Finite Group Databasebeta

← #13161957

Subgroups — #13161957

Order 1728.

788 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C2×C3×#3754C211yesmaximal, characteristic, complemented
3C2×C3×#3753C211yesmaximal, characteristic
4#11723677C211yesmaximal, characteristic, complemented
5#11722078C211yesmaximal, characteristic, complemented
6#11722076C211yesmaximal, characteristic, complemented
7#11722075C211yesmaximal, characteristic, complemented
8#10591476—31nomaximal
9#10591471—31nomaximal
10#10591468—31nomaximal
11C2×C33×D4C2211yescharacteristic
12C22×C3×C3⋊Dic3—21no—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15C3×#3757—21no—
16C3×#3754—22no—
17C3×#3753—22no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#88814—21no—
23#88814—21no—
24D4×C3⋊Dic3D311yescharacteristic, complemented
25C2×#3754D311yescharacteristic, complemented
26C2×#3753D311yescharacteristic
27C3×D4×Dic3—31no—
28C3×D4×Dic3—31no—
29C3×D4×Dic3—61no—
30C2×C3×#214—31no—
31C2×C3×#214—31no—
32C2×C3×#214—61no—
33C2×C3×#213—31no—
34C2×C3×#213—31no—
35C2×C3×#213—61no—
36#63864—31no—
37#63863—31no—
38#63860—31no—
39#63572—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—31no—
42#63567—31no—
43#63566—31no—
44#63565—31no—
45#63564—31no—
46#63563—31no—
47#63562—31no—
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C33×D4—21no—
51C33×D4D412yes—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—42no—
56C2×C3×C3⋊Dic3D411yescharacteristic
57C2×C3×C3⋊Dic3D411yescharacteristic
58C3×C4⋊(C3⋊D3)—21no—
59C3×C4⋊(C3⋊D3)—21no—
60C3×#634—21no—
61C3×#634—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C3×C3⋊(C3⋊C8)—21no—
64#5165—91no—
65#5153—91no—
66#5016—91no—
67C2×C32×D4—21no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—41no—
71C2×C32×D4C2×D311yescharacteristic
72C2×C32×D4C2×D311yescharacteristic
73C2×C32×D4C2×D311yescharacteristic
74C22×C3⋊Dic3—21no—
75C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
76C2×#634C2×D311yescharacteristic
77C22×C3×Dic3—61no—
78C22×C3×Dic3—61no—
79C22×C3×Dic3—62no—
80C2×C3×D12—31no—
81C2×C3×D12—31no—
82C2×C3×D12—61no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86#3757—21no—
87#3754—22no—
88#3753—22no—
89C4⋊(C3⋊Dic3)C2×D311yescharacteristic
90C4×C3⋊Dic3C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
93C3×#217—61no—
94C3×#217—61no—
95C3×#217—62no—
96C3×#214—64no—
97C3×#214—62no—
98C3×#214—62no—
99C3×#213—64no—
100C3×#213—62no—
101C3×#213—62no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—61no—
105C3×C4×Dic3—31no—
106C3×C4×Dic3—31no—
107C3×C4×Dic3—61no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—31no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—61no—
113C2×C3×C3⋊C8—61no—
114#3722—61no—
115#3722—61no—
116(C2×D3)⋊Dic3—61no—
117(C2×D3)⋊Dic3—61no—
118(C2×D3)⋊Dic3—61no—
119(C2×D3)⋊Dic3—61no—
120C22×C33—22no—
121C22×C33—41no—
122C22×C33—41no—
123C22×C33C2×D411yescharacteristic
124C2×C3×C3⋊D3—41no—
125C2×C3×C3⋊D3—41no—
126C2×C3×C3⋊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—
135C3×C4×D4—91no—
136#947—91no—
137#936—91no—
138D4×Dic3—31no—
139D4×Dic3—31no—
140D4×Dic3—61no—
141C2×#214—31no—
142C2×#214—31no—
