Finite Group Databasebeta

← C3×#10590877 #13157840

Subgroups — C3×#10590877 #13157840

Order 1728.

770 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3744C211yesmaximal, characteristic, complemented
2C2×C32×#204C211yesmaximal, characteristic, complemented
3C3×#63911C211yesmaximal, characteristic
4C3×#63866C211yesmaximal, characteristic, complemented
5C2×C3×#3717C211yesmaximal, characteristic, complemented
6C2×C3×#3716C211yesmaximal, characteristic, complemented
7C32×C8×Dic3C211yesmaximal, characteristic, complemented
8C3×#5175—31nomaximal
9C3×#4693—31nomaximal
10#10590877C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#626—22no—
16C2×C33×C8C2211yescharacteristic
17C3×#3744—22no—
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C32×#204—22no—
21C3×Dic3⋊Dic3—22no—
22C3×#3717—22no—
23C3×#3716—22no—
24C2×C32×SD16—31no—
25C2×#3744C2×C311yescharacteristic, complemented
26C3×#944—31no—
27C2×C3×#215—31no—
28C3×#940—31no—
29C2×C3×#214—31no—
30C2×C3×#204—21no—
31C2×C3×#204—31no—
32C2×C3×#204—31no—
33C2×C3×#204—31no—
34C2×C3×#204—31no—
35C2×C3×#204—61no—
36C2×C3×#204D311yescharacteristic, complemented
37C2×C3×#204C2×C311yescharacteristic, complemented
38C3×#874—31no—
39C3×#869—31no—
40#63911C2×C311yescharacteristic
41#63866C2×C311yescharacteristic, complemented
42C2×#3717C2×C311yescharacteristic, complemented
43C2×#3716C2×C311yescharacteristic, complemented
44C32×C4×C8—31no—
45C3×C8×Dic3—21no—
46C3×C8×Dic3D311yescharacteristic, complemented
47C3×C8×Dic3C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D412yes—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×#626—44no—
63C33×C8D412yescomplemented
64C32×C3⋊C8D412yescomplemented
65C3×#492—91no—
66#5175—31no—
67#4693—31no—
68C2×C32×Q8—31no—
69C2×C32×D4—31no—
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×#634C22×C311yescharacteristic
72C2×C3×#67—62no—
73C2×C3×D12—21no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×C3×D12—62no—
78C2×C3×D12C2×D311yescharacteristic
79C2×C3×D12C22×C311yescharacteristic
80C2×C3×Dic6—21no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C2×#626—22no—
87C32×SD16—62no—
88C2×C32×C8—21no—
89C2×C32×C8—21no—
90C2×C32×C8—21no—
91C2×C32×C8—31no—
92C2×C32×C8—41no—
93C2×C32×C8C2×D311yescharacteristic
94C2×C32×C8C2×D311yescharacteristic
95C2×C32×C8C22×C311yescharacteristic
96C32×C42—31no—
97#3744—22no—
98C3×#215—62no—
99C3×#214—62no—
100C3×C4⋊Dic3—62no—
101C3×Dic3⋊C4—62no—
102C3×C4×Dic3—21no—
103C3×C4×Dic3C2×D311yescharacteristic
104C3×C4×Dic3C22×C311yescharacteristic
105C2×C3×C3⋊C8—21no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C22×C311yescharacteristic
108C3×#204—24no—
109C3×#204—22no—
110C3×#204—22no—
111C3×#204—64no—
112C3×#204—62no—
113C3×#204—62no—
114C3×#204—62no—
115C3×#204—62no—
116Dic3⋊Dic3—22no—
117#3717—22no—
118#3716—22no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—42no—
122C2×C32×D3—41no—
123C2×C32×D3—42no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2×D411yescharacteristic
126C3×C3⋊Dic3—41no—
127C32×Dic3—41no—
128C32×Dic3SD1614yescomplemented
129C2×C3×SD16—31no—
130C2×C3×SD16—31no—
131C2×C3×SD16—61no—
132C2×C3×SD16—91no—
