Finite Group Databasebeta

← #13168714

Subgroups — #13168714

Order 1728.

1192 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723679C211yesmaximal, characteristic, complemented
2D3×#3750C212yesmaximal, complemented
3#11723390C211yesmaximal, characteristic, complemented
4C3×C4×#625C211yesmaximal, characteristic
5C3×#63851C212yesmaximal, complemented
6#11723032C211yesmaximal, characteristic, complemented
7#11723015C211yesmaximal, characteristic, complemented
8#11722980C212yesmaximal, complemented
9#11722976C212yesmaximal, complemented
10#11722925C211yesmaximal, characteristic
11#11722925C211yesmaximal, characteristic
12#10594959—32nomaximal
13#10594534—31nomaximal
14C2×C32×C4×D3C2212yes—
15C2×#6293C2211yescharacteristic
16C2×D3×C3⋊Dic3C2212yes—
17C2×D3×C3⋊Dic3C2212yes—
18C2×C3×#627C2212yes—
19C2×C3×#625C2211yescharacteristic
20#88857C2211yescharacteristic
21C3×#3750C2212yes—
22C3×C4×C3⋊Dic3C2211yescharacteristic
23C32×Dic3⋊C4C2211yescharacteristic
24#88815C2211yescharacteristic
25#88815C2211yescharacteristic
26#88815C2212yes—
27#88807C2212yes—
28#88807C2211yescharacteristic
29#88806C2212yes—
30Dic3×C3⋊Dic3C2211yescharacteristic
31Dic3×C3⋊Dic3C2211yescharacteristic
32C3×#3724C2211yescharacteristic
33C3×Dic3⋊Dic3C2212yes—
34C3×(C2×D3)⋊Dic3C2212yes—
35C3×(C2×D3)⋊Dic3C2212yes—
36C3×(C2×D3)⋊Dic3C2212yes—
37C2×#3750—32no—
38#64096—31no—
39C3×C4×#67—32no—
40C3×#924—32no—
41C3×#896—32no—
42#63982—32no—
43#63982—32no—
44#67⋊Dic3—32no—
45#67⋊Dic3—32no—
46#63902—31no—
47#63902—31no—
48C4×#625D311yescharacteristic
49#63899—32no—
50#63896—32no—
51#63887—32no—
52#63887—32no—
53D3×Dic3⋊C4—32no—
54D3×Dic3⋊C4—32no—
55D3×Dic3⋊C4—62no—
56#63873—31no—
57#63870—31no—
58#63870—31no—
59#63867—32no—
60#63851D312yescomplemented
61#63846—32no—
62#63839—32no—
63#63837—32no—
64#63837—61no—
65C22×C32×D3C2312yes—
66C2×C33×C4C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2311yescharacteristic
76C32×C4×D3—42no—
77#6293—22no—
78D3×C3⋊Dic3—24no—
79D3×C3⋊Dic3—42no—
80C3×#627—42no—
81C3×#625Q814yescomplemented
82#5636—92no—
83#5539—91no—
84C22×C32×C4—32no—
85C22×C3⋊Dic3—32no—
86C22×C3⋊Dic3—32no—
87C2×#634—31no—
88C2×C3×#67—32no—
89C2×C3×C4×D3—22no—
90C2×C3×C4×D3C2×D312yes—
91C2×C3×C4×D3C2×D312yes—
92C2×C3×Dic6—32no—
93C2×C3×Dic6—32no—
94C2×#627—32no—
95C2×#627—61no—
96C2×#627C2×D312yes—
97C2×#625C2×D311yescharacteristic
98C2×D3×Dic3—32no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102C2×D3×Dic3—62no—
103C2×D3×Dic3—62no—
104C32×C4⋊C4—31no—
105#3750—32no—
106#3750—31no—
107#3750—32no—
108#3750—31no—
109#3750—32no—
110#3750—61no—
111#3750—62no—
112#3750—121no—
