Finite Group Databasebeta

← C3×#10590860 #13157761

Subgroups — C3×#10590860 #13157761

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×#3744C211yesmaximal, characteristic, complemented
2C3×#63894C211yesmaximal, characteristic, complemented
3C3×#63860C211yesmaximal, characteristic, complemented
4C2×C3×#3714C211yesmaximal, characteristic
5C32×#821C211yesmaximal, characteristic, complemented
6C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
7C3×#63567C211yesmaximal, characteristic, complemented
8C3×#4844—31nomaximal
9C3×#4787—31nomaximal
10#10590860C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C3×#624—21no—
15C2×C33×C8C2211yescharacteristic
16C3×#3744—22no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3722—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×(C2×D3)⋊Dic3—21no—
23C3×#3714—22no—
24C2×#3744C2×C311yescharacteristic, complemented
25C3×#936—31no—
26C2×C3×#213—31no—
27C2×C3×D24—31no—
28C2×C3×#204—31no—
29C2×C3×#204—31no—
30C2×C3×#204—61no—
31C3×#893—31no—
32C3×#875—31no—
33C3×C4×D12—31no—
34#63894C2×C311yescharacteristic, complemented
35#63860C2×C311yescharacteristic, complemented
36C2×#3714C2×C311yescharacteristic
37C32×C4⋊C8—31no—
38C32×#102—31no—
39C3×#821—21no—
40C3×#821D311yescharacteristic, complemented
41C3×#821C2×C311yescharacteristic, complemented
42C3×#816—31no—
43C3×Dic3⋊C8—21no—
44C3×Dic3⋊C8D311yescharacteristic, complemented
45C3×Dic3⋊C8C2×C311yescharacteristic, complemented
46C3×#810—31no—
47#63567C2×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)D412yes—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×D12—21no—
60C32×D12—21no—
61C3×#624—42no—
62C33×C8—21no—
63C32×C3⋊C8—21no—
64C3×#463—91no—
65#4844—31no—
66#4787—31no—
67C2×C32×D4—31no—
68C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
69C2×#634C22×C311yescharacteristic
70C2×C3×D12—21no—
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×D12C2×D311yescharacteristic
75C2×C3×D12C22×C311yescharacteristic
76C2×C3×C4×D3—61no—
77C2×C3×C4×D3—61no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—61no—
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#3744—22no—
93C3×#217—61no—
94C3×#217—61no—
95C3×#213—62no—
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×#204—62no—
111C3×#204—62no—
112C3×#204—64no—
113#3722—21no—
114(C2×D3)⋊Dic3—21no—
115(C2×D3)⋊Dic3—21no—
116#3714—22no—
117C22×C33C2×D411yescharacteristic
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C2×C32×D3—41no—
122C2×C32×D3—41no—
123C2×C32×D3—41no—
124C33×C4C2×D411yescharacteristic
125C33×C4#4111yescharacteristic
126C3×C3⋊Dic3—41no—
127C32×Dic3—21no—
128C32×Dic3—21no—
129C32×Dic3—41no—
130C2×C3×SD16—91no—
131C2×C3×D8—91no—
132C3×#124—91no—
133C3×C4×D4—91no—
134#936—31no—
135C2×#213—31no—
136C2×D24—31no—
137C2×#204—31no—
138C2×#204—31no—
139C2×#204—61no—
140#893—31no—
141#875—31no—
142C4×D12—31no—
143C3×C4⋊C8—31no—
144C3×C4⋊C8—31no—
145C3×C4⋊C8—61no—
146C3×#103—91no—
147C3×#102—31no—
148C3×#102—31no—
149C3×#102—61no—
150#821C3×D311yescharacteristic, complemented
151#816—31no—
152Dic3⋊C8C3×D311yescharacteristic, complemented
153#810—31no—
154C23×C32—61no—
155C22×C3⋊D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—41no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—121no—
162C32×D4—61no—
163C32×D4—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×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C23×C311yescharacteristic
174C2×C3⋊Dic3—21no—
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)C3×D412yes—
