Finite Group Databasebeta

← C3×#10590770 #13157756

Subgroups — C3×#10590770 #13157756

Order 1728.

674 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3745C211yesmaximal, characteristic
2C3×#63894C211yesmaximal, characteristic, complemented
3C3×#63864C211yesmaximal, characteristic, complemented
4C2×C3×#3717C211yesmaximal, characteristic, complemented
5C32×#816C211yesmaximal, characteristic, complemented
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8C3×#4844—31nomaximal
9C3×#4819—31nomaximal
10#10590770C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×#624—21no—
15C2×C33×C8C2211yescharacteristic
16C3×#3745—22no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3722—21no—
21C3×#3722—21no—
22C3×#3722—21no—
23C3×#3717—22no—
24C2×#3745C2×C311yescharacteristic
25C3×#947—31no—
26C2×C3×#215—31no—
27C2×C3×D24—31no—
28C2×C3×D24—31no—
29C2×C3×D24—61no—
30C2×C3×#204—31no—
31C3×#893—31no—
32C3×#875—31no—
33C3×C4×D12—31no—
34#63894C2×C311yescharacteristic, complemented
35#63864C2×C311yescharacteristic, complemented
36C2×#3717C2×C311yescharacteristic, complemented
37C32×C4⋊C8—31no—
38C32×#103—31no—
39C3×#821—31no—
40C3×#816—21no—
41C3×#816D311yescharacteristic, complemented
42C3×#816C2×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—
50C2×C33×C4C2311yescharacteristic
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)D412yes—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×#624—42no—
62C33×C8—21no—
63C32×C3⋊C8—21no—
64C3×#463—91no—
65#4844—31no—
66#4819—31no—
67C2×C32×Q8—31no—
68C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×C3×D12—31no—
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×D12—61no—
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3—61no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C2×#624—21no—
82C2×C32×C8—21no—
83C2×C32×C8—21no—
84C2×C32×C8—21no—
85C2×C32×C8—31no—
86C2×C32×C8—41no—
87C2×C32×C8C2×D311yescharacteristic
88C2×C32×C8C2×D311yescharacteristic
89C2×C32×C8C22×C311yescharacteristic
90C32×C4⋊C4—31no—
91C32×C42—31no—
92#3745—22no—
93C3×#215—62no—
94C3×(C2×D3)⋊C4—61no—
95C3×(C2×D3)⋊C4—61no—
96C3×(C2×D3)⋊C4—61no—
97C3×(C2×D3)⋊C4—61no—
98C3×(C2×D3)⋊C4—61no—
99C3×(C2×D3)⋊C4—61no—
100C3×C4⋊Dic3—21no—
101C3×C4⋊Dic3C2×D311yescharacteristic
102C3×C4⋊Dic3C22×C311yescharacteristic
103C3×C4×Dic3—21no—
104C3×C4×Dic3C2×D311yescharacteristic
105C3×C4×Dic3C22×C311yescharacteristic
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C22×C311yescharacteristic
109C3×D24—62no—
110C3×D24—62no—
111C3×D24—64no—
112C3×#204—62no—
113#3722—21no—
114#3722—21no—
115#3722—21no—
116#3717—22no—
117C22×C33C2×D411yescharacteristic
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—41no—
124C33×C4C2×D411yescharacteristic
125C33×C4#4111yescharacteristic
126C32×Dic3—21no—
127C32×Dic3—21no—
128C32×Dic3—41no—
129C32×Dic3—41no—
130C2×C3×SD16—91no—
131C2×C3×D8—91no—
132C3×#124—91no—
133C3×C4×D4—91no—
134#947—31no—
135C2×#215—31no—
136C2×D24—31no—
137C2×D24—31no—
138C2×D24—61no—
139C2×#204—31no—
140#893—31no—
141#875—31no—
142C4×D12—31no—
143C3×C4⋊C8—31no—
144C3×C4⋊C8—31no—
145C3×C4⋊C8—61no—
146C3×#103—31no—
