Finite Group Databasebeta

← C3×#10591536 #13158568

Subgroups — C3×#10591536 #13158568

Order 1728.

706 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3755C211yesmaximal, characteristic
2C32×Q8×Dic3C211yesmaximal, characteristic, complemented
3C3×#63894C211yesmaximal, characteristic
4C32×#835C211yesmaximal, characteristic, complemented
5C3×#63574C211yesmaximal, characteristic, complemented
6C3×#63564C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×C4×#215—31nomaximal
9C3×#4788—31nomaximal
10#10591536C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#624—21no—
14C3×#3755C414yescomplemented
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3—21no—
20C32×C4×Dic3C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3722—21no—
23C32×C4×Q8—31no—
24C2×#3755C2×C311yescharacteristic
25C3×Q8×Dic3—21no—
26C3×Q8×Dic3D311yescharacteristic, complemented
27C3×Q8×Dic3C2×C311yescharacteristic, complemented
28C2×C3×#215—31no—
29C2×C3×#215—31no—
30C2×C3×#215—61no—
31C3×#893—31no—
32C3×C4×D12—31no—
33#63894C2×C311yescharacteristic
34C32×#103—31no—
35C3×#835—21no—
36C3×#835D311yescharacteristic, complemented
37C3×#835C2×C311yescharacteristic, complemented
38C3×#821—31no—
39C3×C8×Dic3—31no—
40C3×#809—31no—
41C3×#808—31no—
42C3×#808—31no—
43C3×C4×C3⋊C8—31no—
44#63574C2×C311yescharacteristic, complemented
45#63564C2×C311yescharacteristic
46Dic3×C3⋊C8C2×C311yescharacteristic, complemented
47C22×C3×C3⋊D3—21no—
48C33×Q8—21no—
49C33×Q8C2×C412yescomplemented
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)C2×C412yes—
54C2×C32×Dic3—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3D411yescharacteristic
57C2×C32×Dic3D411yescharacteristic
58C3×#624—42no—
59C3×C3⋊(C3⋊C8)C2×C412yescomplemented
60C32×C3⋊C8—21no—
61C3×C4×SD16—91no—
62C4×#215—31no—
63#4788—31no—
64C2×C32×Q8—21no—
65C2×C32×Q8—21no—
66C2×C32×Q8—21no—
67C2×C32×Q8—41no—
68C2×C32×Q8C2×D311yescharacteristic
69C2×C32×Q8C2×D311yescharacteristic
70C2×C32×Q8C22×C311yescharacteristic
71C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×C4×D3—61no—
76C2×C3×C4×D3—61no—
77C2×#624—21no—
78C2×C32×C8—31no—
79C32×C4⋊C4—31no—
80C32×C4⋊C4—31no—
81C32×C4⋊C4—61no—
82C32×C42—31no—
83C32×C42—61no—
84#3755C3×C414yescomplemented
85C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
86C3×#215—34no—
87C3×#215—34no—
88C3×#215—64no—
89C3×(C2×D3)⋊C4—61no—
90C3×(C2×D3)⋊C4—61no—
91C3×C4⋊Dic3—22no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4⋊Dic3C22×C311yescharacteristic
99C3×C4⋊Dic3C22×C311yescharacteristic
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—61no—
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C2×C3×C3⋊C8C22×C311yescharacteristic
112#3722—21no—
113C22×C33C2×D411yescharacteristic
114C2×C3×C3⋊D3—22no—
115C2×C3×C3⋊D3—41no—
116C2×C3×C3⋊D3—41no—
117C33×C4—22no—
118C33×C4C22×C411yescharacteristic
119C33×C4#4111yescharacteristic
120C32×Dic3—21no—
121C32×Dic3—41no—
122C32×Dic3—41no—
123C32×Dic3SD1612yescomplemented
124C2×C3×SD16—91no—
125C3×C4×Q8—31no—
126C3×C4×Q8—31no—
127C3×C4×Q8—61no—
128C3×C4×D4—91no—
129Q8×Dic3C3×D311yescharacteristic, complemented
