Finite Group Databasebeta

← #13178203

Subgroups — #13178203

Order 1728.

1104 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64090C211yesmaximal, characteristic
2C3×#64088C212yesmaximal, complemented
3#11723724C212yesmaximal
4#11723721C211yesmaximal, characteristic, complemented
5#11723715C211yesmaximal, characteristic, complemented
6C4×#6296C211yesmaximal, characteristic, complemented
7#11723689C211yesmaximal, characteristic, complemented
8#11723683C212yesmaximal, complemented
9#11723679C211yesmaximal, characteristic, complemented
10#11723671C211yesmaximal, characteristic, complemented
11#11723669C212yesmaximal, complemented
12#10594155—31nomaximal
13#10594095—32nomaximal
14C2×C3×#638C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×#6296C2211yescharacteristic
18C2×#6294C2212yes—
19C33×C22⋊C4C2211yescharacteristic
20C3×#3757C2212yes—
21C3×#3757C2212yes—
22C3×#3752C2211yescharacteristic
23C3×C4⋊(C3⋊Dic3)C2212yes—
24C3×#3750C2212yes—
25C3×#3750C2212yes—
26C3×#3750C2211yescharacteristic
27C3×C4×C3⋊Dic3C2212yes—
28C3×C4×C3⋊Dic3C2211yescharacteristic
29#88815C2212yes—
30#88815C2212yes—
31#88815C2212yes—
32#88815C2211yescharacteristic
33#88815C2211yescharacteristic
34#88815C2211yescharacteristic
35#88814C2212yes—
36#88813C2212yes—
37#64090D311yescharacteristic
38#64088D312yescomplemented
39C3×#67⋊C4—32no—
40C3×#67⋊C4—61no—
41C3×#879—32no—
42C3×#879—32no—
43C3×#879—62no—
44#63982—31no—
45#63975—32no—
46#63969—32no—
47#63969—32no—
48#63964—32no—
49#63957—31no—
50#63955—32no—
51C4×#627—32no—
52C4×#627—31no—
53#63900—31no—
54D3×C4⋊Dic3—32no—
55#63878—32no—
56#63877—32no—
57#63870—31no—
58#63867—32no—
59#63865—31no—
60#63851—32no—
61#63847—32no—
62#63846—32no—
63#63845—32no—
64#63836—32no—
65C23×C33C2311yescharacteristic
66C22×C3×C3⋊D3C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C3×#638—22no—
70C2×C3×C3⋊Dic3—22no—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2312yes—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C3×C4×C3⋊D3—22no—
79#6296—22no—
80#6294—24no—
81#5708—92no—
82#5541—91no—
83C2×#638C2×D311yescharacteristic
84C22×C3⋊Dic3C2×D311yescharacteristic
85C2×C4×C3⋊D3C2×D311yescharacteristic
86C2×C3×#67—32no—
87C2×C3×#67—61no—
88C22×C3×Dic3—32no—
89C22×C3×Dic3—61no—
90C2×C3×C4×D3—32no—
91C2×C3×C4×D3—61no—
92C2×#627—32no—
93C2×#627—31no—
94C2×#624—32no—
95C2×D3×Dic3—32no—
96C2×D3×Dic3—32no—
97C32×C22⋊C4—22no—
98C32×C22⋊C4—21no—
99C32×C22⋊C4—41no—
100C32×C22⋊C4C2×D311yescharacteristic
101C32×C22⋊C4C2×D312yes—
102#3757C2×D312yes—
103#3757C2×D312yes—
104#3752C2×D311yescharacteristic
105C4⋊(C3⋊Dic3)C2×D312yes—
