Finite Group Databasebeta

← #13156035

Subgroups — #13156035

Order 1728.

734 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723499C211yesmaximal, characteristic, complemented
2#11723434C211yesmaximal, characteristic, complemented
3C2×C3×#3716C211yesmaximal, characteristic, complemented
4C2×C3×#3713C211yesmaximal, characteristic
5#11722076C211yesmaximal, characteristic, complemented
6#11722050C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8#10591476—31nomaximal
9#10591434—31nomaximal
10#10590750—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×D3×C3⋊Dic3—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19(C2×D3)⋊(C3⋊Dic3)—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22C3×#3716—22no—
23C3×#3713—22no—
24#64145—31no—
25D4×C3⋊Dic3—31no—
26C2×C3×#214—31no—
27C2×C3×#213—31no—
28C2×C3×#213—31no—
29C2×C3×#204—31no—
30#63898—31no—
31#63898—31no—
32#63898—61no—
33#63863—31no—
34#63863—31no—
35#63863—61no—
36C2×#3716D311yescharacteristic, complemented
37C2×#3713D311yescharacteristic
38#63641—31no—
39C3×#832—31no—
40C3×#821—31no—
41#63572—31no—
42#63572—31no—
43#63572—61no—
44#63567—31no—
45#63566—31no—
46#63565—31no—
47#63563D311yescharacteristic, complemented
48C22×C32×D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)D411yescharacteristic
52C2×C3⋊(C3⋊Dic3)D411yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C3×#634—21no—
56C3×#634—21no—
57C32×D12—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×D12D412yes—
61D3×C3⋊Dic3—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64#5165—91no—
65#5153—91no—
66#4787—91no—
67C2×C32×D4—31no—
68C2×C32×D4—31no—
69C22×C3⋊Dic3—61no—
70C2×#634C2×D311yescharacteristic
71C2×C3×D12—21no—
72C2×C3×D12—21no—
73C2×C3×D12C2×D311yescharacteristic
74C2×C3×D12C2×D311yescharacteristic
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×Dic6—31no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—61no—
80C2×D3×Dic3—61no—
81C2×D3×Dic3—61no—
82C2×D3×Dic3—62no—
83C2×C32×C8—31no—
84#3757—61no—
85#3757—61no—
86#3757—61no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88C4×C3⋊Dic3—31no—
89C4×C3⋊Dic3—31no—
90C4×C3⋊Dic3—31no—
91C4×C3⋊Dic3—61no—
92C4×C3⋊Dic3—61no—
93C4×C3⋊Dic3—61no—
94C4×C3⋊Dic3—121no—
95C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
96C3×#214—62no—
97C3×#213—62no—
98C3×#213—62no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3—61no—
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C3×#204—62no—
109(C2×D3)⋊Dic3—61no—
110(C2×D3)⋊Dic3—61no—
111(C2×D3)⋊Dic3—61no—
112(C2×D3)⋊Dic3—61no—
113(C2×D3)⋊Dic3—61no—
114(C2×D3)⋊Dic3—61no—
115(C2×D3)⋊Dic3—62no—
116(C2×D3)⋊Dic3—62no—
117(C2×D3)⋊Dic3—62no—
118#3716—22no—
119#3713—22no—
120C22×C33C2×D411yescharacteristic
121C2×C32×D3—22no—
122C2×C32×D3—41no—
123C2×C32×D3—41no—
124C2×C32×D3—41no—
125C2×C32×D3—41no—
126C2×C32×D3—41no—
127C33×C4C2×D411yescharacteristic
128C33×C4#4111yescharacteristic
129C3⋊(C3⋊Dic3)—21no—
130C3⋊(C3⋊Dic3)—21no—
131C3×C3⋊Dic3—41no—
132C3×C3⋊Dic3—41no—
133C2×C3×SD16—91no—
134C2×C3×D8—91no—
135#947—91no—
136#936—91no—
137#936—91no—
138#936—181no—
139D4×Dic3—91no—
140D4×Dic3—91no—