130C2×#215—31no—
131C2×#215—31no—
132C2×#215—61no—
133#893—31no—
134C4×D12—31no—
135C3×#106—91no—
136C3×#103—31no—
137C3×#103—31no—
138C3×#103—61no—
139C3×#102—91no—
140C3×C4×C8—91no—
141#835C3×D311yescharacteristic, complemented
142#821—31no—
143C8×Dic3—31no—
144#809—31no—
145#808—31no—
146#808—31no—
147C4×C3⋊C8—31no—
148C22×C3⋊D3—21no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—121no—
152C32×Q8—21no—
153C32×Q8—22no—
154C32×Q8—21no—
155C32×Q8—21no—
156C32×Q8—22no—
157C32×Q8—22no—
158C32×Q8—22no—
159C32×Q8—42no—
160C32×Q8—41no—
161C32×Q8—41no—
162C32×Q8—42no—
163C32×Q8C4×D312yescomplemented
164C32×Q8C4×D312yescomplemented
165C32×Q8C2×C3×C412yescomplemented
166C2×C32×C4—21no—
167C2×C32×C4—22no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—31no—
174C2×C32×C4—31no—
175C2×C32×C4—41no—
176C2×C32×C4—41no—
177C2×C32×C4—41no—
178C2×C32×C4—42no—
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)C2×C3×C412yes—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—21no—
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×Dic3#6711yescharacteristic
196C2×C3×Dic3C3×D411yescharacteristic
197C2×C3×Dic3C3×D411yescharacteristic
198C3×D12—32no—
199C3×D12—32no—
200C3×D12—62no—
201C3×D12—61no—
202C3×D12—61no—
203C3×D12—121no—
204C3×C4×D3—122no—
205C3×C4×D3—122no—
206#624—42no—
207C32×C8—61no—
208C3⋊(C3⋊C8)C2×C3×C412yescomplemented
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—32no—
212C3×C3⋊C8—32no—
213C3×C3⋊C8—41no—
214C3×C3⋊C8—62no—
215C4×SD16—91noSylow
216C2×C33C4×D411yescharacteristic
217C2×C33C2×SD1611yescharacteristic
218C2×C33#13511yescharacteristic
219C3×C3⋊D3—42no—
220C2×C3×Q8—21no—
221C2×C3×Q8—21no—
222C2×C3×Q8—21no—
223C2×C3×Q8—41no—
224C2×C3×Q8D3211yescharacteristic
225C2×C3×Q8C2×C3×D311yescharacteristic
226C2×C3×Q8C2×C3×D311yescharacteristic
227C2×C3×D4—91no—
228C22×C3×C4—181no—
229C2×D12—31no—
230C2×D12—31no—
231C2×D12—61no—
232C2×C4×D3—61no—
233C2×C4×D3—61no—
234C3×SD16—94no—
235C2×C3×C8—31no—
236C2×C3×C8—31no—
237C2×C3×C8—61no—
238C2×C3×C8—91no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—61no—
244C3×C4⋊C4—61no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4—62no—
248C3×C22⋊C4—181no—
249C3×C42—31no—
250C3×C42—31no—
251C3×C42—61no—
252C3×C42—61no—
253C3×C42—61no—
254C3×C42—62no—
255#215—34no—
256#215—34no—
257#215—64no—
258(C2×D3)⋊C4—61no—
259(C2×D3)⋊C4—61no—
260C4⋊Dic3—21no—
261C4⋊Dic3C2×C3×D311yescharacteristic
262C4⋊Dic3C2×C3×D311yescharacteristic
263C4×Dic3—21no—
264C4×Dic3C2×C3×D311yescharacteristic
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8C2×C3×D311yescharacteristic
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32D3×D411yescharacteristic
274C22×C32C2×#6711yescharacteristic
275C22×C32C2×C3×D411yescharacteristic
276C2×C3⋊D3—22no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—122no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
284C2×C3×D3—121no—
285C2×C3×D3—121no—
286C2×C3×D3—241no—
287C2×C3×D3—241no—
288C32×C4—22no—
289C32×C4—24no—
290C32×C4—22no—
291C32×C4—22no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—32no—
