Finite Group Databasebeta

← C3×#10591579 #13158652

Subgroups — C3×#10591579 #13158652

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
1C3×#64156C211yesmaximal, characteristic
2C2×C32×#216C211yesmaximal, characteristic, complemented
3C3×#63843C211yesmaximal, characteristic, complemented
4C2×C3×#3717C211yesmaximal, characteristic
5C32×#835C211yesmaximal, characteristic, complemented
6C3×#63572C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic
8C3×#5185—31nomaximal
9C3×#4798—31nomaximal
10#10591579C311yesmaximal, characteristic, direct product factor
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C3×#3752—21no—
15C3×#3752—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×#216—22no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C3×#3717—22no—
24C2×C32×Q16—31no—
25#64156C2×C311yescharacteristic
26C3×#947—31no—
27C3×#947—31no—
28C3×#947—61no—
29C3×Q8×Dic3—31no—
30C2×C3×#216—21no—
31C2×C3×#216D311yescharacteristic, complemented
32C2×C3×#216C2×C311yescharacteristic, complemented
33C2×C3×#215—31no—
34C2×C3×#204—31no—
35C3×#887—31no—
36#63843C2×C311yescharacteristic, complemented
37C2×#3717C2×C311yescharacteristic
38C32×#103—31no—
39C3×#835—21no—
40C3×#835D311yescharacteristic, complemented
41C3×#835C2×C311yescharacteristic, complemented
42C3×#821—31no—
43C3×Dic3⋊C8—31no—
44C3×Dic3⋊C8—31no—
45C3×#809—31no—
46#63572C2×C311yescharacteristic, complemented
47#63564C2×C311yescharacteristic
48C22×C3×C3⋊D3—21no—
49C33×Q8—21no—
50C33×Q8—21no—
51C2×C33×C4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3D411yescharacteristic
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C32×Dic6—21no—
60C32×Dic6D412yes—
61C32×C3⋊C8—21no—
62C32×C3⋊C8—21no—
63C3×#464—91no—
64#5185—31no—
65#4798—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×C4⋊(C3⋊D3)C22×C311yescharacteristic
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6C2×D311yescharacteristic
80C2×C3×Dic6C22×C311yescharacteristic
81C32×Q16—62no—
82C2×C32×C8—31no—
83C2×C32×C8—31no—
84C32×C4⋊C4—31no—
85#3752—21no—
86#3752—21no—
87C4×C3⋊Dic3C22×C311yescharacteristic
88C3×#216—24no—
89C3×#216—22no—
90C3×#216—22no—
91C3×#215—62no—
92C3×(C2×D3)⋊C4—61no—
93C3×(C2×D3)⋊C4—61no—
94C3×(C2×D3)⋊C4—61no—
95C3×(C2×D3)⋊C4—61no—
96C3×(C2×D3)⋊C4—62no—
97C3×(C2×D3)⋊C4—62no—
98C3×C4⋊Dic3—21no—
99C3×C4⋊Dic3—61no—
100C3×C4⋊Dic3C2×D311yescharacteristic
101C3×C4⋊Dic3C22×C311yescharacteristic
102C3×Dic3⋊C4—61no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3—31no—
105C3×C4×Dic3—61no—
106C3×C4×Dic3—61no—
107C3×C4×Dic3—61no—
108C2×C3×C3⋊C8—21no—
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113C2×C3×C3⋊C8C22×C311yescharacteristic
114C3×#204—62no—
115Dic3⋊Dic3—21no—
116Dic32—21no—
117#3717—22no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C33×C4—41no—
123C33×C4C2×D411yescharacteristic
124C33×C4#4111yescharacteristic
125C3×C3⋊Dic3—21no—
126C3×C3⋊Dic3—21no—
127C32×Dic3—22no—
128C32×Dic3—41no—
129C2×C3×Q16—31no—
130C2×C3×Q16—31no—
131C2×C3×Q16—61no—
132C2×C3×SD16—91no—
133C3×#124—91no—
134C3×C4×Q8—91no—
135#947—31no—
136#947—31no—
137#947—61no—
138Q8×Dic3—31no—
139C2×#216C3×D311yescharacteristic, complemented
140C2×#215—31no—
141C2×#204—31no—
142#887—31no—
143C3×C4⋊C8—91no—
144C3×#103—31no—
145C3×#103—31no—
146C3×#103—61no—
147C3×#102—91no—
148#835C3×D311yescharacteristic, complemented
149#821—31no—
150Dic3⋊C8—31no—
151Dic3⋊C8—31no—
152#809—31no—
153C22×C3⋊D3—21no—
154C22×C3×D3—62no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
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—
173C2×C32×C4—21no—
174C2×C32×C4—22no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—41no—
