Finite Group Databasebeta

← #13161994

Subgroups — #13161994

Order 1728.

720 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64156C211yesmaximal, characteristic, complemented
2C2×C3×#3755C211yesmaximal, characteristic, complemented
3#11723691C211yesmaximal, characteristic, complemented
4C2×#88810C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722078C211yesmaximal, characteristic, complemented
7#11722075C211yesmaximal, characteristic, complemented
8#10591570—31nomaximal
9#10591568—31nomaximal
10#10591567—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×#6294—21no—
15C3×#3755—22no—
16C3×#3752—21no—
17C3×#3752—21no—
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#88810—22no—
24#64156D311yescharacteristic, complemented
25C2×#3755D311yescharacteristic, complemented
26C3×#947—31no—
27C3×#947—31no—
28C3×#947—61no—
29C2×C3×#215—31no—
30C2×C3×#215—31no—
31C2×C3×#215—61no—
32#63898—31no—
33#63894—31no—
34Dic3×D12—31no—
35C2×#3714—31no—
36C2×#3714—31no—
37C2×#3713—31no—
38#63662D311yescharacteristic, complemented
39C3×#835—31no—
40C3×#835—31no—
41C3×#835—61no—
42#63572—31no—
43Dic3⋊(C3⋊C8)—31no—
44Dic3⋊(C3⋊C8)—31no—
45#63564—31no—
46#63563—31no—
47#63562—31no—
48C22×C3×C3⋊D3—21no—
49C22×C3×C3⋊D3—21no—
50C33×Q8—21no—
51C33×Q8—21no—
52C2×C33×C4—21no—
53C2×C33×C4C2311yescharacteristic
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C2×C3×C3⋊Dic3D411yescharacteristic
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C3×C4⋊(C3⋊D3)—21no—
60C3×C4⋊(C3⋊D3)D412yes—
61#6294—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C3×C3⋊(C3⋊C8)—21no—
64#5165—91no—
65#5057—91no—
66#4819—91no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—21no—
70C2×C32×Q8—41no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C2×D311yescharacteristic
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—61no—
81C2×C3×D12—61no—
82C2×#624—61no—
83C2×D3×Dic3—61no—
84C2×D3×Dic3—61no—
85#3755—22no—
86#3752—21no—
87#3752—21no—
88C4⋊(C3⋊Dic3)C2×D311yescharacteristic
89C4×C3⋊Dic3C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#215—62no—
93C3×#215—62no—
94C3×#215—64no—
95C3×(C2×D3)⋊C4—61no—
96C3×(C2×D3)⋊C4—61no—
97C3×(C2×D3)⋊C4—61no—
98C3×(C2×D3)⋊C4—61no—
99C3×(C2×D3)⋊C4—62no—
100C3×(C2×D3)⋊C4—62no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—61no—
104C3×C4×Dic3—31no—
105C3×C4×Dic3—31no—
106C3×C4×Dic3—61no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—31no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—61no—
112C2×C3×C3⋊C8—61no—
113#3722—61no—
114(C2×D3)⋊Dic3—61no—
115(C2×D3)⋊Dic3—61no—
116#3714—62no—
117#3714—62no—
118#3713—62no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—41no—
125C2×C3×C3⋊D3—41no—
126C33×C4—41no—
127C33×C4C2×D411yescharacteristic
128C33×C4#4111yescharacteristic
129C3×C3⋊Dic3—21no—
130C3×C3⋊Dic3—21no—
131C3×C3⋊Dic3—41no—
132C2×C3×SD16—91no—
133C3×#124—91no—
134#947—31no—
135#947—31no—
136#947—61no—
137C2×#215—31no—
138C2×#215—31no—
139C2×#215—61no—
140D4×Dic3—91no—
141C2×#213—91no—
142C2×#213—91no—
143C2×D24—91no—
144#893—91no—