106#3750C2×D312yes—
107#3750C2×D312yes—
108#3750C2×D311yescharacteristic
109C4×C3⋊Dic3C2×D312yes—
110C4×C3⋊Dic3C2×D311yescharacteristic
111C3×#217—32no—
112C3×#217—32no—
113C3×#217—32no—
114C3×#217—32no—
115C3×#217—62no—
116C3×#217—62no—
117C3×(C2×D3)⋊C4—32no—
118C3×(C2×D3)⋊C4—61no—
119C3×C4⋊Dic3—32no—
120C3×C4⋊Dic3—32no—
121C3×C4⋊Dic3—62no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—62no—
128C3×Dic3⋊C4—62no—
129C3×Dic3⋊C4—61no—
130C3×C4×Dic3—32no—
131C3×C4×Dic3—32no—
132C3×C4×Dic3—32no—
133C3×C4×Dic3—62no—
134C3×C4×Dic3—61no—
135#3724—32no—
136#3724—32no—
137#3724—32no—
138#3724—31no—
139#3724—31no—
140#3724—31no—
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—
150#3722—32no—
151(C2×D3)⋊Dic3—32no—
152(C2×D3)⋊Dic3—32no—
153Dic32—32no—
154Dic32—32no—
155Dic32—32no—
156C22×C33—21no—
157C22×C33—21no—
158C22×C33—21no—
159C22×C33C2411yescharacteristic
160C2×C3×C3⋊D3—21no—
161C2×C3×C3⋊D3—21no—
162C2×C3×C3⋊D3#4112yes—
163C33×C4—21no—
164C33×C4—21no—
165C3×C3⋊Dic3—21no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3—22no—
168C3×C3⋊Dic3—22no—
169C3×C3⋊Dic3—21no—
170C3×C3⋊Dic3—22no—
171C3×C3⋊Dic3#4112yes—
172C3×#126—92no—
173C3×C4×D4—91no—
174#935—92no—
175#926—91no—
176C2×C4⋊Dic3—91no—
177#924—91no—
178#924—91no—
179#896—92no—
180#892—92no—
181D3×C4⋊C4—92no—
182#890—92no—
183#889—92no—
184#889—92no—
185#889—92no—
186#883—92no—
187#883—92no—
188#67⋊C4—32no—
189#67⋊C4—61no—
190#879—32no—
191#879—32no—
192#879—62no—
193#878—92no—
194#877—92no—
195#877—92no—
196#872—92no—
197#870—91no—
198C4×Dic6—92no—
199C4×Dic6—91no—
200C23×C32—21no—
201C23×C32—22no—
202C23×C32—41no—
203C23×C32C22×D311yescharacteristic
204C23×C32C22×D312yes—
205C22×C3⋊D3C22×D311yescharacteristic
206C22×C3×D3—32no—
207C22×C3×D3—61no—
208C2×C32×C4—22no—
209C2×C32×C4—21no—
210C2×C32×C4—22no—
211C2×C32×C4—21no—
212C2×C32×C4—41no—
213C2×C32×C4—41no—
214C2×C32×C4C22×D311yescharacteristic
215C2×C32×C4C22×D312yes—
216C2×C32×C4C22×D311yescharacteristic
217C2×C32×C4C22×D312yes—
218#638—22no—
219C2×C3⋊Dic3—22no—
220C2×C3⋊Dic3C22×D311yescharacteristic
221C2×C3⋊Dic3C22×D312yes—
222C2×C3⋊Dic3C22×D312yes—
223C2×C3⋊Dic3C22×D312yes—
224C2×C3⋊Dic3C22×D311yescharacteristic
225C2×C3⋊Dic3C22×D312yes—
226C2×C3⋊Dic3C22×D311yescharacteristic
227C4×C3⋊D3—22no—
228C3×#67—64no—
229C3×#67—64no—
230C2×C3×Dic3—32no—
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—61no—
242C2×C3×Dic3—61no—
243C2×C3×Dic3—62no—
244C2×C3×Dic3—62no—
245C2×C3×Dic3—62no—
246C2×C3×Dic3—62no—
247C2×C3×Dic3—61no—
248C2×C3×Dic3—64no—
249C2×C3×Dic3—122no—
250C3×C4×D3—64no—
251C3×C4×D3—122no—