141D4×Dic3—181no—
142C2×#214—31no—
143C2×#213—31no—
144C2×#213—31no—
145C2×#204—31no—
146#893—91no—
147C3×#102—91no—
148#835—91no—
149#832—31no—
150#821—31no—
151#816—91no—
152Dic3⋊C8—91no—
153Dic3⋊C8—91no—
154Dic3⋊C8—91no—
155Dic3⋊C8—181no—
156#810—91no—
157C23×C32—61no—
158C23×C32—61no—
159C22×C3×D3—22no—
160C22×C3×D3—21no—
161C22×C3×D3—21no—
162C22×C3×D3—22no—
163C22×C3×D3—21no—
164C22×C3×D3—21no—
165C32×D4—32no—
166C32×D4—61no—
167C32×D4—61no—
168C32×D4—61no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—41no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—31no—
182C2×C3⋊Dic3—31no—
183C2×C3⋊Dic3—31no—
184C2×C3⋊Dic3—61no—
185C2×C3⋊Dic3—61no—
186C2×C3⋊Dic3—61no—
187C2×C3⋊Dic3—61no—
188C2×C3⋊Dic3—61no—
189C2×C3⋊Dic3—61no—
190C2×C3⋊Dic3—121no—
191C2×C3⋊Dic3—121no—
192C2×C3⋊Dic3—122no—
193#634—21no—
194#634—21no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C3×D12—22no—
202C3×D12—21no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—22no—
209C3×D12—41no—
210C3×D12—41no—
211C3×D12#6712yes—
212C3×D12#6712yes—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—121no—
218C3×Dic6—121no—
219D3×Dic3—124no—
220D3×Dic3—122no—
221D3×Dic3—122no—
222C32×C8—61no—
223C3⋊(C3⋊C8)—21no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—41no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8—121no—
230#463—271noSylow
231C2×C33#12111yescharacteristic
232C2×C33#13511yescharacteristic
233C2×C33AGL(1,ℤ/8)11yescharacteristic
234C32×D3—42no—
235C32×D3—81no—
236C2×C3×Q8—91no—
237C2×C3×D4—31no—
238C2×C3×D4—31no—
239C2×C3×D4—31no—
240C2×C3×D4—31no—
241C2×C3×D4—61no—
242C2×C3×D4—61no—
243C22×Dic3—181no—
244C22×Dic3—181no—
245C22×Dic3—182no—
246C2×D12D3211yescharacteristic
247C2×D12D3211yescharacteristic
248C2×C4×D3—181no—
249C2×Dic6—31no—
250C2×Dic6—31no—
251C2×Dic6—61no—
252C3×SD16—182no—
253C3×D8—182no—
254C2×C3×C8—31no—
255C2×C3×C8—31no—
256C2×C3×C8—61no—
257C2×C3×C8—91no—
258C3×C4⋊C4—91no—
259#217—181no—
260#217—181no—
261#217—181no—
262#217—181no—
263#217—181no—
264#217—181no—
265#217—182no—
266#217—182no—
267#217—182no—
268#214—62no—
269#213—62no—
270#213—62no—
271(C2×D3)⋊C4—181no—
272(C2×D3)⋊C4—181no—
273(C2×D3)⋊C4—181no—
274C4⋊Dic3—31no—
275C4⋊Dic3—31no—
276C4⋊Dic3—61no—
277C4×Dic3—91no—
278C4×Dic3—91no—
279C4×Dic3—91no—
280C4×Dic3—181no—
281C4×Dic3—181no—
282C4×Dic3—181no—
283C4×Dic3—361no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—61no—
287C2×C3⋊C8D3211yescharacteristic
288#204—62no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—41no—
293C22×C32—62no—
294C22×C32—121no—
295C22×C32—121no—
296C22×C32—121no—
297C22×C32—121no—
298C22×C32—121no—
299C22×C32D3×D411yescharacteristic
300C22×C32D3×D411yescharacteristic
301C22×C32D3×D411yescharacteristic
302C2×C3×D3—24no—
303C2×C3×D3—22no—
304C2×C3×D3—22no—
305C2×C3×D3—42no—
306C2×C3×D3—42no—
307C2×C3×D3—41no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3—41no—
311C2×C3×D3—42no—
312C2×C3×D3—42no—
313C2×C3×D3—42no—
314C2×C3×D3—41no—
315C2×C3×D3—41no—
316C2×C3×D3—41no—
317C2×C3×D3—41no—
318C2×C3×D3—41no—
319C2×C3×D3—41no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—21no—