145C4×D12—91no—
146C3×#103—91no—
147#835—31no—
148#835—31no—
149#835—61no—
150#832—91no—
151#821—91no—
152Dic3⋊C8—91no—
153Dic3⋊C8—91no—
154#809—91no—
155C4⋊(C3⋊C8)—91no—
156C22×C3⋊D3—21no—
157C22×C3⋊D3—21no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—62no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—121no—
164C32×Q8—21no—
165C32×Q8—21no—
166C32×Q8—22no—
167C32×Q8—22no—
168C32×Q8—21no—
169C32×Q8—21no—
170C32×Q8—21no—
171C32×Q8—21no—
172C32×Q8—41no—
173C32×Q8—41no—
174C32×Q8—41no—
175C32×Q8—41no—
176C32×Q8—42no—
177C32×Q8—42no—
178C2×C32×C4—21no—
179C2×C32×C4—22no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—41no—
186C2×C32×C4—41no—
187C2×C32×C4—41no—
188C2×C32×C4—42no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C3⋊Dic3—21no—
193C2×C3⋊Dic3#6711yescharacteristic
194C2×C3⋊Dic3#6711yescharacteristic
195C4⋊(C3⋊D3)—21no—
196C4⋊(C3⋊D3)—21no—
197C4⋊(C3⋊D3)—21no—
198C4⋊(C3⋊D3)#6712yes—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—121no—
208C3×D12—32no—
209C3×D12—32no—
210C3×D12—62no—
211C3×D12—61no—
212C3×D12—61no—
213C3×D12—61no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—61no—
217C3×D12—121no—
218C3×D12—121no—
219C3×D12—121no—
220#624—122no—
221D3×Dic3—122no—
222D3×Dic3—122no—
223C3⋊(C3⋊C8)—21no—
224C3⋊(C3⋊C8)—21no—
225C3×C3⋊C8—62no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—61no—
229C3×C3⋊C8—61no—
230C3×C3⋊C8—121no—
231#463—271noSylow
232C2×C33#12111yescharacteristic
233C2×C33#13511yescharacteristic
234C2×C33AGL(1,ℤ/8)11yescharacteristic
235C3×C3⋊D3—42no—
236C3×C3⋊D3—81no—
237C2×C3×Q8—21no—
238C2×C3×Q8—21no—
239C2×C3×Q8—21no—
240C2×C3×Q8—41no—
241C2×C3×Q8D3211yescharacteristic
242C2×C3×Q8D3211yescharacteristic
243C2×C3×Q8D3211yescharacteristic
244C2×C3×D4—91no—
245C2×C3×D4—91no—
246C22×Dic3—181no—
247C2×D12—31no—
248C2×D12—31no—
249C2×D12—31no—
250C2×D12—31no—
251C2×D12—61no—
252C2×D12—61no—
253C2×C4×D3—181no—
254C2×C4×D3—181no—
255C3×SD16—182no—
256C2×C3×C8—91no—
257C2×C3×C8—91no—
258C3×C4⋊C4—91no—
259C3×C22⋊C4—181no—
260C3×C22⋊C4—181no—
261C3×C42—91no—
262#217—181no—
263#215—62no—
264#215—62no—
265#215—64no—
266#213—182no—
267#213—182no—
268(C2×D3)⋊C4—61no—
269(C2×D3)⋊C4—61no—
270(C2×D3)⋊C4—61no—
271(C2×D3)⋊C4—61no—
272(C2×D3)⋊C4—62no—
273(C2×D3)⋊C4—62no—
274(C2×D3)⋊C4—181no—
275(C2×D3)⋊C4—181no—
276C4⋊Dic3—31no—
277C4⋊Dic3—31no—
278C4⋊Dic3—61no—
279C4×Dic3—31no—
280C4×Dic3—31no—
281C4×Dic3—61no—
282C2×C3⋊C8—31no—
283C2×C3⋊C8—31no—
284C2×C3⋊C8—31no—
285C2×C3⋊C8—31no—
286C2×C3⋊C8—61no—
287C2×C3⋊C8—61no—
288D24—182no—
289C22×C32—21no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—41no—
293C22×C32D3×D411yescharacteristic
294C22×C32D3×D411yescharacteristic
295C22×C32C2×#6711yescharacteristic
296C2×C3⋊D3—22no—
297C2×C3⋊D3—41no—
298C2×C3⋊D3—41no—
299C2×C3⋊D3—41no—
300C2×C3⋊D3—41no—
301C2×C3⋊D3—41no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C2×C3×D3—122no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
308C2×C3×D3—121no—
309C2×C3×D3—122no—
310C2×C3×D3—122no—
311C2×C3×D3—122no—
312C2×C3×D3—121no—
