Finite Group Databasebeta

← C3×#10591490 #13158503

Subgroups — C3×#10591490 #13158503

Order 1728.

866 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×#3754C211yesmaximal, characteristic, complemented
2C32×#940C211yesmaximal, characteristic
3C2×C32×#214C211yesmaximal, characteristic, complemented
4C3×#63866C211yesmaximal, characteristic, complemented
5C2×C3×#3716C211yesmaximal, characteristic, complemented
6C2×C3×#3715C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×#5175—31nomaximal
9C3×#5076—31nomaximal
10#10591490C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×#67—22no—
14C2×C32×D12C2211yescharacteristic
15C2×C32×Dic6C2211yescharacteristic
16C3×#3754—22no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#214—22no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—22no—
22C3×#3716—22no—
23C3×#3715—22no—
24C2×C32×SD16—31no—
25C32×#127—31no—
26C2×#3754C2×C311yescharacteristic, complemented
27C3×#944—31no—
28C2×C3×#215—31no—
29C3×#940—21no—
30C3×#940D311yescharacteristic
31C3×#940C2×C311yescharacteristic
32C2×C3×#214—21no—
33C2×C3×#214—31no—
34C2×C3×#214—31no—
35C2×C3×#214—31no—
36C2×C3×#214—31no—
37C2×C3×#214—61no—
38C2×C3×#214D311yescharacteristic, complemented
39C2×C3×#214C2×C311yescharacteristic, complemented
40C2×C3×#204—31no—
41C3×#869—31no—
42#63866C2×C311yescharacteristic, complemented
43C2×#3716C2×C311yescharacteristic, complemented
44C2×#3715C2×C311yescharacteristic, complemented
45C3×C8×Dic3—31no—
46C3×C4×C3⋊C8—31no—
47Dic3×C3⋊C8C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C32×D3—21no—
50C33×D4—21no—
51C33×D4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C3×#634—21no—
55C3×#634—21no—
56C32×#67—44no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3D412yes—
59C32×D12—21no—
60C32×D12—21no—
61C32×Dic6—21no—
62C32×Dic6—21no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64C32×C3⋊C8D412yescomplemented
65C3×#492—91no—
66#5175—31no—
67#5076—31no—
68C2×C32×Q8—31no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4—31no—
73C2×C32×D4—41no—
74C2×C32×D4—62no—
75C2×C32×D4C2×D311yescharacteristic
76C2×C32×D4C2×D311yescharacteristic
77C2×C32×D4C22×C311yescharacteristic
78C2×#634C22×C311yescharacteristic
79C2×C3×#67—24no—
80C2×C3×#67—22no—
81C2×C3×#67—22no—
82C2×C3×D12—21no—
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×D12C22×C311yescharacteristic
85C2×C3×Dic6—21no—
86C2×C3×Dic6—31no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—61no—
89C2×C3×Dic6C2×D311yescharacteristic
90C2×C3×Dic6C22×C311yescharacteristic
91C32×SD16—62no—
92C2×C32×C8—31no—
93C32×C42—31no—
94#3754—22no—
95C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
96C3×#215—62no—
97C3×#214—24no—
98C3×#214—22no—
99C3×#214—22no—
100C3×#214—62no—
101C3×#214—62no—
102C3×#214—62no—
103C3×#214—64no—
104C3×#214—62no—
105C3×C4⋊Dic3—62no—
106C3×Dic3⋊C4—62no—
107C3×C4×Dic3—21no—
108C3×C4×Dic3C2×D311yescharacteristic
109C3×C4×Dic3C22×C311yescharacteristic
110C2×C3×C3⋊C8—21no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—31no—
113C2×C3×C3⋊C8—61no—
114C2×C3×C3⋊C8C2×D311yescharacteristic
115C2×C3×C3⋊C8C22×C311yescharacteristic
116C3×#204—62no—
117Dic3⋊Dic3—22no—
118#3716—22no—
119#3715—22no—
120C22×C33—41no—
121C22×C33—42no—
122C22×C33C2×D411yescharacteristic
123C2×C32×D3—41no—
124C2×C32×D3—42no—
125C33×C4C2×D411yescharacteristic
