Finite Group Databasebeta

← C3×#10591578 #13158581

Subgroups — C3×#10591578 #13158581

Order 1728.

664 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#947C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic, complemented
3C3×Dic3×Dic6C211yesmaximal, characteristic, complemented
4C2×C3×#3715C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic
8C3×#5185—31nomaximal
9C3×#5054—31nomaximal
10#10591578C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×#216—22no—
17C32×(C2×D3)⋊C4—21no—
18C32×(C2×D3)⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C3×#3715—22no—
24C2×C32×Q16—31no—
25C32×#124—31no—
26C3×#947—21no—
27C3×#947D311yescharacteristic
28C3×#947C2×C311yescharacteristic
29C3×Q8×Dic3—31no—
30C2×C3×#216—21no—
31C2×C3×#216D311yescharacteristic, complemented
32C2×C3×#216C2×C311yescharacteristic, complemented
33C2×C3×#215—31no—
34C2×C3×#214—31no—
35C3×C4×Dic6—31no—
36Dic3×Dic6C2×C311yescharacteristic, complemented
37C2×#3715C2×C311yescharacteristic
38#63662C2×C311yescharacteristic, complemented
39C3×#835—31no—
40C3×#835—31no—
41C3×#835—61no—
42C3×#832—31no—
43C3×#821—31no—
44C3×Dic3⋊C8—31no—
45C3×C4⋊(C3⋊C8)—31no—
46Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
47#63563C2×C311yescharacteristic
48C22×C32×D3—21no—
49C33×Q8—21no—
50C33×Q8—21no—
51C2×C33×C4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C32×D12—21no—
58C32×D12—21no—
59C32×Dic6—21no—
60C32×Dic6D412yes—
61C3×C3⋊(C3⋊C8)—21no—
62C32×C3⋊C8—21no—
63C3×#464—91no—
64#5185—31no—
65#5054—31no—
66C2×C32×Q8—21no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—31no—
70C2×C32×Q8—41no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C22×C311yescharacteristic
74C2×C32×D4—31no—
75C2×C3×D12—21no—
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×D12C22×C311yescharacteristic
78C2×C3×Dic6—21no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C32×Q16—62no—
82C2×C32×C8—31no—
83C32×C22⋊C4—61no—
84C32×C22⋊C4—61no—
85C32×C42—31no—
86C4⋊(C3⋊Dic3)C22×C311yescharacteristic
87C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
88C3×#216—24no—
89C3×#216—22no—
90C3×#216—22no—
91C3×#215—62no—
92C3×#214—62no—
93C3×(C2×D3)⋊C4—22no—
94C3×(C2×D3)⋊C4—22no—
95C3×(C2×D3)⋊C4—21no—
96C3×(C2×D3)⋊C4—21no—
97C3×(C2×D3)⋊C4—21no—
98C3×(C2×D3)⋊C4—21no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3—61no—
102C3×C4⋊Dic3—61no—
103C3×Dic3⋊C4—61no—
104C3×C4×Dic3—21no—
105C3×C4×Dic3—61no—
106C3×C4×Dic3—61no—
107C3×C4×Dic3C2×D311yescharacteristic
108C3×C4×Dic3C22×C311yescharacteristic
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—61no—
113C2×C3×C3⋊C8C2×D311yescharacteristic
114C2×C3×C3⋊C8C22×C311yescharacteristic
115Dic3⋊Dic3—21no—
116Dic32—21no—
117#3715—22no—
118C22×C33C2×D411yescharacteristic
119C2×C32×D3—41no—
120C2×C32×D3—41no—
121C2×C32×D3—41no—
122C33×C4—41no—
123C33×C4C2×D411yescharacteristic
124C33×C4#4111yescharacteristic
125C3×C3⋊Dic3—41no—
126C32×Dic3—22no—
127C32×Dic3—21no—
128C32×Dic3—21no—
129C2×C3×Q16—31no—
130C2×C3×Q16—31no—
131C2×C3×Q16—61no—
132C2×C3×SD16—91no—
133C3×#124—31no—
134C3×#124—31no—