113#3750C2×D312yes—
114C4×C3⋊Dic3—31no—
115C4×C3⋊Dic3—31no—
116C4×C3⋊Dic3C2×D311yescharacteristic
117C3×#217—32no—
118C3×#217—32no—
119C3×#217—32no—
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—32no—
122C3×(C2×D3)⋊C4—32no—
123C3×C4⋊Dic3—32no—
124C3×Dic3⋊C4—21no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—32no—
129C3×Dic3⋊C4—62no—
130C3×Dic3⋊C4C2×D312yes—
131C3×C4×Dic3—32no—
132C3×C4×Dic3—61no—
133#3724—32no—
134#3724—31no—
135#3724—31no—
136#3724—32no—
137#3724—32no—
138#3724—62no—
139#3724C2×D311yescharacteristic
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—32no—
145Dic3⋊Dic3—32no—
146Dic3⋊Dic3—32no—
147Dic3⋊Dic3—61no—
148Dic3⋊Dic3—62no—
149Dic3⋊Dic3C2×D312yes—
150(C2×D3)⋊Dic3C2×D312yes—
151(C2×D3)⋊Dic3C2×D312yes—
152(C2×D3)⋊Dic3C2×D312yes—
153Dic32—32no—
154Dic32—32no—
155Dic32—61no—
156Dic32—61no—
157C22×C33C2411yescharacteristic
158C2×C32×D3—22no—
159C2×C32×D3—22no—
160C2×C32×D3C2×Q814yes—
161C33×C4—21no—
162C3⋊(C3⋊Dic3)—21no—
163C3⋊(C3⋊Dic3)#4112yes—
164C3×C3⋊Dic3—22no—
165C3×C3⋊Dic3—21no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3C2×Q812yes—
168C32×Dic3—21no—
169C32×Dic3—21no—
170C32×Dic3—22no—
171C3×C22⋊Q8—92no—
172C3×C4×D4—91no—
173#946—92no—
174#944—92no—
175D4×Dic3—91no—
176D4×Dic3—91no—
177C4×#67—32no—
178#924—32no—
179C2×Dic3⋊C4—92no—
180C2×Dic3⋊C4—92no—
181C2×Dic3⋊C4—182no—
182#896—32no—
183#896—92no—
184D3×C4⋊C4—92no—
185#890—91no—
186#889—92no—
187#889—92no—
188#888—91no—
189#887—92no—
190#887—181no—
191#67⋊C4—92no—
192#67⋊C4—92no—
193#878—92no—
194#878—92no—
195#878—92no—
196#878—92no—
197#870—91no—
198C4×Dic6—91no—
199C23×C32—32no—
200C22×C3×D3—22no—
201C22×C3×D3C22×D312yes—
202C22×C3×D3C22×D312yes—
203C2×C32×C4—21no—
204C2×C32×C4—22no—
205C2×C32×C4—31no—
206C2×C32×C4—32no—
207C2×C32×C4—31no—
208C2×C32×C4—41no—
209C2×C32×C4—122no—
210C2×C32×C4C22×D312yes—
211C2×C32×C4C22×D311yescharacteristic
212C2×C3⋊Dic3—32no—
213C2×C3⋊Dic3—31no—
214C2×C3⋊Dic3—32no—
215C2×C3⋊Dic3—31no—
216C2×C3⋊Dic3—61no—
217C2×C3⋊Dic3—62no—
218C2×C3⋊Dic3—61no—
219C2×C3⋊Dic3—62no—
220C2×C3⋊Dic3—64no—
221C2×C3⋊Dic3—121no—
222C2×C3⋊Dic3—121no—
223C2×C3⋊Dic3—122no—
224C2×C3⋊Dic3C22×D311yescharacteristic
225C2×C3⋊Dic3C22×D312yes—
226C2×C3⋊Dic3C22×D312yes—
227C2×C3⋊Dic3C22×D311yescharacteristic
228#634—62no—
229C3×#67—38no—
230C2×C3×Dic3—21no—
231C2×C3×Dic3—21no—
232C2×C3×Dic3—22no—
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—61no—
240C2×C3×Dic3—61no—
241C2×C3×Dic3—62no—
242C2×C3×Dic3—62no—
243C2×C3×Dic3C22×D312yes—
244C2×C3×Dic3C22×D312yes—
245C2×C3×Dic3C22×D312yes—
246C2×C3×Dic3C22×D312yes—
247C3×C4×D3—42no—
