Finite Group Databasebeta

← #13168479

Subgroups — #13168479

Order 1728.

1194 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723679C211yesmaximal, characteristic, complemented
2C4×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3#11723432C212yesmaximal, complemented
4C3×#63915C211yesmaximal, characteristic, complemented
5C3×#63879C212yesmaximal, complemented
6#11722989C212yesmaximal, complemented
7#11722976C211yesmaximal, characteristic, complemented
8#11722953C211yesmaximal, characteristic
9Dic3×#627C211yesmaximal, characteristic, complemented
10Dic3×#625C211yesmaximal, characteristic, complemented
11#11722913C212yesmaximal, complemented
12#10595033—31nomaximal
13#10594087—32nomaximal
14C2×C32×C4×D3C2211yescharacteristic
15C2×D3×C3⋊Dic3C2211yescharacteristic
16C2×D3×C3⋊Dic3C2211yescharacteristic
17C2×C3×#627C2211yescharacteristic
18C2×C3×#625C2211yescharacteristic
19C2×C3×#625C2212yes—
20C2×C3×D3×Dic3C2212yes—
21C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
22C3×#3750C2212yes—
23C3×C4×C3⋊Dic3C2211yescharacteristic
24C32×(C2×D3)⋊C4C2212yes—
25#88815C2212yes—
26#88815C2211yescharacteristic
27#88815C2211yescharacteristic
28#88807C2212yes—
29Dic3⋊(C3⋊Dic3)C2212yes—
30(C2×D3)⋊(C3⋊Dic3)C2212yes—
31(C2×D3)⋊(C3⋊Dic3)C2212yes—
32Dic3×C3⋊Dic3C2211yescharacteristic
33Dic3×C3⋊Dic3C2211yescharacteristic
34C3×(C2×D3)⋊Dic3C2212yes—
35C3×(C2×D3)⋊Dic3C2212yes—
36C3×Dic32C2212yes—
37C2×C4×C3⋊Dic3—31no—
38#64088—32no—
39C3×C4×#67—32no—
40C3×#885—32no—
41C3×#67⋊C4—32no—
42#63982—31no—
43#63980—31no—
44Dic3×#67—32no—
45#63970—32no—
46#63969—32no—
47#63962—32no—
48#63955—32no—
49C4×#627—31no—
50#63915D311yescharacteristic, complemented
51C4×#625—31no—
52#63891—32no—
53#63887—32no—
54#63879D312yescomplemented
55#63877—32no—
56C4×D3×Dic3—32no—
57C4×D3×Dic3—61no—
58#63870—31no—
59#63867—32no—
60#63862—32no—
61#63862—32no—
62#63862—62no—
63#63855—32no—
64Dic3×Dic6—32no—
65C22×C32×D3C2312yes—
66C22×C32×D3C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2311yescharacteristic
76C32×C4×D3—22no—
77D3×C3⋊Dic3—22no—
78D3×C3⋊Dic3—22no—
79C3×#627—22no—
80C3×#625—24no—
81C3×#625—22no—
82C3×D3×Dic3—24no—
83#5680—91no—
84#5538—92no—
85C22×C32×C4—31no—
86C22×C3⋊Dic3—31no—
87C22×C3⋊Dic3—31no—
88C2×C3×#67—32no—
89C2×C3×#67—32no—
90C2×C3×#67—32no—
91C22×C3×Dic3—32no—
92C2×C3×C4×D3—21no—
93C2×C3×C4×D3—32no—
94C2×C3×C4×D3C2×D312yes—
95C2×C3×Dic6—32no—
96C2×#627C2×D311yescharacteristic
97C2×#625C2×D312yes—
98C2×#625C2×D311yescharacteristic
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—61no—
102C2×D3×Dic3—61no—
103C2×D3×Dic3C2×D312yes—
104C32×C22⋊C4—32no—
105#3757—32no—
106#3757—32no—
107C4⋊(C3⋊Dic3)—32no—
108#3750—32no—
109#3750C2×D312yes—
