Finite Group Databasebeta

← #13155978

Subgroups — #13155978

Order 1728.

746 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723691C211yesmaximal, characteristic, complemented
2C2×#88789C211yesmaximal, characteristic, complemented
3C3×#63914C211yesmaximal, characteristic, complemented
4C2×C3×#3714C211yesmaximal, characteristic, complemented
5#11722051C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8#10591429—31nomaximal
9#10591428—31nomaximal
10#10590749—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6294—21no—
15C2×C3×#625—21no—
16C2×C3×#625—21no—
17C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×C4×C3⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88814—21no—
22#88789—22no—
23C3×#3714—22no—
24C2×#3753—31no—
25C3×#940—31no—
26C3×#940—31no—
27C2×C3×#213—31no—
28C2×C3×D24—31no—
29#63914D311yescharacteristic, complemented
30#63898—31no—
31#63894—31no—
32Dic3×D12—31no—
33C2×#3714D311yescharacteristic, complemented
34C2×#3713—31no—
35C2×#3713—31no—
36C2×#3713—61no—
37#63659—31no—
38#63641—31no—
39C3×#832—31no—
40C3×#821—31no—
41Dic3⋊(C3⋊C8)—31no—
42Dic3⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)—61no—
44#63563D311yescharacteristic, complemented
45#63562—31no—
46#63562—31no—
47#63562—61no—
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)D412yescomplemented
57C32×D12—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×D12—21no—
61#6294—42no—
62C3×#625—42no—
63C3×#625—42no—
64C3⋊(C3⋊(C3⋊C8))—21no—
65C32×C3⋊C8—21no—
66#5150—91no—
67#5072—91no—
68#4786—91no—
69C2×C32×D4—31no—
70C2×C32×D4—31no—
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C2×C3×#67—61no—
73C2×C3×#67—61no—
74C2×C3×#67—61no—
75C2×C3×#67—61no—
76C2×C3×D12—21no—
77C2×C3×D12—21no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—61no—
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×D12C2×D311yescharacteristic
85C2×#625—21no—
86C2×#625—21no—
87C2×#624—61no—
88C2×D3×Dic3—61no—
89C2×D3×Dic3—61no—
90C2×C32×C8—31no—
91#3753—62no—
92C4⋊(C3⋊Dic3)C2×D311yescharacteristic
93C4×C3⋊Dic3C2×D311yescharacteristic
94C2×C3⋊(C3⋊C8)—31no—
95C2×C3⋊(C3⋊C8)—31no—
96C2×C3⋊(C3⋊C8)—31no—
97C2×C3⋊(C3⋊C8)—61no—
98C2×C3⋊(C3⋊C8)—61no—
99C2×C3⋊(C3⋊C8)—61no—
100C2×C3⋊(C3⋊C8)—121no—
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—21no—
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C3×D24—62no—
112#3722—61no—
113(C2×D3)⋊Dic3—61no—
114(C2×D3)⋊Dic3—61no—
115#3714—22no—
116#3713—62no—
117#3713—62no—
118#3713—64no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C32×D3—41no—
124C2×C32×D3—41no—
125C2×C32×D3—41no—
126C2×C32×D3—41no—
127C2×C32×D3—41no—
128C2×C32×D3—41no—
129C33×C4C2×D411yescharacteristic
130C33×C4#4111yescharacteristic
131C3×C3⋊Dic3—21no—
132C3×C3⋊Dic3—41no—
133C3×C3⋊Dic3D812yescomplemented
134C2×C3×D8—91no—
135C3×#127—91no—
136#940—31no—
137#940—31no—
138D4×Dic3—91no—
139C2×#213—31no—
140C2×#213—91no—
141C2×#213—91no—
142C2×#213—91no—
143C2×#213—181no—
144C2×D24—31no—
145#893—91no—
146C4×D12—91no—
147C3×#102—91no—
148#832—31no—
149#832—91no—
150#832—91no—
151#832—181no—
152#821—31no—
153Dic3⋊C8—91no—
154Dic3⋊C8—91no—
155Dic3⋊C8—181no—
156#809—91no—
157C4⋊(C3⋊C8)—91no—
