Finite Group Databasebeta

← #13178222

Subgroups — #13178222

Order 1728.

1328 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64092C212yesmaximal, complemented
2C3×#64091C211yesmaximal, characteristic, complemented
3#11723728C211yesmaximal, characteristic, complemented
4#11723727C212yesmaximal, complemented
5#11723724C211yesmaximal, characteristic, complemented
6#11723696C212yesmaximal, complemented
7#11723695C212yesmaximal, complemented
8#11723687C212yesmaximal, complemented
9#11723675C212yesmaximal, complemented
10#10594244—31nomaximal
11#10594233—32nomaximal
12C23×C3×C3⋊D3C2211yescharacteristic
13C2×C3×#638C2211yescharacteristic
14C2×C3×#638C2212yes—
15C2×C3×C4⋊(C3⋊D3)C2212yes—
16C2×C3×C4×C3⋊D3C2212yescomplemented
17C2×#6295C2212yes—
18C2×#6295C2212yes—
19C2×#6295C2212yescomplemented
20C2×#6295C2211yescharacteristic, complemented
21C2×#6294C2211yescharacteristic, complemented
22C33×C22⋊C4C2211yescharacteristic
23C3×#3757C2212yes—
24C3×#3752C2212yes—
25C3×#3752C2212yescomplemented
26C3×C4⋊(C3⋊Dic3)C2212yescomplemented
27C3×#3750C2212yescomplemented
28#88815C2212yescomplemented
29#88814C2212yes—
30#88814C2212yes—
31#88814C2211yescharacteristic, complemented
32#88813C2211yescharacteristic, complemented
33#64092D312yescomplemented
34#64091D311yescharacteristic, complemented
35C3×#883—32no—
36C3×#883—32no—
37C3×#883—62no—
38C3×#882—32no—
39C3×#882—61no—
40#63981—32no—
41#63979—32no—
42#63979—31no—
43#63977—32no—
44#63976—32no—
45#63975—31no—
46#63969—32no—
47#63916—32no—
48#63912—32no—
49#63910—32no—
50#63909—32no—
51#63908—32no—
52#63906—32no—
53#63906—32no—
54#63886—32no—
55#63885—32no—
56#63861—32no—
57#63858—32no—
58#63848—32no—
59C23×C33C2311yescharacteristic
60C22×C3×C3⋊D3—42no—
61C22×C3×C3⋊D3—41no—
62C22×C3×C3⋊D3C2312yes—
63C22×C3×C3⋊D3C2312yes—
64C22×C3×C3⋊D3C2311yescharacteristic
65C2×C33×C4C2312yes—
66C3×#638—42no—
67C3×#638—41no—
68C2×C3×C3⋊Dic3C2312yes—
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×C4⋊(C3⋊D3)—42no—
73C3×C4×C3⋊D3—42no—
74#6295—42no—
75#6295—42no—
76#6295—41no—
77#6295—42no—
78#6294—41no—
79#5666—92no—
80#5595—91no—
81C23×C3⋊D3C2×D311yescharacteristic
82C23×C3×D3—32no—
83C23×C3×D3—61no—
84C2×#638C2×D311yescharacteristic
85C2×#638C2×D312yes—
86C2×C4⋊(C3⋊D3)C2×D312yes—
87C2×C4×C3⋊D3C2×D312yescomplemented
88C2×C3×#67—32no—
89C2×C3×#67—32no—
90C2×C3×#67—32no—
91C2×C3×#67—61no—
92C2×C3×#67—62no—