110C4×C3⋊Dic3—32no—
111C4×C3⋊Dic3—31no—
112C4×C3⋊Dic3—31no—
113C4×C3⋊Dic3—31no—
114C4×C3⋊Dic3—62no—
115C4×C3⋊Dic3—61no—
116C4×C3⋊Dic3—121no—
117C4×C3⋊Dic3C2×D311yescharacteristic
118C3×#217—32no—
119C3×#217—32no—
120C3×(C2×D3)⋊C4—22no—
121C3×(C2×D3)⋊C4—32no—
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4C2×D312yes—
124C3×(C2×D3)⋊C4C2×D312yes—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—62no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—32no—
130C3×C4×Dic3—32no—
131C3×C4×Dic3—61no—
132#3724—32no—
133#3724—31no—
134#3724—31no—
135Dic3⋊Dic3—32no—
136Dic3⋊Dic3—32no—
137Dic3⋊Dic3—32no—
138Dic3⋊Dic3—32no—
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—62no—
144Dic3⋊Dic3—62no—
145(C2×D3)⋊Dic3—32no—
146(C2×D3)⋊Dic3—32no—
147(C2×D3)⋊Dic3—32no—
148(C2×D3)⋊Dic3—32no—
149(C2×D3)⋊Dic3—62no—
150(C2×D3)⋊Dic3—62no—
151(C2×D3)⋊Dic3C2×D312yes—
152(C2×D3)⋊Dic3C2×D312yes—
153Dic32—32no—
154Dic32—32no—
155Dic32—61no—
156Dic32—61no—
157Dic32C2×D312yes—
158C22×C33C2411yescharacteristic
159C2×C32×D3—21no—
160C2×C32×D3—21no—
161C2×C32×D3—22no—
162C2×C32×D3—22no—
163C2×C32×D3—22no—
164C2×C32×D3#4112yes—
165C33×C4—21no—
166C3⋊(C3⋊Dic3)—21no—
167C3⋊(C3⋊Dic3)—21no—
168C3×C3⋊Dic3—21no—
169C3×C3⋊Dic3—22no—
170C3×C3⋊Dic3—22no—
171C3×C3⋊Dic3#4112yes—
172C32×Dic3—22no—
173C32×Dic3#4112yes—
174C3×#124—91no—
175C3×C4×D4—92no—
176#946—91no—
177Q8×Dic3—91no—
178#939—92no—
179#938—91no—
180#935—92no—
181D4×Dic3—91no—
182C4×#67—32no—
183C4×#67—92no—
184#926—92no—
185C2×C4×Dic3—92no—
186C2×C4×Dic3—181no—
187#898—92no—
188#889—92no—
189#886—92no—
190#885—32no—
191#883—92no—
192#67⋊C4—32no—
193#879—92no—
194#879—92no—
195#879—182no—
196#878—92no—
197#877—92no—
198#872—92no—
199C42×D3—91no—
200#870—91no—
201C4×Dic6—92no—
202C23×C32—31no—
203C23×C32—32no—
204C22×C3×D3—21no—
205C22×C3×D3—22no—
206C22×C3×D3C22×D312yes—
207C22×C3×D3C22×D312yes—
208C22×C3×D3C22×D312yes—
209C2×C32×C4—21no—
210C2×C32×C4—22no—
211C2×C32×C4—31no—
212C2×C32×C4—32no—
213C2×C32×C4—41no—
214C2×C32×C4—62no—
215C2×C32×C4C22×D312yes—
216C2×C32×C4C22×D311yescharacteristic
217C2×C3⋊Dic3—32no—
218C2×C3⋊Dic3—32no—
219C2×C3⋊Dic3—31no—
220C2×C3⋊Dic3—31no—
221C2×C3⋊Dic3—61no—
222C2×C3⋊Dic3—61no—
223C2×C3⋊Dic3—62no—
224C2×C3⋊Dic3—62no—
225C2×C3⋊Dic3—62no—
226C2×C3⋊Dic3—62no—
227C2×C3⋊Dic3—121no—
228C2×C3⋊Dic3—121no—
229C2×C3⋊Dic3C22×D312yes—
230C2×C3⋊Dic3C22×D311yescharacteristic
231C2×C3⋊Dic3C22×D312yes—
232C2×C3⋊Dic3C22×D311yescharacteristic
233C3×#67—64no—
234C3×#67—64no—
235C3×#67—64no—
236C2×C3×Dic3—21no—
237C2×C3×Dic3—22no—
238C2×C3×Dic3—32no—
239C2×C3×Dic3—32no—
240C2×C3×Dic3—32no—
241C2×C3×Dic3—32no—
242C2×C3×Dic3—32no—
243C2×C3×Dic3—32no—
244C2×C3×Dic3—61no—
245C2×C3×Dic3—61no—