133C2×C3×SD16—91no—
134C2×C3×SD16—91no—
135C3×C4⋊Q8—91no—
136C3×#127—91no—
137#944—31no—
138C2×#215—31no—
139#940—31no—
140C2×#214—31no—
141C2×#204—31no—
142C2×#204—31no—
143C2×#204—31no—
144C2×#204—31no—
145C2×#204—61no—
146C2×#204C3×D311yescharacteristic, complemented
147#874—31no—
148#869—31no—
149C3×C4×C8—31no—
150C3×C4×C8—31no—
151C3×C4×C8—61no—
152C8×Dic3C3×D311yescharacteristic, complemented
153C23×C32—61no—
154C22×C3⋊D3—21no—
155C22×C3×D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3—41no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—121no—
161C32×Q8—61no—
162C32×Q8—61no—
163C32×D4—61no—
164C32×D4—61no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—32no—
169C2×C32×C4—41no—
170C2×C32×C4—61no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C23×C311yescharacteristic
174C2×C3⋊Dic3—21no—
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)—21no—
177#634—21no—
178#634—21no—
179C3×#67—124no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—41no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—121no—
187C2×C3×Dic3D1212yes—
188C2×C3×Dic3C3×D412yes—
189C3×D12—22no—
190C3×D12—21no—
191C3×D12—21no—
192C3×D12—21no—
193C3×D12—21no—
194C3×D12—41no—
195C3×D12—62no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—121no—
201C3×D12—124no—
202C3×Dic6—22no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—41no—
208C3×Dic6—62no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—121no—
214#626—44no—
215C32×C8—22no—
216C32×C8—22no—
217C32×C8—22no—
218C32×C8—32no—
219C32×C8—42no—
220C32×C8D1212yescomplemented
221C32×C8#6712yescomplemented
222C32×C8C3×D412yescomplemented
223C3×C3⋊C8—22no—
224C3×C3⋊C8D1212yescomplemented
225C3×C3⋊C8C3×D412yescomplemented
226#492—91noSylow
227C2×C33#12711yescharacteristic
228C2×C33C2×SD1612yes—
229C3×C3⋊D3—81no—
230C32×D3—81no—
231C2×C3×Q8—31no—
232C2×C3×Q8—31no—
233C2×C3×Q8—61no—
234C2×C3×Q8—91no—
235C2×C3×D4—31no—
236C2×C3×D4—31no—
237C2×C3×D4—61no—
238C2×C3×D4—91no—
239C2×C3×D4—182no—
240C2×#67—62no—
241C2×D12—31no—
242C2×D12—31no—
243C2×D12—61no—
244C2×D12—62no—
245C2×D12C2×C3×D311yescharacteristic
246C2×Dic6—31no—
247C2×Dic6—31no—
248C2×Dic6—61no—
249C2×Dic6C2×C3×D311yescharacteristic
250C3×SD16—64no—
251C3×SD16—62no—
252C3×SD16—62no—
253C3×SD16—182no—
254C3×SD16—182no—
255C3×SD16—182no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—31no—
260C2×C3×C8—31no—
261C2×C3×C8—41no—
262C2×C3×C8—61no—
263C2×C3×C8D3211yescharacteristic
264C2×C3×C8C2×C3×D311yescharacteristic
265C2×C3×C8C2×C3×D311yescharacteristic
266C3×C4⋊C4—182no—
267C3×C42—31no—
268C3×C42—31no—
269C3×C42—61no—
270#215—62no—
271#214—62no—
272C4⋊Dic3—62no—
273Dic3⋊C4—62no—
274C4×Dic3C2×C3×D311yescharacteristic
275C2×C3⋊C8C2×C3×D311yescharacteristic
276#204—22no—
277#204—64no—
278#204—62no—
279#204—62no—
280#204—62no—
281#204—62no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—41no—
286C22×C32—121no—
287C22×C32—122no—
288C22×C32C2×D1211yescharacteristic
289C22×C32D3×D411yescharacteristic
290C22×C32C2×C3×D411yescharacteristic
291C2×C3⋊D3—41no—
292C2×C3⋊D3—42no—
293C2×C3×D3—42no—
294C2×C3×D3—42no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C2×C3×D3—42no—