126C33×C4C2×D411yescharacteristic
127C3×C3⋊Dic3—41no—
128C32×Dic3—41no—
129C32×Dic3SD1614yescomplemented
130C2×C3×SD16—31no—
131C2×C3×SD16—31no—
132C2×C3×SD16—61no—
133C2×C3×SD16—91no—
134C2×C3×SD16—91no—
135C2×C3×SD16—91no—
136C3×C4⋊Q8—91no—
137C3×#127—31no—
138C3×#127—31no—
139C3×#127—61no—
140#944—31no—
141C2×#215—31no—
142#940C3×D311yescharacteristic
143C2×#214—31no—
144C2×#214—31no—
145C2×#214—31no—
146C2×#214—31no—
147C2×#214—61no—
148C2×#214C3×D311yescharacteristic, complemented
149C2×#204—31no—
150#869—31no—
151C3×C4×C8—91no—
152C8×Dic3—31no—
153C4×C3⋊C8—31no—
154C23×C32—21no—
155C23×C32—22no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32—41no—
159C23×C32—41no—
160C23×C32—42no—
161C23×C32—61no—
162C22×C3×D3—22no—
163C22×C3×D3—21no—
164C22×C3×D3—21no—
165C32×Q8—61no—
166C32×Q8—61no—
167C32×D4—21no—
168C32×D4—21no—
169C32×D4—22no—
170C32×D4—22no—
171C32×D4—21no—
172C32×D4—21no—
173C32×D4—21no—
174C32×D4—21no—
175C32×D4—41no—
176C32×D4—41no—
177C32×D4—41no—
178C32×D4—41no—
179C32×D4—42no—
180C32×D4—42no—
181C32×D4—61no—
182C32×D4—61no—
183C32×D4—124no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—32no—
188C2×C32×C4—41no—
189C2×C32×C4—61no—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C23×C311yescharacteristic
193C2×C3⋊Dic3—21no—
194#634—21no—
195#634—21no—
196C3×#67—48no—
197C3×#67—44no—
198C3×#67—44no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—22no—
202C2×C3×Dic3—41no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—121no—
206C2×C3×Dic3#6712yes—
207C2×C3×Dic3C3×D412yes—
208C3×D12—21no—
209C3×D12—21no—
210C3×D12—21no—
211C3×D12—21no—
212C3×D12—41no—
213C3×D12—41no—
214C3×Dic6—22no—
215C3×Dic6—21no—
216C3×Dic6—21no—
217C3×Dic6—21no—
218C3×Dic6—21no—
219C3×Dic6—41no—
220C3×Dic6—62no—
221C3×Dic6—61no—
222C3×Dic6—61no—
223C3×Dic6—61no—
224C3×Dic6—61no—
225C3×Dic6—121no—
226C32×C8—32no—
227C3⋊(C3⋊C8)C3×D412yescomplemented
228C3×C3⋊C8—22no—
229C3×C3⋊C8—32no—
230C3×C3⋊C8—32no—
231C3×C3⋊C8—62no—
232C3×C3⋊C8#6712yescomplemented
233C3×C3⋊C8C3×D412yescomplemented
234#492—91noSylow
235C2×C33—81no—
236C2×C33#12711yescharacteristic
237C2×C33C2×SD1612yes—
238C32×D3—81no—
239C2×C3×Q8—31no—
240C2×C3×Q8—31no—
241C2×C3×Q8—61no—
242C2×C3×Q8—91no—
243C2×C3×D4—21no—
244C2×C3×D4—21no—
245C2×C3×D4—21no—
246C2×C3×D4—31no—
247C2×C3×D4—31no—
248C2×C3×D4—41no—
249C2×C3×D4—61no—
250C2×C3×D4—62no—
251C2×C3×D4—62no—
252C2×C3×D4—64no—
253C2×C3×D4D3211yescharacteristic
254C2×C3×D4C2×C3×D311yescharacteristic
255C2×C3×D4C2×C3×D311yescharacteristic
256C2×#67—22no—
257C2×D12C2×C3×D311yescharacteristic
258C2×Dic6—31no—
259C2×Dic6—31no—
260C2×Dic6—61no—
261C2×Dic6C2×C3×D311yescharacteristic
262C3×SD16—62no—
263C3×SD16—62no—
264C3×SD16—64no—
265C3×SD16—182no—
266C3×SD16—182no—
267C3×SD16—182no—
268C2×C3×C8—31no—
269C2×C3×C8—31no—
270C2×C3×C8—61no—
271C2×C3×C8—91no—
272C3×C4⋊C4—182no—
273C3×C42—31no—
274C3×C42—31no—
275C3×C42—61no—
276#215—62no—
277#214—22no—
278#214—62no—
279#214—62no—
280#214—62no—
281#214—64no—
282#214—62no—
283C4⋊Dic3—62no—
284Dic3⋊C4—62no—
285C4×Dic3C2×C3×D311yescharacteristic
286C2×C3⋊C8—31no—
287C2×C3⋊C8—31no—
288C2×C3⋊C8—61no—
289C2×C3⋊C8C2×C3×D311yescharacteristic
290#204—62no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—21no—
294C22×C32—41no—
295C22×C32—42no—
296C22×C32—42no—
297C22×C32—42no—
298C22×C32—44no—
299C22×C32—42no—