299C32×C4—41no—
300C32×C4—41no—
301C32×C4—42no—
302C32×C4—42no—
303C32×C4—44no—
304C32×C4—61no—
305C32×C4—121no—
306C32×C4—121no—
307C32×C4C2×C4×D311yescharacteristic
308C32×C4C2×C4×D311yescharacteristic
309C32×C4#23511yescharacteristic
310C32×C4#23711yescharacteristic
311C32×C4C22×C3×C411yescharacteristic
312C32×C4C3×#4111yescharacteristic
313C3×Dic3—21no—
314C3×Dic3—21no—
315C3×Dic3—22no—
316C3×Dic3—41no—
317C3×Dic3—42no—
318C3×Dic3—41no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—81no—
323C3×Dic3#21512yescomplemented
324C3×Dic3C3×SD1612yescomplemented
325C2×SD16—91no—
326C4×Q8—31no—
327C4×D4—91no—
328#106—91no—
329#103—31no—
330#102—91no—
331C4×C8—91no—
332C33C4×SD1611yescharacteristic, complemented, Sylow
333C23×C3—181no—
334C22×D3—61no—
335C22×D3—61no—
336C22×D3—121no—
337C3×Q8—22no—
338C3×Q8—21no—
339C3×Q8—21no—
340C3×Q8—21no—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—41no—
345C3×Q8—41no—
346C3×Q8—42no—
347C3×Q8—42no—
348C3×Q8#62412yescomplemented
349C3×Q8C3×C4×D312yescomplemented
350C3×Q8C3×C4×D312yescomplemented
351C3×D4—92no—
352C3×D4—181no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—41no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—62no—
371C2×C3×C4—61no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—121no—
376C2×C3×C4—362no—
377C2×C3×C4C2×D3211yescharacteristic
378C2×C3×C4C22×C3×D311yescharacteristic
379C2×C3×C4C22×C3×D311yescharacteristic
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3C3×#6711yescharacteristic
383C2×Dic3C3×#6711yescharacteristic
384D12—32no—
385D12—32no—
386D12—62no—
387D12—61no—
388D12—61no—
389D12—121no—
390C4×D3—122no—
391C4×D3—122no—
392C3×C8—61no—
393C3×C8—61no—
394C3×C8—92no—
395C3×C8—121no—
396C3⋊C8—21no—
397C3⋊C8—32no—
398C3⋊C8—32no—
399C3⋊C8—62no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—41no—
412C2×C32#67⋊C411yescharacteristic
413C2×C32D3×SD1611yescharacteristic
414C2×C32#91911yescharacteristic
415C2×C32C4×#6711yescharacteristic
416C2×C32C2×#21511yescharacteristic
417C2×C32#95011yescharacteristic
418C2×C32C3×C4×D411yescharacteristic
419C2×C32C2×C3×SD1611yescharacteristic
420C2×C32C3×#13511yescharacteristic
421C3⋊D3—42no—
422C3×D3—122no—
423C3×D3—122no—
424C3×D3—242no—
425C2×Q8C3×D3211yescharacteristic
426C2×D4—91no—
427C22×C4—181no—
428SD16—94no—
429C2×C8—31no—
430C2×C8—91no—
431C4⋊C4—31no—
432C4⋊C4—31no—
433C4⋊C4—61no—
434C22⋊C4—181no—
435C42—31no—
436C42—61no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—41no—
441C22×C3—182no—
442C22×C3—361no—
443C22×C3—361no—
444C22×C3D3×#6711yescharacteristic
445C22×C3C3×D3×D411yescharacteristic
446C22×C3C2×C3×#6711yescharacteristic
447C2×D3—62no—
448C2×D3—62no—
449C2×D3—122no—
450C2×D3—121no—
451C2×D3—121no—
452C2×D3—121no—
453C2×D3—121no—
454C2×D3—241no—
455C2×D3—241no—
456C3×C4—24no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—22no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—32no—
467C3×C4—32no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4—44no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—62no—
474C3×C4—61no—