298C2×C3×D3—82no—
299C2×C3×D3—122no—
300C2×C3×D3—121no—
301C2×C3×D3—122no—
302C2×C3×D3—122no—
303C2×C3×D3—121no—
304C2×C3×D3—242no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—34no—
312C32×C4—41no—
313C32×C4—41no—
314C32×C4—121no—
315C32×C4C2×D1211yescharacteristic
316C32×C4C2×D1211yescharacteristic
317C32×C4D3×D411yescharacteristic
318C32×C4C2×#6711yescharacteristic
319C32×C4C2×C3×D411yescharacteristic
320C32×C4C2×C3×D411yescharacteristic
321C3⋊Dic3—41no—
322C3×Dic3—24no—
323C3×Dic3—42no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3—122no—
327C3×Dic3—121no—
328C3×Dic3—121no—
329C3×Dic3#20414yescomplemented
330C3×Dic3C3×SD1614yescomplemented
331C2×SD16—31no—
332C2×SD16—91no—
333C2×SD16—91no—
334C2×SD16—91no—
335C4⋊Q8—91no—
336#127—91no—
337C4×C8—31no—
338C33#49211yescharacteristic, complemented, Sylow
339C23×C3—61no—
340C23×C3—61no—
341C23×C3—121no—
342C23×C3—181no—
343C22×D3—21no—
344C22×D3—61no—
345C22×D3—61no—
346C22×D3—121no—
347C3×Q8—62no—
348C3×Q8—61no—
349C3×Q8—61no—
350C3×Q8—61no—
351C3×Q8—61no—
352C3×Q8—121no—
353C3×Q8—181no—
354C3×Q8—181no—
355C3×D4—62no—
356C3×D4—61no—
357C3×D4—61no—
358C3×D4—61no—
359C3×D4—61no—
360C3×D4—121no—
361C3×D4—181no—
362C3×D4—181no—
363C3×D4—364no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—32no—
368C2×C3×C4—32no—
369C2×C3×C4—41no—
370C2×C3×C4—62no—
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
378#67—124no—
379C2×Dic3—21no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—121no—
383C2×Dic3C3×D1212yes—
384D12—21no—
385D12—21no—
386D12—62no—
387D12—61no—
388D12—61no—
389D12—61no—
390D12—61no—
391D12—121no—
392D12—124no—
393Dic6—21no—
394Dic6—21no—
395Dic6—62no—
396Dic6—61no—
397Dic6—61no—
398Dic6—61no—
399Dic6—61no—
400Dic6—121no—
401C3×C8—22no—
402C3×C8—22no—
403C3×C8—22no—
404C3×C8—32no—
405C3×C8—32no—
406C3×C8—42no—
407C3×C8—62no—
408C3×C8#62612yescomplemented
409C3×C8C3×D1212yescomplemented
410C3×C8C3×#6712yescomplemented
411C3⋊C8C3×D1212yescomplemented
412C2×C32—22no—
413C2×C32—22no—
414C2×C32—22no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—41no—
419C2×C32—42no—
420C2×C32—241no—
421C2×C32#87411yescharacteristic
422C2×C32C2×#20412yes—
423C2×C32D3×SD1612yes—
424C2×C32#94011yescharacteristic
425C2×C32C3×#12711yescharacteristic
426C2×C32C2×C3×SD1612yes—
427C3⋊D3—81no—
428C3×D3—82no—
429C3×D3—81no—
430C3×D3—81no—
431C3×D3—242no—
432C3×D3—241no—
433C3×D3—241no—
434C2×Q8—31no—
435C2×Q8—91no—
436C2×D4—31no—
437C2×D4—91no—
438C2×D4—182no—
439SD16—62no—
440SD16—182no—
441SD16—182no—
442SD16—182no—
443C2×C8—31no—
444C2×C8C3×D3211yescharacteristic
445C4⋊C4—182no—
446C42—31no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—41no—
451C22×C3—122no—
452C22×C3—122no—
453C22×C3—122no—
454C22×C3—121no—
455C22×C3—121no—
456C22×C3—242no—
457C22×C3—362no—
458C22×C3—361no—
459C22×C3D3×D1211yescharacteristic
460C22×C3C2×C3×D1211yescharacteristic
461C22×C3C3×D3×D411yescharacteristic
462C2×D3—42no—
463C2×D3—41no—
464C2×D3—122no—
465C2×D3—122no—
466C2×D3—122no—
467C2×D3—121no—
468C2×D3—121no—
469C2×D3—242no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—34no—
