Finite Group Databasebeta

← C3×#10590765 #13157751

Subgroups — C3×#10590765 #13157751

Order 1728.

746 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3745C211yesmaximal, characteristic, complemented
2C3×#63911C211yesmaximal, characteristic, complemented
3C3×#63894C211yesmaximal, characteristic, complemented
4C2×C3×#3714C211yesmaximal, characteristic, complemented
5C32×#821C211yesmaximal, characteristic, complemented
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8C3×#4843—31nomaximal
9C3×#4786—31nomaximal
10#10590765C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C3×#626—21no—
15C2×C3×#626—21no—
16C2×C3×#624—21no—
17C2×C33×C8C2211yescharacteristic
18C3×#3745—22no—
19C32×C4⋊Dic3C2211yescharacteristic
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3722—21no—
23C3×#3714—22no—
24C2×#3745C2×C311yescharacteristic, complemented
25C3×#940—31no—
26C2×C3×#213—31no—
27C2×C3×D24—31no—
28C2×C3×D24—31no—
29C2×C3×D24—31no—
30C2×C3×D24—61no—
31C3×#893—31no—
32C3×#874—31no—
33C3×C4×D12—31no—
34#63911C2×C311yescharacteristic, complemented
35#63894C2×C311yescharacteristic, complemented
36C2×#3714C2×C311yescharacteristic, complemented
37C32×C4⋊C8—31no—
38C32×#102—31no—
39C3×#821—21no—
40C3×#821—31no—
41C3×#821D311yescharacteristic, complemented
42C3×#821C2×C311yescharacteristic, complemented
43C3×Dic3⋊C8—21no—
44C3×Dic3⋊C8D311yescharacteristic, complemented
45C3×Dic3⋊C8C2×C311yescharacteristic, complemented
46C3×#809—31no—
47#63564C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C3×C3⋊D3—21no—
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)D412yescomplemented
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×D12—21no—
60C32×D12—21no—
61C3×#626—42no—
62C3×#626—42no—
63C3×#624—42no—
64C33×C8—21no—
65C32×C3⋊C8—21no—
66C3×#459—91no—
67#4843—31no—
68#4786—31no—
69C2×C32×D4—31no—
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
71C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
72C2×C3×#67—61no—
73C2×C3×#67—61no—
74C2×C3×D12—21no—
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—61no—
80C2×C3×D12—61no—
81C2×C3×D12—61no—
82C2×C3×D12—61no—
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×D12C22×C311yescharacteristic
85C2×C3×C4×D3—61no—
86C2×C3×C4×D3—61no—
87C2×#626—21no—
88C2×#626—21no—
89C2×#624—21no—
90C2×C32×C8—21no—
91C2×C32×C8—21no—
92C2×C32×C8—21no—
93C2×C32×C8—31no—
94C2×C32×C8—41no—
95C2×C32×C8C2×D311yescharacteristic
96C2×C32×C8C2×D311yescharacteristic
97C2×C32×C8C22×C311yescharacteristic
98C32×C4⋊C4—31no—
99C32×C42—31no—
100#3745—22no—
101C3×#213—62no—
102C3×(C2×D3)⋊C4—61no—
103C3×(C2×D3)⋊C4—61no—
104C3×C4⋊Dic3—21no—
105C3×C4⋊Dic3C2×D311yescharacteristic
106C3×C4⋊Dic3C22×C311yescharacteristic
107C3×C4×Dic3—21no—
108C3×C4×Dic3C2×D311yescharacteristic