246C2×C3×Dic3—62no—
247C2×C3×Dic3—62no—
248C2×C3×Dic3—64no—
249C2×C3×Dic3C22×D312yes—
250C2×C3×Dic3C22×D312yes—
251C2×C3×Dic3C22×D312yes—
252C3×C4×D3—24no—
253C3×C4×D3—24no—
254C3×C4×D3—64no—
255C3×Dic6—64no—
256#627—22no—
257#625—24no—
258#625—22no—
259D3×Dic3—24no—
260D3×Dic3—64no—
261D3×Dic3—64no—
262D3×Dic3—64no—
263D3×Dic3—64no—
264#529—271noSylow
265C2×C33C2×#4111yescharacteristic
266C2×C33C2×#4111yescharacteristic
267C2×C33C2×#4111yescharacteristic
268C32×D3—22no—
269C32×D3—42no—
270C2×C3×Q8—91no—
271C2×C3×D4—91no—
272C2×C3×D4—92no—
273C22×C3×C4—32no—
274C22×C3×C4—61no—
275C22×C3×C4—92no—
276C2×#67—32no—
277C2×#67—32no—
278C2×#67—32no—
279C22×Dic3—32no—
280C22×Dic3—92no—
281C22×Dic3—92no—
282C22×Dic3—181no—
283C22×Dic3—181no—
284C2×C4×D3—32no—
285C2×C4×D3—91no—
286C2×C4×D3—91no—
287C2×C4×D3D3211yescharacteristic
288C2×Dic6—32no—
289C3×C4⋊C4—92no—
290C3×C22⋊C4—32no—
291C3×C22⋊C4—32no—
292C3×C22⋊C4—62no—
293C3×C22⋊C4—92no—
294C3×C22⋊C4—92no—
295C3×C42—92no—
296C3×C42—91no—
297#217—32no—
298#217—32no—
299#217—92no—
300#217—92no—
301#217—92no—
302#217—92no—
303#217—182no—
304#217—182no—
305(C2×D3)⋊C4—32no—
306(C2×D3)⋊C4—32no—
307(C2×D3)⋊C4—92no—
308(C2×D3)⋊C4—92no—
309(C2×D3)⋊C4D3212yes—
310C4⋊Dic3—91no—
311C4⋊Dic3—91no—
312C4⋊Dic3—92no—
313C4⋊Dic3—92no—
314C4⋊Dic3—92no—
315C4⋊Dic3—92no—
316C4⋊Dic3—182no—
317Dic3⋊C4—32no—
318Dic3⋊C4—32no—
319Dic3⋊C4—62no—
320Dic3⋊C4—92no—
321Dic3⋊C4—92no—
322Dic3⋊C4—92no—
323Dic3⋊C4—92no—
324Dic3⋊C4—92no—
325Dic3⋊C4—92no—
326Dic3⋊C4—92no—
327Dic3⋊C4—182no—
328C4×Dic3—32no—
329C4×Dic3—32no—
330C4×Dic3—32no—
331C4×Dic3—61no—
332C4×Dic3—92no—
333C4×Dic3—92no—
334C4×Dic3—91no—
335C4×Dic3—92no—
336C4×Dic3—91no—
337C4×Dic3—91no—
338C4×Dic3—181no—
339C4×Dic3—181no—
340C4×Dic3—182no—
341C4×Dic3—181no—
342C4×Dic3—361no—
343C22×C32—21no—
344C22×C32—22no—
345C22×C32—32no—
346C22×C32—41no—
347C22×C32—61no—
348C22×C32—61no—
349C22×C32—62no—
350C22×C32—62no—
351C22×C32—62no—
352C22×C32C23×D312yes—
353C22×C32C23×D311yescharacteristic
354C2×C3×D3—22no—
355C2×C3×D3—22no—
356C2×C3×D3—22no—
357C2×C3×D3—22no—
358C2×C3×D3—22no—
359C2×C3×D3—22no—
360C2×C3×D3—22no—
361C2×C3×D3—22no—
362C2×C3×D3—22no—
363C2×C3×D3—22no—
364C2×C3×D3—22no—
365C2×C3×D3—42no—
366C2×C3×D3—42no—
367C2×C3×D3—42no—
368C2×C3×D3#23514yes—
369C32×C4—21no—
370C32×C4—22no—
371C32×C4—22no—
372C32×C4—32no—
373C32×C4—42no—
374C32×C4—42no—
375C32×C4—62no—
376C3⋊Dic3—21no—
377C3⋊Dic3—22no—
378C3⋊Dic3—22no—
379C3⋊Dic3—61no—
380C3⋊Dic3—61no—
381C3⋊Dic3—62no—
382C3⋊Dic3—62no—
383C3⋊Dic3—62no—
384C3⋊Dic3—62no—
385C3⋊Dic3—122no—
386C3⋊Dic3—122no—
387C3⋊Dic3—122no—
388C3⋊Dic3—122no—
389C3⋊Dic3#23712yes—
390C3×Dic3—22no—
391C3×Dic3—22no—
392C3×Dic3—22no—
393C3×Dic3—34no—