93C2×C3×D12—32no—
94C2×C3×D12—32no—
95C2×C3×D12—62no—
96C2×C3×C4×D3—32no—
97C2×C3×C4×D3—32no—
98C2×C3×C4×D3—62no—
99C2×#626—32no—
100C2×#626—32no—
101C2×#626—32no—
102C2×#626—32no—
103C2×#626—32no—
104C2×#626—32no—
105C2×#626—32no—
106C2×#625—32no—
107C2×#625—32no—
108C2×#625—31no—
109C2×#625—32no—
110C2×#624—31no—
111C2×D3×Dic3—32no—
112C32×C22⋊C4—21no—
113C32×C22⋊C4—22no—
114C32×C22⋊C4—41no—
115C32×C22⋊C4C2×D312yes—
116C32×C22⋊C4C2×D311yescharacteristic
117#3757C2×D312yes—
118#3752C2×D312yes—
119#3752C2×D312yescomplemented
120C4⋊(C3⋊Dic3)C2×D312yescomplemented
121#3750C2×D312yescomplemented
122C3×#217—32no—
123C3×#217—32no—
124C3×#217—62no—
125C3×(C2×D3)⋊C4—32no—
126C3×(C2×D3)⋊C4—32no—
127C3×(C2×D3)⋊C4—32no—
128C3×(C2×D3)⋊C4—32no—
129C3×(C2×D3)⋊C4—62no—
130C3×(C2×D3)⋊C4—62no—
131C3×C4⋊Dic3—32no—
132C3×C4⋊Dic3—32no—
133C3×C4⋊Dic3—62no—
134C3×Dic3⋊C4—32no—
135C3×Dic3⋊C4—32no—
136C3×Dic3⋊C4—62no—
137#3724—32no—
138Dic3⋊Dic3—32no—
139Dic3⋊Dic3—32no—
140#3722—32no—
141#3722—32no—
142#3722—31no—
143(C2×D3)⋊Dic3—32no—
144(C2×D3)⋊Dic3—32no—
145(C2×D3)⋊Dic3—32no—
146(C2×D3)⋊Dic3—32no—
147(C2×D3)⋊Dic3—32no—
148Dic32—31no—
149Dic32—32no—
150C22×C33—21no—
151C22×C33—22no—
152C22×C33C2411yescharacteristic
153C2×C3×C3⋊D3—22no—
154C2×C3×C3⋊D3—22no—
155C2×C3×C3⋊D3—21no—
156C2×C3×C3⋊D3—22no—
157C2×C3×C3⋊D3—22no—
158C2×C3×C3⋊D3—22no—
159C2×C3×C3⋊D3—22no—
160C2×C3×C3⋊D3—22no—
161C2×C3×C3⋊D3—44no—
162C33×C4—22no—
163C3×C3⋊Dic3—22no—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3—21no—
167C3×#123—92no—
168C3×#120—91no—
169#953—92no—
170#940—92no—
171#940—91no—
172#939—92no—
173#938—92no—
174#937—92no—
175#935—91no—
176#930—92no—
177#930—92no—
178#898—92no—
179#898—92no—
180#895—92no—
181#895—92no—
182#884—92no—
183#884—92no—
184#884—92no—
185#883—32no—
186#883—32no—
187#883—62no—
188#883—92no—
189#882—32no—
190#882—61no—
191#882—92no—
192#879—92no—
193C23×C32—21no—
194C23×C32—22no—
195C23×C32—41no—
196C23×C32C22×D312yes—
197C23×C32C22×D311yescharacteristic
198C22×C3⋊D3—42no—
199C22×C3⋊D3—41no—
200C22×C3⋊D3C22×D312yes—
201C22×C3⋊D3C22×D311yescharacteristic
202C22×C3⋊D3C22×D312yes—
203C22×C3×D3—32no—
204C22×C3×D3—32no—
205C22×C3×D3—32no—
206C22×C3×D3—32no—
207C22×C3×D3—32no—
208C22×C3×D3—62no—
209C22×C3×D3—62no—
210C22×C3×D3—61no—
211C22×C3×D3—122no—
212C22×C3×D3—122no—
213C22×C3×D3—122no—
214C22×C3×D3—124no—
215C22×C3×D3—122no—