135C3×#124—61no—
136C3×C4×Q8—91no—
137#947C3×D311yescharacteristic
138Q8×Dic3—31no—
139C2×#216C3×D311yescharacteristic, complemented
140C2×#215—31no—
141C2×#214—31no—
142C4×Dic6—31no—
143C3×C4⋊C8—91no—
144C3×#103—91no—
145C3×#102—91no—
146#835—31no—
147#835—31no—
148#835—61no—
149#832—31no—
150#821—31no—
151Dic3⋊C8—31no—
152C4⋊(C3⋊C8)—31no—
153C23×C32—61no—
154C22×C3×D3—22no—
155C22×C3×D3—21no—
156C22×C3×D3—21no—
157C32×Q8—21no—
158C32×Q8—21no—
159C32×Q8—22no—
160C32×Q8—22no—
161C32×Q8—21no—
162C32×Q8—21no—
163C32×Q8—21no—
164C32×Q8—21no—
165C32×Q8—32no—
166C32×Q8—41no—
167C32×Q8—41no—
168C32×Q8—41no—
169C32×Q8—41no—
170C32×Q8—42no—
171C32×Q8—42no—
172C32×Q8—61no—
173C32×D4—61no—
174C32×D4—61no—
175C2×C32×C4—21no—
176C2×C32×C4—22no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—31no—
183C2×C32×C4—31no—
184C2×C32×C4—41no—
185C2×C32×C4—41no—
186C2×C32×C4—41no—
187C2×C32×C4—42no—
188C2×C32×C4—61no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C23×C311yescharacteristic
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—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—121no—
201C2×C3×Dic3#6711yescharacteristic
202C2×C3×Dic3#6711yescharacteristic
203C2×C3×Dic3C3×D411yescharacteristic
204C2×C3×Dic3C3×D411yescharacteristic
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—21no—
209C3×D12—41no—
210C3×D12—41no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—22no—
214C3×Dic6—41no—
215C3×Dic6#6712yes—
216C3×Dic6C3×D412yes—
217C32×C8—61no—
218C3⋊(C3⋊C8)—21no—
219C3×C3⋊C8—22no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—21no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224C3×C3⋊C8—121no—
225#464—91noSylow
226C2×C33#12111yescharacteristic
227C2×C33#13511yescharacteristic
228C2×C33#13711yescharacteristic
229C32×D3—81no—
230C2×C3×Q8—21no—
231C2×C3×Q8—21no—
232C2×C3×Q8—21no—
233C2×C3×Q8—31no—
234C2×C3×Q8—31no—
235C2×C3×Q8—41no—
236C2×C3×Q8—61no—
237C2×C3×Q8D3211yescharacteristic
238C2×C3×Q8C2×C3×D311yescharacteristic
239C2×C3×Q8C2×C3×D311yescharacteristic
240C2×C3×D4—31no—
241C2×C3×D4—31no—
242C2×C3×D4—61no—
243C2×D12C2×C3×D311yescharacteristic
244C2×Dic6C2×C3×D311yescharacteristic
245C3×Q16—62no—
246C3×Q16—62no—
247C3×Q16—64no—
248C3×SD16—182no—
249C2×C3×C8—31no—
250C2×C3×C8—31no—
251C2×C3×C8—61no—
252C2×C3×C8—91no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4—181no—
255C3×C22⋊C4—61no—
256C3×C22⋊C4—61no—
257C3×C22⋊C4—61no—
258C3×C22⋊C4—61no—
259C3×C22⋊C4—62no—
260C3×C22⋊C4—62no—
261C3×C42—31no—
262C3×C42—31no—
263C3×C42—61no—
264C3×C42—181no—
265#216—22no—
266#215—62no—
267#214—62no—
268(C2×D3)⋊C4—21no—
269(C2×D3)⋊C4—21no—
270C4⋊Dic3—31no—
271C4⋊Dic3—31no—
272C4⋊Dic3—61no—
273C4⋊Dic3—61no—
274Dic3⋊C4—61no—
275C4×Dic3—61no—
276C4×Dic3—61no—
277C4×Dic3C2×C3×D311yescharacteristic
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—61no—
281C2×C3⋊C8C2×C3×D311yescharacteristic
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—41no—
286C22×C32—121no—
287C22×C32—121no—
288C22×C32—121no—
289C22×C32D3×D411yescharacteristic
290C22×C32C2×#6711yescharacteristic