252#627—64no—
253#627—62no—
254#624—64no—
255D3×Dic3—64no—
256D3×Dic3—64no—
257#552—271noSylow
258C2×C33—41no—
259C2×C33C2×#4111yescharacteristic
260C2×C33C2×#4111yescharacteristic
261C2×C332−1+411yescharacteristic
262C3×C3⋊D3—22no—
263C2×C3×D4—91no—
264C22×C3×C4—91no—
265C22×C3×C4—91no—
266C2×#67—32no—
267C2×#67—61no—
268C22×Dic3—32no—
269C22×Dic3—61no—
270C22×Dic3—92no—
271C2×C4×D3—32no—
272C2×C4×D3—61no—
273C2×C4×D3—92no—
274C2×C4×D3—92no—
275C2×Dic6—92no—
276C2×Dic6—91no—
277C3×C4⋊C4—91no—
278C3×C4⋊C4—92no—
279C3×C4⋊C4—92no—
280C3×C4⋊C4—92no—
281C3×C22⋊C4—22no—
282C3×C22⋊C4—21no—
283C3×C22⋊C4—41no—
284C3×C22⋊C4—92no—
285C3×C22⋊C4—92no—
286C3×C22⋊C4—91no—
287C3×C22⋊C4D3211yescharacteristic
288C3×C22⋊C4D3212yes—
289C3×C42—91no—
290C3×C42—92no—
291#217—32no—
292#217—32no—
293#217—32no—
294#217—32no—
295#217—62no—
296#217—62no—
297#217—92no—
298(C2×D3)⋊C4—32no—
299(C2×D3)⋊C4—61no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—92no—
302C4⋊Dic3—32no—
303C4⋊Dic3—32no—
304C4⋊Dic3—62no—
305C4⋊Dic3—92no—
306C4⋊Dic3—91no—
307C4⋊Dic3—91no—
308C4⋊Dic3—91no—
309C4⋊Dic3—92no—
310C4⋊Dic3—92no—
311Dic3⋊C4—32no—
312Dic3⋊C4—32no—
313Dic3⋊C4—32no—
314Dic3⋊C4—32no—
315Dic3⋊C4—32no—
316Dic3⋊C4—62no—
317Dic3⋊C4—62no—
318Dic3⋊C4—61no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321Dic3⋊C4—92no—
322Dic3⋊C4—92no—
323Dic3⋊C4—92no—
324Dic3⋊C4—92no—
325Dic3⋊C4—92no—
326Dic3⋊C4—92no—
327Dic3⋊C4—92no—
328C4×Dic3—32no—
329C4×Dic3—32no—
330C4×Dic3—32no—
331C4×Dic3—62no—
332C4×Dic3—61no—
333C4×Dic3—92no—
334C4×Dic3—92no—
335C4×Dic3—92no—
336C22×C32—21no—
337C22×C32—22no—
338C22×C32—21no—
339C22×C32—22no—
340C22×C32—22no—
341C22×C32—21no—
342C22×C32—22no—
343C22×C32—22no—
344C22×C32—21no—
345C22×C32—41no—
346C22×C32—42no—
347C22×C32—42no—
348C22×C32—41no—
349C22×C32—41no—
350C22×C32—42no—
351C22×C32—42no—
352C22×C32—81no—
353C22×C32—81no—
354C22×C32C23×D311yescharacteristic
355C22×C32C23×D312yes—
356C2×C3⋊D3—21no—
357C2×C3⋊D3—21no—
358C2×C3⋊D3#23512yes—
359C2×C3×D3—34no—
360C2×C3×D3—62no—
361C2×C3×D3—62no—
362C2×C3×D3—62no—
363C2×C3×D3—121no—
364C2×C3×D3—121no—
365C32×C4—21no—
366C32×C4—22no—
367C32×C4—22no—
368C32×C4—21no—
369C32×C4—41no—
370C32×C4—42no—
371C32×C4—41no—
372C32×C4—42no—
373C32×C4—81no—
374C32×C4—81no—
375C3⋊Dic3—21no—
376C3⋊Dic3—21no—
377C3⋊Dic3—22no—
378C3⋊Dic3—22no—
379C3⋊Dic3—22no—
380C3⋊Dic3—22no—
381C3⋊Dic3#23712yes—
382C3×Dic3—34no—
383C3×Dic3—62no—
384C3×Dic3—62no—
385C3×Dic3—62no—
386C3×Dic3—62no—
387C3×Dic3—62no—
388C3×Dic3—62no—
389C3×Dic3—62no—
390C3×Dic3—62no—
391C3×Dic3—62no—
392C3×Dic3—62no—
393C3×Dic3—62no—
