Finite Group Databasebeta

← C3×#10590360 #13157376

Subgroups — C3×#10590360 #13157376

Order 1728.

714 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#634C211yesmaximal, characteristic, complemented
2C32×C4×D12C211yesmaximal, characteristic
3C2×C3×#3716C211yesmaximal, characteristic
4C32×#803C211yesmaximal, characteristic, complemented
5C3×#63575C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic
8C3×#5018—31nomaximal
9C3×#4710—31nomaximal
10#10590360C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3—21no—
14C33×C42C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3750—21no—
17C3×C4×C3⋊Dic3—21no—
18C32×(C2×D3)⋊C4—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3716C414yescomplemented
23C32×C4×D4—31no—
24C4×#634C2×C311yescharacteristic, complemented
25C2×C3×#214—31no—
26C2×C3×#204—31no—
27C3×C4×D12—21no—
28C3×C4×D12D311yescharacteristic
29C3×C4×D12C2×C311yescharacteristic
30C3×C4×Dic6—31no—
31C3×C4×Dic6—31no—
32C3×C4×Dic6—61no—
33C2×#3716C2×C311yescharacteristic
34C32×#97—31no—
35C3×#832—31no—
36C3×#821—31no—
37C3×#818—31no—
38C3×#816—31no—
39C3×#810—31no—
40C3×#807—31no—
41C3×#803—21no—
42C3×#803D311yescharacteristic, complemented
43C3×#803C2×C311yescharacteristic, complemented
44#63575C2×C311yescharacteristic, complemented
45#63567C2×C311yescharacteristic, complemented
46#63563C2×C311yescharacteristic
47C22×C32×D3—21no—
48C2×C33×C4D411yescharacteristic
49C2×C33×C4D411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634C2×C412yescomplemented
55C2×C32×Dic3—21no—
56C32×D12—21no—
57C32×D12C2×C412yes—
58C32×C4×D3—42no—
59C32×C3⋊C8—21no—
60C32×C3⋊C8C2×C412yescomplemented
61C3×#440—91no—
62#5018—31no—
63#4710—31no—
64C2×C32×D4—31no—
65C22×C32×C4—61no—
66C2×#634C22×C311yescharacteristic
67C2×C3×D12—21no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×D12C22×C311yescharacteristic
70C2×C3×C4×D3—22no—
71C2×C3×C4×D3—21no—
72C2×C3×C4×D3—21no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—31no—
75C2×C3×Dic6—61no—
76C2×C32×C8—31no—
77C2×C32×C8—31no—
78C32×C4⋊C4—31no—
79C32×C22⋊C4—61no—
80C32×C42—21no—
81C32×C42—21no—
82C32×C42—21no—
83C32×C42—41no—
84C32×C42C2×D311yescharacteristic
85C32×C42C2×D311yescharacteristic
86C32×C42C22×C311yescharacteristic
87C4⋊(C3⋊Dic3)C22×C311yescharacteristic
88#3750—21no—
89C4×C3⋊Dic3—21no—
90C3×#214—34no—
91C3×(C2×D3)⋊C4—22no—
92C3×(C2×D3)⋊C4—21no—
93C3×(C2×D3)⋊C4—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—61no—
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4⋊Dic3C22×C311yescharacteristic
100C3×Dic3⋊C4—62no—
101C3×Dic3⋊C4—61no—
102C3×Dic3⋊C4—61no—
103C3×C4×Dic3—62no—
104C3×C4×Dic3—61no—
105C3×C4×Dic3—61no—
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8—21no—
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C2×C3×C3⋊C8C22×C311yescharacteristic
111C2×C3×C3⋊C8C22×C311yescharacteristic
112C3×#204—34no—
113#3716C3×C414yescomplemented
114C22×C33C2×D411yescharacteristic
115C2×C32×D3—22no—
116C2×C32×D3—41no—
117C2×C32×D3—41no—
118C33×C4—21no—
119C33×C4—21no—
120C33×C4C22×C411yescharacteristic
121C33×C4#4111yescharacteristic
