Finite Group Databasebeta

← C3×#10590577 #13157472

Subgroups — C3×#10590577 #13157472

Order 1728.

682 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#897C211yesmaximal, characteristic, complemented
2C3×#63909C211yesmaximal, characteristic
3C2×C3×#3717C211yesmaximal, characteristic, complemented
4C2×C3×D3×C3⋊C8C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic, complemented
6C32×#808C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8C3×#5048—31nomaximal
9C3×#4864—31nomaximal
10#10590577C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3×#626—21no—
15C33×C4⋊C4C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3—21no—
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×#3722—21no—
22C3×#3717—22no—
23C3×D3×C3⋊C8—22no—
24C32×C22⋊Q8—31no—
25C2×C3×#215—31no—
26C3×#930—31no—
27C22×C3×C3⋊C8—31no—
28C2×C3×#204—31no—
29C2×C3×D3×C8—31no—
30C3×#897—21no—
31C3×#897D311yescharacteristic, complemented
32C3×#897C2×C311yescharacteristic, complemented
33C3×#895—31no—
34#63909C2×C311yescharacteristic
35C2×#3717C2×C311yescharacteristic, complemented
36C2×D3×C3⋊C8C2×C311yescharacteristic, complemented
37C32×#106—31no—
38#63636C2×C311yescharacteristic, complemented
39C3×#835—31no—
40C3×#810—31no—
41C3×#809—31no—
42C3×#809—31no—
43C3×#809—61no—
44C3×#808—21no—
45C3×#808D311yescharacteristic, complemented
46C3×#808C2×C311yescharacteristic, complemented
47#63565C2×C311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C22×C32×D3D411yescharacteristic
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3D411yescharacteristic
57C32×C4×D3—21no—
58C32×C4×D3—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×#626—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8D412yescomplemented
64C3×#465—91no—
65#5048—31no—
66#4864—31no—
67C2×C32×Q8—31no—
68C22×C32×C4—31no—
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×C3×#67—61no—
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×D12—61no—
75C2×C3×C4×D3—21no—
76C2×C3×C4×D3C2×D311yescharacteristic
77C2×C3×C4×D3C22×C311yescharacteristic
78C2×C3×Dic6—21no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C2×#626—21no—
82C2×C32×C8—31no—
83C32×C4⋊C4—21no—
84C32×C4⋊C4—21no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4—31no—
87C32×C4⋊C4—41no—
88C32×C4⋊C4—61no—
89C32×C4⋊C4C2×D311yescharacteristic
90C32×C4⋊C4C2×D311yescharacteristic
91C32×C4⋊C4C22×C311yescharacteristic
92C32×C22⋊C4—61no—
93C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
94C3×#215—62no—
95C3×(C2×D3)⋊C4—22no—
96C3×(C2×D3)⋊C4—21no—
97C3×(C2×D3)⋊C4—21no—
98C3×(C2×D3)⋊C4—61no—
99C3×(C2×D3)⋊C4—61no—
100C3×C4⋊Dic3—22no—
101C3×C4⋊Dic3—21no—
102C3×C4⋊Dic3—21no—
103C3×C4⋊Dic3—21no—
104C3×C4⋊Dic3C2×D311yescharacteristic
105C3×C4⋊Dic3C22×C311yescharacteristic
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—62no—
110C2×C3×C3⋊C8—61no—
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113C3×#204—62no—
114C3×D3×C8—62no—
115#3722—21no—
116#3717—22no—
117D3×C3⋊C8—22no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C32×D3—21no—
123C2×C32×D3—21no—
124C2×C32×D3SD1612yescomplemented
125C33×C4—41no—
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C32×Dic3—21no—
129C32×Dic3—41no—
130C32×Dic3—41no—
131C2×C3×SD16—91no—
132C22×C3×C8—91no—
133C3×C22⋊Q8—31no—
134C3×C22⋊Q8—31no—
135C3×C22⋊Q8—61no—
136C3×#121—91no—
137C2×#215—31no—