158C23×C32—61no—
159C23×C32—61no—
160C22×C3⋊D3—21no—
161C22×C3×D3—21no—
162C22×C3×D3—21no—
163C22×C3×D3—21no—
164C22×C3×D3—21no—
165C22×C3×D3—41no—
166C22×C3×D3—41no—
167C22×C3×D3—61no—
168C22×C3×D3—61no—
169C22×C3×D3—121no—
170C32×D4—61no—
171C32×D4—61no—
172C32×D4—61no—
173C32×D4—61no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—41no—
178C2×C32×C4C22×D311yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3#6711yescharacteristic
183C2×C3⋊Dic3#6711yescharacteristic
184C4⋊(C3⋊D3)—21no—
185C4⋊(C3⋊D3)#6712yescomplemented
186C3×#67—122no—
187C3×#67—122no—
188C3×#67—122no—
189C3×#67—122no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—121no—
199C3×D12—22no—
200C3×D12—22no—
201C3×D12—21no—
202C3×D12—21no—
203C3×D12—21no—
204C3×D12—21no—
205C3×D12—21no—
206C3×D12—21no—
207C3×D12—21no—
208C3×D12—21no—
209C3×D12—32no—
210C3×D12—32no—
211C3×D12—41no—
212C3×D12—41no—
213C3×D12—62no—
214C3×D12—61no—
215C3×D12—61no—
216C3×D12—121no—
217#625—42no—
218#625—42no—
219#624—122no—
220D3×Dic3—122no—
221D3×Dic3—122no—
222C32×C8—61no—
223C3⋊(C3⋊C8)—61no—
224C3⋊(C3⋊C8)—62no—
225C3⋊(C3⋊C8)—61no—
226C3⋊(C3⋊C8)—61no—
227C3⋊(C3⋊C8)—121no—
228C3⋊(C3⋊C8)—121no—
229C3⋊(C3⋊C8)—122no—
230C3×C3⋊C8—21no—
231C3×C3⋊C8—21no—
232C3×C3⋊C8—41no—
233#459—271noSylow
234C2×C33#12111yescharacteristic
235C2×C33C2×D811yescharacteristic
236C2×C33AGL(1,ℤ/8)11yescharacteristic
237C3×C3⋊D3—42no—
238C32×D3—81no—
239C32×D3—81no—
240C2×C3×D4—31no—
241C2×C3×D4—31no—
242C2×C3×D4—31no—
243C2×C3×D4—31no—
244C2×C3×D4—61no—
245C2×C3×D4—61no—
246C2×C3×D4—91no—
247C2×C3×D4—181no—
248C2×C3×D4—181no—
249C2×#67—61no—
250C2×#67—61no—
251C2×#67—61no—
252C2×#67—61no—
253C22×Dic3—181no—
254C2×D12—31no—
255C2×D12—31no—
256C2×D12—61no—
257C2×D12D3211yescharacteristic
258C2×D12D3211yescharacteristic
259C2×C4×D3—181no—
260C2×C4×D3—181no—
261C3×D8—182no—
262C2×C3×C8—31no—
263C2×C3×C8—31no—
264C2×C3×C8—61no—
265C3×C4⋊C4—91no—
266C3×C42—91no—
267#217—181no—
268#213—62no—
269#213—182no—
270#213—182no—
271#213—182no—
272#213—184no—
273(C2×D3)⋊C4—181no—
274(C2×D3)⋊C4—181no—
275C4⋊Dic3—31no—
276C4⋊Dic3—31no—
277C4⋊Dic3—61no—
278C4×Dic3—31no—
279C4×Dic3—31no—
280C4×Dic3—61no—
281C2×C3⋊C8—91no—
282C2×C3⋊C8—91no—
283C2×C3⋊C8—91no—
284C2×C3⋊C8—181no—
285C2×C3⋊C8—181no—
286C2×C3⋊C8—181no—
287C2×C3⋊C8—361no—
288C2×C3⋊C8D3211yescharacteristic
289D24—62no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32—121no—
295C22×C32—121no—
296C22×C32—121no—
297C22×C32—121no—
298C22×C32—121no—
299C22×C32—121no—
300C22×C32D3×D411yescharacteristic
301C22×C32D3×D411yescharacteristic
302C22×C32C2×#6711yescharacteristic
303C2×C3⋊D3—22no—
304C2×C3⋊D3—41no—
305C2×C3⋊D3—41no—
306C2×C3×D3—42no—
307C2×C3×D3—42no—
308C2×C3×D3—41no—
309C2×C3×D3—41no—
310C2×C3×D3—41no—
311C2×C3×D3—41no—
312C2×C3×D3—41no—
313C2×C3×D3—41no—
314C2×C3×D3—41no—
315C2×C3×D3—41no—
316C2×C3×D3—41no—
317C2×C3×D3—41no—
318C2×C3×D3—41no—
319C2×C3×D3—41no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—81no—
323C2×C3×D3—81no—
324C2×C3×D3—81no—
325C2×C3×D3—81no—
326C2×C3×D3—122no—
327C2×C3×D3—121no—
328C2×C3×D3—121no—
329C2×C3×D3—121no—
330C2×C3×D3—121no—
331C2×C3×D3—241no—