300C22×C32—41no—
301C22×C32—41no—
302C22×C32—41no—
303C22×C32—44no—
304C22×C32—42no—
305C22×C32—42no—
306C22×C32—82no—
307C22×C32—82no—
308C22×C32—84no—
309C22×C32—122no—
310C22×C32—121no—
311C22×C32D3×D411yescharacteristic
312C22×C32C2×#6711yescharacteristic
313C22×C32C2×C3×D411yescharacteristic
314C2×C3×D3—42no—
315C2×C3×D3—44no—
316C2×C3×D3—42no—
317C2×C3×D3—41no—
318C2×C3×D3—41no—
319C2×C3×D3—42no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—34no—
327C32×C4—41no—
328C32×C4—41no—
329C32×C4—121no—
330C32×C4D3×D411yescharacteristic
331C32×C4D3×D411yescharacteristic
332C32×C4D3×D411yescharacteristic
333C32×C4C2×#6711yescharacteristic
334C32×C4C2×C3×D411yescharacteristic
335C32×C4C2×C3×D411yescharacteristic
336C3⋊Dic3—41no—
337C3×Dic3—24no—
338C3×Dic3—42no—
339C3×Dic3—41no—
340C3×Dic3—41no—
341C3×Dic3—122no—
342C3×Dic3—121no—
343C3×Dic3—121no—
344C3×Dic3#21414yescomplemented
345C3×Dic3C3×SD1614yescomplemented
346C2×SD16—31no—
347C2×SD16—91no—
348C2×SD16—91no—
349C2×SD16—91no—
350C4⋊Q8—91no—
351#127—31no—
352C4×C8—91no—
353C33#49211yescharacteristic, complemented, Sylow
354C23×C3—22no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—42no—
361C23×C3—62no—
362C23×C3—61no—
363C23×C3—61no—
364C22×D3—21no—
365C3×Q8—62no—
366C3×Q8—61no—
367C3×Q8—61no—
368C3×Q8—61no—
369C3×Q8—61no—
370C3×Q8—121no—
371C3×Q8—181no—
372C3×Q8—181no—
373C3×D4—22no—
374C3×D4—22no—
375C3×D4—21no—
376C3×D4—21no—
377C3×D4—21no—
378C3×D4—21no—
379C3×D4—21no—
380C3×D4—21no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—42no—
386C3×D4—42no—
387C3×D4—61no—
388C3×D4—61no—
389C3×D4—61no—
390C3×D4—61no—
391C3×D4—121no—
392C3×D4—121no—
393C3×D4—124no—
394C3×D4—124no—
395C3×D4—128no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—32no—
400C2×C3×C4—32no—
401C2×C3×C4—41no—
402C2×C3×C4—62no—
403C2×C3×C4—61no—
404C2×C3×C4—61no—
405C2×C3×C4—121no—
406C2×C3×C4—181no—
407C2×C3×C4C2×D3211yescharacteristic
408C2×C3×C4C22×C3×D311yescharacteristic
409C2×C3×C4C22×C3×D311yescharacteristic
410#67—44no—
411C2×Dic3—21no—
412C2×Dic3—61no—
413C2×Dic3—61no—
414C2×Dic3—121no—
415C2×Dic3C3×#6712yes—
416D12—21no—
417D12—21no—
418Dic6—21no—
419Dic6—21no—
420Dic6—62no—
421Dic6—61no—
422Dic6—61no—
423Dic6—61no—
424Dic6—61no—
425Dic6—121no—
426C3×C8—32no—
427C3×C8—32no—
428C3×C8—62no—
429C3×C8—92no—
430C3⋊C8—32no—
431C3⋊C8—32no—
432C3⋊C8—62no—
433C3⋊C8C3×#6712yescomplemented
434C2×C32—22no—
435C2×C32—22no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—22no—
439C2×C32—21no—
440C2×C32—41no—
441C2×C32—42no—
442C2×C32—82no—
443C2×C32—82no—
444C2×C32—82no—
445C2×C32—84no—
446C2×C32—81no—
447C2×C32—81no—
448C2×C32—81no—
449C2×C32—241no—
450C2×C32D3×SD1612yes—
451C2×C32C2×#21412yes—
452C2×C32#94011yescharacteristic
453C2×C32#94011yescharacteristic
454C2×C32C3×#12711yescharacteristic
455C2×C32C2×C3×SD1612yes—
456C3×D3—82no—
457C3×D3—81no—
458C3×D3—81no—
459C2×Q8—31no—
460C2×Q8—91no—
461C2×D4—31no—
462C2×D4—62no—
463C2×D4C3×D3211yescharacteristic
464SD16—62no—
465SD16—182no—
466SD16—182no—
467SD16—182no—
468C2×C8—31no—
469C2×C8—91no—
470C4⋊C4—182no—
471C42—31no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—44no—
479C22×C3—44no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—42no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—82no—
488C22×C3—82no—
489C22×C3—84no—
490C22×C3—122no—