143C2×#214—61no—
144C2×#213—31no—
145C2×#213—31no—
146C2×#213—61no—
147#875—91no—
148#835—91no—
149#832—91no—
150#821—91no—
151#816—91no—
152Dic3⋊C8—91no—
153Dic3⋊C8—91no—
154#810—91no—
155#809—91no—
156C4⋊(C3⋊C8)—91no—
157C23×C32—21no—
158C23×C32—22no—
159C23×C32—21no—
160C23×C32—21no—
161C23×C32—41no—
162C23×C32—41no—
163C23×C32—42no—
164C22×C3⋊D3—21no—
165C22×C3×D3—61no—
166C22×C3×D3—61no—
167C22×C3×D3—121no—
168C32×D4—21no—
169C32×D4—22no—
170C32×D4—21no—
171C32×D4—21no—
172C32×D4—22no—
173C32×D4—22no—
174C32×D4—22no—
175C32×D4—42no—
176C32×D4—41no—
177C32×D4—41no—
178C32×D4—42no—
179C32×D4D1212yes—
180C32×D4#6712yes—
181C32×D4#6712yes—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—41no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C3⋊Dic3—21no—
190C2×C3⋊Dic3—21no—
191C2×C3⋊Dic3—42no—
192C2×C3⋊Dic3#6711yescharacteristic
193C2×C3⋊Dic3#6711yescharacteristic
194C4⋊(C3⋊D3)—21no—
195C4⋊(C3⋊D3)—21no—
196#634—21no—
197#634—21no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—121no—
210C2×C3×Dic3—124no—
211C2×C3×Dic3—122no—
212C2×C3×Dic3—122no—
213C3×D12—62no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—61no—
217C3×D12—61no—
218C3×D12—121no—
219C3×Dic6—62no—
220C3×Dic6—61no—
221C3×Dic6—61no—
222C3×Dic6—61no—
223C3×Dic6—61no—
224C3×Dic6—121no—
225C3⋊(C3⋊C8)—21no—
226C3⋊(C3⋊C8)—21no—
227C3×C3⋊C8—62no—
228C3×C3⋊C8—62no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—61no—
231C3×C3⋊C8—61no—
232C3×C3⋊C8—61no—
233#463—271noSylow
234C2×C33—42no—
235C2×C33#12111yescharacteristic
236C2×C33#13511yescharacteristic
237C2×C33AGL(1,ℤ/8)11yescharacteristic
238C3×C3⋊D3—81no—
239C2×C3×Q8—91no—
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—41no—
244C2×C3×D4—91no—
245C2×C3×D4D3211yescharacteristic
246C2×C3×D4D3211yescharacteristic
247C2×C3×D4D3211yescharacteristic
248C22×C3×C4—181no—
249C22×Dic3—61no—
250C22×Dic3—61no—
251C22×Dic3—62no—
252C2×D12—31no—
253C2×D12—31no—
254C2×D12—61no—
255C2×Dic6—31no—
256C2×Dic6—31no—
257C2×Dic6—61no—
258C3×SD16—182no—
259C3×D8—182no—
260C2×C3×C8—91no—
261C2×C3×C8—91no—
262C3×C4⋊C4—91no—
263C3×C22⋊C4—181no—
264C3×C42—91no—
265#217—61no—
266#217—61no—
267#217—62no—
268#217—181no—
269#217—181no—
270#214—64no—
271#214—62no—
272#214—62no—
273#213—64no—
274#213—62no—
275#213—62no—
276(C2×D3)⋊C4—181no—
277(C2×D3)⋊C4—181no—
278(C2×D3)⋊C4—181no—
279(C2×D3)⋊C4—181no—
280C4⋊Dic3—31no—
281C4⋊Dic3—31no—
282C4⋊Dic3—61no—
283C4×Dic3—31no—
284C4×Dic3—31no—
285C4×Dic3—61no—
286C2×C3⋊C8—31no—
287C2×C3⋊C8—31no—
288C2×C3⋊C8—31no—
289C2×C3⋊C8—31no—
290C2×C3⋊C8—61no—
291C2×C3⋊C8—61no—
292C22×C32—22no—
293C22×C32—24no—
294C22×C32—22no—
295C22×C32—22no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—41no—
300C22×C32—42no—
301C22×C32—42no—
302C22×C32—44no—
303C22×C32—41no—
304C22×C32—41no—
305C22×C32—42no—
306C22×C32—42no—
307C22×C32—41no—
308C22×C32—41no—
309C22×C32—41no—
310C22×C32—41no—
311C22×C32—81no—
312C22×C32—81no—
313C22×C32—81no—
314C22×C32—81no—
315C22×C32—82no—
316C22×C32—82no—
317C22×C32D3×D411yescharacteristic
318C22×C32D3×D411yescharacteristic
319C22×C32C2×#6711yescharacteristic
320C2×C3⋊D3—41no—
321C2×C3⋊D3—41no—