109C3×C4×Dic3C22×C311yescharacteristic
110C2×C3×C3⋊C8—21no—
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113C3×D24—62no—
114C3×D24—62no—
115C3×D24—64no—
116C3×D24—62no—
117#3722—21no—
118#3714—22no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—41no—
125C2×C3×C3⋊D3—41no—
126C2×C32×D3—41no—
127C2×C32×D3—41no—
128C2×C32×D3—41no—
129C33×C4C2×D411yescharacteristic
130C33×C4#4111yescharacteristic
131C32×Dic3—21no—
132C32×Dic3—41no—
133C32×Dic3D812yescomplemented
134C2×C3×D8—91no—
135C2×C3×D8—91no—
136C3×#127—91no—
137C3×C4×D4—91no—
138#940—31no—
139C2×#213—31no—
140C2×D24—31no—
141C2×D24—31no—
142C2×D24—31no—
143C2×D24—61no—
144#893—31no—
145#874—31no—
146C4×D12—31no—
147C3×C4⋊C8—31no—
148C3×C4⋊C8—31no—
149C3×C4⋊C8—61no—
150C3×#102—31no—
151C3×#102—31no—
152C3×#102—61no—
153C3×#102—91no—
154#821—31no—
155#821C3×D311yescharacteristic, complemented
156Dic3⋊C8C3×D311yescharacteristic, complemented
157#809—31no—
158C23×C32—61no—
159C22×C3⋊D3—21no—
160C22×C3⋊D3—21no—
161C22×C3×D3—21no—
162C22×C3×D3—21no—
163C22×C3×D3—41no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C22×C3×D3—61no—
167C22×C3×D3—61no—
168C22×C3×D3—121no—
169C22×C3×D3—121no—
170C32×D4—61no—
171C32×D4—61no—
172C2×C32×C4—21no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—31no—
176C2×C32×C4—31no—
177C2×C32×C4—41no—
178C2×C32×C4—61no—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C23×C311yescharacteristic
182C4⋊(C3⋊D3)—21no—
183C4⋊(C3⋊D3)—21no—
184C4⋊(C3⋊D3)—21no—
185C4⋊(C3⋊D3)C3×D412yescomplemented
186C3×#67—122no—
187C3×#67—122no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—41no—
193C2×C3×Dic3D1211yescharacteristic
194C2×C3×Dic3D1211yescharacteristic
195C2×C3×Dic3C3×D411yescharacteristic
196C2×C3×Dic3C3×D411yescharacteristic
197C3×D12—22no—
198C3×D12—21no—
199C3×D12—21no—
200C3×D12—21no—
201C3×D12—21no—
202C3×D12—32no—
203C3×D12—32no—
204C3×D12—41no—
205C3×D12—62no—
206C3×D12—62no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—61no—
210C3×D12—61no—
211C3×D12—61no—
212C3×D12—61no—
213C3×D12—121no—
214C3×D12—121no—
215C3×D12—122no—
216C3×D12—122no—
217C3×C4×D3—122no—
218C3×C4×D3—122no—
219#626—42no—
220#626—42no—
221#624—42no—
222C32×C8—21no—
223C32×C8—22no—
224C32×C8—21no—
225C32×C8—21no—
226C32×C8—41no—
227C32×C8—41no—
228C32×C8—42no—
229C32×C8—61no—
230C3×C3⋊C8—21no—
231C3×C3⋊C8—21no—
232C3×C3⋊C8—41no—
233#459—91noSylow
234C2×C33#12111yescharacteristic
235C2×C33C2×D811yescharacteristic
236C2×C33AGL(1,ℤ/8)11yescharacteristic
237C3×C3⋊D3—42no—
238C3×C3⋊D3—81no—
239C32×D3—81no—
240C2×C3×D4—31no—
241C2×C3×D4—31no—
242C2×C3×D4—61no—
243C2×C3×D4—91no—
244C2×C3×D4—91no—
245C2×C3×D4—181no—
246C2×C3×D4—181no—
247C22×C3×C4—181no—
248C2×#67—61no—
249C2×#67—61no—
250C2×D12—31no—
251C2×D12—31no—
252C2×D12—31no—
253C2×D12—31no—