138#930—31no—
139C22×C3⋊C8—31no—
140C2×#204—31no—
141C2×D3×C8—31no—
142#897C3×D311yescharacteristic, complemented
143#895—31no—
144C3×#106—31no—
145C3×#106—31no—
146C3×#106—61no—
147C3×#103—91no—
148C3×#102—91no—
149#835—31no—
150#810—31no—
151#809—31no—
152#809—31no—
153#809—61no—
154#808C3×D311yescharacteristic, complemented
155C23×C32—31no—
156C22×C3⋊D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—121no—
161C22×C3×D3#6711yescharacteristic
162C22×C3×D3C3×D411yescharacteristic
163C32×Q8—61no—
164C32×Q8—61no—
165C2×C32×C4—21no—
166C2×C32×C4—22no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—31no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4—41no—
176C2×C32×C4—42no—
177C2×C32×C4—61no—
178C2×C32×C4—61no—
179C2×C32×C4—61no—
180C2×C32×C4—61no—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C23×C311yescharacteristic
184C4⋊(C3⋊D3)—21no—
185C4⋊(C3⋊D3)—21no—
186C3×#67—122no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—41no—
194C2×C3×Dic3#6711yescharacteristic
195C2×C3×Dic3C3×D411yescharacteristic
196C3×D12—62no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—61no—
201C3×D12—121no—
202C3×D12—122no—
203C3×C4×D3—21no—
204C3×C4×D3—21no—
205C3×C4×D3—21no—
206C3×C4×D3—21no—
207C3×C4×D3—41no—
208C3×C4×D3—41no—
209C3×Dic6—21no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—41no—
214C3×Dic6—41no—
215#626—42no—
216C32×C8—32no—
217C3⋊(C3⋊C8)—21no—
218C3×C3⋊C8—22no—
219C3×C3⋊C8—61no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—121no—
222C3×C3⋊C8D1212yescomplemented
223C3×C3⋊C8C3×D412yescomplemented
224#465—91noSylow
225C2×C33#12111yescharacteristic
226C2×C33C2×SD1611yescharacteristic
227C2×C33#13511yescharacteristic
228C3×C3⋊D3—81no—
229C32×D3—22no—
230C2×C3×Q8—31no—
231C2×C3×Q8—31no—
232C2×C3×Q8—61no—
233C2×C3×D4—91no—
234C2×C3×D4—181no—
235C22×C3×C4—31no—
236C22×C3×C4—31no—
237C22×C3×C4—61no—
238C2×#67—61no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—61no—
242C2×D12—61no—
243C2×C4×D3C2×C3×D311yescharacteristic
244C2×Dic6C2×C3×D311yescharacteristic
245C3×SD16—182no—
246C2×C3×C8—31no—
247C2×C3×C8—31no—
248C2×C3×C8—61no—
249C2×C3×C8—91no—
250C2×C3×C8—182no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—31no—
255C3×C4⋊C4—31no—
256C3×C4⋊C4—41no—
257C3×C4⋊C4—61no—
258C3×C4⋊C4—61no—
259C3×C4⋊C4—61no—
260C3×C4⋊C4—62no—
261C3×C4⋊C4D3211yescharacteristic
262C3×C4⋊C4C2×C3×D311yescharacteristic
263C3×C4⋊C4C2×C3×D311yescharacteristic
264C3×C22⋊C4—61no—
265C3×C22⋊C4—61no—
266C3×C22⋊C4—62no—
267C3×C22⋊C4—181no—
268#215—62no—
269(C2×D3)⋊C4—21no—
270(C2×D3)⋊C4—61no—
271(C2×D3)⋊C4—61no—
272C4⋊Dic3—21no—
273C4⋊Dic3C2×C3×D311yescharacteristic
274C2×C3⋊C8—31no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—61no—
277C2×C3⋊C8—62no—
278C2×C3⋊C8C2×C3×D311yescharacteristic
279#204—62no—
280D3×C8—62no—
281C22×C32—21no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—32no—
285C22×C32—41no—
286C22×C32—61no—
287C22×C32—61no—
288C22×C32D3×D411yescharacteristic
289C22×C32C2×#6711yescharacteristic
290C22×C32C2×C3×D411yescharacteristic
291C2×C3⋊D3—41no—
292C2×C3⋊D3—41no—
293C2×C3⋊D3—41no—
294C2×C3×D3—21no—
295C2×C3×D3—21no—
296C2×C3×D3—21no—
297C2×C3×D3—21no—
298C2×C3×D3—22no—
299C2×C3×D3—41no—
300C2×C3×D3—41no—
301C2×C3×D3—122no—
302C2×C3×D3—121no—
303C2×C3×D3—121no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
308C2×C3×D3—241no—
