Finite Group Databasebeta

← C3×#10591404 #13158440

Subgroups — C3×#10591404 #13158440

Order 1728.

802 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3753C211yesmaximal, characteristic
2C32×D4×Dic3C211yesmaximal, characteristic, complemented
3C3×#63894C211yesmaximal, characteristic, complemented
4C32×#832C211yesmaximal, characteristic, complemented
5C3×#63574C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8C3×#5014—31nomaximal
9C3×#4785—31nomaximal
10#10591404C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C22×C32×Dic3—21no—
14C2×C3×#624—21no—
15C3×#3753C414yescomplemented
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×#217—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3722—21no—
23C32×C4×D4—31no—
24C2×#3753C2×C311yescharacteristic
25C3×D4×Dic3—21no—
26C3×D4×Dic3D311yescharacteristic, complemented
27C3×D4×Dic3C2×C311yescharacteristic, complemented
28C2×C3×#213—31no—
29C2×C3×#213—31no—
30C2×C3×#213—61no—
31C3×#893—31no—
32C3×C4×D12—31no—
33#63894C2×C311yescharacteristic, complemented
34C32×#102—31no—
35C3×#832—21no—
36C3×#832D311yescharacteristic, complemented
37C3×#832C2×C311yescharacteristic, complemented
38C3×#821—31no—
39C3×#815—31no—
40C3×#809—31no—
41C3×#808—31no—
42C3×#808—31no—
43C3×#803—31no—
44#63574C2×C311yescharacteristic, complemented
45#63564C2×C311yescharacteristic, complemented
46#63554C2×C311yescharacteristic, complemented
47C23×C33—21no—
48C22×C3×C3⋊D3—21no—
49C33×D4—21no—
50C33×D4C2×C412yescomplemented
51C2×C33×C4C2311yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)C2×C412yescomplemented
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—42no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C3×#624—42no—
60C3×C3⋊(C3⋊C8)C2×C412yescomplemented
61C32×C3⋊C8—21no—
62C3×#442—91no—
63#5014—31no—
64#4785—31no—
65C2×C32×D4—21no—
66C2×C32×D4—21no—
67C2×C32×D4—21no—
68C2×C32×D4—41no—
69C2×C32×D4C2×D311yescharacteristic
70C2×C32×D4C2×D311yescharacteristic
71C2×C32×D4C22×C311yescharacteristic
72C22×C32×C4—61no—
73C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
74C22×C3×Dic3—22no—
75C22×C3×Dic3—21no—
76C22×C3×Dic3—21no—
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—61no—
80C2×C3×C4×D3—61no—
81C2×C3×C4×D3—61no—
82C2×#624—21no—
83C2×C32×C8—31no—
84C32×C4⋊C4—31no—
85C32×C4⋊C4—31no—
86C32×C22⋊C4—61no—
87C32×C42—31no—
88#3753C3×C414yescomplemented
89C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
90C3×#217—22no—
91C3×#217—21no—
92C3×#217—21no—
93C3×#213—34no—
94C3×#213—34no—
95C3×#213—64no—
96C3×(C2×D3)⋊C4—61no—
97C3×(C2×D3)⋊C4—61no—
98C3×C4⋊Dic3—21no—
99C3×C4⋊Dic3—21no—
100C3×C4⋊Dic3C2×D311yescharacteristic
101C3×C4⋊Dic3C2×D311yescharacteristic
102C3×C4⋊Dic3C22×C311yescharacteristic
103C3×C4⋊Dic3C22×C311yescharacteristic
104C3×C4×Dic3—21no—
105C3×C4×Dic3C2×D311yescharacteristic
106C3×C4×Dic3C22×C311yescharacteristic
107C2×C3×C3⋊C8—21no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—31no—
110C2×C3×C3⋊C8—61no—
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113#3722—21no—
114C22×C33—22no—
115C22×C33—41no—
116C22×C33—41no—
117C22×C33C2×D411yescharacteristic
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C33×C4C22×C411yescharacteristic
122C33×C4#4111yescharacteristic
123C32×Dic3—21no—
124C32×Dic3—21no—
125C32×Dic3—41no—
126C32×Dic3—41no—
127C2×C3×D8—91no—
128C3×C4×D4—31no—
129C3×C4×D4—31no—
130C3×C4×D4—61no—
131C3×C4×D4—91no—
132D4×Dic3C3×D311yescharacteristic, complemented