394C3×Dic3—62no—
395C3×Dic3—122no—
396C3×Dic3—122no—
397C3×Dic3—122no—
398C3×Dic3—122no—
399C3×Dic3—121no—
400#126—92no—
401#125—271no—
402#125—272no—
403#123—272no—
404C22⋊Q8—271no—
405C22⋊Q8—271no—
406C4×Q8—271no—
407C4×D4—91no—
408#117—272no—
409#117—271no—
410C2×C4⋊C4—271no—
411C33#55211yescharacteristic, complemented, Sylow
412C23×C3—21no—
413C23×C3—22no—
414C23×C3—41no—
415C23×C3—91no—
416C23×C3C2×D3211yescharacteristic
417C23×C3C2×D3212yes—
418C22×D3—32no—
419C22×D3—61no—
420C3×D4—182no—
421C2×C3×C4—22no—
422C2×C3×C4—21no—
423C2×C3×C4—22no—
424C2×C3×C4—21no—
425C2×C3×C4—41no—
426C2×C3×C4—41no—
427C2×C3×C4—91no—
428C2×C3×C4—91no—
429C2×C3×C4—92no—
430C2×C3×C4—92no—
431C2×C3×C4—91no—
432C2×C3×C4—92no—
433C2×C3×C4—92no—
434C2×C3×C4—182no—
435C2×C3×C4—182no—
436C2×C3×C4C2×D3211yescharacteristic
437C2×C3×C4C2×D3212yes—
438C2×C3×C4C2×D3211yescharacteristic
439C2×C3×C4C2×D3212yes—
440#67—64no—
441#67—64no—
442C2×Dic3—32no—
443C2×Dic3—32no—
444C2×Dic3—32no—
445C2×Dic3—32no—
446C2×Dic3—32no—
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—62no—
458C2×Dic3—62no—
459C2×Dic3—61no—
460C2×Dic3—64no—
461C2×Dic3—122no—
462C2×Dic3—184no—
463C4×D3—64no—
464C4×D3—122no—
465C4×D3—184no—
466C4×D3—184no—
467Dic6—184no—
468Dic6—182no—
469C2×C32—22no—
470C2×C32—21no—
471C2×C32—22no—
472C2×C32—21no—
473C2×C32—22no—
474C2×C32—21no—
475C2×C32—41no—
476C2×C32—41no—
477C2×C32—41no—
478C2×C32—41no—
479C2×C32—42no—
480C2×C32—42no—
481C2×C32—82no—
482C2×C32—82no—
483C2×C32C2×#23511yescharacteristic
484C2×C32C2×#23711yescharacteristic
485C2×C32D3×#4112yes—
486C2×C32D3×#4112yes—
487C2×C32#101012yes—
488C2×C32#101011yescharacteristic
489C3⋊D3—22no—
490C3×D3—64no—
491C3×D3—122no—
492C2×Q8—271no—
493C2×D4—91no—
494C22×C4—91no—
495C22×C4—91no—
496C22×C4—272no—
497C4⋊C4—92no—
498C4⋊C4—92no—
499C4⋊C4—91no—
500C4⋊C4—92no—
501C4⋊C4—271no—
502C4⋊C4—271no—
503C4⋊C4—271no—
504C4⋊C4—272no—
505C4⋊C4—272no—
506C4⋊C4—272no—
507C22⋊C4—92no—
508C22⋊C4—92no—
509C22⋊C4—91no—
510C22⋊C4—272no—
511C22⋊C4#111311yescharacteristic
512C42—92no—
513C42—91no—
514C42—272no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—21no—
518C22×C3—22no—
519C22×C3—21no—
520C22×C3—22no—
521C22×C3—21no—
522C22×C3—22no—
523C22×C3—21no—
524C22×C3—41no—
525C22×C3—41no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—41no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—92no—
534C22×C3—181no—
535C22×C3—181no—
536C22×C3C22×D3212yes—
537C22×C3C22×D3211yescharacteristic
538C2×D3—34no—
539C2×D3—62no—
540C2×D3—62no—
541C2×D3—62no—
542C2×D3—121no—
543C2×D3—121no—
544C3×C4—22no—
