Finite Group Databasebeta

← #13178695

Subgroups — #13178695

Order 1728.

1160 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64144C212yesmaximal
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic
3C2×#88815C211yesmaximal, characteristic, complemented
4#11723721C211yesmaximal, characteristic, complemented
5#11723721C211yesmaximal, characteristic, complemented
6#11723716C212yesmaximal, complemented
7#11723715C212yesmaximal, complemented
8#11723715C211yesmaximal, characteristic, complemented
9#11723679C211yesmaximal, characteristic, complemented
10#11723669C212yesmaximal, complemented
11#11723665C211yesmaximal, characteristic, complemented
12#10595120—31nomaximal
13#10595117—32nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2212yes—
18C2×#6296C2211yescharacteristic
19C3×#3757C2211yescharacteristic
20C3×#3757C2211yescharacteristic
21C3×#3757C2212yes—
22C3×#3757C2212yes—
23C3×#3757C2212yes—
24C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
25C3×#3750C2212yes—
26C3×#3750C2212yes—
27C3×C4×C3⋊Dic3C2211yescharacteristic
28#88815C2211yescharacteristic
29#88815C2212yes—
30#88815C2212yes—
31#88815C2212yes—
32#88815C2212yes—
33#88815C2211yescharacteristic
34#88815C2211yescharacteristic
35#88813C2212yes—
36#88813C2212yes—
37#64144D312yes—
38D4×C3⋊Dic3D311yescharacteristic
39C3×#935—32no—
40C3×#935—32no—
41C3×#935—62no—
42C3×D4×Dic3—32no—
43C3×D4×Dic3—61no—
44#63982—31no—
45#63982—31no—
46C2×#3724—31no—
47C2×Dic3⋊Dic3—32no—
48#63964—32no—
49#63964—32no—
50#63957—32no—
51#63957—31no—
52#63956—32no—
53#63956—32no—
54#63956—32no—
55#63955—32no—
56#63955—32no—
57#63955—32no—
58#63870—31no—
59#63867—32no—
60#63847—32no—
61#63846—32no—
62#63845—32no—
63#63838—31no—
64#63837—32no—
65C23×C33C2311yescharacteristic
66C23×C33C2311yescharacteristic
67C33×D4—22no—
68C2×C33×C4C2311yescharacteristic
69C2×C3×C3⋊Dic3—24no—
70C2×C3×C3⋊Dic3—22no—
71C2×C3×C3⋊Dic3—22no—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2312yes—
76C2×C3×C3⋊Dic3C2312yes—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C2×C3×C3⋊Dic3C2311yescharacteristic
80#6296—22no—
81#5650—92no—
82#5541—91no—
83C2×C32×D4—21no—
84C2×C32×D4—22no—
85C2×C32×D4—41no—
86C2×C32×D4C2×D312yes—
87C2×C32×D4C2×D311yescharacteristic
88C22×C3⋊Dic3C2×D311yescharacteristic
89C22×C3⋊Dic3C2×D311yescharacteristic
90C22×C3⋊Dic3C2×D312yes—
91C22×C3×Dic3—32no—
92C22×C3×Dic3—32no—
93C22×C3×Dic3—32no—
94C22×C3×Dic3—32no—
95C22×C3×Dic3—61no—
96C22×C3×Dic3—61no—
97C22×C3×Dic3—62no—
98C2×#627—32no—
99C2×#627—31no—
100#3757C2×D311yescharacteristic
101#3757C2×D311yescharacteristic
102#3757C2×D312yes—
103#3757C2×D312yes—
104#3757C2×D312yes—