254C2×D12—61no—
255C2×D12—61no—
256C2×D12—61no—
257C2×D12—61no—
258C2×D12C2×C3×D311yescharacteristic
259C2×C4×D3—61no—
260C2×C4×D3—61no—
261C3×D8—182no—
262C3×D8—182no—
263C2×C3×C8—21no—
264C2×C3×C8—21no—
265C2×C3×C8—21no—
266C2×C3×C8—31no—
267C2×C3×C8—31no—
268C2×C3×C8—41no—
269C2×C3×C8—61no—
270C2×C3×C8D3211yescharacteristic
271C2×C3×C8C2×C3×D311yescharacteristic
272C2×C3×C8C2×C3×D311yescharacteristic
273C3×C4⋊C4—31no—
274C3×C4⋊C4—31no—
275C3×C4⋊C4—61no—
276C3×C22⋊C4—181no—
277C3×C42—31no—
278C3×C42—31no—
279C3×C42—61no—
280#213—62no—
281(C2×D3)⋊C4—61no—
282(C2×D3)⋊C4—61no—
283C4⋊Dic3C2×C3×D311yescharacteristic
284C4×Dic3C2×C3×D311yescharacteristic
285C2×C3⋊C8C2×C3×D311yescharacteristic
286D24—62no—
287D24—62no—
288D24—64no—
289D24—62no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32—121no—
295C22×C32—121no—
296C22×C32—121no—
297C22×C32C2×D1211yescharacteristic
298C22×C32D3×D411yescharacteristic
299C22×C32C2×C3×D411yescharacteristic
300C2×C3⋊D3—22no—
301C2×C3⋊D3—41no—
302C2×C3⋊D3—41no—
303C2×C3⋊D3—41no—
304C2×C3⋊D3—41no—
305C2×C3⋊D3—41no—
306C2×C3×D3—42no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3—41no—
311C2×C3×D3—41no—
312C2×C3×D3—41no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—81no—
316C2×C3×D3—81no—
317C2×C3×D3—122no—
318C2×C3×D3—121no—
319C2×C3×D3—121no—
320C2×C3×D3—121no—
321C2×C3×D3—121no—
322C2×C3×D3—122no—
323C2×C3×D3—121no—
324C2×C3×D3—121no—
325C2×C3×D3—121no—
326C2×C3×D3—121no—
327C2×C3×D3—121no—
328C2×C3×D3—121no—
329C2×C3×D3—241no—
330C2×C3×D3—241no—
331C2×C3×D3—241no—
332C2×C3×D3—241no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—32no—
340C32×C4—41no—
341C32×C4—41no—
342C32×C4—61no—
343C32×C4—121no—
344C32×C4#23511yescharacteristic
345C32×C4D3×D411yescharacteristic
346C32×C4D3×D411yescharacteristic
347C32×C4#23911yescharacteristic
348C32×C4C2×C3×D411yescharacteristic
349C32×C4C3×#4111yescharacteristic
350C3×Dic3—21no—
351C3×Dic3—21no—
352C3×Dic3—22no—
353C3×Dic3—41no—
354C3×Dic3—41no—
355C3×Dic3—41no—
356C3×Dic3—81no—
357C3×Dic3D2412yescomplemented
358C3×Dic3C3×D812yescomplemented
359C2×D8—91no—
360C2×D8—91no—
361#127—91no—
362C4×D4—91no—
363C4⋊C8—31no—
364#102—31no—
365#102—91no—
366C33#45911yescharacteristic, complemented, Sylow
367C23×C3—61no—
368C23×C3—61no—
369C23×C3—121no—
370C23×C3—181no—
371C23×C3—181no—
372C22×D3—21no—
373C22×D3—61no—
374C22×D3—61no—
375C22×D3—61no—
376C22×D3—61no—
377C22×D3—121no—
378C22×D3—121no—
379C3×D4—62no—
380C3×D4—61no—
381C3×D4—61no—
382C3×D4—61no—
383C3×D4—61no—
384C3×D4—92no—
385C3×D4—121no—
386C3×D4—181no—
387C3×D4—181no—
388C3×D4—181no—
389C3×D4—362no—
390C3×D4—362no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—31no—
395C2×C3×C4—31no—
396C2×C3×C4—31no—
397C2×C3×C4—31no—
398C2×C3×C4—41no—
399C2×C3×C4—61no—