216C2×C32×C4—22no—
217C2×C32×C4—22no—
218C2×C32×C4—22no—
219C2×C32×C4—42no—
220C2×C32×C4C22×D312yes—
221C2×C32×C4C22×D312yes—
222C2×C32×C4C22×D312yes—
223#638—42no—
224#638—41no—
225C2×C3⋊Dic3C22×D312yes—
226C2×C3⋊Dic3C22×D311yescharacteristic
227C2×C3⋊Dic3C22×D312yes—
228C2×C3⋊Dic3C22×D312yes—
229C4⋊(C3⋊D3)—42no—
230C4×C3⋊D3—42no—
231C3×#67—122no—
232C3×#67—122no—
233C3×#67—122no—
234C3×#67—124no—
235C3×#67—122no—
236C2×C3×Dic3—32no—
237C2×C3×Dic3—32no—
238C2×C3×Dic3—32no—
239C2×C3×Dic3—32no—
240C2×C3×Dic3—32no—
241C2×C3×Dic3—32no—
242C2×C3×Dic3—32no—
243C2×C3×Dic3—62no—
244C2×C3×Dic3—62no—
245C2×C3×Dic3—62no—
246C2×C3×Dic3—61no—
247C3×D12—122no—
248C3×D12—122no—
249C3×D12—124no—
250C3×C4×D3—122no—
251C3×C4×D3—122no—
252C3×C4×D3—124no—
253#626—122no—
254#626—122no—
255#626—122no—
256#626—122no—
257#626—122no—
258#626—122no—
259#626—122no—
260#625—122no—
261#625—122no—
262#625—121no—
263#625—122no—
264#624—121no—
265D3×Dic3—122no—
266#560—271noSylow
267C2×C33—41no—
268C2×C33H(5,𝔽2)12yes—
269C2×C33H(5,𝔽2)11yescharacteristic
270C3×C3⋊D3—42no—
271C3×C3⋊D3—42no—
272C3×C3⋊D3—41no—
273C24×C3—91no—
274C23×D3—32no—
275C23×D3—61no—
276C2×C3×D4—91no—
277C2×C3×D4—92no—
278C2×C3×D4—92no—
279C22×C3×C4—92no—
280C2×#67—32no—
281C2×#67—32no—
282C2×#67—32no—
283C2×#67—61no—
284C2×#67—62no—
285C2×#67—92no—
286C2×#67—92no—
287C2×#67—92no—
288C2×#67—92no—
289C2×#67—92no—
290C2×#67—92no—
291C2×#67—92no—
292C22×Dic3—91no—
293C2×D12—32no—
294C2×D12—32no—
295C2×D12—62no—
296C2×D12—92no—
297C2×D12—92no—
298C2×D12—91no—
299C2×D12—92no—
300C2×C4×D3—32no—
301C2×C4×D3—32no—
302C2×C4×D3—62no—
303C2×C4×D3—92no—
304C3×C4⋊C4—92no—
305C3×C4⋊C4—92no—
306C3×C22⋊C4—21no—
307C3×C22⋊C4—22no—
308C3×C22⋊C4—41no—
309C3×C22⋊C4—92no—
310C3×C22⋊C4—92no—
311C3×C22⋊C4—92no—
312C3×C22⋊C4D3212yes—
313C3×C22⋊C4D3211yescharacteristic
314#217—32no—
315#217—32no—
316#217—62no—
317#217—92no—
318#217—92no—
319#217—91no—
320(C2×D3)⋊C4—32no—
321(C2×D3)⋊C4—32no—
322(C2×D3)⋊C4—32no—
323(C2×D3)⋊C4—32no—
324(C2×D3)⋊C4—62no—
325(C2×D3)⋊C4—62no—
326(C2×D3)⋊C4—92no—
327(C2×D3)⋊C4—92no—
328(C2×D3)⋊C4—92no—
329(C2×D3)⋊C4—92no—
330(C2×D3)⋊C4—92no—
331C4⋊Dic3—32no—
332C4⋊Dic3—32no—
333C4⋊Dic3—62no—
334C4⋊Dic3—92no—
335Dic3⋊C4—32no—
336Dic3⋊C4—32no—
337Dic3⋊C4—62no—
338Dic3⋊C4—92no—
339Dic3⋊C4—92no—
340C4×Dic3—91no—
341C4×Dic3—92no—
342C22×C32—21no—
343C22×C32—22no—
344C22×C32—22no—
345C22×C32—22no—