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106#3750C2×D312yes—
107#3750C2×D312yes—
108C4×C3⋊Dic3C2×D311yescharacteristic
109C3×#217—32no—
110C3×#217—32no—
111C3×#217—32no—
112C3×#217—32no—
113C3×#217—32no—
114C3×#217—32no—
115C3×#217—32no—
116C3×#217—32no—
117C3×#217—61no—
118C3×#217—61no—
119C3×#217—62no—
120C3×#217—62no—
121C3×#217—62no—
122C3×C4⋊Dic3—32no—
123C3×C4⋊Dic3—61no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—62no—
129C3×Dic3⋊C4—62no—
130C3×C4×Dic3—32no—
131C3×C4×Dic3—61no—
132#3724—31no—
133#3724—32no—
134#3724—32no—
135#3724—32no—
136#3724—32no—
137#3724—31no—
138#3724—31no—
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—32no—
145Dic3⋊Dic3—32no—
146Dic3⋊Dic3—32no—
147Dic3⋊Dic3—32no—
148Dic3⋊Dic3—32no—
149Dic3⋊Dic3—32no—
150Dic32—32no—
151Dic32—32no—
152Dic32—32no—
153Dic32—32no—
154Dic32—32no—
155Dic32—32no—
156C22×C33—21no—
157C22×C33—21no—
158C22×C33—21no—
159C22×C33—21no—
160C22×C33—21no—
161C22×C33#4112yes—
162C22×C33C2411yescharacteristic
163C33×C4—21no—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3—21no—
168C3×C3⋊Dic3—21no—
169C3×C3⋊Dic3—21no—
170C3×C3⋊Dic3—22no—
171C3×C3⋊Dic3#4112yes—
172C3×#123—92no—
173C3×C4×D4—91no—
174#935—32no—
175#935—32no—
176#935—62no—
177D4×Dic3—32no—
178D4×Dic3—61no—
179#926—92no—
180#926—91no—
181C2×C4⋊Dic3—91no—
182#924—91no—
183#924—91no—
184C2×Dic3⋊C4—92no—
185#890—92no—
186#889—92no—
187#889—92no—
188#889—92no—
189#887—92no—
190#879—92no—
191#879—92no—
192#879—92no—
193#878—92no—
194#878—92no—
195#877—92no—
196#877—92no—
197#877—92no—
198#870—91no—
199C4×Dic6—91no—
200C23×C32—21no—
201C23×C32—22no—
202C23×C32—21no—
203C23×C32—22no—
204C23×C32—41no—
205C23×C32—41no—
206C23×C32C22×D311yescharacteristic
207C23×C32C22×D312yes—
208C23×C32C22×D311yescharacteristic
209C23×C32C22×D312yes—
210C32×D4—22no—
211C32×D4—24no—
212C32×D4—24no—
213C32×D4—44no—
214C32×D4—44no—
215C2×C32×C4—21no—
216C2×C32×C4—22no—
217C2×C32×C4—41no—
218C2×C32×C4C22×D312yes—
219C2×C32×C4C22×D311yescharacteristic
220C2×C3⋊Dic3—24no—
221C2×C3⋊Dic3—22no—
222C2×C3⋊Dic3—22no—
223C2×C3⋊Dic3C22×D311yescharacteristic
224C2×C3⋊Dic3C22×D312yes—
225C2×C3⋊Dic3C22×D311yescharacteristic
226C2×C3⋊Dic3C22×D312yes—
227C2×C3⋊Dic3C22×D312yes—
228C2×C3⋊Dic3C22×D312yes—
229C2×C3⋊Dic3C22×D311yescharacteristic
230C2×C3⋊Dic3C22×D311yescharacteristic
231C2×C3×Dic3—32no—
232C2×C3×Dic3—32no—
233C2×C3×Dic3—32no—
234C2×C3×Dic3—32no—
235C2×C3×Dic3—32no—
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—61no—
244C2×C3×Dic3—61no—
245C2×C3×Dic3—62no—
246C2×C3×Dic3—62no—
247C2×C3×Dic3—62no—
248C2×C3×Dic3—62no—
249C2×C3×Dic3—61no—
250C2×C3×Dic3—61no—
251C2×C3×Dic3—64no—
252C2×C3×Dic3—64no—
253C2×C3×Dic3—64no—