309C2×C3×D3—241no—
310C2×C3×D3#21512yescomplemented
311C2×C3×D3C3×SD1612yescomplemented
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—41no—
319C32×C4—41no—
320C32×C4—41no—
321C32×C4—42no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4—61no—
325C32×C4—81no—
326C32×C4—81no—
327C32×C4—82no—
328C32×C4—121no—
329C32×C4—121no—
330C32×C4C2×D1211yescharacteristic
331C32×C4D3×D411yescharacteristic
332C32×C4#23711yescharacteristic
333C32×C4#23911yescharacteristic
334C32×C4C2×C3×D411yescharacteristic
335C32×C4C3×#4111yescharacteristic
336C3×Dic3—21no—
337C3×Dic3—21no—
338C3×Dic3—41no—
339C3×Dic3—42no—
340C3×Dic3—41no—
341C3×Dic3—41no—
342C3×Dic3—41no—
343C3×Dic3—41no—
344C3×Dic3—81no—
345C2×SD16—91no—
346C22×C8—91no—
347C22⋊Q8—31no—
348#121—91no—
349#106—31no—
350#103—91no—
351#102—91no—
352C33#46511yescharacteristic, complemented, Sylow
353C23×C3—31no—
354C23×C3—31no—
355C23×C3—61no—
356C23×C3—181no—
357C22×D3—61no—
358C22×D3—61no—
359C22×D3—121no—
360C22×D3C3×#6711yescharacteristic
361C3×Q8—61no—
362C3×Q8—61no—
363C3×Q8—61no—
364C3×Q8—61no—
365C3×Q8—121no—
366C3×Q8—121no—
367C3×D4—181no—
368C3×D4—181no—
369C3×D4—362no—
370C2×C3×C4—22no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—31no—
378C2×C3×C4—31no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—42no—
382C2×C3×C4—41no—
383C2×C3×C4—61no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4—62no—
389C2×C3×C4—61no—
390C2×C3×C4—61no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—121no—
394C2×C3×C4—121no—
395C2×C3×C4—121no—
396C2×C3×C4C2×D3211yescharacteristic
397C2×C3×C4C22×C3×D311yescharacteristic
398C2×C3×C4C22×C3×D311yescharacteristic
399#67—122no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3C3×#6711yescharacteristic
403D12—62no—
404D12—61no—
405D12—61no—
406D12—61no—
407D12—61no—
408D12—121no—
409D12—122no—
410C4×D3—21no—
411C4×D3—21no—
412Dic6—21no—
413Dic6—21no—
414C3×C8—32no—
415C3×C8—32no—
416C3×C8—62no—
417C3×C8—181no—
418C3⋊C8—61no—
419C3⋊C8—61no—
420C3⋊C8—121no—
421C3⋊C8C3×D1212yescomplemented
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—41no—
432C2×C32—41no—
433C2×C32—41no—
434C2×C32—62no—
435C2×C32#89511yescharacteristic
436C2×C32D3×SD1611yescharacteristic
437C2×C32#91611yescharacteristic
438C2×C32#93911yescharacteristic
439C2×C32C2×#21511yescharacteristic
440C2×C32#95011yescharacteristic
441C2×C32C3×#12111yescharacteristic
442C2×C32C2×C3×SD1611yescharacteristic
443C2×C32C3×#13511yescharacteristic
444C3⋊D3—81no—
445C3×D3—22no—
446C3×D3—22no—
447C3×D3—42no—
448C3×D3—242no—
449C3×D3—241no—
450C3×D3—241no—
451C2×Q8—31no—
452C2×D4—91no—
453C2×D4—181no—
454C22×C4—31no—
455SD16—182no—
456C2×C8—31no—
457C2×C8—91no—
458C2×C8—182no—
459C4⋊C4—31no—
460C4⋊C4—61no—
461C4⋊C4C3×D3211yescharacteristic
462C22⋊C4—61no—
463C22⋊C4—181no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—32no—
468C22×C3—32no—
469C22×C3—41no—
470C22×C3—62no—
471C22×C3—61no—
472C22×C3—61no—
473C22×C3—61no—
474C22×C3—61no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3—361no—
478C22×C3—361no—
479C22×C3—361no—
480C22×C3D3×#6711yescharacteristic
481C22×C3C3×D3×D411yescharacteristic
482C22×C3C2×C3×#6711yescharacteristic
483C2×D3—21no—
484C2×D3—21no—
485C2×D3—122no—
486C2×D3—121no—
487C2×D3—121no—
488C2×D3—121no—
489C2×D3—121no—
490C2×D3—121no—
491C2×D3—121no—
492C2×D3—241no—
