—
327C3⋊Dic3—21no—
328C3⋊Dic3—41no—
329C3×Dic3—22no—
330C3×Dic3—22no—
331C3×Dic3—42no—
332C3×Dic3—41no—
333C3×Dic3—41no—
334C3×Dic3—61no—
335C3×Dic3—61no—
336C3×Dic3—61no—
337C3×Dic3—61no—
338C3×Dic3—81no—
339C3×Dic3—121no—
340C3×Dic3—121no—
341C3×Dic3—122no—
342C3×Dic3—121no—
343C3×Dic3—121no—
344C2×Q16—91no—
345C2×SD16—31no—
346#124—91no—
347C4×Q8—91no—
348C4⋊C8—91no—
349#103—91no—
350#102—31no—
351C33#46411yescharacteristic, complemented, Sylow
352C23×C3—22no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—41no—
357C23×C3—41no—
358C23×C3—42no—
359C3×Q8—32no—
360C3×Q8—32no—
361C3×Q8—62no—
362C3×Q8—61no—
363C3×Q8—61no—
364C3×Q8—121no—
365C3×Q8—181no—
366C3×Q8—181no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—21no—
370C3×D4—21no—
371C3×D4—21no—
372C3×D4—21no—
373C3×D4—21no—
374C3×D4—21no—
375C3×D4—41no—
376C3×D4—41no—
377C3×D4—41no—
378C3×D4—41no—
379C3×D4—42no—
380C3×D4—42no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4—61no—
389C2×C3×C4—91no—
390C2×C3×C4—91no—
391C2×C3×C4—121no—
392C2×C3×C4—121no—
393C2×C3×C4—181no—
394C2×C3×C4C2×D3211yescharacteristic
395C2×C3×C4C22×C3×D311yescharacteristic
396C2×C3×C4C22×C3×D311yescharacteristic
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—31no—
400C2×Dic3—31no—
401C2×Dic3—31no—
402C2×Dic3—31no—
403C2×Dic3—61no—
404C2×Dic3—61no—
405C2×Dic3—62no—
406C2×Dic3—61no—
407C2×Dic3—61no—
408Dic6—21no—
409Dic6—61no—
410Dic6—61no—
411Dic6—61no—
412Dic6—61no—
413Dic6—121no—
414Dic6—121no—
415Dic6C3×#6712yes—
416C3×C8—62no—
417C3×C8—61no—
418C3×C8—61no—
419C3×C8—61no—
420C3×C8—61no—
421C3×C8—121no—
422C3⋊C8—21no—
423C3⋊C8—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—82no—
437C2×C32—82no—
438C2×C32—82no—
439C2×C32—84no—
440C2×C32—81no—
441C2×C32—81no—
442C2×C32—81no—
443C2×C32#88411yescharacteristic
444C2×C32#91211yescharacteristic
445C2×C32#91511yescharacteristic
446C2×C32#91511yescharacteristic
447C2×C32#91611yescharacteristic
448C2×C32#93911yescharacteristic
449C2×C32C3×#12111yescharacteristic
450C2×C32C3×#13511yescharacteristic
451C2×C32C3×#13711yescharacteristic
452C2×Q8—31no—
453C2×Q8—91no—
454C2×D4C3×D3211yescharacteristic
455Q16—182no—
456SD16—62no—
457C2×C8—31no—
458C2×C8—31no—
459C4⋊C4—31no—
460C4⋊C4—181no—
461C22⋊C4—181no—
462C22⋊C4—181no—
463C42—91no—
464C42—181no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—44no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3—82no—
489C22×C3—82no—
490C22×C3#381111yescharacteristic
491C22×C3C3×D3×D411yescharacteristic
492C22×C3C3×D3×D411yescharacteristic
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—62no—
502C3×C4—62no—
503C3×C4—122no—
504C3×C4—121no—
505C3×C4—121no—
506C3×C4—181no—
507C3×C4—181no—
508C3×C4—241no—
509C3×C4—361no—
510C3×C4#380411yescharacteristic
511C3×C4D3×#6711yescharacteristic
512C3×C4C3×#23511yescharacteristic
513C3×C4C3×D3×D411yescharacteristic
514C3×C4C3×#23711yescharacteristic
515C3×C4C2×C3×#6711yescharacteristic
516Dic3—22no—
517Dic3—41no—
518Dic3—61no—
519Dic3—61no—
520Dic3—61no—
521Dic3—61no—
522Dic3—121no—
523Dic3—121no—
524Dic3—122no—
525Dic3—121no—
526Dic3—121no—
527C32—21no—
528C32—21no—
529C32—21no—
530C32—41no—
531C32#476711yescharacteristic, complemented
532C32#516611yescharacteristic, complemented
533C32C3×#46411yescharacteristic, complemented
534C23—21no—
535Q8—32no—
536Q8—61no—
537Q8—181no—
538Q8—181no—
539D4—21no—
540D4—21no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—91no—
544C2×C4—91no—
545C2×C4—181no—
546C2×C4C2×C3×D3211yescharacteristic
547C8—61no—
548C8—61no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—41no—
559C2×C3—41no—
560C2×C3—41no—
561C2×C3—82no—
562C2×C3—82no—
563C2×C3—82no—
564C2×C3—84no—
565C2×C3—81no—
566C2×C3—81no—
567C2×C3—81no—
568C2×C3#6393111yescharacteristic
569C2×C3#6393611yescharacteristic
570C2×C3#6397111yescharacteristic
571C2×C3C3×#88411yescharacteristic
572C2×C3C3×#91211yescharacteristic
573C2×C3C3×#91511yescharacteristic
574C2×C3C3×#91511yescharacteristic
575C2×C3C3×#91611yescharacteristic
576C2×C3C3×#93911yescharacteristic
577C22—41no—
578C22—41no—
579C22—41no—
580C22C3×#381111yescharacteristic
581C4—62no—
582C4—121no—
583C4—181no—
584C4—181no—
585C4—361no—
586C4C3×#380411yescharacteristic
587C4C3×D3×#6711yescharacteristic
588C3—21no—
589C3—21no—
590C3—21no—
591C3—41no—
592C3#1059144311yescharacteristic, direct product factor
593C3C3×#476711yescharacteristic, complemented
594C3C3×#516611yescharacteristic, complemented
595C2—81no—
596C2C3×#6393111yescharacteristic
597C2C3×#6393611yescharacteristic
598C2C3×#6397111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
Fitting, prime radicalC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×C3×D4D32
hypocenterC32C3×#464
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
3. C3×#63843C2semidirect
4. C2×C3×#3719C2semidirect
6. C3×#63572C2semidirect
7. C3×#63567C2semidirect
10. #10591443C3direct (N × H)
36. #63843C2×C3semidirect
37. C2×#3719C2×C3semidirect
46. #63572C2×C3semidirect
47. #63567C2×C3semidirect
351. C33#464semidirect
531. C32#4767semidirect
532. C32#5166semidirect
533. C32C3×#464semidirect
592. C3#10591443direct (N × H)
593. C3C3×#4767semidirect
594. C3C3×#5166semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C2×C32×#2142
C3×#638432
C2×C3×#37192
C32×#8322
C3×#635722
C3×#635672
C3×#51663
C3×#47673
#105914433