323C32×C4—21no—
324C32×C4—21no—
325C32×C4—21no—
326C32×C4—41no—
327C32×C4—41no—
328C32×C4#23511yescharacteristic
329C32×C4D3×D411yescharacteristic
330C32×C4#23711yescharacteristic
331C32×C4#23711yescharacteristic
332C32×C4C2×#6711yescharacteristic
333C32×C4C2×#6711yescharacteristic
334C3⋊Dic3—41no—
335C3⋊Dic3—41no—
336C3⋊Dic3—61no—
337C3⋊Dic3—61no—
338C3⋊Dic3—62no—
339C3⋊Dic3—62no—
340C3⋊Dic3—61no—
341C3⋊Dic3—61no—
342C3⋊Dic3—61no—
343C3⋊Dic3—61no—
344C3⋊Dic3—121no—
345C3⋊Dic3—121no—
346C3⋊Dic3—121no—
347C3⋊Dic3—121no—
348C3⋊Dic3—122no—
349C3⋊Dic3—122no—
350C3×Dic3—122no—
351C3×Dic3—122no—
352C3×Dic3—121no—
353C3×Dic3—121no—
354C3×Dic3—121no—
355C3×Dic3—121no—
356C2×SD16—91no—
357C2×D8—91no—
358#124—271no—
359C4×D4—271no—
360C4⋊C8—271no—
361#103—271no—
362#102—91no—
363C33#46311yescharacteristic, complemented, Sylow
364C23×C3—62no—
365C23×C3—62no—
366C23×C3—61no—
367C23×C3—61no—
368C23×C3—61no—
369C23×C3—61no—
370C22×D3—21no—
371C22×D3—21no—
372C3×Q8—181no—
373C3×Q8—181no—
374C3×D4—32no—
375C3×D4—32no—
376C3×D4—62no—
377C3×D4—62no—
378C3×D4—61no—
379C3×D4—61no—
380C3×D4—61no—
381C3×D4—61no—
382C3×D4—61no—
383C3×D4—61no—
384C3×D4—121no—
385C3×D4—121no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—41no—
390C2×C3×C4—181no—
391C2×C3×C4—181no—
392C2×C3×C4C2×D3211yescharacteristic
393C2×C3×C4C2×D3211yescharacteristic
394C2×C3×C4C2×D3211yescharacteristic
395C2×Dic3—62no—
396C2×Dic3—62no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399C2×Dic3—61no—
400C2×Dic3—61no—
401C2×Dic3—91no—
402C2×Dic3—91no—
403C2×Dic3—91no—
404C2×Dic3—91no—
405C2×Dic3—91no—
406C2×Dic3—91no—
407C2×Dic3—181no—
408C2×Dic3—181no—
409C2×Dic3—181no—
410C2×Dic3—181no—
411C2×Dic3—181no—
412C2×Dic3—181no—
413C2×Dic3—361no—
414C2×Dic3—361no—
415C2×Dic3—364no—
416C2×Dic3—362no—
417C2×Dic3—362no—
418D12—21no—
419D12—21no—
420D12—21no—
421D12#62512yes—
422C4×D3—362no—
423Dic6—61no—
424Dic6—61no—
425Dic6—61no—
426Dic6—61no—
427Dic6—121no—
428Dic6—121no—
429C3×C8—61no—
430C3×C8—61no—
431C3×C8—121no—
432C3×C8—181no—
433C3⋊C8—21no—
434C3⋊C8—61no—
435C3⋊C8—61no—
436C3⋊C8—121no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—41no—
447C2×C32—41no—
448C2×C32—41no—
449C2×C32—122no—
450C2×C32—241no—
451C2×C32#88411yescharacteristic
452C2×C32#91111yescharacteristic
453C2×C32#91111yescharacteristic
454C2×C32#91211yescharacteristic
455C2×C32#91411yescharacteristic
456C2×C32#91611yescharacteristic
457C2×C32#91611yescharacteristic
458C2×C32#93911yescharacteristic
459C2×C32#93911yescharacteristic
460C3×D3—44no—
461C3×D3—42no—
462C3×D3—42no—
463C3×D3—82no—
464C3×D3—81no—
465C3×D3—81no—
466C2×Q8—91no—
467C2×D4—31no—
468C2×D4—31no—
469C22×C4—541no—
470SD16—182no—
471D8—182no—
472C2×C8—31no—
473C2×C8—91no—
474C4⋊C4—91no—
475C22⋊C4—541no—
476C22⋊C4—541no—
477C22⋊C4—541no—
478C42—271no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—41no—
483C22×C3—64no—
484C22×C3—62no—
485C22×C3—62no—
486C22×C3—122no—
487C22×C3—122no—
488C22×C3—122no—
489C22×C3—122no—
490C22×C3—122no—
491C22×C3—121no—
492C22×C3—121no—
493C22×C3—121no—
494C22×C3—121no—