313C2×C3×D3—121no—
314C2×C3×D3—121no—
315C2×C3×D3—121no—
316C2×C3×D3—121no—
317C2×C3×D3—121no—
318C2×C3×D3—241no—
319C2×C3×D3—241no—
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—42no—
329C32×C4—42no—
330C32×C4—42no—
331C32×C4—44no—
332C32×C4—41no—
333C32×C4—41no—
334C32×C4—41no—
335C32×C4#23511yescharacteristic
336C32×C4D3×D411yescharacteristic
337C32×C4D3×D411yescharacteristic
338C32×C4#23711yescharacteristic
339C32×C4#23711yescharacteristic
340C32×C4C2×#6711yescharacteristic
341C3⋊Dic3—21no—
342C3⋊Dic3—21no—
343C3⋊Dic3—41no—
344C3×Dic3—61no—
345C3×Dic3—61no—
346C3×Dic3—61no—
347C3×Dic3—61no—
348C3×Dic3—121no—
349C3×Dic3—121no—
350C3×Dic3—121no—
351C3×Dic3—121no—
352C3×Dic3—241no—
353C2×SD16—91no—
354C2×D8—271no—
355#124—91no—
356C4×D4—271no—
357C4⋊C8—271no—
358#103—91no—
359#102—271no—
360C33#46311yescharacteristic, complemented, Sylow
361C23×C3—181no—
362C23×C3—181no—
363C22×D3—61no—
364C22×D3—61no—
365C22×D3—62no—
366C22×D3—61no—
367C22×D3—61no—
368C22×D3—121no—
369C3×Q8—22no—
370C3×Q8—22no—
371C3×Q8—21no—
372C3×Q8—21no—
373C3×Q8—21no—
374C3×Q8—21no—
375C3×Q8—21no—
376C3×Q8—21no—
377C3×Q8—41no—
378C3×Q8—41no—
379C3×Q8—41no—
380C3×Q8—41no—
381C3×Q8—42no—
382C3×Q8—42no—
383C3×D4—92no—
384C3×D4—181no—
385C3×D4—181no—
386C3×D4—181no—
387C2×C3×C4—22no—
388C2×C3×C4—21no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—21no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—42no—
397C2×C3×C4—41no—
398C2×C3×C4—91no—
399C2×C3×C4—91no—
400C2×C3×C4—181no—
401C2×C3×C4C2×D3211yescharacteristic
402C2×C3×C4C2×D3211yescharacteristic
403C2×C3×C4C2×D3211yescharacteristic
404C2×Dic3—31no—
405C2×Dic3—31no—
406C2×Dic3—31no—
407C2×Dic3—31no—
408C2×Dic3—61no—
409C2×Dic3—61no—
410C2×Dic3—61no—
411C2×Dic3—61no—
412C2×Dic3—121no—
413C2×Dic3—362no—
414D12—32no—
415D12—32no—
416D12—62no—
417D12—61no—
418D12—61no—
419D12—61no—
420D12—61no—
421D12—61no—
422D12—61no—
423D12—121no—
424D12—121no—
425D12—121no—
426C4×D3—362no—
427C4×D3—362no—
428C3×C8—181no—
429C3×C8—181no—
430C3⋊C8—62no—
431C3⋊C8—61no—
432C3⋊C8—61no—
433C3⋊C8—61no—
434C3⋊C8—61no—
435C3⋊C8—121no—
436C2×C32—21no—
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—41no—
446C2×C32—41no—
447C2×C32—41no—
448C2×C32#88411yescharacteristic
449C2×C32#91411yescharacteristic
450C2×C32#91411yescharacteristic
451C2×C32#91611yescharacteristic
452C2×C32#91911yescharacteristic
453C2×C32#93811yescharacteristic
454C2×C32#93911yescharacteristic
455C2×C32#94911yescharacteristic
456C2×C32#95011yescharacteristic
457C3⋊D3—42no—
458C3⋊D3—81no—
459C3×D3—122no—
460C3×D3—122no—
461C3×D3—242no—
462C3×D3—242no—
463C3×D3—241no—
464C3×D3—241no—
465C2×Q8#111311yescharacteristic
466C2×D4—91no—
467C2×D4—91no—
468C22×C4—541no—
469SD16—182no—
470D8—542no—
471C2×C8—91no—
472C2×C8—91no—
473C4⋊C4—91no—
474C22⋊C4—181no—
475C22⋊C4—181no—
476C22⋊C4—541no—
477C42—91no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—41no—
482C22×C3—182no—
483C22×C3—361no—
484C22×C3—361no—
485C22×C3—361no—
486C22×C3—361no—
487C22×C3—361no—
488C22×C3D3×#6711yescharacteristic