248C3×C4×D3—42no—
249C3×C4×D3—44no—
250C3×Dic6—122no—
251C3×Dic6—122no—
252#627—42no—
253#627—64no—
254#627—64no—
255#625Dic614yescomplemented
256D3×Dic3—64no—
257D3×Dic3—64no—
258D3×Dic3—68no—
259D3×Dic3—122no—
260D3×Dic3—122no—
261D3×Dic3—124no—
262#554—271noSylow
263C2×C33C22×Q811yescharacteristic
264C2×C33C2×#4111yescharacteristic
265C2×C33H(5,𝔽2)11yescharacteristic
266C32×D3—24no—
267C2×C3×Q8—92no—
268C2×C3×D4—91no—
269C22×C3×C4—32no—
270C22×C3×C4—32no—
271C22×C3×C4—62no—
272C2×#67—32no—
273C22×Dic3—92no—
274C22×Dic3—92no—
275C22×Dic3—92no—
276C22×Dic3—92no—
277C22×Dic3—182no—
278C22×Dic3—182no—
279C2×C4×D3—92no—
280C2×C4×D3—92no—
281C2×C4×D3D3212yes—
282C2×Dic6—32no—
283C2×Dic6—32no—
284C2×Dic6—92no—
285C2×Dic6—91no—
286C2×Dic6—181no—
287C3×C4⋊C4—32no—
288C3×C4⋊C4—61no—
289C3×C4⋊C4—91no—
290C3×C4⋊C4—92no—
291C3×C4⋊C4—92no—
292C3×C22⋊C4—92no—
293C3×C22⋊C4—92no—
294C3×C22⋊C4—92no—
295C3×C42—91no—
296#217—32no—
297#217—32no—
298#217—32no—
299(C2×D3)⋊C4—32no—
300(C2×D3)⋊C4—32no—
301(C2×D3)⋊C4—32no—
302C4⋊Dic3—32no—
303C4⋊Dic3—91no—
304C4⋊Dic3—91no—
305C4⋊Dic3—92no—
306C4⋊Dic3—91no—
307C4⋊Dic3—92no—
308Dic3⋊C4—32no—
309Dic3⋊C4—32no—
310Dic3⋊C4—32no—
311Dic3⋊C4—32no—
312Dic3⋊C4—62no—
313Dic3⋊C4—92no—
314Dic3⋊C4—92no—
315Dic3⋊C4—92no—
316Dic3⋊C4—92no—
317Dic3⋊C4—92no—
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321Dic3⋊C4—92no—
322Dic3⋊C4—91no—
323Dic3⋊C4—92no—
324Dic3⋊C4—92no—
325Dic3⋊C4—181no—
326Dic3⋊C4—182no—
327Dic3⋊C4—182no—
328Dic3⋊C4—181no—
329Dic3⋊C4—182no—
330Dic3⋊C4—361no—
331Dic3⋊C4D3211yescharacteristic
332C4×Dic3—32no—
333C4×Dic3—61no—
334C4×Dic3—92no—
335C4×Dic3—92no—
336C4×Dic3—91no—
337C4×Dic3—91no—
338C4×Dic3—181no—
339C4×Dic3—181no—
340C22×C32—21no—
341C22×C32—22no—
342C22×C32—34no—
343C22×C32—41no—
344C22×C32—62no—
345C22×C32—62no—
346C22×C32C23×D311yescharacteristic
347C22×C32C23×D312yes—
348C2×C3×D3—24no—
349C2×C3×D3—24no—
350C2×C3×D3—22no—
351C2×C3×D3—22no—
352C2×C3×D3—22no—
353C2×C3×D3—22no—
354C2×C3×D3—24no—
355C2×C3×D3C2×Dic614yes—
356C2×C3×D3D3×Q814yes—
357C32×C4—21no—
358C32×C4—22no—
359C32×C4—42no—
360C32×C4—41no—
361C32×C4—61no—
362C32×C4—61no—
363C32×C4—62no—
364C32×C4—81no—
365C3⋊Dic3—21no—
366C3⋊Dic3—22no—
367C3⋊Dic3—22no—
368C3⋊Dic3—32no—
369C3⋊Dic3—34no—
370C3⋊Dic3—64no—
371C3⋊Dic3—62no—
372C3⋊Dic3—61no—
373C3⋊Dic3—62no—
374C3⋊Dic3—122no—
375C3⋊Dic3—122no—
376C3⋊Dic3—121no—
377C3⋊Dic3—241no—
378C3⋊Dic3C2×Dic612yes—
379C3×Dic3—22no—
380C3×Dic3—22no—
381C3×Dic3—22no—
382C3×Dic3—22no—
383C3×Dic3—22no—
384C3×Dic3—34no—
385C3×Dic3—42no—
386C3×Dic3—41no—
387C3×Dic3—62no—
388C3×Dic3—64no—
389C3×Dic3—62no—