493C2×D3—241no—
494C2×D3C3×#21512yescomplemented
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—42no—
502C3×C4—41no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—61no—
508C3×C4—61no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4—82no—
512C3×C4—121no—
513C3×C4—122no—
514C3×C4—121no—
515C3×C4—121no—
516C3×C4—121no—
517C3×C4—121no—
518C3×C4—241no—
519C3×C4#379611yescharacteristic
520C3×C4D3×D1211yescharacteristic
521C3×C4C2×C3×D1211yescharacteristic
522C3×C4C3×D3×D411yescharacteristic
523C3×C4C3×#23711yescharacteristic
524C3×C4C3×#23911yescharacteristic
525Dic3—21no—
526Dic3—41no—
527Dic3—41no—
528C32—21no—
529C32—21no—
530C32—21no—
531C32—41no—
532C32#486411yescharacteristic, complemented
533C32#504811yescharacteristic, complemented
534C32C3×#46511yescharacteristic, complemented
535C23—31no—
536C23—181no—
537Q8—61no—
538Q8—61no—
539D4—181no—
540D4—181no—
541D4—362no—
542C2×C4—21no—
543C2×C4—31no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4C2×C3×D3211yescharacteristic
549C8—32no—
550C8—181no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—41no—
561C2×C3—41no—
562C2×C3—41no—
563C2×C3—62no—
564C2×C3—62no—
565C2×C3—122no—
566C2×C3—721no—
567C2×C3#6388611yescharacteristic
568C2×C3#6393511yescharacteristic
569C2×C3D3×#21511yescharacteristic
570C2×C3C3×#89511yescharacteristic
571C2×C3C3×D3×SD1611yescharacteristic
572C2×C3C3×#91611yescharacteristic
573C2×C3C3×#93911yescharacteristic
574C2×C3C2×C3×#21511yescharacteristic
575C2×C3C3×#95011yescharacteristic
576D3—22no—
577D3—242no—
578D3—241no—
579D3—241no—
580C22—32no—
581C22—61no—
582C22—61no—
583C22—361no—
584C22—361no—
585C22—361no—
586C22C3×D3×#6711yescharacteristic
587C4—41no—
588C4—61no—
589C4—121no—
590C4—121no—
591C4C3×#379611yescharacteristic
592C4C3×D3×D1211yescharacteristic
593C3—21no—
594C3—21no—
595C3—21no—
596C3—41no—
597C3#1059057711yescharacteristic, direct product factor
598C3C3×#486411yescharacteristic, complemented
599C3C3×#504811yescharacteristic, complemented
600C2—62no—
601C2—721no—
602C2C3×#6388611yescharacteristic
603C2C3×#6393511yescharacteristic
604C2C3×D3×#21511yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#63909C2
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#465
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×#897C2semidirect
3. C2×C3×#3717C2semidirect
4. C2×C3×D3×C3⋊C8C2semidirect
5. C3×#63636C2semidirect
6. C32×#808C2semidirect
7. C3×#63565C2semidirect
10. #10590577C3direct (N × H)
31. C3×#897D3semidirect
32. C3×#897C2×C3semidirect
35. C2×#3717C2×C3semidirect
36. C2×D3×C3⋊C8C2×C3semidirect
38. #63636C2×C3semidirect
45. C3×#808D3semidirect
46. C3×#808C2×C3semidirect
47. #63565C2×C3semidirect
63. C32×C3⋊C8D4semidirect
124. C2×C32×D3SD16semidirect
142. #897C3×D3semidirect
154. #808C3×D3semidirect
222. C3×C3⋊C8D12semidirect
223. C3×C3⋊C8C3×D4semidirect
310. C2×C3×D3#215semidirect
311. C2×C3×D3C3×SD16semidirect
352. C33#465semidirect
421. C3⋊C8C3×D12semidirect
494. C2×D3C3×#215semidirect
532. C32#4864semidirect
533. C32#5048semidirect
534. C32C3×#465semidirect
597. C3#10590577direct (N × H)
598. C3C3×#4864semidirect
599. C3C3×#5048semidirect

Maximal subgroups

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

subgroupindex
C32×#8972
C3×#639092
C2×C3×#37172
C2×C3×D3×C3⋊C82
C3×#636362
C32×#8082
C3×#635652
C3×#50483
C3×#48643
#105905773