181C2×C32×C4—41no—
182C2×C32×C4—41no—
183C2×C32×C4—42no—
184C2×C32×C4—61no—
185C2×C32×C4—61no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C23×C311yescharacteristic
189C2×C3⋊Dic3C3×D411yescharacteristic
190C2×C3⋊Dic3C3×D411yescharacteristic
191C4⋊(C3⋊D3)—21no—
192C4⋊(C3⋊D3)—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—41no—
202C2×C3×Dic3—41no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—61no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—121no—
210C3×D12—121no—
211C3×Dic6—21no—
212C3×Dic6—21no—
213C3×Dic6—22no—
214C3×Dic6—41no—
215C3×Dic6#6712yes—
216C3×Dic6C3×D412yes—
217C32×C8—61no—
218C32×C8—61no—
219C3×C3⋊C8—22no—
220C3×C3⋊C8—21no—
221C3×C3⋊C8—21no—
222C3×C3⋊C8—21no—
223C3×C3⋊C8—21no—
224C3×C3⋊C8—41no—
225#464—91noSylow
226C2×C33#12111yescharacteristic
227C2×C33#13511yescharacteristic
228C2×C33#13711yescharacteristic
229C3×C3⋊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—91no—
241C2×D12—31no—
242C2×D12—31no—
243C2×D12—61no—
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—31no—
252C2×C3×C8—31no—
253C2×C3×C8—61no—
254C2×C3×C8—61no—
255C3×C4⋊C4—31no—
256C3×C4⋊C4—31no—
257C3×C4⋊C4—61no—
258C3×C4⋊C4—181no—
259C3×C22⋊C4—181no—
260C3×C22⋊C4—181no—
261C3×C42—91no—
262C3×C42—181no—
263#216—22no—
264#215—62no—
265(C2×D3)⋊C4—61no—
266(C2×D3)⋊C4—61no—
267(C2×D3)⋊C4—61no—
268(C2×D3)⋊C4—61no—
269(C2×D3)⋊C4—62no—
270(C2×D3)⋊C4—62no—
271C4⋊Dic3—61no—
272C4⋊Dic3C2×C3×D311yescharacteristic
273Dic3⋊C4—61no—
274C4×Dic3—31no—
275C4×Dic3—31no—
276C4×Dic3—61no—
277C4×Dic3—61no—
278C4×Dic3—61no—
279C2×C3⋊C8C2×C3×D311yescharacteristic
280C2×C3⋊C8C2×C3×D311yescharacteristic
281#204—62no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—41no—
286C22×C32D3×D411yescharacteristic
287C22×C32D3×D411yescharacteristic
288C22×C32C2×C3×D411yescharacteristic
289C2×C3⋊D3—41no—
290C2×C3⋊D3—41no—
291C2×C3⋊D3—41no—
292C2×C3×D3—122no—
293C2×C3×D3—122no—
294C2×C3×D3—122no—
295C2×C3×D3—121no—
296C2×C3×D3—121no—
297C2×C3×D3—121no—
298C2×C3×D3—121no—
299C2×C3×D3—121no—
300C2×C3×D3—121no—
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—62no—
317C32×C4—121no—
318C32×C4#23511yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4#23711yescharacteristic
321C32×C4C2×#6711yescharacteristic
322C32×C4C2×C3×D411yescharacteristic
323C32×C4C3×#4111yescharacteristic
324C3⋊Dic3—21no—
325C3⋊Dic3—21no—
326C3×Dic3—22no—
327C3×Dic3—22no—
328C3×Dic3—42no—
329C3×Dic3—41no—
330C3×Dic3—41no—
331C3×Dic3—61no—
332C3×Dic3—61no—
333C3×Dic3—61no—
334C3×Dic3—61no—
335C3×Dic3—81no—
336C3×Dic3—121no—
337C3×Dic3—121no—
338C2×Q16—31no—
339C2×SD16—91no—
340#124—91no—
341C4×Q8—91no—
342C4⋊C8—91no—
343#103—31no—
344#102—91no—
345C33#46411yescharacteristic, complemented, Sylow
346C23×C3—181no—
347C22×D3—62no—
348C22×D3—61no—
349C22×D3—61no—
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—181no—
371C3×D4—181no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
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—91no—
388C2×C3×C4—91no—
389C2×C3×C4—121no—
390C2×C3×C4—121no—
391C2×C3×C4C2×D3211yescharacteristic
392C2×C3×C4C22×C3×D311yescharacteristic
393C2×C3×C4C22×C3×D311yescharacteristic
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—31no—
397C2×Dic3—31no—
398C2×Dic3—31no—
399C2×Dic3—31no—
400C2×Dic3—61no—
401C2×Dic3—61no—
402D12—61no—
403D12—61no—
404D12—61no—
405D12—61no—
406D12—121no—
407D12—121no—
408Dic6—21no—
409Dic6C3×#6712yes—
410C3×C8—62no—
411C3×C8—61no—
412C3×C8—61no—
413C3×C8—61no—
414C3×C8—61no—
415C3×C8—121no—
416C3⋊C8—21no—
417C3⋊C8—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—21no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32—41no—