491C22×C3—124no—
492C22×C3—122no—
493C22×C3—121no—
494C22×C3—122no—
495C22×C3—121no—
496C22×C3D3×#6711yescharacteristic
497C22×C3C3×D3×D411yescharacteristic
498C22×C3C2×C3×#6711yescharacteristic
499C2×D3—42no—
500C2×D3—41no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—21no—
505C3×C4—21no—
506C3×C4—21no—
507C3×C4—34no—
508C3×C4—34no—
509C3×C4—41no—
510C3×C4—41no—
511C3×C4—64no—
512C3×C4—122no—
513C3×C4—121no—
514C3×C4—121no—
515C3×C4—361no—
516C3×C4D3×#6711yescharacteristic
517C3×C4#381111yescharacteristic
518C3×C4C3×D3×D411yescharacteristic
519C3×C4C3×D3×D411yescharacteristic
520C3×C4C3×D3×D411yescharacteristic
521C3×C4C2×C3×#6711yescharacteristic
522Dic3—41no—
523Dic3—122no—
524Dic3—121no—
525Dic3—121no—
526Dic3C3×#21414yescomplemented
527C32—21no—
528C32—21no—
529C32—21no—
530C32—41no—
531C32#507611yescharacteristic, complemented
532C32#517511yescharacteristic, complemented
533C32C3×#49211yescharacteristic, complemented
534C23—21no—
535C23—61no—
536Q8—61no—
537Q8—61no—
538Q8—181no—
539Q8—181no—
540D4—21no—
541D4—21no—
542D4—61no—
543D4—61no—
544D4—124no—
545C2×C4—32no—
546C2×C4—61no—
547C2×C4—181no—
548C2×C4C2×C3×D3211yescharacteristic
549C8—32no—
550C8—92no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—22no—
555C2×C3—22no—
556C2×C3—22no—
557C2×C3—41no—
558C2×C3—42no—
559C2×C3—82no—
560C2×C3—82no—
561C2×C3—82no—
562C2×C3—84no—
563C2×C3—81no—
564C2×C3—81no—
565C2×C3—81no—
566C2×C3—242no—
567C2×C3—241no—
568C2×C3—241no—
569C2×C3D3×#21412yes—
570C2×C3#6397911yescharacteristic
571C2×C3C3×D3×SD1612yes—
572C2×C3C2×C3×#21412yes—
573C2×C3C3×#94011yescharacteristic
574C2×C3C3×#94011yescharacteristic
575D3—81no—
576C22—41no—
577C22—42no—
578C22—121no—
579C22—122no—
580C22C3×D3×#6711yescharacteristic
581C4—34no—
582C4—121no—
583C4—361no—
584C4C3×D3×#6711yescharacteristic
585C4C3×#381111yescharacteristic
586C3—21no—
587C3—21no—
588C3—21no—
589C3—41no—
590C3#1059149011yescharacteristic, direct product factor
591C3C3×#507611yescharacteristic, complemented
592C3C3×#517511yescharacteristic, complemented
593C2—81no—
594C2—241no—
595C2C3×D3×#21412yes—
596C2C3×#6397911yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC32×#940C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#492
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3754C2semidirect
3. C2×C32×#214C2semidirect
4. C3×#63866C2semidirect
5. C2×C3×#3716C2semidirect
6. C2×C3×#3715C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10591490C3direct (N × H)
26. C2×#3754C2×C3semidirect
38. C2×C3×#214D3semidirect
39. C2×C3×#214C2×C3semidirect
42. #63866C2×C3semidirect
43. C2×#3716C2×C3semidirect
44. C2×#3715C2×C3semidirect
47. Dic3×C3⋊C8C2×C3semidirect
63. C3×C3⋊(C3⋊C8)D4semidirect
64. C32×C3⋊C8D4semidirect
129. C32×Dic3SD16semidirect
148. C2×#214C3×D3semidirect
227. C3⋊(C3⋊C8)C3×D4semidirect
232. C3×C3⋊C8#67semidirect
233. C3×C3⋊C8C3×D4semidirect
344. C3×Dic3#214semidirect
345. C3×Dic3C3×SD16semidirect
353. C33#492semidirect
433. C3⋊C8C3×#67semidirect
526. Dic3C3×#214semidirect
531. C32#5076semidirect
532. C32#5175semidirect
533. C32C3×#492semidirect
590. C3#10591490direct (N × H)
591. C3C3×#5076semidirect
592. C3C3×#5175semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37542
C32×#9402
C2×C32×#2142
C3×#638662
C2×C3×#37162
C2×C3×#37152
C3×Dic3×C3⋊C82
C3×#51753
C3×#50763
#105914903