147C3×#103—31no—
148C3×#103—61no—
149C3×#102—91no—
150#821—31no—
151#816C3×D311yescharacteristic, complemented
152Dic3⋊C8C3×D311yescharacteristic, complemented
153#809—31no—
154C22×C3⋊D3—21no—
155C22×C3⋊D3—21no—
156C22×C3×D3—61no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—121no—
161C22×C3×D3—121no—
162C32×Q8—61no—
163C32×Q8—61no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—31no—
168C2×C32×C4—31no—
169C2×C32×C4—41no—
170C2×C32×C4—61no—
171C2×C32×C4—61no—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C23×C311yescharacteristic
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)C3×D412yes—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—41no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3D1211yescharacteristic
188C2×C3×Dic3D1211yescharacteristic
189C2×C3×Dic3C3×D411yescharacteristic
190C2×C3×Dic3C3×D411yescharacteristic
191C3×D12—32no—
192C3×D12—32no—
193C3×D12—62no—
194C3×D12—62no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—61no—
201C3×D12—121no—
202C3×D12—121no—
203C3×C4×D3—122no—
204C3×C4×D3—122no—
205C3×Dic6—22no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—21no—
209C3×Dic6—21no—
210C3×Dic6—41no—
211#624—42no—
212C32×C8—21no—
213C32×C8—22no—
214C32×C8—21no—
215C32×C8—21no—
216C32×C8—41no—
217C32×C8—41no—
218C32×C8—42no—
219C32×C8—61no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—21no—
222C3×C3⋊C8—41no—
223#463—91noSylow
224C2×C33#12111yescharacteristic
225C2×C33#13511yescharacteristic
226C2×C33AGL(1,ℤ/8)11yescharacteristic
227C3×C3⋊D3—42no—
228C3×C3⋊D3—81no—
229C2×C3×Q8—31no—
230C2×C3×Q8—31no—
231C2×C3×Q8—61no—
232C2×C3×D4—91no—
233C2×C3×D4—91no—
234C22×C3×C4—181no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—31no—
238C2×D12—31no—
239C2×D12—61no—
240C2×D12—61no—
241C2×C4×D3—61no—
242C2×C4×D3—61no—
243C2×Dic6C2×C3×D311yescharacteristic
244C3×SD16—182no—
245C3×D8—182no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—31no—
250C2×C3×C8—31no—
251C2×C3×C8—41no—
252C2×C3×C8—61no—
253C2×C3×C8D3211yescharacteristic
254C2×C3×C8C2×C3×D311yescharacteristic
255C2×C3×C8C2×C3×D311yescharacteristic
256C3×C4⋊C4—31no—
257C3×C4⋊C4—31no—
258C3×C4⋊C4—61no—
259C3×C22⋊C4—181no—
260C3×C22⋊C4—181no—
261C3×C22⋊C4—181no—
262C3×C42—31no—
263C3×C42—31no—
264C3×C42—61no—
265#215—62no—
266(C2×D3)⋊C4—61no—
267(C2×D3)⋊C4—61no—
268(C2×D3)⋊C4—61no—
269(C2×D3)⋊C4—61no—
270(C2×D3)⋊C4—61no—
271(C2×D3)⋊C4—61no—
272C4⋊Dic3C2×C3×D311yescharacteristic
273C4×Dic3C2×C3×D311yescharacteristic
274C2×C3⋊C8C2×C3×D311yescharacteristic
275D24—62no—
276D24—62no—
277D24—64no—
278#204—62no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—41no—
283C22×C32C2×D1211yescharacteristic
284C22×C32D3×D411yescharacteristic
285C22×C32C2×C3×D411yescharacteristic
286C2×C3⋊D3—22no—
287C2×C3⋊D3—41no—
288C2×C3⋊D3—41no—
289C2×C3⋊D3—41no—
290C2×C3⋊D3—41no—
291C2×C3⋊D3—41no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C2×C3×D3—122no—
295C2×C3×D3—121no—
296C2×C3×D3—121no—
297C2×C3×D3—121no—
298C2×C3×D3—121no—
299C2×C3×D3—122no—
300C2×C3×D3—121no—
301C2×C3×D3—121no—
302C2×C3×D3—121no—
303C2×C3×D3—121no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C2×C3×D3—241no—