430C2×C32#88411yescharacteristic
431C2×C32#91611yescharacteristic
432C2×C32#91811yescharacteristic
433C2×C32#91811yescharacteristic
434C2×C32#91911yescharacteristic
435C2×C32#93911yescharacteristic
436C2×C32C3×#12111yescharacteristic
437C2×C32C3×#13511yescharacteristic
438C2×C32C3×#13711yescharacteristic
439C3⋊D3—81no—
440C3×D3—242no—
441C3×D3—241no—
442C3×D3—241no—
443C2×Q8—31no—
444C2×Q8C3×D3211yescharacteristic
445C2×D4—91no—
446Q16—62no—
447SD16—182no—
448C2×C8—31no—
449C2×C8—31no—
450C4⋊C4—31no—
451C4⋊C4—181no—
452C22⋊C4—181no—
453C22⋊C4—181no—
454C42—91no—
455C42—181no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—41no—
460C22×C3—361no—
461C22×C3—361no—
462C22×C3—361no—
463C22×C3#381111yescharacteristic
464C22×C3C3×D3×D411yescharacteristic
465C22×C3C3×D3×D411yescharacteristic
466C2×D3—122no—
467C2×D3—122no—
468C2×D3—122no—
469C2×D3—121no—
470C2×D3—121no—
471C2×D3—121no—
472C2×D3—121no—
473C2×D3—121no—
474C2×D3—121no—
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—62no—
491C3×C4—62no—
492C3×C4—122no—
493C3×C4—121no—
494C3×C4—121no—
495C3×C4—181no—
496C3×C4—181no—
497C3×C4—241no—
498C3×C4#380411yescharacteristic
499C3×C4D3×#6711yescharacteristic
500C3×C4C3×#23511yescharacteristic
501C3×C4C3×D3×D411yescharacteristic
502C3×C4C3×#23711yescharacteristic
503C3×C4C2×C3×#6711yescharacteristic
504Dic3—22no—
505Dic3—41no—
506Dic3—61no—
507Dic3—61no—
508Dic3—61no—
509Dic3—61no—
510Dic3—121no—
511Dic3—121no—
512C32—21no—
513C32—21no—
514C32—21no—
515C32—41no—
516C32#479811yescharacteristic, complemented
517C32#518511yescharacteristic, complemented
518C32C3×#46411yescharacteristic, complemented
519C23—181no—
520Q8—21no—
521Q8—21no—
522Q8—32no—
523Q8—61no—
524D4—181no—
525D4—181no—
526C2×C4—21no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—91no—
530C2×C4—91no—
531C2×C4C2×C3×D3211yescharacteristic
532C8—61no—
533C8—61no—
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—721no—
547C2×C3#6394511yescharacteristic
548C2×C3#6395011yescharacteristic
549C2×C3#6397111yescharacteristic
550C2×C3C3×#88411yescharacteristic
551C2×C3C3×#91611yescharacteristic
552C2×C3C3×#91811yescharacteristic
553C2×C3C3×#91811yescharacteristic
554C2×C3C3×#91911yescharacteristic
555C2×C3C3×#93911yescharacteristic
556D3—242no—
557D3—241no—
558D3—241no—
559C22—361no—
560C22—361no—
561C22—361no—
562C22C3×#381111yescharacteristic
563C4—41no—
564C4—62no—
565C4—121no—
566C4—181no—
567C4—181no—
568C4C3×#380411yescharacteristic
569C4C3×D3×#6711yescharacteristic
570C3—21no—
571C3—21no—
572C3—21no—
573C3—41no—
574C3#1059157911yescharacteristic, direct product factor
575C3C3×#479811yescharacteristic, complemented
576C3C3×#518511yescharacteristic, complemented
577C2—721no—
578C2C3×#6394511yescharacteristic
579C2C3×#6395011yescharacteristic
580C2C3×#6397111yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
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×#63843C2semidirect
5. C32×#835C2semidirect
6. C3×#63572C2semidirect
10. #10591579C3direct (N × H)
31. C2×C3×#216D3semidirect
32. C2×C3×#216C2×C3semidirect
36. #63843C2×C3semidirect
40. C3×#835D3semidirect
41. C3×#835C2×C3semidirect
46. #63572C2×C3semidirect
139. C2×#216C3×D3semidirect
148. #835C3×D3semidirect
345. C33#464semidirect
516. C32#4798semidirect
517. C32#5185semidirect
518. C32C3×#464semidirect
574. C3#10591579direct (N × H)
575. C3C3×#4798semidirect
576. C3C3×#5185semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C2×C32×#2162
C3×#638432
C2×C3×#37172
C32×#8352
C3×#635722
C3×#635642
C3×#51853
C3×#47983
#105915793