133C2×#213—31no—
134C2×#213—31no—
135C2×#213—61no—
136#893—31no—
137C4×D12—31no—
138C3×#106—91no—
139C3×#102—31no—
140C3×#102—31no—
141C3×#102—61no—
142C3×#102—91no—
143C3×#97—91no—
144#832C3×D311yescharacteristic, complemented
145#821—31no—
146#815—31no—
147#809—31no—
148#808—31no—
149#808—31no—
150#803—31no—
151C23×C32—21no—
152C23×C32—22no—
153C23×C32—21no—
154C23×C32—21no—
155C23×C32—41no—
156C23×C32—41no—
157C23×C32—42no—
158C22×C3⋊D3—21no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—121no—
162C32×D4—21no—
163C32×D4—22no—
164C32×D4—21no—
165C32×D4—21no—
166C32×D4—22no—
167C32×D4—22no—
168C32×D4—22no—
169C32×D4—42no—
170C32×D4—41no—
171C32×D4—41no—
172C32×D4—42no—
173C32×D4C4×D312yescomplemented
174C32×D4C4×D312yescomplemented
175C32×D4C2×C3×C412yescomplemented
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—31no—
180C2×C32×C4—31no—
181C2×C32×C4—41no—
182C2×C32×C4—61no—
183C2×C32×C4—61no—
184C2×C32×C4—122no—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C23×C311yescharacteristic
188C4⋊(C3⋊D3)—21no—
189C4⋊(C3⋊D3)C2×C3×C412yescomplemented
190C2×C3×Dic3—22no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—41no—
198C2×C3×Dic3—44no—
199C2×C3×Dic3—42no—
200C2×C3×Dic3—42no—
201C2×C3×Dic3#6711yescharacteristic
202C2×C3×Dic3#6711yescharacteristic
203C2×C3×Dic3C3×D411yescharacteristic
204C2×C3×Dic3C3×D411yescharacteristic
205C3×D12—32no—
206C3×D12—32no—
207C3×D12—62no—
208C3×D12—61no—
209C3×D12—61no—
210C3×D12—121no—
211C3×C4×D3—122no—
212C3×C4×D3—122no—
213#624—42no—
214C32×C8—61no—
215C3⋊(C3⋊C8)C2×C3×C412yescomplemented
216C3×C3⋊C8—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—32no—
219C3×C3⋊C8—32no—
220C3×C3⋊C8—41no—
221C3×C3⋊C8—62no—
222#442—91noSylow
223C2×C33—42no—
224C2×C33C4×D411yescharacteristic
225C2×C33AGL(1,ℤ/8)11yescharacteristic
226C2×C33AGL(1,ℤ/8)11yescharacteristic
227C3×C3⋊D3—42no—
228C2×C3×D4—21no—
229C2×C3×D4—21no—
230C2×C3×D4—21no—
231C2×C3×D4—41no—
232C2×C3×D4—91no—
233C2×C3×D4D3211yescharacteristic
234C2×C3×D4C2×C3×D311yescharacteristic
235C2×C3×D4C2×C3×D311yescharacteristic
236C22×C3×C4—61no—
237C22×C3×C4—61no—
238C22×C3×C4—62no—
239C22×C3×C4—181no—
240C22×Dic3—21no—
241C2×D12—31no—
242C2×D12—31no—
243C2×D12—61no—
244C2×C4×D3—61no—
245C2×C4×D3—61no—
246C3×D8—94no—
247C2×C3×C8—31no—
248C2×C3×C8—31no—
249C2×C3×C8—61no—
250C2×C3×C8—91no—
251C3×C4⋊C4—31no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—31no—
254C3×C4⋊C4—31no—
255C3×C4⋊C4—61no—
256C3×C4⋊C4—61no—
257C3×C22⋊C4—61no—
258C3×C22⋊C4—61no—
259C3×C22⋊C4—62no—
260C3×C22⋊C4—181no—
261C3×C42—31no—
262C3×C42—31no—
263C3×C42—61no—
264#217—21no—
265#213—34no—
266#213—34no—
267#213—64no—
268(C2×D3)⋊C4—61no—
269(C2×D3)⋊C4—61no—
270C4⋊Dic3C2×C3×D311yescharacteristic
271C4⋊Dic3C2×C3×D311yescharacteristic
272C4×Dic3C2×C3×D311yescharacteristic
273C2×C3⋊C8—31no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—61no—
276C2×C3⋊C8C2×C3×D311yescharacteristic
277C22×C32—22no—
278C22×C32—24no—
279C22×C32—22no—
280C22×C32—22no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—41no—
285C22×C32—42no—
286C22×C32—42no—
287C22×C32—44no—
288C22×C32—41no—
289C22×C32—41no—
290C22×C32—42no—
291C22×C32—42no—
292C22×C32—41no—
293C22×C32—41no—
294C22×C32—41no—
295C22×C32—41no—
296C22×C32—81no—
297C22×C32—81no—
298C22×C32—81no—
299C22×C32—81no—
300C22×C32—82no—
301C22×C32—82no—
302C22×C32D3×D411yescharacteristic
303C22×C32C2×#6711yescharacteristic
304C22×C32C2×C3×D411yescharacteristic