307C2×C3×D3—241no—
308C2×C3×D3—241no—
309C2×C3×D3—241no—
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—61no—
319C32×C4—61no—
320C32×C4—121no—
321C32×C4—121no—
322C32×C4#23511yescharacteristic
323C32×C4D3×D411yescharacteristic
324C32×C4D3×D411yescharacteristic
325C32×C4#23911yescharacteristic
326C32×C4C2×C3×D411yescharacteristic
327C32×C4C3×#4111yescharacteristic
328C3×Dic3—22no—
329C3×Dic3—21no—
330C3×Dic3—21no—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3—41no—
334C3×Dic3—42no—
335C3×Dic3—41no—
336C3×Dic3—41no—
337C3×Dic3—41no—
338C3×Dic3—41no—
339C3×Dic3—81no—
340C2×SD16—91no—
341C2×D8—91no—
342#124—91no—
343C4×D4—91no—
344C4⋊C8—31no—
345#103—31no—
346#102—91no—
347C33#46311yescharacteristic, complemented, Sylow
348C23×C3—181no—
349C23×C3—181no—
350C22×D3—61no—
351C22×D3—61no—
352C22×D3—61no—
353C22×D3—61no—
354C22×D3—121no—
355C22×D3—121no—
356C3×Q8—62no—
357C3×Q8—61no—
358C3×Q8—61no—
359C3×Q8—61no—
360C3×Q8—61no—
361C3×Q8—121no—
362C3×D4—92no—
363C3×D4—181no—
364C3×D4—181no—
365C3×D4—181no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—31no—
370C2×C3×C4—31no—
371C2×C3×C4—31no—
372C2×C3×C4—31no—
373C2×C3×C4—41no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—121no—
381C2×C3×C4—121no—
382C2×C3×C4—362no—
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C22×C3×D311yescharacteristic
385C2×C3×C4C22×C3×D311yescharacteristic
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3C3×D1211yescharacteristic
389C2×Dic3C3×D1211yescharacteristic
390D12—32no—
391D12—32no—
392D12—62no—
393D12—62no—
394D12—61no—
395D12—61no—
396D12—61no—
397D12—61no—
398D12—61no—
399D12—61no—
400D12—121no—
401D12—121no—
402C4×D3—122no—
403C4×D3—122no—
404Dic6—21no—
405Dic6—21no—
406C3×C8—22no—
407C3×C8—21no—
408C3×C8—21no—
409C3×C8—21no—
410C3×C8—41no—
411C3×C8—41no—
412C3×C8—42no—
413C3×C8—61no—
414C3×C8—61no—
415C3×C8—121no—
416C3⋊C8—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—41no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32#88411yescharacteristic
430C2×C32#89511yescharacteristic
431C2×C32#90411yescharacteristic
432C2×C32#90811yescharacteristic
433C2×C32#91411yescharacteristic
434C2×C32#91911yescharacteristic
435C2×C32C3×#12111yescharacteristic
436C2×C32C3×#13511yescharacteristic
437C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
438C3⋊D3—42no—
439C3⋊D3—81no—
440C3×D3—122no—
441C3×D3—122no—
442C3×D3—242no—
443C3×D3—242no—
444C3×D3—241no—
445C3×D3—241no—
446C2×Q8—31no—
447C2×D4—91no—
448C2×D4—91no—
449C22×C4—181no—
450SD16—182no—
451D8—182no—
452C2×C8—31no—
453C2×C8C3×D3211yescharacteristic
454C4⋊C4—31no—
455C22⋊C4—181no—
456C22⋊C4—181no—
457C22⋊C4—181no—
458C42—31no—
459C22×C3—21no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—41no—
463C22×C3—182no—
464C22×C3—361no—
465C22×C3—361no—
466C22×C3—361no—
467C22×C3—361no—
468C22×C3—361no—
469C22×C3D3×D1211yescharacteristic
470C22×C3C2×C3×D1211yescharacteristic
471C22×C3C3×D3×D411yescharacteristic
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—122no—
475C2×D3—121no—
476C2×D3—121no—
477C2×D3—121no—
478C2×D3—121no—
479C2×D3—122no—