291C22×C32C2×C3×D411yescharacteristic
292C2×C3×D3—42no—
293C2×C3×D3—42no—
294C2×C3×D3—42no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C2×C3×D3—41no—
298C2×C3×D3—41no—
299C2×C3×D3—41no—
300C2×C3×D3—41no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—41no—
308C32×C4—41no—
309C32×C4—42no—
310C32×C4—42no—
311C32×C4—42no—
312C32×C4—44no—
313C32×C4—41no—
314C32×C4—41no—
315C32×C4—41no—
316C32×C4—61no—
317C32×C4—61no—
318C32×C4—62no—
319C32×C4D3×D411yescharacteristic
320C32×C4#23711yescharacteristic
321C32×C4#23711yescharacteristic
322C32×C4C2×#6711yescharacteristic
323C32×C4C2×C3×D411yescharacteristic
324C32×C4C3×#4111yescharacteristic
325C3⋊Dic3—41no—
326C3×Dic3—22no—
327C3×Dic3—21no—
328C3×Dic3—21no—
329C3×Dic3—22no—
330C3×Dic3—21no—
331C3×Dic3—21no—
332C3×Dic3—42no—
333C3×Dic3—41no—
334C3×Dic3—41no—
335C3×Dic3—121no—
336C3×Dic3—121no—
337C3×Dic3—241no—
338C2×Q16—31no—
339C2×SD16—91no—
340#124—31no—
341C4×Q8—91no—
342C4⋊C8—91no—
343#103—91no—
344#102—91no—
345C33#46411yescharacteristic, complemented, Sylow
346C23×C3—62no—
347C23×C3—61no—
348C23×C3—61no—
349C22×D3—21no—
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—21no—
353C3×Q8—21no—
354C3×Q8—21no—
355C3×Q8—21no—
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—32no—
359C3×Q8—32no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—42no—
365C3×Q8—42no—
366C3×Q8—62no—
367C3×Q8—61no—
368C3×Q8—61no—
369C3×Q8—121no—
370C3×D4—61no—
371C3×D4—61no—
372C3×D4—61no—
373C3×D4—61no—
374C3×D4—121no—
375C3×D4—121no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—31no—
384C2×C3×C4—31no—
385C2×C3×C4—31no—
386C2×C3×C4—31no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—42no—
390C2×C3×C4—41no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—121no—
396C2×C3×C4—181no—
397C2×C3×C4C2×D3211yescharacteristic
398C2×C3×C4C22×C3×D311yescharacteristic
399C2×C3×C4C22×C3×D311yescharacteristic
400C2×Dic3—21no—
401C2×Dic3—61no—
402C2×Dic3—61no—
403C2×Dic3—121no—
404C2×Dic3C3×#6711yescharacteristic
405C2×Dic3C3×#6711yescharacteristic
406D12—21no—
407D12—21no—
408Dic6—21no—
409Dic6C3×#6712yes—
410C3×C8—62no—
411C3×C8—61no—
412C3×C8—61no—
413C3×C8—181no—
414C3⋊C8—21no—
415C3⋊C8—61no—
416C3⋊C8—61no—
417C3⋊C8—121no—
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—21no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32—41no—
430C2×C32—241no—
431C2×C32#91811yescharacteristic
432C2×C32#91911yescharacteristic
433C2×C32#93811yescharacteristic
434C2×C32#93911yescharacteristic
435C2×C32#95011yescharacteristic
436C2×C32#95111yescharacteristic
437C2×C32C3×#12111yescharacteristic
438C2×C32C3×#13511yescharacteristic
439C2×C32C3×#13711yescharacteristic
440C3×D3—82no—
441C3×D3—81no—
442C3×D3—81no—
443C2×Q8—31no—
444C2×Q8C3×D3211yescharacteristic
445C2×D4—31no—
446Q16—62no—
447SD16—182no—
448C2×C8—31no—
449C2×C8—91no—
450C4⋊C4—91no—
451C4⋊C4—181no—
452C22⋊C4—61no—
453C22⋊C4—61no—
454C42—31no—
455C42—181no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—41no—
460C22×C3—122no—
461C22×C3—122no—
462C22×C3—122no—
463C22×C3—121no—