177#634—21no—
178#634—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—41no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—121no—
187C2×C3×Dic3D1211yescharacteristic
188C2×C3×Dic3D1211yescharacteristic
189C2×C3×Dic3C3×D411yescharacteristic
190C2×C3×Dic3C3×D411yescharacteristic
191C3×D12—22no—
192C3×D12—21no—
193C3×D12—21no—
194C3×D12—21no—
195C3×D12—21no—
196C3×D12—32no—
197C3×D12—32no—
198C3×D12—41no—
199C3×D12—62no—
200C3×D12—61no—
201C3×D12—61no—
202C3×D12—121no—
203C3×C4×D3—122no—
204C3×C4×D3—122no—
205C3×Dic6—62no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C3×Dic6—121no—
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—
228C32×D3—81no—
229C2×C3×Q8—91no—
230C2×C3×D4—31no—
231C2×C3×D4—31no—
232C2×C3×D4—61no—
233C2×C3×D4—91no—
234C22×C3×C4—181no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—61no—
238C2×D12C2×C3×D311yescharacteristic
239C2×C4×D3—61no—
240C2×C4×D3—61no—
241C2×Dic6—31no—
242C2×Dic6—31no—
243C2×Dic6—61no—
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#217—61no—
266#217—61no—
267#213—62no—
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—
276#204—62no—
277#204—62no—
278#204—64no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—41no—
283C22×C32—121no—
284C22×C32—121no—
285C22×C32—121no—
286C22×C32C2×D1211yescharacteristic
287C22×C32D3×D411yescharacteristic
288C22×C32C2×C3×D411yescharacteristic
289C2×C3⋊D3—22no—
290C2×C3⋊D3—41no—
291C2×C3⋊D3—41no—
292C2×C3×D3—42no—
293C2×C3×D3—41no—
294C2×C3×D3—41no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C2×C3×D3—41no—
298C2×C3×D3—41no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—81no—
302C2×C3×D3—81no—
303C2×C3×D3—122no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
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#23511yescharacteristic
322C32×C4D3×D411yescharacteristic
323C32×C4D3×D411yescharacteristic
324C32×C4#23911yescharacteristic
325C32×C4C2×C3×D411yescharacteristic
326C32×C4C3×#4111yescharacteristic
327C3⋊Dic3—41no—
328C3×Dic3—22no—
329C3×Dic3—21no—
330C3×Dic3—21no—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3—41no—
334C3×Dic3—41no—
335C3×Dic3—41no—
336C3×Dic3—81no—
337C3×Dic3—122no—
338C3×Dic3—121no—
339C3×Dic3—121no—
340C2×SD16—91no—
341C2×D8—91no—
342#124—91no—
343C4×D4—91no—
344C4⋊C8—31no—
345#103—91no—
346#102—31no—
347C33#46311yescharacteristic, complemented, Sylow
348C23×C3—61no—
349C23×C3—61no—
350C23×C3—121no—
351C23×C3—181no—
352C22×D3—21no—
353C22×D3—61no—
354C22×D3—61no—
355C22×D3—121no—
356C3×Q8—181no—
357C3×Q8—181no—
358C3×D4—62no—
359C3×D4—61no—
360C3×D4—61no—
361C3×D4—61no—
362C3×D4—61no—
363C3×D4—92no—
364C3×D4—121no—
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—121no—
379C2×C3×C4—181no—
380C2×C3×C4—362no—
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C22×C3×D311yescharacteristic
383C2×C3×C4C22×C3×D311yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—61no—
386C2×Dic3—61no—
387C2×Dic3—121no—
388C2×Dic3C3×D1211yescharacteristic
389C2×Dic3C3×D1211yescharacteristic
390D12—21no—
391D12—21no—
392D12—32no—
393D12—32no—
394D12—62no—
395D12—61no—
396D12—61no—
397D12—121no—
398C4×D3—122no—
399C4×D3—122no—
400Dic6—62no—
401Dic6—61no—
402Dic6—61no—
403Dic6—61no—
404Dic6—61no—
405Dic6—121no—
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—241no—
430C2×C32#88411yescharacteristic
431C2×C32#89511yescharacteristic
432C2×C32#90411yescharacteristic
433C2×C32#90811yescharacteristic
434C2×C32#91111yescharacteristic