322C2×C3⋊D3—41no—
323C2×C3×D3—122no—
324C2×C3×D3—121no—
325C2×C3×D3—121no—
326C2×C3×D3—121no—
327C2×C3×D3—121no—
328C2×C3×D3—121no—
329C2×C3×D3—121no—
330C2×C3×D3—241no—
331C2×C3×D3—241no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—41no—
339C32×C4—41no—
340C32×C4C2×D1211yescharacteristic
341C32×C4#23511yescharacteristic
342C32×C4#23711yescharacteristic
343C32×C4#23711yescharacteristic
344C32×C4C2×#6711yescharacteristic
345C32×C4C2×#6711yescharacteristic
346C3⋊Dic3—21no—
347C3⋊Dic3—21no—
348C3⋊Dic3—41no—
349C3⋊Dic3—41no—
350C3×Dic3—62no—
351C3×Dic3—61no—
352C3×Dic3—61no—
353C3×Dic3—61no—
354C3×Dic3—61no—
355C3×Dic3—121no—
356C3×Dic3—122no—
357C3×Dic3—122no—
358C3×Dic3—121no—
359C3×Dic3—121no—
360C3×Dic3—121no—
361C3×Dic3—121no—
362C2×SD16—91no—
363C2×D8—91no—
364#124—271no—
365C4×D4—91no—
366C4⋊C8—271no—
367#103—271no—
368#102—271no—
369C33#46311yescharacteristic, complemented, Sylow
370C23×C3—22no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—41no—
375C23×C3—41no—
376C23×C3—42no—
377C23×C3—181no—
378C22×D3—61no—
379C22×D3—61no—
380C22×D3—121no—
381C3×Q8—181no—
382C3×Q8—181no—
383C3×D4—22no—
384C3×D4—21no—
385C3×D4—21no—
386C3×D4—21no—
387C3×D4—22no—
388C3×D4—22no—
389C3×D4—22no—
390C3×D4—41no—
391C3×D4—41no—
392C3×D4—42no—
393C3×D4—42no—
394C3×D4—181no—
395C3×D4—181no—
396C3×D4#62512yes—
397C3×D4#62612yes—
398C3×D4#62612yes—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—41no—
403C2×C3×C4—91no—
404C2×C3×C4—91no—
405C2×C3×C4—181no—
406C2×C3×C4—181no—
407C2×C3×C4—362no—
408C2×C3×C4C2×D3211yescharacteristic
409C2×C3×C4C2×D3211yescharacteristic
410C2×C3×C4C2×D3211yescharacteristic
411C2×Dic3—31no—
412C2×Dic3—31no—
413C2×Dic3—31no—
414C2×Dic3—31no—
415C2×Dic3—61no—
416C2×Dic3—61no—
417C2×Dic3—62no—
418C2×Dic3—61no—
419C2×Dic3—61no—
420C2×Dic3—61no—
421C2×Dic3—61no—
422C2×Dic3—121no—
423C2×Dic3—124no—
424C2×Dic3—122no—
425C2×Dic3—122no—
426D12—62no—
427D12—61no—
428D12—61no—
429D12—61no—
430D12—61no—
431D12—121no—
432Dic6—62no—
433Dic6—61no—
434Dic6—61no—
435Dic6—61no—
436Dic6—61no—
437Dic6—121no—
438C3×C8—181no—
439C3×C8—181no—
440C3⋊C8—62no—
441C3⋊C8—62no—
442C3⋊C8—61no—
443C3⋊C8—61no—
444C3⋊C8—61no—
445C3⋊C8—61no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—21no—
454C2×C32—21no—
455C2×C32—41no—
456C2×C32—41no—
457C2×C32—41no—
458C2×C32—42no—
459C2×C32—44no—
460C2×C32—42no—
461C2×C32—42no—
462C2×C32—82no—
463C2×C32—82no—
464C2×C32—84no—
465C2×C32#91111yescharacteristic
466C2×C32#91211yescharacteristic
467C2×C32#91411yescharacteristic
468C2×C32#91611yescharacteristic
469C2×C32#93011yescharacteristic
470C2×C32#93211yescharacteristic
471C2×C32#93911yescharacteristic
472C2×C32#93911yescharacteristic
473C2×C32#95011yescharacteristic
474C3⋊D3—81no—
475C3×D3—242no—
476C3×D3—241no—
477C3×D3—241no—
478C2×Q8—91no—
479C2×D4—91no—
480C2×D4#111311yescharacteristic
481C22×C4—181no—
482SD16—182no—
483D8—182no—
484C2×C8—91no—
485C2×C8—91no—
486C4⋊C4—91no—
487C22⋊C4—181no—
488C22⋊C4—541no—
489C22⋊C4—541no—
490C42—91no—
491C22×C3—24no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—21no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—44no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3—82no—