489C22×C3D3×#6711yescharacteristic
490C22×C3#381111yescharacteristic
491C2×D3—62no—
492C2×D3—62no—
493C2×D3—122no—
494C2×D3—121no—
495C2×D3—121no—
496C2×D3—121no—
497C2×D3—121no—
498C2×D3—122no—
499C2×D3—122no—
500C2×D3—122no—
501C2×D3—121no—
502C2×D3—121no—
503C2×D3—121no—
504C2×D3—121no—
505C2×D3—121no—
506C2×D3—121no—
507C2×D3—241no—
508C2×D3—241no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—42no—
516C3×C4—42no—
517C3×C4—42no—
518C3×C4—44no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—41no—
524C3×C4—181no—
525C3×C4—181no—
526C3×C4—361no—
527C3×C4#380411yescharacteristic
528C3×C4#380411yescharacteristic
529C3×C4#380511yescharacteristic
530C3×C4D3×#6711yescharacteristic
531C3×C4D3×#6711yescharacteristic
532C3×C4#381111yescharacteristic
533Dic3—61no—
534Dic3—61no—
535Dic3—61no—
536Dic3—61no—
537Dic3—121no—
538Dic3—121no—
539Dic3—121no—
540Dic3—121no—
541Dic3—241no—
542C32—21no—
543C32—21no—
544C32—21no—
545C32—41no—
546C32#481911yescharacteristic, complemented
547C32#505711yescharacteristic, complemented
548C32#516511yescharacteristic, complemented
549C23—181no—
550C23—181no—
551Q8—21no—
552Q8—21no—
553D4—92no—
554D4—181no—
555D4—181no—
556D4—181no—
557C2×C4—21no—
558C2×C4—91no—
559C2×C4—91no—
560C2×C4—181no—
561C2×C4—1082no—
562C2×C4C2×#111311yescharacteristic
563C8—181no—
564C8—181no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—41no—
577C2×C3—362no—
578C2×C3—721no—
579C2×C3#6393911yescharacteristic
580C2×C3#6393911yescharacteristic
581C2×C3#6394111yescharacteristic
582C2×C3#6394711yescharacteristic
583C2×C3#6394911yescharacteristic
584C2×C3#6395011yescharacteristic
585C2×C3#6395811yescharacteristic
586C2×C3#6397011yescharacteristic
587C2×C3#6397111yescharacteristic
588D3—122no—
589D3—122no—
590D3—242no—
591D3—242no—
592D3—241no—
593D3—241no—
594C22—182no—
595C22—361no—
596C22—361no—
597C22—361no—
598C22—361no—
599C22—361no—
600C22#8904911yescharacteristic
601C4—41no—
602C4—181no—
603C4—181no—
604C4—361no—
605C4#8904611yescharacteristic
606C4#8904911yescharacteristic
607C3—21no—
608C3—21no—
609C3—21no—
610C3—41no—
611C3#1059156711yescharacteristic, complemented
612C3#1059156811yescharacteristic, complemented
613C3#1059157011yescharacteristic, complemented
614C2—362no—
615C2—721no—
616C2#1172370911yescharacteristic
617C2#1172371311yescharacteristic
618C2#1172372511yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#88810C2
hypercenterC2×Q8#1113
hypocenterC33#463
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64156C2semidirect
2. C2×C3×#3755C2semidirect
3. #11723691C2semidirect
5. C3×#63662C2semidirect
6. #11722078C2semidirect
7. #11722075C2semidirect
24. #64156D3semidirect
25. C2×#3755D3semidirect
38. #63662D3semidirect
360. C33#463semidirect
546. C32#4819semidirect
547. C32#5057semidirect
548. C32#5165semidirect
611. C3#10591567semidirect
612. C3#10591568semidirect
613. C3#10591570semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C2×C3×#37552
#117236912
C2×#888102
C3×#636622
#117220782
#117220752
#105915703
#105915683
#105915673