346C22×C32—22no—
347C22×C32—22no—
348C22×C32—22no—
349C22×C32—21no—
350C22×C32—41no—
351C22×C32—42no—
352C22×C32—42no—
353C22×C32—42no—
354C22×C32—42no—
355C22×C32—42no—
356C22×C32—82no—
357C22×C32C23×D311yescharacteristic
358C22×C32C23×D312yes—
359C2×C3⋊D3—22no—
360C2×C3⋊D3—22no—
361C2×C3⋊D3—21no—
362C2×C3⋊D3—22no—
363C2×C3⋊D3—22no—
364C2×C3⋊D3—22no—
365C2×C3⋊D3—22no—
366C2×C3⋊D3—22no—
367C2×C3⋊D3—44no—
368C2×C3×D3—64no—
369C2×C3×D3—64no—
370C2×C3×D3—62no—
371C2×C3×D3—62no—
372C2×C3×D3—62no—
373C2×C3×D3—62no—
374C2×C3×D3—62no—
375C2×C3×D3—62no—
376C2×C3×D3—62no—
377C2×C3×D3—62no—
378C2×C3×D3—62no—
379C2×C3×D3—62no—
380C2×C3×D3—62no—
381C2×C3×D3—62no—
382C2×C3×D3—62no—
383C2×C3×D3—62no—
384C2×C3×D3—62no—
385C2×C3×D3—122no—
386C2×C3×D3—122no—
387C2×C3×D3—122no—
388C2×C3×D3—122no—
389C2×C3×D3—122no—
390C2×C3×D3—121no—
391C2×C3×D3—128no—
392C2×C3×D3—128no—
393C32×C4—22no—
394C32×C4—22no—
395C32×C4—22no—
396C32×C4—42no—
397C32×C4—42no—
398C32×C4—42no—
399C32×C4—82no—
400C3⋊Dic3—22no—
401C3⋊Dic3—21no—
402C3⋊Dic3—22no—
403C3⋊Dic3—22no—
404C3×Dic3—62no—
405C3×Dic3—62no—
406C3×Dic3—62no—
407C3×Dic3—62no—
408C3×Dic3—62no—
409C3×Dic3—62no—
410C3×Dic3—62no—
411C3×Dic3—122no—
412C3×Dic3—122no—
413C3×Dic3—121no—
414C3×Dic3—122no—
415#127—271no—
416#126—272no—
417#123—92no—
418#123—271no—
419#121—272no—
420#121—272no—
421#121—272no—
422#120—91no—
423#120—272no—
424C33#56011yescharacteristic, complemented, Sylow
425C23×C3—21no—
426C23×C3—22no—
427C23×C3—41no—
428C23×C3—92no—
429C23×C3—91no—
430C23×C3—92no—
431C23×C3—361no—
432C23×C3—362no—
433C23×C3C2×D3211yescharacteristic
434C23×C3C2×D3212yes—
435C22×D3—32no—
436C22×D3—32no—
437C22×D3—32no—
438C22×D3—32no—
439C22×D3—32no—
440C22×D3—62no—
441C22×D3—62no—
442C22×D3—61no—
443C22×D3—122no—
444C22×D3—122no—
445C22×D3—122no—
446C22×D3—124no—
447C22×D3—122no—
448C3×D4—362no—
449C3×D4—361no—
450C3×D4—362no—
451C2×C3×C4—22no—
452C2×C3×C4—22no—
453C2×C3×C4—22no—
454C2×C3×C4—42no—
455C2×C3×C4—91no—
456C2×C3×C4—92no—
457C2×C3×C4—92no—
458C2×C3×C4—92no—
459C2×C3×C4—362no—
460C2×C3×C4C2×D3212yes—
461C2×C3×C4C2×D3212yes—
462C2×C3×C4C2×D3212yes—
463#67—122no—
464#67—122no—
465#67—122no—
466#67—124no—
467#67—122no—
468#67—362no—
469#67—362no—
470#67—362no—
471#67—362no—
472#67—362no—
473#67—362no—
474#67—362no—
475C2×Dic3—32no—
476C2×Dic3—32no—
477C2×Dic3—32no—
478C2×Dic3—32no—
479C2×Dic3—32no—
480C2×Dic3—32no—