515C22×C3—82no—
516C22×C3—361no—
517C22×C3—361no—
518C22×C3—361no—
519C22×C3D3×#6711yescharacteristic
520C22×C3D3×#6711yescharacteristic
521C22×C3#381111yescharacteristic
522C2×D3—122no—
523C2×D3—121no—
524C2×D3—121no—
525C2×D3—121no—
526C2×D3—121no—
527C2×D3—121no—
528C2×D3—121no—
529C2×D3—241no—
530C2×D3—241no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—21no—
536C3×C4—21no—
537C3×C4—41no—
538C3×C4—41no—
539C3×C4—181no—
540C3×C4—181no—
541C3×C4—361no—
542C3×C4—361no—
543C3×C4#380411yescharacteristic
544C3×C4#380411yescharacteristic
545C3×C4#380511yescharacteristic
546C3×C4C2×#62511yescharacteristic
547C3×C4C2×#62611yescharacteristic
548C3×C4C2×#62611yescharacteristic
549Dic3—62no—
550Dic3—61no—
551Dic3—61no—
552Dic3—61no—
553Dic3—61no—
554Dic3—121no—
555Dic3—122no—
556Dic3—122no—
557Dic3—121no—
558Dic3—121no—
559Dic3—121no—
560Dic3—121no—
561C32—21no—
562C32—21no—
563C32—21no—
564C32—41no—
565C32#501611yescharacteristic, complemented
566C32#515311yescharacteristic, complemented
567C32#516511yescharacteristic, complemented
568C23—21no—
569C23—181no—
570Q8—181no—
571Q8—181no—
572D4—21no—
573D4—181no—
574D4—181no—
575D4#629512yes—
576C2×C4—91no—
577C2×C4—91no—
578C2×C4—181no—
579C2×C4—181no—
580C2×C4—362no—
581C2×C4C2×#111311yescharacteristic
582C8—181no—
583C8—181no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—44no—
594C2×C3—42no—
595C2×C3—42no—
596C2×C3—42no—
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—41no—
600C2×C3—82no—
601C2×C3—82no—
602C2×C3—84no—
603C2×C3—721no—
604C2×C3#6393211yescharacteristic
605C2×C3#6393311yescharacteristic
606C2×C3#6393411yescharacteristic
607C2×C3#6393511yescharacteristic
608C2×C3#6393611yescharacteristic
609C2×C3#6393711yescharacteristic
610C2×C3#6397811yescharacteristic
611C2×C3#6397811yescharacteristic
612C2×C3#6398011yescharacteristic
613D3—242no—
614D3—241no—
615D3—241no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—361no—
620C22—361no—
621C22—361no—
622C22#8904911yescharacteristic
623C4—181no—
624C4—181no—
625C4—361no—
626C4—361no—
627C4#8904611yescharacteristic
628C4C2×#629511yescharacteristic
629C3—21no—
630C3—21no—
631C3—21no—
632C3—41no—
633C3#1059146811yescharacteristic, complemented
634C3#1059147111yescharacteristic, complemented
635C3#1059147611yescharacteristic, complemented
636C2—42no—
637C2—721no—
638C2#1172370611yescharacteristic
639C2#1172370711yescharacteristic
640C2#1172371711yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
hypercenterC2×D4#1113
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×D4×C3⋊Dic3C2semidirect
2. C2×C3×#3754C2semidirect
4. #11723677C2semidirect
5. #11722078C2semidirect
6. #11722076C2semidirect
7. #11722075C2semidirect
24. D4×C3⋊Dic3D3semidirect
25. C2×#3754D3semidirect
369. C33#463semidirect
565. C32#5016semidirect
566. C32#5153semidirect
567. C32#5165semidirect
633. C3#10591468semidirect
634. C3#10591471semidirect
635. C3#10591476semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊Dic32
C2×C3×#37542
C2×C3×#37532
#117236772
#117220782
#117220762
#117220752
#105914763
#105914713
#105914683