400C2×C3×C4—61no—
401C2×C3×C4—61no—
402C2×C3×C4—61no—
403C2×C3×C4—121no—
404C2×C3×C4—362no—
405C2×C3×C4C2×D3211yescharacteristic
406C2×C3×C4C22×C3×D311yescharacteristic
407C2×C3×C4C22×C3×D311yescharacteristic
408#67—122no—
409#67—122no—
410C2×Dic3—21no—
411C2×Dic3C3×D1211yescharacteristic
412C2×Dic3C3×D1211yescharacteristic
413D12—21no—
414D12—21no—
415D12—32no—
416D12—32no—
417D12—62no—
418D12—62no—
419D12—61no—
420D12—61no—
421D12—61no—
422D12—61no—
423D12—61no—
424D12—61no—
425D12—121no—
426D12—121no—
427D12—122no—
428D12—122no—
429C4×D3—122no—
430C4×D3—122no—
431C3×C8—22no—
432C3×C8—21no—
433C3×C8—21no—
434C3×C8—21no—
435C3×C8—41no—
436C3×C8—41no—
437C3×C8—42no—
438C3×C8—61no—
439C3×C8—61no—
440C3×C8—121no—
441C3⋊C8—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—41no—
452C2×C32—41no—
453C2×C32—41no—
454C2×C32—241no—
455C2×C32#88411yescharacteristic
456C2×C32#89511yescharacteristic
457C2×C32C2×D2411yescharacteristic
458C2×C32#90811yescharacteristic
459C2×C32D3×D811yescharacteristic
460C2×C32#91411yescharacteristic
461C2×C32C3×#12111yescharacteristic
462C2×C32C2×C3×D811yescharacteristic
463C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
464C3⋊D3—42no—
465C3⋊D3—81no—
466C3×D3—82no—
467C3×D3—81no—
468C3×D3—81no—
469C3×D3—122no—
470C3×D3—122no—
471C3×D3—242no—
472C3×D3—242no—
473C3×D3—241no—
474C3×D3—241no—
475C2×D4—31no—
476C2×D4—91no—
477C2×D4—91no—
478C2×D4—181no—
479C2×D4—181no—
480C22×C4—181no—
481D8—182no—
482D8—182no—
483C2×C8—31no—
484C2×C8C3×D3211yescharacteristic
485C4⋊C4—31no—
486C22⋊C4—181no—
487C42—31no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—41no—
492C22×C3—122no—
493C22×C3—121no—
494C22×C3—121no—
495C22×C3—121no—
496C22×C3—121no—
497C22×C3—121no—
498C22×C3—121no—
499C22×C3—182no—
500C22×C3—241no—
501C22×C3—241no—
502C22×C3—361no—
503C22×C3—361no—
504C22×C3—361no—
505C22×C3—361no—
506C22×C3—361no—
507C22×C3D3×D1211yescharacteristic
508C22×C3C2×C3×D1211yescharacteristic
509C22×C3C3×D3×D411yescharacteristic
510C2×D3—41no—
511C2×D3—41no—
512C2×D3—41no—
513C2×D3—62no—
514C2×D3—62no—
515C2×D3—122no—
516C2×D3—121no—
517C2×D3—121no—
518C2×D3—121no—
519C2×D3—121no—
520C2×D3—122no—
521C2×D3—121no—
522C2×D3—121no—
523C2×D3—121no—
524C2×D3—121no—
525C2×D3—121no—
526C2×D3—121no—
527C2×D3—241no—
528C2×D3—241no—
529C2×D3—241no—
530C2×D3—241no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—32no—
538C3×C4—32no—
539C3×C4—41no—
540C3×C4—41no—
541C3×C4—62no—
542C3×C4—61no—
543C3×C4—61no—
544C3×C4—121no—
545C3×C4—121no—
546C3×C4—121no—
547C3×C4—241no—
548C3×C4#379911yescharacteristic
549C3×C4#381111yescharacteristic
550C3×C4C3×#23511yescharacteristic
551C3×C4C3×D3×D411yescharacteristic
552C3×C4C3×D3×D411yescharacteristic
553C3×C4C3×#23911yescharacteristic
554Dic3—21no—
555Dic3—41no—