122C3×C3⋊Dic3—22no—
123C3×C3⋊Dic3—41no—
124C32×Dic3—41no—
125C2×C3×SD16—91no—
126C3×C4×Q8—91no—
127C3×C4×D4—31no—
128C3×C4×D4—31no—
129C3×C4×D4—61no—
130C2×#214—31no—
131C2×#204—31no—
132C4×D12C3×D311yescharacteristic
133C4×Dic6—31no—
134C4×Dic6—31no—
135C4×Dic6—61no—
136C3×#107—91no—
137C3×#103—91no—
138C3×#102—91no—
139C3×#97—31no—
140C3×#97—31no—
141C3×#97—61no—
142#832—31no—
143#821—31no—
144#818—31no—
145#816—31no—
146#810—31no—
147#807—31no—
148#803C3×D311yescharacteristic, complemented
149C23×C32—61no—
150C22×C3×D3—22no—
151C22×C3×D3—21no—
152C22×C3×D3—21no—
153C32×D4—32no—
154C32×D4—61no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—41no—
165C2×C32×C4—41no—
166C2×C32×C4—41no—
167C2×C32×C4—61no—
168C2×C32×C4—122no—
169C2×C32×C4D1211yescharacteristic
170C2×C32×C4D1211yescharacteristic
171C2×C32×C4#6711yescharacteristic
172C2×C32×C4#6711yescharacteristic
173C2×C32×C4C3×D411yescharacteristic
174C2×C32×C4C3×D411yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C23×C311yescharacteristic
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—21no—
180#634—21no—
181#634C2×C3×C412yescomplemented
182C2×C3×Dic3—22no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—62no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C3×D12—22no—
192C3×D12—21no—
193C3×D12—21no—
194C3×D12—22no—
195C3×D12C4×D312yes—
196C3×D12C2×C3×C412yes—
197C3×C4×D3—44no—
198C3×C4×D3—42no—
199C3×C4×D3—42no—
200C3×Dic6—32no—
201C3×Dic6—32no—
202C3×Dic6—62no—
203C3×Dic6—62no—
204C3×Dic6—61no—
205C3×Dic6—61no—
206C32×C8—32no—
207C32×C8—61no—
208C3×C3⋊C8—22no—
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—22no—
212C3×C3⋊C8C4×D312yescomplemented
213C3×C3⋊C8C2×C3×C412yescomplemented
214#440—91noSylow
215C2×C33C4×D411yescharacteristic
216C2×C33AGL(1,ℤ/8)11yescharacteristic
217C2×C33#13711yescharacteristic
218C32×D3—42no—
219C2×C3×Q8—91no—
220C2×C3×D4—31no—
221C2×C3×D4—31no—
222C2×C3×D4—61no—
223C22×C3×C4—62no—
224C22×C3×C4—61no—
225C22×C3×C4—61no—
226C2×D12C2×C3×D311yescharacteristic
227C2×C4×D3—21no—
228C2×Dic6—31no—
229C2×Dic6—31no—
230C2×Dic6—61no—
231C3×SD16—94no—
232C2×C3×C8—31no—
233C2×C3×C8—31no—
234C2×C3×C8—31no—
235C2×C3×C8—31no—
236C2×C3×C8—61no—
237C2×C3×C8—61no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—61no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4—181no—
243C3×C22⋊C4—62no—
244C3×C22⋊C4—61no—
245C3×C22⋊C4—61no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—41no—
250C3×C42—181no—
251C3×C42D3211yescharacteristic
252C3×C42C2×C3×D311yescharacteristic
253C3×C42C2×C3×D311yescharacteristic
254#214—34no—
255(C2×D3)⋊C4—21no—
256C4⋊Dic3—31no—
257C4⋊Dic3—31no—
258C4⋊Dic3—61no—
259C4⋊Dic3C2×C3×D311yescharacteristic
260Dic3⋊C4—62no—
261Dic3⋊C4—61no—
262Dic3⋊C4—61no—
263C4×Dic3—62no—
264C4×Dic3—61no—
265C4×Dic3—61no—
266C2×C3⋊C8C2×C3×D311yescharacteristic
267C2×C3⋊C8C2×C3×D311yescharacteristic
268#204—34no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32—62no—
274C22×C32—121no—
275C22×C32—121no—
276C22×C32C2×D1211yescharacteristic
277C22×C32C2×#6711yescharacteristic