481C2×Dic3—32no—
482C2×Dic3—62no—
483C2×Dic3—62no—
484C2×Dic3—62no—
485C2×Dic3—61no—
486C2×Dic3—361no—
487D12—122no—
488D12—122no—
489D12—124no—
490D12—362no—
491D12—361no—
492D12—362no—
493D12—362no—
494C4×D3—122no—
495C4×D3—122no—
496C4×D3—124no—
497C4×D3—362no—
498C2×C32—22no—
499C2×C32—21no—
500C2×C32—22no—
501C2×C32—22no—
502C2×C32—22no—
503C2×C32—42no—
504C2×C32—41no—
505C2×C32—41no—
506C2×C32—42no—
507C2×C32—42no—
508C2×C32—82no—
509C2×C32—82no—
510C2×C32#100411yescharacteristic
511C2×C32#100412yes—
512C2×C32#100412yes—
513C2×C32#100912yes—
514C2×C32#100912yes—
515C3⋊D3—42no—
516C3⋊D3—42no—
517C3⋊D3—41no—
518C3×D3—124no—
519C3×D3—124no—
520C3×D3—122no—
521C3×D3—122no—
522C3×D3—122no—
523C3×D3—122no—
524C3×D3—122no—
525C3×D3—241no—
526C24—91no—
527C2×D4—91no—
528C2×D4—92no—
529C2×D4—92no—
530C2×D4—272no—
531C2×D4—272no—
532C2×D4—271no—
533C2×D4—272no—
534C22×C4—92no—
535C22×C4—271no—
536C4⋊C4—92no—
537C4⋊C4—92no—
538C4⋊C4—272no—
539C22⋊C4—92no—
540C22⋊C4—92no—
541C22⋊C4—92no—
542C22⋊C4—272no—
543C22⋊C4—272no—
544C22⋊C4—271no—
545C22⋊C4#111311yescharacteristic
546C42—271no—
547C22×C3—22no—
548C22×C3—22no—
549C22×C3—21no—
550C22×C3—22no—
551C22×C3—22no—
552C22×C3—22no—
553C22×C3—22no—
554C22×C3—21no—
555C22×C3—42no—
556C22×C3—42no—
557C22×C3—42no—
558C22×C3—42no—
559C22×C3—42no—
560C22×C3—41no—
561C22×C3—82no—
562C22×C3—182no—
563C22×C3—181no—
564C22×C3—182no—
565C22×C3—182no—
566C22×C3—182no—
567C22×C3—182no—
568C22×C3—182no—
569C22×C3—182no—
570C22×C3—364no—
571C22×C3C22×D3212yes—
572C22×C3C22×D3211yescharacteristic
573C2×D3—64no—
574C2×D3—64no—
575C2×D3—62no—
576C2×D3—62no—
577C2×D3—62no—
578C2×D3—62no—
579C2×D3—62no—
580C2×D3—62no—
581C2×D3—62no—
582C2×D3—62no—
583C2×D3—62no—
584C2×D3—62no—
585C2×D3—62no—
586C2×D3—62no—
587C2×D3—62no—
588C2×D3—62no—
589C2×D3—62no—
590C2×D3—122no—
591C2×D3—122no—
592C2×D3—122no—
593C2×D3—122no—
594C2×D3—122no—
595C2×D3—121no—
596C2×D3—128no—
597C2×D3—128no—
598C3×C4—22no—
599C3×C4—22no—
600C3×C4—22no—
601C3×C4—42no—
602C3×C4—42no—
603C3×C4—42no—
604C3×C4—82no—
605C3×C4—182no—
606C3×C4—181no—
607C3×C4—182no—
608C3×C4—182no—
609Dic3—62no—
610Dic3—62no—
611Dic3—62no—
612Dic3—62no—
613Dic3—62no—
614Dic3—62no—
615Dic3—62no—
616Dic3—122no—
617Dic3—121no—
618Dic3—122no—
619Dic3—122no—
620C32—22no—
621C32—21no—
622C32—41no—
623C32#559511yescharacteristic, complemented
624C32#566612yescomplemented