305C2×C3⋊D3—22no—
306C2×C3⋊D3—41no—
307C2×C3⋊D3—41no—
308C2×C3×D3—62no—
309C2×C3×D3—62no—
310C2×C3×D3—122no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C2×C3×D3—121no—
314C2×C3×D3—121no—
315C2×C3×D3—241no—
316C2×C3×D3—241no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—41no—
324C32×C4—41no—
325C32×C4—61no—
326C32×C4—61no—
327C32×C4—121no—
328C32×C4—121no—
329C32×C4C2×C4×D311yescharacteristic
330C32×C4C2×C4×D311yescharacteristic
331C32×C4#23511yescharacteristic
332C32×C4#23711yescharacteristic
333C32×C4C22×C3×C411yescharacteristic
334C32×C4C3×#4111yescharacteristic
335C3×Dic3—22no—
336C3×Dic3—21no—
337C3×Dic3—21no—
338C3×Dic3—21no—
339C3×Dic3—21no—
340C3×Dic3—41no—
341C3×Dic3—42no—
342C3×Dic3—41no—
343C3×Dic3—41no—
344C3×Dic3—41no—
345C3×Dic3—41no—
346C3×Dic3—81no—
347C2×D8—91no—
348C4×D4—31no—
349C4×D4—91no—
350#106—91no—
351#102—31no—
352#102—91no—
353#97—91no—
354C33#44211yescharacteristic, complemented, Sylow
355C23×C3—22no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—42no—
362C23×C3—181no—
363C22×D3—61no—
364C22×D3—61no—
365C22×D3—121no—
366C3×D4—22no—
367C3×D4—21no—
368C3×D4—21no—
369C3×D4—21no—
370C3×D4—22no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—41no—
374C3×D4—41no—
375C3×D4—42no—
376C3×D4—42no—
377C3×D4—92no—
378C3×D4—181no—
379C3×D4#62412yescomplemented
380C3×D4C3×C4×D312yescomplemented
381C3×D4C3×C4×D312yescomplemented
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—31no—
386C2×C3×C4—31no—
387C2×C3×C4—31no—
388C2×C3×C4—31no—
389C2×C3×C4—41no—
390C2×C3×C4—61no—
391C2×C3×C4—61no—
392C2×C3×C4—62no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—61no—
396C2×C3×C4—61no—
397C2×C3×C4—121no—
398C2×C3×C4—124no—
399C2×C3×C4—122no—
400C2×C3×C4—122no—
401C2×C3×C4—362no—
402C2×C3×C4C2×D3211yescharacteristic
403C2×C3×C4C22×C3×D311yescharacteristic
404C2×C3×C4C22×C3×D311yescharacteristic
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—42no—
408C2×Dic3C3×#6711yescharacteristic
409C2×Dic3C3×#6711yescharacteristic
410D12—32no—
411D12—32no—
412D12—62no—
413D12—61no—
414D12—61no—
415D12—121no—
416C4×D3—122no—
417C4×D3—122no—
418C3×C8—61no—
419C3×C8—61no—
420C3×C8—92no—
421C3×C8—121no—
422C3⋊C8—21no—
423C3⋊C8—32no—
424C3⋊C8—32no—
425C3⋊C8—62no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32—41no—
438C2×C32—42no—
439C2×C32—44no—
440C2×C32—42no—
441C2×C32—42no—
442C2×C32—82no—
443C2×C32—82no—
444C2×C32—84no—
445C2×C32#67⋊C411yescharacteristic
446C2×C32#91111yescharacteristic
447C2×C32#91411yescharacteristic
448C2×C32C4×#6711yescharacteristic
449C2×C32#93211yescharacteristic
450C2×C32#94911yescharacteristic
451C2×C32C3×C4×D411yescharacteristic
452C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
453C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
454C3⋊D3—42no—
455C3×D3—122no—
456C3×D3—122no—
457C3×D3—242no—
458C2×D4—91no—
459C2×D4C3×D3211yescharacteristic
460C22×C4—61no—
461C22×C4—181no—
462D8—94no—
463C2×C8—31no—
464C2×C8—91no—
465C4⋊C4—31no—
466C4⋊C4—31no—
467C22⋊C4—61no—
468C22⋊C4—181no—
469C42—31no—
470C22×C3—24no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—44no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—82no—
494C22×C3—82no—
495C22×C3—182no—
496C22×C3—361no—
497C22×C3—361no—
498C22×C3D3×#6711yescharacteristic
499C22×C3C3×D3×D411yescharacteristic
500C22×C3C2×C3×#6711yescharacteristic
501C2×D3—62no—
502C2×D3—62no—
503C2×D3—122no—
504C2×D3—121no—
505C2×D3—121no—
506C2×D3—121no—