254C2×C3×Dic3—64no—
255C2×C3×Dic3—122no—
256C2×C3×Dic3—122no—
257C2×C3×Dic3—124no—
258#627—64no—
259#627—62no—
260#552—271noSylow
261C2×C33—22no—
262C2×C33—41no—
263C2×C33C2×#4111yescharacteristic
264C2×C33C2×#4111yescharacteristic
265C2×C332−1+411yescharacteristic
266C2×C3×D4—21no—
267C2×C3×D4—22no—
268C2×C3×D4—41no—
269C2×C3×D4D3212yes—
270C2×C3×D4D3211yescharacteristic
271C22×C3×C4—92no—
272C22×C3×C4—91no—
273C22×C3×C4—91no—
274C22×Dic3—32no—
275C22×Dic3—32no—
276C22×Dic3—32no—
277C22×Dic3—32no—
278C22×Dic3—61no—
279C22×Dic3—61no—
280C22×Dic3—62no—
281C2×Dic6—91no—
282C2×Dic6—92no—
283C3×C4⋊C4—92no—
284C3×C4⋊C4—91no—
285C3×C4⋊C4—92no—
286C3×C22⋊C4—91no—
287C3×C22⋊C4—92no—
288C3×C22⋊C4—92no—
289C3×C22⋊C4—92no—
290C3×C22⋊C4—91no—
291C3×C42—91no—
292#217—32no—
293#217—32no—
294#217—32no—
295#217—32no—
296#217—32no—
297#217—32no—
298#217—32no—
299#217—32no—
300#217—61no—
301#217—61no—
302#217—62no—
303#217—62no—
304#217—62no—
305C4⋊Dic3—32no—
306C4⋊Dic3—61no—
307C4⋊Dic3—91no—
308C4⋊Dic3—91no—
309C4⋊Dic3—91no—
310C4⋊Dic3—92no—
311C4⋊Dic3—92no—
312C4⋊Dic3—92no—
313C4⋊Dic3—92no—
314Dic3⋊C4—32no—
315Dic3⋊C4—32no—
316Dic3⋊C4—32no—
317Dic3⋊C4—32no—
318Dic3⋊C4—62no—
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—92no—
328Dic3⋊C4—92no—
329Dic3⋊C4—92no—
330Dic3⋊C4—92no—
331C4×Dic3—32no—
332C4×Dic3—61no—
333C4×Dic3—92no—
334C4×Dic3—92no—
335C4×Dic3—92no—
336C4×Dic3—92no—
337C4×Dic3—92no—
338C4×Dic3—92no—
339C22×C32—21no—
340C22×C32—21no—
341C22×C32—22no—
342C22×C32—22no—
343C22×C32—21no—
344C22×C32—22no—
345C22×C32—21no—
346C22×C32—22no—
347C22×C32—22no—
348C22×C32—21no—
349C22×C32—22no—
350C22×C32—22no—
351C22×C32—21no—
352C22×C32—24no—
353C22×C32—22no—
354C22×C32—41no—
355C22×C32—42no—
356C22×C32—42no—
357C22×C32—42no—
358C22×C32—41no—
359C22×C32—41no—
360C22×C32—42no—
361C22×C32—41no—
362C22×C32—42no—
363C22×C32—41no—
364C22×C32—42no—
365C22×C32—42no—
366C22×C32—81no—
367C22×C32—81no—
368C22×C32—81no—
369C22×C32—81no—
370C22×C32#23512yes—
371C22×C32#23714yes—
372C22×C32C23×D311yescharacteristic
373C22×C32C23×D312yes—
374C32×C4—21no—
375C32×C4—22no—
376C32×C4—22no—
377C32×C4—42no—
378C32×C4—42no—
379C3⋊Dic3—21no—
380C3⋊Dic3—21no—
381C3⋊Dic3—22no—
382C3⋊Dic3—22no—
383C3⋊Dic3—22no—
384C3⋊Dic3—22no—
385C3⋊Dic3—21no—
386C3⋊Dic3#23712yes—
387C3×Dic3—34no—
388C3×Dic3—62no—
389C3×Dic3—62no—
390C3×Dic3—62no—
391C3×Dic3—62no—
392C3×Dic3—62no—
393C3×Dic3—62no—
394C3×Dic3—62no—
395C3×Dic3—62no—
396C3×Dic3—62no—
397C3×Dic3—62no—
398C3×Dic3—62no—
399C3×Dic3—62no—
400C3×Dic3—121no—
401C3×Dic3—122no—
402C3×Dic3—122no—
403C3×Dic3—121no—
404C3×Dic3—121no—
405C3×Dic3—122no—
406C3×Dic3—122no—
407#126—272no—