625C23—92no—
626C23—92no—
627C23—91no—
628C23—361no—
629C23—362no—
630C23C2×#111311yescharacteristic
631D4—362no—
632D4—361no—
633D4—362no—
634D4—1082no—
635D4—1081no—
636D4—1082no—
637D4—1082no—
638C2×C4—91no—
639C2×C4—92no—
640C2×C4—92no—
641C2×C4—92no—
642C2×C4—362no—
643C2×C4—1081no—
644C2×C4C2×#111312yes—
645C2×C3—21no—
646C2×C3—22no—
647C2×C3—22no—
648C2×C3—22no—
649C2×C3—22no—
650C2×C3—42no—
651C2×C3—42no—
652C2×C3—41no—
653C2×C3—41no—
654C2×C3—42no—
655C2×C3—82no—
656C2×C3—82no—
657C2×C3—362no—
658C2×C3—362no—
659C2×C3—361no—
660C2×C3#6430712yes—
661C2×C3#6430811yescharacteristic
662C2×C3#6431312yes—
663C2×C3#6431412yes—
664C2×C3#6431412yes—
665D3—124no—
666D3—124no—
667D3—122no—
668D3—122no—
669D3—122no—
670D3—122no—
671D3—122no—
672D3—241no—
673C22—21no—
674C22—22no—
675C22—181no—
676C22—182no—
677C22—182no—
678C22—182no—
679C22—182no—
680C22—182no—
681C22—182no—
682C22—182no—
683C22—364no—
684C22C22×#111311yescharacteristic
685C4—22no—
686C4—182no—
687C4—181no—
688C4—182no—
689C4—182no—
690C3—21no—
691C3—22no—
692C3—41no—
693C3#1059423312yescomplemented
694C3#1059424411yescharacteristic, complemented
695C2—41no—
696C2—362no—
697C2—362no—
698C2—361no—
699C2#1172461911yescharacteristic
700C2#1172462312yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
hypercenterC22⋊C4#1113
hypocenterC33#560
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64092C2semidirect
2. C3×#64091C2semidirect
3. #11723728C2semidirect
4. #11723727C2semidirect
5. #11723724C2semidirect
6. #11723696C2semidirect
7. #11723695C2semidirect
8. #11723687C2semidirect
9. #11723675C2semidirect
16. C2×C3×C4×C3⋊D3C22semidirect
19. C2×#6295C22semidirect
20. C2×#6295C22semidirect
21. C2×#6294C22semidirect
25. C3×#3752C22semidirect
26. C3×C4⋊(C3⋊Dic3)C22semidirect
27. C3×#3750C22semidirect
28. #88815C22semidirect
31. #88814C22semidirect
32. #88813C22semidirect
33. #64092D3semidirect
34. #64091D3semidirect
87. C2×C4×C3⋊D3C2×D3semidirect
119. #3752C2×D3semidirect
120. C4⋊(C3⋊Dic3)C2×D3semidirect
121. #3750C2×D3semidirect
424. C33#560semidirect
623. C32#5595semidirect
624. C32#5666semidirect
693. C3#10594233semidirect
694. C3#10594244semidirect

Maximal subgroups

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

subgroupindex
C3×#640922
C3×#640912
#117237282
#117237272
#117237242
#117236962
#117236952
#117236872
#117236752
#105942443
#105942333