507C2×D3—121no—
508C2×D3—241no—
509C2×D3—241no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—62no—
519C3×C4—61no—
520C3×C4—61no—
521C3×C4—61no—
522C3×C4—61no—
523C3×C4—121no—
524C3×C4—122no—
525C3×C4—121no—
526C3×C4—121no—
527C3×C4—121no—
528C3×C4—121no—
529C3×C4—241no—
530C3×C4#380411yescharacteristic
531C3×C4C2×#62411yescharacteristic
532C3×C4C2×C3×C4×D311yescharacteristic
533C3×C4C2×C3×C4×D311yescharacteristic
534C3×C4C3×#23511yescharacteristic
535C3×C4C3×#23711yescharacteristic
536Dic3—21no—
537Dic3—21no—
538Dic3—41no—
539Dic3—41no—
540C32—21no—
541C32—21no—
542C32—21no—
543C32—41no—
544C32#478511yescharacteristic, complemented
545C32#501411yescharacteristic, complemented
546C32C3×#44211yescharacteristic, complemented
547C23—21no—
548C23—181no—
549D4—21no—
550D4—92no—
551D4—181no—
552D4C3×#62412yescomplemented
553C2×C4—31no—
554C2×C4—31no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—122no—
558C2×C4—362no—
559C2×C4C2×C3×D3211yescharacteristic
560C8—61no—
561C8—92no—
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—44no—
572C2×C3—42no—
573C2×C3—42no—
574C2×C3—42no—
575C2×C3—41no—
576C2×C3—41no—
577C2×C3—41no—
578C2×C3—82no—
579C2×C3—82no—
580C2×C3—84no—
581C2×C3—362no—
582C2×C3#6392511yescharacteristic
583C2×C3#6393411yescharacteristic
584C2×C3#6395211yescharacteristic
585C2×C3C3×#67⋊C411yescharacteristic
586C2×C3C3×#91111yescharacteristic
587C2×C3C3×#91411yescharacteristic
588C2×C3C3×C4×#6711yescharacteristic
589C2×C3C3×#93211yescharacteristic
590C2×C3C3×#94911yescharacteristic
591D3—122no—
592D3—122no—
593D3—242no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—182no—
598C22—361no—
599C22—361no—
600C22C3×D3×#6711yescharacteristic
601C4—61no—
602C4—61no—
603C4—121no—
604C4—121no—
605C4C3×#380411yescharacteristic
606C4C2×C3×#62411yescharacteristic
607C3—21no—
608C3—21no—
609C3—21no—
610C3—41no—
611C3#1059140411yescharacteristic, direct product factor
612C3C3×#478511yescharacteristic, complemented
613C3C3×#501411yescharacteristic, complemented
614C2—42no—
615C2—362no—
616C2C3×#6392511yescharacteristic
617C2C3×#6393411yescharacteristic
618C2C3×#6395211yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×D4D32
hypocenterC32C3×#442
LCSG ▷ 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×D4×Dic3C2semidirect
3. C3×#63894C2semidirect
4. C32×#832C2semidirect
5. C3×#63574C2semidirect
6. C3×#63564C2semidirect
7. C3×#63554C2semidirect
10. #10591404C3direct (N × H)
15. C3×#3753C4semidirect
26. C3×D4×Dic3D3semidirect
27. C3×D4×Dic3C2×C3semidirect
33. #63894C2×C3semidirect
36. C3×#832D3semidirect
37. C3×#832C2×C3semidirect
44. #63574C2×C3semidirect
45. #63564C2×C3semidirect
46. #63554C2×C3semidirect
50. C33×D4C2×C4semidirect
53. C3×C4⋊(C3⋊D3)C2×C4semidirect
60. C3×C3⋊(C3⋊C8)C2×C4semidirect
88. #3753C3×C4semidirect
132. D4×Dic3C3×D3semidirect
144. #832C3×D3semidirect
173. C32×D4C4×D3semidirect
174. C32×D4C4×D3semidirect
175. C32×D4C2×C3×C4semidirect
189. C4⋊(C3⋊D3)C2×C3×C4semidirect
215. C3⋊(C3⋊C8)C2×C3×C4semidirect
354. C33#442semidirect
379. C3×D4#624semidirect
380. C3×D4C3×C4×D3semidirect
381. C3×D4C3×C4×D3semidirect
544. C32#4785semidirect
545. C32#5014semidirect
546. C32C3×#442semidirect
552. D4C3×#624semidirect
611. C3#10591404direct (N × H)
612. C3C3×#4785semidirect
613. C3C3×#5014semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37532
C32×D4×Dic32
C3×#638942
C32×#8322
C3×#635742
C3×#635642
C3×#635542
C3×#50143
C3×#47853
#105914043