408#125—271no—
409#125—272no—
410#123—92no—
411C22⋊Q8—271no—
412C22⋊Q8—271no—
413C4×Q8—271no—
414C4×D4—91no—
415#117—271no—
416#117—272no—
417C2×C4⋊C4—271no—
418C33#55211yescharacteristic, complemented, Sylow
419C23×C3—21no—
420C23×C3—22no—
421C23×C3—21no—
422C23×C3—22no—
423C23×C3—41no—
424C23×C3—41no—
425C23×C3C2×D3211yescharacteristic
426C23×C3C2×D3212yes—
427C23×C3C2×D3212yes—
428C23×C3C2×D3211yescharacteristic
429C3×D4—24no—
430C3×D4—24no—
431C3×D4—22no—
432C3×D4—44no—
433C3×D4—44no—
434C2×C3×C4—21no—
435C2×C3×C4—22no—
436C2×C3×C4—41no—
437C2×C3×C4—91no—
438C2×C3×C4—91no—
439C2×C3×C4—91no—
440C2×C3×C4—92no—
441C2×C3×C4—92no—
442C2×C3×C4—92no—
443C2×C3×C4—91no—
444C2×C3×C4—92no—
445C2×C3×C4—182no—
446C2×C3×C4—182no—
447C2×C3×C4—184no—
448C2×C3×C4C2×D3212yes—
449C2×C3×C4C2×D3211yescharacteristic
450C2×Dic3—32no—
451C2×Dic3—32no—
452C2×Dic3—32no—
453C2×Dic3—32no—
454C2×Dic3—32no—
455C2×Dic3—32no—
456C2×Dic3—32no—
457C2×Dic3—32no—
458C2×Dic3—32no—
459C2×Dic3—32no—
460C2×Dic3—32no—
461C2×Dic3—32no—
462C2×Dic3—61no—
463C2×Dic3—61no—
464C2×Dic3—62no—
465C2×Dic3—62no—
466C2×Dic3—62no—
467C2×Dic3—62no—
468C2×Dic3—61no—
469C2×Dic3—61no—
470C2×Dic3—64no—
471C2×Dic3—64no—
472C2×Dic3—64no—
473C2×Dic3—64no—
474C2×Dic3—122no—
475C2×Dic3—122no—
476C2×Dic3—124no—
477Dic6—184no—
478Dic6—182no—
479C2×C32—24no—
480C2×C32—22no—
481C2×C32—22no—
482C2×C32—21no—
483C2×C32—22no—
484C2×C32—21no—
485C2×C32—22no—
486C2×C32—21no—
487C2×C32—41no—
488C2×C32—41no—
489C2×C32—41no—
490C2×C32—44no—
491C2×C32—42no—
492C2×C32—41no—
493C2×C32—42no—
494C2×C32—42no—
495C2×C32—82no—
496C2×C32—82no—
497C2×C32—82no—
498C2×C32C2×#23511yescharacteristic
499C2×C32C2×#23711yescharacteristic
500C2×C32C2×#23712yes—
501C2×C32D3×#4112yes—
502C2×C32#101012yes—
503C2×C32#101011yescharacteristic
504C2×Q8—271no—
505C2×D4#111311yescharacteristic
506C22×C4—92no—
507C22×C4—91no—
508C22×C4—91no—
509C4⋊C4—91no—
510C4⋊C4—92no—
511C4⋊C4—92no—
512C4⋊C4—271no—
513C4⋊C4—271no—
514C4⋊C4—272no—
515C4⋊C4—271no—
516C4⋊C4—272no—
517C4⋊C4—272no—
518C4⋊C4—272no—
519C22⋊C4—91no—
520C22⋊C4—92no—
521C22⋊C4—92no—
522C22⋊C4—91no—
523C22⋊C4—92no—
524C42—91no—
525C42—272no—
526C42—272no—
527C22×C3—22no—
528C22×C3—21no—
529C22×C3—22no—
530C22×C3—21no—
531C22×C3—22no—
532C22×C3—22no—
533C22×C3—21no—
534C22×C3—22no—
535C22×C3—21no—
536C22×C3—22no—
537C22×C3—21no—
538C22×C3—21no—
539C22×C3—22no—
540C22×C3—24no—
541C22×C3—22no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3—42no—
546C22×C3—42no—
547C22×C3—41no—
548C22×C3—42no—
549C22×C3—42no—
550C22×C3—42no—
551C22×C3—42no—
552C22×C3—41no—
553C22×C3—42no—
554C22×C3—81no—
555C22×C3—81no—
556C22×C3—81no—
557C22×C3—81no—