464C22×C3—121no—
465C22×C3—121no—
466C22×C3—121no—
467C22×C3—121no—
468C22×C3—121no—
469C22×C3D3×#6711yescharacteristic
470C22×C3C3×D3×D411yescharacteristic
471C22×C3C2×C3×#6711yescharacteristic
472C2×D3—41no—
473C2×D3—41no—
474C2×D3—41no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—42no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—44no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—61no—
491C3×C4—61no—
492C3×C4—62no—
493C3×C4—61no—
494C3×C4—61no—
495C3×C4—62no—
496C3×C4—121no—
497C3×C4—121no—
498C3×C4—122no—
499C3×C4—361no—
500C3×C4#380511yescharacteristic
501C3×C4D3×#6711yescharacteristic
502C3×C4C3×D3×D411yescharacteristic
503C3×C4C3×#23711yescharacteristic
504C3×C4C3×#23711yescharacteristic
505C3×C4C2×C3×#6711yescharacteristic
506Dic3—22no—
507Dic3—21no—
508Dic3—21no—
509Dic3—121no—
510Dic3—121no—
511Dic3—241no—
512C32—21no—
513C32—21no—
514C32—21no—
515C32—41no—
516C32#505411yescharacteristic, complemented
517C32#518511yescharacteristic, complemented
518C32C3×#46411yescharacteristic, complemented
519C23—61no—
520Q8—21no—
521Q8—21no—
522Q8—32no—
523Q8—61no—
524D4—61no—
525D4—61no—
526C2×C4—21no—
527C2×C4—31no—
528C2×C4—31no—
529C2×C4—61no—
530C2×C4—181no—
531C2×C4C2×C3×D3211yescharacteristic
532C8—61no—
533C8—181no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—41no—
546C2×C3—242no—
547C2×C3—241no—
548C2×C3—241no—
549C2×C3#6394311yescharacteristic
550C2×C3#6394911yescharacteristic
551C2×C3#6397011yescharacteristic
552C2×C3C3×#91811yescharacteristic
553C2×C3C3×#91911yescharacteristic
554C2×C3C3×#93811yescharacteristic
555C2×C3C3×#93911yescharacteristic
556C2×C3C3×#95011yescharacteristic
557C2×C3C3×#95111yescharacteristic
558D3—81no—
559C22—121no—
560C22—121no—
561C22—121no—
562C22C3×D3×#6711yescharacteristic
563C4—41no—
564C4—61no—
565C4—61no—
566C4—62no—
567C4—361no—
568C4C3×#380511yescharacteristic
569C4C3×D3×#6711yescharacteristic
570C3—21no—
571C3—21no—
572C3—21no—
573C3—41no—
574C3#1059157811yescharacteristic, direct product factor
575C3C3×#505411yescharacteristic, complemented
576C3C3×#518511yescharacteristic, complemented
577C2—241no—
578C2C3×#6394311yescharacteristic
579C2C3×#6394911yescharacteristic
580C2C3×#6397011yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC2×C3×Q8D32
hypocenterC32C3×#464
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C32×#216C2semidirect
3. C3×Dic3×Dic6C2semidirect
5. C3×#63662C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
10. #10591578C3direct (N × H)
31. C2×C3×#216D3semidirect
32. C2×C3×#216C2×C3semidirect
36. Dic3×Dic6C2×C3semidirect
38. #63662C2×C3semidirect
46. Dic3⋊(C3⋊C8)C2×C3semidirect
139. C2×#216C3×D3semidirect
345. C33#464semidirect
516. C32#5054semidirect
517. C32#5185semidirect
518. C32C3×#464semidirect
574. C3#10591578direct (N × H)
575. C3C3×#5054semidirect
576. C3C3×#5185semidirect

Maximal subgroups

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

subgroupindex
C32×#9472
C2×C32×#2162
C3×Dic3×Dic62
C2×C3×#37152
C3×#636622
C3×Dic3⋊(C3⋊C8)2
C3×#635632
C3×#51853
C3×#50543
#105915783