278C22×C32C2×C3×D411yescharacteristic
279C2×C3×D3—24no—
280C2×C3×D3—22no—
281C2×C3×D3—22no—
282C2×C3×D3—42no—
283C2×C3×D3—42no—
284C2×C3×D3—41no—
285C2×C3×D3—41no—
286C2×C3×D3—41no—
287C2×C3×D3—41no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—22no—
291C32×C4—22no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—41no—
303C32×C4—41no—
304C32×C4—41no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—41no—
308C32×C4—42no—
309C32×C4—42no—
310C32×C4—121no—
311C32×C4C2×C4×D311yescharacteristic
312C32×C4C2×C4×D311yescharacteristic
313C32×C4#23511yescharacteristic
314C32×C4#23511yescharacteristic
315C32×C4C22×C3×C411yescharacteristic
316C32×C4C3×#4111yescharacteristic
317C3⋊Dic3—22no—
318C3⋊Dic3—41no—
319C3×Dic3—42no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—64no—
323C3×Dic3—62no—
324C3×Dic3—62no—
325C3×Dic3—122no—
326C3×Dic3—121no—
327C3×Dic3—121no—
328C2×SD16—91no—
329C4×Q8—91no—
330C4×D4—31no—
331#107—91no—
332#103—91no—
333#102—91no—
334#97—31no—
335C33#44011yescharacteristic, complemented, Sylow
336C23×C3—62no—
337C23×C3—61no—
338C23×C3—61no—
339C22×D3—21no—
340C3×Q8—92no—
341C3×Q8—181no—
342C3×D4—32no—
343C3×D4—32no—
344C3×D4—62no—
345C3×D4—62no—
346C3×D4—61no—
347C3×D4—61no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—62no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—124no—
364C2×C3×C4—122no—
365C2×C3×C4—122no—
366C2×C3×C4—181no—
367C2×C3×C4—181no—
368C2×C3×C4#62611yescharacteristic
369C2×C3×C4#62611yescharacteristic
370C2×C3×C4C3×D1211yescharacteristic
371C2×C3×C4C3×D1211yescharacteristic
372C2×C3×C4C3×#6711yescharacteristic
373C2×C3×C4C3×#6711yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C22×C3×D311yescharacteristic
376C2×C3×C4C22×C3×D311yescharacteristic
377C2×Dic3—21no—
378C2×Dic3—62no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—62no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384D12—21no—
385D12C3×C4×D312yes—
386C4×D3—42no—
387Dic6—32no—
388Dic6—32no—
389Dic6—62no—
390Dic6—62no—
391Dic6—61no—
392Dic6—61no—
393C3×C8—32no—
394C3×C8—32no—
395C3×C8—62no—
396C3×C8—62no—
397C3×C8—61no—
398C3×C8—61no—
399C3⋊C8—21no—
400C3⋊C8C3×C4×D312yescomplemented
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—41no—
411C2×C32—41no—
412C2×C32—41no—
413C2×C32—122no—
414C2×C32C4×D1211yescharacteristic
415C2×C32#90811yescharacteristic
416C2×C32#90911yescharacteristic
417C2×C32C4×#6711yescharacteristic
418C2×C32#93211yescharacteristic
419C2×C32#95111yescharacteristic
420C2×C32C3×C4×D411yescharacteristic
421C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
422C2×C32C3×#13711yescharacteristic
423C3×D3—44no—
424C3×D3—42no—
425C3×D3—42no—
426C2×Q8—91no—
427C2×D4—31no—
428C22×C4—61no—
429SD16—94no—
430C2×C8—31no—
431C2×C8—31no—
432C4⋊C4—31no—
433C4⋊C4—91no—
434C4⋊C4—181no—
435C22⋊C4—61no—
436C42—181no—
437C42C3×D3211yescharacteristic
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—41no—
442C22×C3—64no—
443C22×C3—62no—
444C22×C3—62no—
445C22×C3—122no—
446C22×C3—122no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3—121no—