332C2×C3×D3—241no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—21no—
337C32×C4—21no—
338C32×C4—21no—
339C32×C4—41no—
340C32×C4—41no—
341C32×C4#23511yescharacteristic
342C32×C4D3×D411yescharacteristic
343C32×C4D3×D411yescharacteristic
344C32×C4#23711yescharacteristic
345C32×C4#23711yescharacteristic
346C32×C4C2×#6711yescharacteristic
347C3⋊Dic3—21no—
348C3⋊Dic3—41no—
349C3⋊Dic3#21312yescomplemented
350C3×Dic3—32no—
351C3×Dic3—32no—
352C3×Dic3—62no—
353C3×Dic3—61no—
354C3×Dic3—61no—
355C3×Dic3—121no—
356C3×Dic3—121no—
357C3×Dic3—121no—
358C3×Dic3—241no—
359C2×D8—91no—
360C2×D8—271no—
361#127—91no—
362C4×D4—271no—
363C4⋊C8—271no—
364#102—91no—
365#102—271no—
366C33#45911yescharacteristic, complemented, Sylow
367C23×C3—61no—
368C23×C3—61no—
369C23×C3—61no—
370C23×C3—61no—
371C23×C3—121no—
372C23×C3—121no—
373C23×C3—181no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—61no—
377C22×D3—61no—
378C22×D3—121no—
379C3×D4—62no—
380C3×D4—62no—
381C3×D4—61no—
382C3×D4—61no—
383C3×D4—61no—
384C3×D4—61no—
385C3×D4—61no—
386C3×D4—61no—
387C3×D4—61no—
388C3×D4—61no—
389C3×D4—92no—
390C3×D4—121no—
391C3×D4—121no—
392C3×D4—181no—
393C3×D4—362no—
394C3×D4—362no—
395C2×C3×C4—21no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—41no—
399C2×C3×C4—91no—
400C2×C3×C4—91no—
401C2×C3×C4—181no—
402C2×C3×C4C2×D3211yescharacteristic
403C2×C3×C4C2×D3211yescharacteristic
404C2×C3×C4C2×D3211yescharacteristic
405#67—122no—
406#67—122no—
407#67—122no—
408#67—122no—
409C2×Dic3—31no—
410C2×Dic3—31no—
411C2×Dic3—31no—
412C2×Dic3—31no—
413C2×Dic3—61no—
414C2×Dic3—61no—
415C2×Dic3—61no—
416C2×Dic3—61no—
417C2×Dic3—121no—
418C2×Dic3—362no—
419D12—21no—
420D12—21no—
421D12—21no—
422D12—21no—
423D12—32no—
424D12—32no—
425D12—62no—
426D12—61no—
427D12—61no—
428D12—121no—
429C4×D3—362no—
430C4×D3—362no—
431C3×C8—61no—
432C3×C8—61no—
433C3×C8—121no—
434C3⋊C8—21no—
435C3⋊C8—182no—
436C3⋊C8—181no—
437C3⋊C8—181no—
438C3⋊C8—181no—
439C3⋊C8—361no—
440C3⋊C8—361no—
441C3⋊C8—362no—
442C2×C32—21no—
443C2×C32—21no—
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—41no—
452C2×C32—41no—
453C2×C32—41no—
454C2×C32—241no—
455C2×C32—241no—
456C2×C32#88411yescharacteristic
457C2×C32D3×D811yescharacteristic
458C2×C32D3×D811yescharacteristic
459C2×C32#91111yescharacteristic
460C2×C32#91411yescharacteristic
461C2×C32C2×#21311yescharacteristic
462C2×C32#93211yescharacteristic
463C2×C32#93811yescharacteristic
464C2×C32#93911yescharacteristic
465C3⋊D3—42no—
466C3×D3—82no—
467C3×D3—82no—
468C3×D3—81no—
469C3×D3—81no—
470C3×D3—81no—
471C3×D3—81no—
472C3×D3—122no—
473C3×D3—122no—
474C3×D3—242no—
475C2×D4—31no—
476C2×D4—31no—
477C2×D4—91no—
478C2×D4—181no—
479C2×D4—181no—
480C22×C4—541no—
481D8—182no—
482D8—542no—
483C2×C8—31no—
484C2×C8—271no—
485C4⋊C4—91no—
486C22⋊C4—541no—
487C42—91no—
488C22×C3—21no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—41no—
492C22×C3—122no—
493C22×C3—122no—
494C22×C3—121no—
495C22×C3—121no—
496C22×C3—121no—
497C22×C3—121no—
498C22×C3—121no—
499C22×C3—121no—
500C22×C3—121no—
501C22×C3—121no—
502C22×C3—121no—
503C22×C3—121no—
504C22×C3—121no—
505C22×C3—121no—
506C22×C3—182no—
507C22×C3—241no—
508C22×C3—241no—
509C22×C3—241no—