495C22×C3—121no—
496C22×C3—121no—
497C22×C3—121no—
498C22×C3—121no—
499C22×C3—121no—
500C22×C3—121no—
501C22×C3#380211yescharacteristic
502C22×C3#381111yescharacteristic
503C22×C3#381111yescharacteristic
504C2×D3—22no—
505C2×D3—41no—
506C2×D3—41no—
507C2×D3—41no—
508C2×D3—41no—
509C2×D3—41no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—21no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—361no—
519C3×C4—361no—
520C3×C4#379511yescharacteristic
521C3×C4#380411yescharacteristic
522C3×C4#380511yescharacteristic
523C3×C4C2×#62511yescharacteristic
524C3×C4D3×#6711yescharacteristic
525C3×C4D3×#6711yescharacteristic
526Dic3—122no—
527Dic3—122no—
528Dic3—121no—
529Dic3—121no—
530Dic3—121no—
531Dic3—121no—
532Dic3—182no—
533Dic3—182no—
534Dic3—181no—
535Dic3—181no—
536Dic3—181no—
537Dic3—181no—
538Dic3—181no—
539Dic3—181no—
540Dic3—361no—
541Dic3—361no—
542Dic3—361no—
543Dic3—361no—
544Dic3—362no—
545Dic3—362no—
546C32—21no—
547C32—21no—
548C32—21no—
549C32—41no—
550C32#478711yescharacteristic, complemented
551C32#515311yescharacteristic, complemented
552C32#516511yescharacteristic, complemented
553C23—61no—
554C23—61no—
555Q8—181no—
556Q8—181no—
557D4—32no—
558D4—61no—
559D4—61no—
560D4—61no—
561C2×C4—181no—
562C2×C4—181no—
563C2×C4—271no—
564C2×C4—271no—
565C2×C4—1082no—
566C2×C4D3311yescharacteristic
567C8—61no—
568C8—181no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—41no—
579C2×C3—41no—
580C2×C3—41no—
581C2×C3—124no—
582C2×C3—122no—
583C2×C3—122no—
584C2×C3—242no—
585C2×C3—241no—
586C2×C3—241no—
587C2×C3#6379811yescharacteristic
588C2×C3#6380911yescharacteristic
589C2×C3#6388511yescharacteristic
590C2×C3#6392711yescharacteristic
591C2×C3#6393611yescharacteristic
592C2×C3#6393611yescharacteristic
593C2×C3#6393711yescharacteristic
594C2×C3#6397111yescharacteristic
595C2×C3#6398011yescharacteristic
596D3—42no—
597D3—81no—
598C22—62no—
599C22—121no—
600C22—121no—
601C22—121no—
602C22—121no—
603C22—121no—
604C22#8895411yescharacteristic
605C4—361no—
606C4—361no—
607C4—541no—
608C4—541no—
609C4D3×#62511yescharacteristic
610C4#8896111yescharacteristic
611C3—21no—
612C3—21no—
613C3—21no—
614C3—41no—
615C3#1059075011yescharacteristic, complemented
616C3#1059143411yescharacteristic, complemented
617C3#1059147611yescharacteristic, complemented
618C2—122no—
619C2—241no—
620C2#1172286111yescharacteristic
621C2#1172287711yescharacteristic
622C2#1172295411yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×#3713C2
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723499C2semidirect
2. #11723434C2semidirect
3. C2×C3×#3716C2semidirect
5. #11722076C2semidirect
6. #11722050C2semidirect
7. C3×#63563C2semidirect
36. C2×#3716D3semidirect
47. #63563D3semidirect
363. C33#463semidirect
550. C32#4787semidirect
551. C32#5153semidirect
552. C32#5165semidirect
615. C3#10590750semidirect
616. C3#10591434semidirect
617. C3#10591476semidirect

Maximal subgroups

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

subgroupindex
#117234992
#117234342
C2×C3×#37162
C2×C3×#37132
#117220762
#117220502
C3×#635632
#105914763
#105914343
#105907503