545C3×C4—21no—
546C3×C4—22no—
547C3×C4—21no—
548C3×C4—41no—
549C3×C4—42no—
550C3×C4—41no—
551C3×C4—42no—
552C3×C4—81no—
553C3×C4—81no—
554C3×C4—92no—
555C3×C4—182no—
556C3×C4—182no—
557C3×C4—181no—
558C3×C4—182no—
559C3×C4—182no—
560C3×C4—181no—
561Dic3—34no—
562Dic3—62no—
563Dic3—62no—
564Dic3—62no—
565Dic3—62no—
566Dic3—62no—
567Dic3—62no—
568Dic3—62no—
569Dic3—62no—
570Dic3—62no—
571Dic3—62no—
572Dic3—62no—
573Dic3—62no—
574Dic3—121no—
575Dic3—122no—
576Dic3—122no—
577Dic3—122no—
578Dic3—122no—
579C32—22no—
580C32—21no—
581C32—41no—
582C32#554111yescharacteristic, complemented
583C32#570812yescomplemented
584C23—91no—
585C23C2×#111311yescharacteristic
586Q8—542no—
587D4—182no—
588C2×C4—91no—
589C2×C4—92no—
590C2×C4—92no—
591C2×C4—91no—
592C2×C4—91no—
593C2×C4—92no—
594C2×C4—92no—
595C2×C4—182no—
596C2×C4—182no—
597C2×C4—544no—
598C2×C4C2×#111311yescharacteristic
599C2×C4C2×#111311yescharacteristic
600C2×C3—21no—
601C2×C3—22no—
602C2×C3—22no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—22no—
606C2×C3—42no—
607C2×C3—42no—
608C2×C3—41no—
609C2×C3—41no—
610C2×C3—41no—
611C2×C3—41no—
612C2×C3—82no—
613C2×C3—82no—
614C2×C3—182no—
615C2×C3D3×#23512yes—
616C2×C3#6430611yescharacteristic
617C2×C3#6430912yes—
618C2×C3#6430911yescharacteristic
619C2×C3D3×#23712yes—
620C2×C3#6431211yescharacteristic
621D3—64no—
622D3—122no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—92no—
627C22—181no—
628C22—181no—
629C22C22×#111311yescharacteristic
630C4—21no—
631C4—21no—
632C4—92no—
633C4—182no—
634C4—182no—
635C4—181no—
636C4—182no—
637C4—181no—
638C4—182no—
639C3—21no—
640C3—22no—
641C3—41no—
642C3#1059409512yescomplemented
643C3#1059415511yescharacteristic, complemented
644C2—41no—
645C2—182no—
646C2#1172461811yescharacteristic
647C2#1172462011yescharacteristic
648C2#1172462211yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×C3×#638C22
hypercenterC22⋊C4#1113
hypocenterC33#552
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64088C2semidirect
4. #11723721C2semidirect
5. #11723715C2semidirect
6. C4×#6296C2semidirect
7. #11723689C2semidirect
8. #11723683C2semidirect
9. #11723679C2semidirect
10. #11723671C2semidirect
11. #11723669C2semidirect
38. #64088D3semidirect
411. C33#552semidirect
582. C32#5541semidirect
583. C32#5708semidirect
642. C3#10594095semidirect
643. C3#10594155semidirect

Maximal subgroups

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

subgroupindex
C3×#640902
C3×#640882
#117237242
#117237212
#117237152
C4×#62962
#117236892
#117236832
#117236792
#117236712
#117236692
#105941553
#105940953