475C3×C4—61no—
476C3×C4—121no—
477C3×C4—122no—
478C3×C4—121no—
479C3×C4—121no—
480C3×C4—121no—
481C3×C4—121no—
482C3×C4—241no—
483C3×C4#380411yescharacteristic
484C3×C4C2×#62411yescharacteristic
485C3×C4C2×C3×C4×D311yescharacteristic
486C3×C4C2×C3×C4×D311yescharacteristic
487C3×C4C3×#23511yescharacteristic
488C3×C4C3×#23711yescharacteristic
489Dic3—21no—
490Dic3—41no—
491Dic3—41no—
492Dic3C3×#21512yescomplemented
493C32—21no—
494C32—21no—
495C32—21no—
496C32—41no—
497C32#478811yescharacteristic, complemented
498C32C4×#21511yescharacteristic, complemented
499C32C3×C4×SD1611yescharacteristic, complemented
500C23—181no—
501Q8—21no—
502Q8C3×#62412yescomplemented
503D4—92no—
504D4—181no—
505C2×C4—21no—
506C2×C4—31no—
507C2×C4—31no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—362no—
511C2×C4C2×C3×D3211yescharacteristic
512C8—61no—
513C8—92no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—362no—
527C2×C3D3×#21511yescharacteristic
528C2×C3#6394911yescharacteristic
529C2×C3#6395211yescharacteristic
530C2×C3C3×#67⋊C411yescharacteristic
531C2×C3C3×D3×SD1611yescharacteristic
532C2×C3C3×#91911yescharacteristic
533C2×C3C3×C4×#6711yescharacteristic
534C2×C3C2×C3×#21511yescharacteristic
535C2×C3C3×#95011yescharacteristic
536D3—122no—
537D3—122no—
538D3—242no—
539C22—182no—
540C22—361no—
541C22—361no—
542C22C3×D3×#6711yescharacteristic
543C4—22no—
544C4—32no—
545C4—61no—
546C4—121no—
547C4—121no—
548C4C3×#380411yescharacteristic
549C4C2×C3×#62411yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059153611yescharacteristic, direct product factor
555C3C3×#478811yescharacteristic, complemented
556C3C3×C4×#21511yescharacteristic, complemented
557C2—362no—
558C2C3×D3×#21511yescharacteristic
559C2C3×#6394911yescharacteristic
560C2C3×#6395211yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×Q8D32
hypocenterC32C3×C4×SD16
LCSG ▷ 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. C32×Q8×Dic3C2semidirect
4. C32×#835C2semidirect
5. C3×#63574C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10591536C3direct (N × H)
14. C3×#3755C4semidirect
26. C3×Q8×Dic3D3semidirect
27. C3×Q8×Dic3C2×C3semidirect
36. C3×#835D3semidirect
37. C3×#835C2×C3semidirect
44. #63574C2×C3semidirect
46. Dic3×C3⋊C8C2×C3semidirect
49. C33×Q8C2×C4semidirect
59. C3×C3⋊(C3⋊C8)C2×C4semidirect
84. #3755C3×C4semidirect
123. C32×Dic3SD16semidirect
129. Q8×Dic3C3×D3semidirect
141. #835C3×D3semidirect
163. C32×Q8C4×D3semidirect
164. C32×Q8C4×D3semidirect
165. C32×Q8C2×C3×C4semidirect
208. C3⋊(C3⋊C8)C2×C3×C4semidirect
323. C3×Dic3#215semidirect
324. C3×Dic3C3×SD16semidirect
332. C33C4×SD16semidirect
348. C3×Q8#624semidirect
349. C3×Q8C3×C4×D3semidirect
350. C3×Q8C3×C4×D3semidirect
492. Dic3C3×#215semidirect
497. C32#4788semidirect
498. C32C4×#215semidirect
499. C32C3×C4×SD16semidirect
502. Q8C3×#624semidirect
554. C3#10591536direct (N × H)
555. C3C3×#4788semidirect
556. C3C3×C4×#215semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37552
C32×Q8×Dic32
C3×#638942
C32×#8352
C3×#635742
C3×#635642
C3×Dic3×C3⋊C82
C3×C4×#2153
C3×#47883
#105915363