450C22×C3—121no—
451C22×C3C2×#62611yescharacteristic
452C22×C3C2×C3×D1211yescharacteristic
453C22×C3C2×C3×#6711yescharacteristic
454C2×D3—22no—
455C2×D3—41no—
456C2×D3—41no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—122no—
480C3×C4—121no—
481C3×C4—121no—
482C3×C4—182no—
483C3×C4—361no—
484C3×C4#379811yescharacteristic
485C3×C4C4×D3211yescharacteristic
486C3×C4C2×C3×C4×D311yescharacteristic
487C3×C4C2×C3×C4×D311yescharacteristic
488C3×C4C3×#23511yescharacteristic
489C3×C4C3×#23511yescharacteristic
490Dic3—41no—
491Dic3—64no—
492Dic3—62no—
493Dic3—62no—
494Dic3—122no—
495Dic3—121no—
496Dic3—121no—
497C32—21no—
498C32—21no—
499C32—21no—
500C32—41no—
501C32#471011yescharacteristic, complemented
502C32#501811yescharacteristic, complemented
503C32C3×#44011yescharacteristic, complemented
504C23—61no—
505Q8—92no—
506Q8—181no—
507D4—32no—
508D4—61no—
509C2×C4—61no—
510C2×C4—122no—
511C2×C4—181no—
512C2×C4—181no—
513C2×C4C3×#62611yescharacteristic
514C2×C4C3×#62611yescharacteristic
515C2×C4C2×C3×D3211yescharacteristic
516C8—32no—
517C8—61no—
518C2×C3—21no—
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—41no—
528C2×C3—41no—
529C2×C3—41no—
530C2×C3—124no—
531C2×C3—122no—
532C2×C3—122no—
533C2×C3#6383011yescharacteristic
534C2×C3#6383111yescharacteristic
535C2×C3C4×#62611yescharacteristic
536C2×C3C3×C4×D1211yescharacteristic
537C2×C3C3×#90811yescharacteristic
538C2×C3C3×#90911yescharacteristic
539C2×C3C3×C4×#6711yescharacteristic
540C2×C3C3×#93211yescharacteristic
541C2×C3C3×#95111yescharacteristic
542D3—42no—
543C22—62no—
544C22—121no—
545C22—121no—
546C22C2×C3×#62611yescharacteristic
547C4—21no—
548C4—21no—
549C4—121no—
550C4—182no—
551C4—361no—
552C4C3×#379811yescharacteristic
553C4C3×C4×D3211yescharacteristic
554C3—21no—
555C3—21no—
556C3—21no—
557C3—41no—
558C3#1059036011yescharacteristic, direct product factor
559C3C3×#471011yescharacteristic, complemented
560C3C3×#501811yescharacteristic, complemented
561C2—122no—
562C2C3×#6383011yescharacteristic
563C2C3×#6383111yescharacteristic
564C2C3×C4×#62611yescharacteristic
subgroupG/H
centerC22×C3C2×#626
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×#440
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
1. C3×C4×#634C2semidirect
4. C32×#803C2semidirect
5. C3×#63575C2semidirect
6. C3×#63567C2semidirect
10. #10590360C3direct (N × H)
22. C3×#3716C4semidirect
24. C4×#634C2×C3semidirect
42. C3×#803D3semidirect
43. C3×#803C2×C3semidirect
44. #63575C2×C3semidirect
45. #63567C2×C3semidirect
54. C3×#634C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
113. #3716C3×C4semidirect
148. #803C3×D3semidirect
181. #634C2×C3×C4semidirect
212. C3×C3⋊C8C4×D3semidirect
213. C3×C3⋊C8C2×C3×C4semidirect
335. C33#440semidirect
400. C3⋊C8C3×C4×D3semidirect
501. C32#4710semidirect
502. C32#5018semidirect
503. C32C3×#440semidirect
558. C3#10590360direct (N × H)
559. C3C3×#4710semidirect
560. C3C3×#5018semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6342
C32×C4×D122
C2×C3×#37162
C32×#8032
C3×#635752
C3×#635672
C3×#635632
C3×#50183
C3×#47103
#105903603