394C3×Dic3—42no—
395C3×Dic3—62no—
396C3×Dic3—62no—
397C3×Dic3—62no—
398C3×Dic3—62no—
399C3×Dic3—62no—
400C3×Dic3—62no—
401C3×Dic3—62no—
402C3×Dic3—122no—
403C3×Dic3—122no—
404C3×Dic3#23714yes—
405#126—272no—
406#125—271no—
407#124—91no—
408#123—272no—
409C22⋊Q8—271no—
410#121—271no—
411C4×Q8—271no—
412C4×D4—92no—
413C4×D4—271no—
414#117—272no—
415C2×C42—271no—
416C33#52911yescharacteristic, complemented, Sylow
417C23×C3—32no—
418C23×C3—32no—
419C23×C3—32no—
420C23×C3—61no—
421C23×C3—62no—
422C22×D3C2×D3212yes—
423C22×D3C2×D3211yescharacteristic
424C3×Q8—182no—
425C3×D4—182no—
426C3×D4—184no—
427C2×C3×C4—21no—
428C2×C3×C4—22no—
429C2×C3×C4—32no—
430C2×C3×C4—32no—
431C2×C3×C4—32no—
432C2×C3×C4—41no—
433C2×C3×C4—61no—
434C2×C3×C4—62no—
435C2×C3×C4—64no—
436C2×C3×C4—64no—
437C2×C3×C4—91no—
438C2×C3×C4—91no—
439C2×C3×C4—92no—
440C2×C3×C4—92no—
441C2×C3×C4—184no—
442C2×C3×C4C2×D3212yes—
443C2×C3×C4C2×D3211yescharacteristic
444#67—64no—
445#67—64no—
446#67—64no—
447C2×Dic3—32no—
448C2×Dic3—32no—
449C2×Dic3—32no—
450C2×Dic3—32no—
451C2×Dic3—32no—
452C2×Dic3—32no—
453C2×Dic3—61no—
454C2×Dic3—61no—
455C2×Dic3—62no—
456C2×Dic3—62no—
457C2×Dic3—64no—
458C2×Dic3—91no—
459C2×Dic3—92no—
460C2×Dic3—92no—
461C2×Dic3—91no—
462C2×Dic3—181no—
463C2×Dic3—181no—
464C2×Dic3—182no—
465C2×Dic3—182no—
466C2×Dic3—184no—
467C2×Dic3—184no—
468C2×Dic3—184no—
469C2×Dic3—184no—
470C2×Dic3—361no—
471C2×Dic3—361no—
472C2×Dic3C2×D3212yes—
473C2×Dic3C2×D3211yescharacteristic
474C4×D3—22no—
475C4×D3—64no—
476C4×D3—182no—
477C4×D3—182no—
478Dic6—64no—
479C2×C32—21no—
480C2×C32—22no—
481C2×C32—21no—
482C2×C32—22no—
483C2×C32—22no—
484C2×C32—21no—
485C2×C32—41no—
486C2×C32—41no—
487C2×C32—41no—
488C2×C32—62no—
489C2×C32—122no—
490C2×C32C2×#23512yes—
491C2×C32C2×#23711yescharacteristic
492C2×C32C2×#23712yes—
493C2×C32D3×#4111yescharacteristic
494C2×C32D3×#4112yes—
495C2×C32D3×#4111yescharacteristic
496C3×D3—24no—
497C3×D3—24no—
498C3×D3—42no—
499C3×D3—42no—
500C3×D3—82no—
501C2×Q8—91no—
502C2×D4—92no—
503C2×D4—91no—
504C22×C4—31no—
505C22×C4—92no—
506C22×C4—271no—
507C22×C4—271no—
508C4⋊C4—92no—
509C4⋊C4—272no—
510C4⋊C4—272no—
511C4⋊C4—272no—
512C4⋊C4—271no—
513C4⋊C4—271no—
514C22⋊C4—32no—
515C22⋊C4—92no—
516C22⋊C4—92no—
517C22⋊C4—272no—
518C22⋊C4—272no—
519C42—92no—
520C42—91no—
521C42—271no—
522C42—271no—
523C42—271no—
524C22×C3—21no—
525C22×C3—22no—
526C22×C3—34no—
527C22×C3—41no—
528C22×C3—62no—
529C22×C3—62no—
530C22×C3—62no—
531C22×C3—62no—
532C22×C3—62no—
533C22×C3—62no—
534C22×C3—62no—
535C22×C3—62no—
536C22×C3—62no—
537C22×C3—62no—
538C22×C3—62no—
539C22×C3—122no—
540C22×C3—122no—
541C22×C3—122no—
542C22×C3C22×D3211yescharacteristic
543C22×C3C22×D3212yes—
544C2×D3—21no—