477C3×C4—34no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—64no—
481C3×C4—122no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4—361no—
485C3×C4D3×D1211yescharacteristic
486C3×C4C2×#62611yescharacteristic
487C3×C4C2×C3×D1211yescharacteristic
488C3×C4C2×C3×D1211yescharacteristic
489C3×C4C3×D3×D411yescharacteristic
490C3×C4C2×C3×#6711yescharacteristic
491Dic3—41no—
492Dic3—122no—
493Dic3—121no—
494Dic3—121no—
495Dic3C3×#20414yescomplemented
496C32—21no—
497C32—21no—
498C32—21no—
499C32—41no—
500C32#469311yescharacteristic, complemented
501C32#517511yescharacteristic, complemented
502C32C3×#49211yescharacteristic, complemented
503C23—61no—
504C23—181no—
505Q8—61no—
506Q8—61no—
507Q8—181no—
508Q8—181no—
509D4—61no—
510D4—61no—
511D4—181no—
512D4—181no—
513D4—364no—
514C2×C4—32no—
515C2×C4—61no—
516C2×C4—181no—
517C2×C4C2×C3×D3211yescharacteristic
518C8—32no—
519C8C3×#62612yescomplemented
520C2×C3—22no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—42no—
527C2×C3—41no—
528C2×C3—242no—
529C2×C3—241no—
530C2×C3—241no—
531C2×C3—721no—
532C2×C3D3×#20412yes—
533C2×C3#6391111yescharacteristic
534C2×C3C3×#87411yescharacteristic
535C2×C3C2×C3×#20412yes—
536C2×C3C3×D3×SD1612yes—
537C2×C3C3×#94011yescharacteristic
538D3—81no—
539D3—242no—
540D3—241no—
541D3—241no—
542C22—122no—
543C22—121no—
544C22—362no—
545C22—361no—
546C22C3×D3×D1211yescharacteristic
547C4—34no—
548C4—121no—
549C4—361no—
550C4C3×D3×D1211yescharacteristic
551C4C2×C3×#62611yescharacteristic
552C3—21no—
553C3—21no—
554C3—21no—
555C3—41no—
556C3#1059087711yescharacteristic, direct product factor
557C3C3×#469311yescharacteristic, complemented
558C3C3×#517511yescharacteristic, complemented
559C2—241no—
560C2—721no—
561C2C3×D3×#20412yes—
562C2C3×#6391111yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypercenterC2×C3×C8D32
hypocenterC32C3×#492
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3744C2semidirect
2. C2×C32×#204C2semidirect
4. C3×#63866C2semidirect
5. C2×C3×#3717C2semidirect
6. C2×C3×#3716C2semidirect
7. C32×C8×Dic3C2semidirect
10. #10590877C3direct (N × H)
25. C2×#3744C2×C3semidirect
36. C2×C3×#204D3semidirect
37. C2×C3×#204C2×C3semidirect
41. #63866C2×C3semidirect
42. C2×#3717C2×C3semidirect
43. C2×#3716C2×C3semidirect
46. C3×C8×Dic3D3semidirect
47. C3×C8×Dic3C2×C3semidirect
63. C33×C8D4semidirect
64. C32×C3⋊C8D4semidirect
128. C32×Dic3SD16semidirect
146. C2×#204C3×D3semidirect
152. C8×Dic3C3×D3semidirect
220. C32×C8D12semidirect
221. C32×C8#67semidirect
222. C32×C8C3×D4semidirect
224. C3×C3⋊C8D12semidirect
225. C3×C3⋊C8C3×D4semidirect
329. C3×Dic3#204semidirect
330. C3×Dic3C3×SD16semidirect
338. C33#492semidirect
408. C3×C8#626semidirect
409. C3×C8C3×D12semidirect
410. C3×C8C3×#67semidirect
411. C3⋊C8C3×D12semidirect
495. Dic3C3×#204semidirect
500. C32#4693semidirect
501. C32#5175semidirect
502. C32C3×#492semidirect
519. C8C3×#626semidirect
556. C3#10590877direct (N × H)
557. C3C3×#4693semidirect
558. C3C3×#5175semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37442
C2×C32×#2042
C3×#639112
C3×#638662
C2×C3×#37172
C2×C3×#37162
C32×C8×Dic32
C3×#51753
C3×#46933
#105908773