510C22×C3—241no—
511C22×C3—361no—
512C22×C3—361no—
513C22×C3#380211yescharacteristic
514C22×C3D3×#6711yescharacteristic
515C22×C3D3×#6711yescharacteristic
516C2×D3—41no—
517C2×D3—41no—
518C2×D3—41no—
519C2×D3—41no—
520C2×D3—41no—
521C2×D3—41no—
522C2×D3—62no—
523C2×D3—62no—
524C2×D3—122no—
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—92no—
540C3×C4—181no—
541C3×C4—361no—
542C3×C4#379511yescharacteristic
543C3×C4#380411yescharacteristic
544C3×C4#380511yescharacteristic
545C3×C4D3×#6711yescharacteristic
546C3×C4D3×#6711yescharacteristic
547C3×C4#381111yescharacteristic
548Dic3—32no—
549Dic3—32no—
550Dic3—62no—
551Dic3—61no—
552Dic3—61no—
553Dic3—121no—
554Dic3—121no—
555Dic3—121no—
556Dic3—241no—
557C32—21no—
558C32—21no—
559C32—21no—
560C32—41no—
561C32#478611yescharacteristic, complemented
562C32#507211yescharacteristic, complemented
563C32#515011yescharacteristic, complemented
564C23—61no—
565C23—61no—
566C23—181no—
567D4—61no—
568D4—61no—
569D4—61no—
570D4—61no—
571D4—92no—
572D4—181no—
573D4—362no—
574D4—362no—
575C2×C4—91no—
576C2×C4—91no—
577C2×C4—181no—
578C2×C4—1082no—
579C2×C4D3311yescharacteristic
580C8—61no—
581C8—541no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—41no—
592C2×C3—41no—
593C2×C3—41no—
594C2×C3—242no—
595C2×C3—242no—
596C2×C3—241no—
597C2×C3—241no—
598C2×C3—241no—
599C2×C3—241no—
600C2×C3—362no—
601C2×C3#6379711yescharacteristic
602C2×C3#6379811yescharacteristic
603C2×C3#6388511yescharacteristic
604C2×C3D3×#21311yescharacteristic
605C2×C3D3×#21311yescharacteristic
606C2×C3#6393211yescharacteristic
607C2×C3#6393411yescharacteristic
608C2×C3#6395811yescharacteristic
609C2×C3#6397011yescharacteristic
610D3—81no—
611D3—81no—
612D3—122no—
613D3—122no—
614D3—242no—
615C22—121no—
616C22—121no—
617C22—121no—
618C22—121no—
619C22—121no—
620C22—121no—
621C22—182no—
622C22—361no—
623C22—361no—
624C22#8895311yescharacteristic
625C4—92no—
626C4—181no—
627C4—361no—
628C4#8895211yescharacteristic
629C4#8896111yescharacteristic
630C3—21no—
631C3—21no—
632C3—21no—
633C3—41no—
634C3#1059074911yescharacteristic, complemented
635C3#1059142811yescharacteristic, complemented
636C3#1059142911yescharacteristic, complemented
637C2—241no—
638C2—241no—
639C2—362no—
640C2#1172283111yescharacteristic
641C2#1172287511yescharacteristic
642C2#1172296511yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#459
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. #11723691C2semidirect
2. C2×#88789C2semidirect
3. C3×#63914C2semidirect
4. C2×C3×#3714C2semidirect
5. #11722051C2semidirect
6. #11722031C2semidirect
7. C3×#63563C2semidirect
29. #63914D3semidirect
33. C2×#3714D3semidirect
44. #63563D3semidirect
56. C3×C4⋊(C3⋊D3)D4semidirect
133. C3×C3⋊Dic3D8semidirect
185. C4⋊(C3⋊D3)#67semidirect
349. C3⋊Dic3#213semidirect
366. C33#459semidirect
561. C32#4786semidirect
562. C32#5072semidirect
563. C32#5150semidirect
634. C3#10590749semidirect
635. C3#10591428semidirect
636. C3#10591429semidirect

Maximal subgroups

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

subgroupindex
#117236912
C2×#887892
C3×#639142
C2×C3×#37142
#117220512
#117220312
C3×#635632
#105914293
#105914283
#105907493