545C2×D3—21no—
546C2×D3—22no—
547C2×D3—22no—
548C2×D3—22no—
549C2×D3#379812yes—
550C3×C4—22no—
551C3×C4—22no—
552C3×C4—21no—
553C3×C4—34no—
554C3×C4—42no—
555C3×C4—42no—
556C3×C4—62no—
557C3×C4—62no—
558C3×C4—62no—
559C3×C4—92no—
560C3×C4—122no—
561C3×C4—181no—
562C3×C4—182no—
563C3×C4—182no—
564Dic3—22no—
565Dic3—34no—
566Dic3—62no—
567Dic3—62no—
568Dic3—62no—
569Dic3—62no—
570Dic3—62no—
571Dic3—62no—
572Dic3—62no—
573Dic3—122no—
574Dic3—122no—
575Dic3—182no—
576Dic3—182no—
577Dic3—181no—
578Dic3—182no—
579Dic3—181no—
580Dic3—182no—
581Dic3—362no—
582Dic3—362no—
583Dic3—362no—
584Dic3—362no—
585Dic3#380512yes—
586C32—22no—
587C32—21no—
588C32—41no—
589C32#553812yescomplemented
590C32#568011yescharacteristic, complemented
591C23—31no—
592C23—32no—
593Q8—182no—
594D4—182no—
595D4—184no—
596C2×C4—31no—
597C2×C4—32no—
598C2×C4—62no—
599C2×C4—91no—
600C2×C4—91no—
601C2×C4—92no—
602C2×C4—92no—
603C2×C4—184no—
604C2×C4—271no—
605C2×C4—271no—
606C2×C4—542no—
607C2×C4—542no—
608C2×C4D3311yescharacteristic
609C2×C3—21no—
610C2×C3—22no—
611C2×C3—21no—
612C2×C3—22no—
613C2×C3—22no—
614C2×C3—21no—
615C2×C3—41no—
616C2×C3—41no—
617C2×C3—41no—
618C2×C3—64no—
619C2×C3—64no—
620C2×C3—122no—
621C2×C3—122no—
622C2×C3—242no—
623C2×C3C2×#379811yescharacteristic
624C2×C3D3×#23512yes—
625C2×C3#6430611yescharacteristic
626C2×C3D3×#23712yes—
627C2×C3D3×#23712yes—
628C2×C3C2×#380511yescharacteristic
629D3—22no—
630D3—42no—
631C22—32no—
632C22—61no—
633C22—61no—
634C22—62no—
635C22—62no—
636C22—62no—
637C22C2×D3311yescharacteristic
638C4—21no—
639C4—32no—
640C4—62no—
641C4—92no—
642C4—181no—
643C4—182no—
644C4—182no—
645C4—541no—
646C4—541no—
647C3—21no—
648C3—22no—
649C3—41no—
650C3#1059408712yescomplemented
651C3#1059503311yescharacteristic, complemented
652C2—62no—
653C2—122no—
654C2#1172437311yescharacteristic
655C2D3×#379811yescharacteristic
656C2D3×#380511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radical#11722953C2
hypercenterC2×C4D33
hypocenterC33#529
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723679C2semidirect
2. C4×D3×C3⋊Dic3C2semidirect
3. #11723432C2semidirect
4. C3×#63915C2semidirect
5. C3×#63879C2semidirect
6. #11722989C2semidirect
7. #11722976C2semidirect
9. Dic3×#627C2semidirect
10. Dic3×#625C2semidirect
11. #11722913C2semidirect
50. #63915D3semidirect
54. #63879D3semidirect
416. C33#529semidirect
589. C32#5538semidirect
590. C32#5680semidirect
650. C3#10594087semidirect
651. C3#10595033semidirect

Maximal subgroups

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

subgroupindex
#117236792
C4×D3×C3⋊Dic32
#117234322
C3×#639152
C3×#638792
#117229892
#117229762
#117229532
Dic3×#6272
Dic3×#6252
#117229132
#105950333
#105940873