480C2×D3—121no—
481C2×D3—121no—
482C2×D3—121no—
483C2×D3—121no—
484C2×D3—121no—
485C2×D3—121no—
486C2×D3—241no—
487C2×D3—241no—
488C2×D3—241no—
489C2×D3—241no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—62no—
499C3×C4—61no—
500C3×C4—61no—
501C3×C4—61no—
502C3×C4—61no—
503C3×C4—121no—
504C3×C4—122no—
505C3×C4—121no—
506C3×C4—121no—
507C3×C4—121no—
508C3×C4—121no—
509C3×C4—241no—
510C3×C4#379911yescharacteristic
511C3×C4#381111yescharacteristic
512C3×C4C3×#23511yescharacteristic
513C3×C4C3×D3×D411yescharacteristic
514C3×C4C3×D3×D411yescharacteristic
515C3×C4C3×#23911yescharacteristic
516Dic3—21no—
517Dic3—21no—
518Dic3—41no—
519Dic3—41no—
520C32—21no—
521C32—21no—
522C32—21no—
523C32—41no—
524C32#481911yescharacteristic, complemented
525C32#484411yescharacteristic, complemented
526C32C3×#46311yescharacteristic, complemented
527C23—181no—
528C23—181no—
529Q8—61no—
530Q8—61no—
531D4—92no—
532D4—181no—
533D4—181no—
534D4—181no—
535C2×C4—31no—
536C2×C4—31no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—362no—
540C2×C4C2×C3×D3211yescharacteristic
541C8—21no—
542C8—61no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—41no—
553C2×C3—41no—
554C2×C3—41no—
555C2×C3—362no—
556C2×C3—721no—
557C2×C3#6379411yescharacteristic
558C2×C3#6380811yescharacteristic
559C2×C3#6391011yescharacteristic
560C2×C3C3×#88411yescharacteristic
561C2×C3C3×#89511yescharacteristic
562C2×C3C3×#90411yescharacteristic
563C2×C3C3×#90811yescharacteristic
564C2×C3C3×#91411yescharacteristic
565C2×C3C3×#91911yescharacteristic
566D3—122no—
567D3—122no—
568D3—242no—
569D3—242no—
570D3—241no—
571D3—241no—
572C22—182no—
573C22—361no—
574C22—361no—
575C22—361no—
576C22—361no—
577C22—361no—
578C22C3×D3×D1211yescharacteristic
579C4—61no—
580C4—61no—
581C4—121no—
582C4—121no—
583C4C3×#379911yescharacteristic
584C4C3×#381111yescharacteristic
585C3—21no—
586C3—21no—
587C3—21no—
588C3—41no—
589C3#1059077011yescharacteristic, direct product factor
590C3C3×#481911yescharacteristic, complemented
591C3C3×#484411yescharacteristic, complemented
592C2—362no—
593C2—721no—
594C2C3×#6379411yescharacteristic
595C2C3×#6380811yescharacteristic
596C2C3×#6391011yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×C3×#3745C2
hypercenterC2×C3×C8D32
hypocenterC32C3×#463
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
2. C3×#63894C2semidirect
3. C3×#63864C2semidirect
4. C2×C3×#3717C2semidirect
5. C32×#816C2semidirect
6. C32×Dic3⋊C8C2semidirect
7. C3×#63564C2semidirect
10. #10590770C3direct (N × H)
34. #63894C2×C3semidirect
35. #63864C2×C3semidirect
36. C2×#3717C2×C3semidirect
41. C3×#816D3semidirect
42. C3×#816C2×C3semidirect
44. C3×Dic3⋊C8D3semidirect
45. C3×Dic3⋊C8C2×C3semidirect
47. #63564C2×C3semidirect
151. #816C3×D3semidirect
152. Dic3⋊C8C3×D3semidirect
347. C33#463semidirect
524. C32#4819semidirect
525. C32#4844semidirect
526. C32C3×#463semidirect
589. C3#10590770direct (N × H)
590. C3C3×#4819semidirect
591. C3C3×#4844semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37452
C3×#638942
C3×#638642
C2×C3×#37172
C32×#8162
C32×Dic3⋊C82
C3×#635642
C3×#48443
C3×#48193
#105907703