558C22×C3#380414yes—
559C22×C3#380512yes—
560C22×C3C22×D3212yes—
561C22×C3C22×D3211yescharacteristic
562C3×C4—22no—
563C3×C4—22no—
564C3×C4—21no—
565C3×C4—42no—
566C3×C4—42no—
567C3×C4—92no—
568C3×C4—182no—
569C3×C4—181no—
570C3×C4—182no—
571C3×C4—181no—
572C3×C4—181no—
573C3×C4—182no—
574C3×C4—182no—
575Dic3—34no—
576Dic3—62no—
577Dic3—62no—
578Dic3—62no—
579Dic3—62no—
580Dic3—62no—
581Dic3—62no—
582Dic3—62no—
583Dic3—62no—
584Dic3—62no—
585Dic3—62no—
586Dic3—62no—
587Dic3—62no—
588Dic3—121no—
589Dic3—121no—
590Dic3—122no—
591Dic3—122no—
592Dic3—121no—
593Dic3—122no—
594Dic3—122no—
595C32—22no—
596C32—21no—
597C32—41no—
598C32#554111yescharacteristic, complemented
599C32#565012yescomplemented
600C23C2×#111311yescharacteristic
601C23C2×#111311yescharacteristic
602Q8—542no—
603D4—22no—
604C2×C4—91no—
605C2×C4—91no—
606C2×C4—91no—
607C2×C4—92no—
608C2×C4—92no—
609C2×C4—92no—
610C2×C4—91no—
611C2×C4—92no—
612C2×C4—182no—
613C2×C4—182no—
614C2×C4—184no—
615C2×C4C2×#111311yescharacteristic
616C2×C3—24no—
617C2×C3—22no—
618C2×C3—21no—
619C2×C3—22no—
620C2×C3—21no—
621C2×C3—22no—
622C2×C3—22no—
623C2×C3—21no—
624C2×C3—42no—
625C2×C3—44no—
626C2×C3—42no—
627C2×C3—42no—
628C2×C3—41no—
629C2×C3—41no—
630C2×C3—41no—
631C2×C3—41no—
632C2×C3—82no—
633C2×C3—82no—
634C2×C3—82no—
635C2×C3#6430912yes—
636C2×C3#6430911yescharacteristic
637C2×C3D3×#23712yes—
638C2×C3#6431211yescharacteristic
639C2×C3C2×#380412yes—
640C2×C3C2×#380511yescharacteristic
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22#8904612yes—
647C22C22×#111311yescharacteristic
648C4—21no—
649C4—92no—
650C4—181no—
651C4—182no—
652C4—181no—
653C4—181no—
654C4—182no—
655C4—182no—
656C4—182no—
657C3—21no—
658C3—22no—
659C3—41no—
660C3#1059511712yescomplemented
661C3#1059512011yescharacteristic, complemented
662C2—22no—
663C2—41no—
664C2#1172462011yescharacteristic
665C2#1172462211yescharacteristic
666C2C2×#8904611yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
Fitting, prime radicalC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#552
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
3. C2×#88815C2semidirect
4. #11723721C2semidirect
5. #11723721C2semidirect
6. #11723716C2semidirect
7. #11723715C2semidirect
8. #11723715C2semidirect
9. #11723679C2semidirect
10. #11723669C2semidirect
11. #11723665C2semidirect
418. C33#552semidirect
598. C32#5541semidirect
599. C32#5650semidirect
660. C3#10595117semidirect
661. C3#10595120semidirect

Maximal subgroups

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

subgroupindex
C3×#641442
C3×D4×C3⋊Dic32
C2×#888152
#117237212
#117237162
#117237152
#117236792
#117236692
#117236652
#105951203
#105951173