556Dic3C3×D2412yescomplemented
557C32—21no—
558C32—21no—
559C32—21no—
560C32—41no—
561C32#478611yescharacteristic, complemented
562C32#484311yescharacteristic, complemented
563C32C3×#45911yescharacteristic, complemented
564C23—61no—
565C23—181no—
566C23—181no—
567D4—61no—
568D4—61no—
569D4—92no—
570D4—181no—
571D4—181no—
572D4—181no—
573D4—362no—
574D4—362no—
575C2×C4—31no—
576C2×C4—31no—
577C2×C4—61no—
578C2×C4—362no—
579C2×C4C2×C3×D3211yescharacteristic
580C8—21no—
581C8—61no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—41no—
592C2×C3—41no—
593C2×C3—41no—
594C2×C3—242no—
595C2×C3—241no—
596C2×C3—241no—
597C2×C3—362no—
598C2×C3—721no—
599C2×C3D3×D2411yescharacteristic
600C2×C3#6379411yescharacteristic
601C2×C3#6391011yescharacteristic
602C2×C3C3×#88411yescharacteristic
603C2×C3C3×#89511yescharacteristic
604C2×C3C2×C3×D2411yescharacteristic
605C2×C3C3×#90811yescharacteristic
606C2×C3C3×D3×D811yescharacteristic
607C2×C3C3×#91411yescharacteristic
608D3—81no—
609D3—122no—
610D3—122no—
611D3—242no—
612D3—242no—
613D3—241no—
614D3—241no—
615C22—121no—
616C22—121no—
617C22—121no—
618C22—182no—
619C22—361no—
620C22—361no—
621C22—361no—
622C22—361no—
623C22—361no—
624C22C3×D3×D1211yescharacteristic
625C4—32no—
626C4—61no—
627C4—121no—
628C4C3×#379911yescharacteristic
629C4C3×#381111yescharacteristic
630C3—21no—
631C3—21no—
632C3—21no—
633C3—41no—
634C3#1059076511yescharacteristic, direct product factor
635C3C3×#478611yescharacteristic, complemented
636C3C3×#484311yescharacteristic, complemented
637C2—241no—
638C2—362no—
639C2—721no—
640C2C3×D3×D2411yescharacteristic
641C2C3×#6379411yescharacteristic
642C2C3×#6391011yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#459
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×#3745C2semidirect
2. C3×#63911C2semidirect
3. C3×#63894C2semidirect
4. C2×C3×#3714C2semidirect
5. C32×#821C2semidirect
6. C32×Dic3⋊C8C2semidirect
7. C3×#63564C2semidirect
10. #10590765C3direct (N × H)
24. C2×#3745C2×C3semidirect
34. #63911C2×C3semidirect
35. #63894C2×C3semidirect
36. C2×#3714C2×C3semidirect
41. C3×#821D3semidirect
42. C3×#821C2×C3semidirect
44. C3×Dic3⋊C8D3semidirect
45. C3×Dic3⋊C8C2×C3semidirect
47. #63564C2×C3semidirect
55. C3×C4⋊(C3⋊D3)D4semidirect
133. C32×Dic3D8semidirect
155. #821C3×D3semidirect
156. Dic3⋊C8C3×D3semidirect
185. C4⋊(C3⋊D3)C3×D4semidirect
357. C3×Dic3D24semidirect
358. C3×Dic3C3×D8semidirect
366. C33#459semidirect
556. Dic3C3×D24semidirect
561. C32#4786semidirect
562. C32#4843semidirect
563. C32C3×#459semidirect
634. C3#10590765direct (N × H)
635. C3C3×#4786semidirect
636. C3C3×#4843semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37452
C3×#639112
C3×#638942
C2×C3×#37142
C32×#8212
C32×Dic3⋊C82
C3×#635642
C3×#48433
C3×#47863
#105907653