435C2×C32#91611yescharacteristic
436C2×C32C3×#12111yescharacteristic
437C2×C32C3×#13511yescharacteristic
438C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
439C3⋊D3—42no—
440C3×D3—82no—
441C3×D3—81no—
442C3×D3—81no—
443C3×D3—122no—
444C3×D3—122no—
445C3×D3—242no—
446C2×Q8—91no—
447C2×D4—31no—
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—122no—
464C22×C3—121no—
465C22×C3—121no—
466C22×C3—121no—
467C22×C3—121no—
468C22×C3—121no—
469C22×C3—121no—
470C22×C3—182no—
471C22×C3—241no—
472C22×C3—241no—
473C22×C3—361no—
474C22×C3—361no—
475C22×C3D3×D1211yescharacteristic
476C22×C3C2×C3×D1211yescharacteristic
477C22×C3C3×D3×D411yescharacteristic
478C2×D3—41no—
479C2×D3—41no—
480C2×D3—41no—
481C2×D3—62no—
482C2×D3—62no—
483C2×D3—122no—
484C2×D3—121no—
485C2×D3—121no—
486C2×D3—121no—
487C2×D3—121no—
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—121no—
505C3×C4—121no—
506C3×C4—241no—
507C3×C4—361no—
508C3×C4#379911yescharacteristic
509C3×C4#381111yescharacteristic
510C3×C4C3×#23511yescharacteristic
511C3×C4C3×D3×D411yescharacteristic
512C3×C4C3×D3×D411yescharacteristic
513C3×C4C3×#23911yescharacteristic
514Dic3—21no—
515Dic3—21no—
516Dic3—41no—
517Dic3—122no—
518Dic3—121no—
519Dic3—121no—
520C32—21no—
521C32—21no—
522C32—21no—
523C32—41no—
524C32#478711yescharacteristic, complemented
525C32#484411yescharacteristic, complemented
526C32C3×#46311yescharacteristic, complemented
527C23—61no—
528C23—181no—
529Q8—181no—
530Q8—181no—
531D4—61no—
532D4—61no—
533D4—92no—
534D4—181no—
535C2×C4—31no—
536C2×C4—31no—
537C2×C4—61no—
538C2×C4—181no—
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—242no—
556C2×C3—241no—
557C2×C3—241no—
558C2×C3—362no—
559C2×C3#6379511yescharacteristic
560C2×C3#6380611yescharacteristic
561C2×C3#6391011yescharacteristic
562C2×C3C3×#88411yescharacteristic
563C2×C3C3×#89511yescharacteristic
564C2×C3C3×#90411yescharacteristic
565C2×C3C3×#90811yescharacteristic
566C2×C3C3×#91111yescharacteristic
567C2×C3C3×#91611yescharacteristic
568D3—81no—
569D3—122no—
570D3—122no—
571D3—242no—
572C22—121no—
573C22—121no—
574C22—121no—
575C22—182no—
576C22—361no—
577C22—361no—
578C22C3×D3×D1211yescharacteristic
579C4—61no—
580C4—61no—
581C4—121no—
582C4—361no—
583C4C3×#379911yescharacteristic
584C4C3×#381111yescharacteristic
585C3—21no—
586C3—21no—
587C3—21no—
588C3—41no—
589C3#1059086011yescharacteristic, direct product factor
590C3C3×#478711yescharacteristic, complemented
591C3C3×#484411yescharacteristic, complemented
592C2—241no—
593C2—362no—
594C2C3×#6379511yescharacteristic
595C2C3×#6380611yescharacteristic
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×#3714C2
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
1. C2×C3×#3744C2semidirect
2. C3×#63894C2semidirect
3. C3×#63860C2semidirect
5. C32×#821C2semidirect
6. C32×Dic3⋊C8C2semidirect
7. C3×#63567C2semidirect
10. #10590860C3direct (N × H)
24. C2×#3744C2×C3semidirect
34. #63894C2×C3semidirect
35. #63860C2×C3semidirect
40. C3×#821D3semidirect
41. C3×#821C2×C3semidirect
44. C3×Dic3⋊C8D3semidirect
45. C3×Dic3⋊C8C2×C3semidirect
47. #63567C2×C3semidirect
150. #821C3×D3semidirect
152. Dic3⋊C8C3×D3semidirect
347. C33#463semidirect
524. C32#4787semidirect
525. C32#4844semidirect
526. C32C3×#463semidirect
589. C3#10590860direct (N × H)
590. C3C3×#4787semidirect
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×#37442
C3×#638942
C3×#638602
C2×C3×#37142
C32×#8212
C32×Dic3⋊C82
C3×#635672
C3×#48443
C3×#47873
#105908603