390C3×Dic3—62no—
391C3×Dic3—62no—
392C3×Dic3—62no—
393C3×Dic3—62no—
394C3×Dic3—62no—
395C3×Dic3—122no—
396C4⋊Q8—271no—
397#125—271no—
398#125—271no—
399C22⋊Q8—92no—
400C22⋊Q8—272no—
401C22⋊Q8—272no—
402C4×Q8—271no—
403C4×D4—91no—
404C4×D4—271no—
405C4×D4—271no—
406C2×C4⋊C4—272no—
407C33#55411yescharacteristic, complemented, Sylow
408C23×C3—32no—
409C23×C3—32no—
410C23×C3—62no—
411C22×D3C2×D3212yes—
412C3×Q8—362no—
413C3×D4—94no—
414C2×C3×C4—21no—
415C2×C3×C4—22no—
416C2×C3×C4—32no—
417C2×C3×C4—32no—
418C2×C3×C4—32no—
419C2×C3×C4—32no—
420C2×C3×C4—41no—
421C2×C3×C4—61no—
422C2×C3×C4—61no—
423C2×C3×C4—62no—
424C2×C3×C4—91no—
425C2×C3×C4—91no—
426C2×C3×C4—92no—
427C2×C3×C4—92no—
428C2×C3×C4—122no—
429C2×C3×C4—122no—
430C2×C3×C4—124no—
431C2×C3×C4C2×D3212yes—
432C2×C3×C4C2×D3211yescharacteristic
433#67—38no—
434C2×Dic3—32no—
435C2×Dic3—32no—
436C2×Dic3—32no—
437C2×Dic3—32no—
438C2×Dic3—32no—
439C2×Dic3—32no—
440C2×Dic3—61no—
441C2×Dic3—61no—
442C2×Dic3—62no—
443C2×Dic3—62no—
444C2×Dic3—92no—
445C2×Dic3—91no—
446C2×Dic3—92no—
447C2×Dic3—91no—
448C2×Dic3—181no—
449C2×Dic3—182no—
450C2×Dic3—181no—
451C2×Dic3—182no—
452C2×Dic3—184no—
453C2×Dic3—184no—
454C2×Dic3—188no—
455C2×Dic3—361no—
456C2×Dic3—361no—
457C2×Dic3—362no—
458C2×Dic3—362no—
459C2×Dic3—364no—
460C2×Dic3C2×D3211yescharacteristic
461C2×Dic3C2×D3211yescharacteristic
462C2×Dic3C2×D3212yes—
463C4×D3—42no—
464C4×D3—184no—
465C4×D3—362no—
466Dic6—122no—
467Dic6—122no—
468Dic6—182no—
469Dic6—184no—
470Dic6—184no—
471C2×C32—22no—
472C2×C32—21no—
473C2×C32—22no—
474C2×C32—21no—
475C2×C32—21no—
476C2×C32—22no—
477C2×C32—41no—
478C2×C32—41no—
479C2×C32—41no—
480C2×C32—64no—
481C2×C32C22×Dic611yescharacteristic
482C2×C32#100412yes—
483C2×C32#100411yescharacteristic
484C2×C32C2×D3×Q812yes—
485C2×C32D3×#4112yes—
486C2×C32D3×#4111yescharacteristic
487C3×D3—28no—
488C3×D3—24no—
489C3×D3—24no—
490C2×Q8—92no—
491C2×Q8—271no—
492C2×D4—91no—
493C22×C4—32no—
494C22×C4—272no—
495C22×C4—272no—
496C4⋊C4—31no—
497C4⋊C4—91no—
498C4⋊C4—92no—
499C4⋊C4—92no—
500C4⋊C4—272no—
501C4⋊C4—272no—
502C4⋊C4—271no—
503C4⋊C4—271no—
504C4⋊C4—271no—
505C4⋊C4—272no—
506C4⋊C4—271no—
507C22⋊C4—92no—
508C22⋊C4—92no—
509C22⋊C4—92no—
510C42—91no—
511C42—271no—
512C42—271no—
513C22×C3—21no—
514C22×C3—22no—
515C22×C3—34no—
516C22×C3—34no—
517C22×C3—41no—
518C22×C3—64no—
519C22×C3—64no—
520C22×C3—64no—
521C22×C3—62no—
522C22×C3—62no—
523C22×C3—62no—
524C22×C3—62no—
525C22×C3C22×D3212yes—
526C22×C3C22×D3211yescharacteristic
527C2×D3—22no—
528C2×D3—22no—
529C2×D3D3×Dic614yes—
530C3×C4—21no—
531C3×C4—22no—
532C3×C4—42no—
533C3×C4—41no—
534C3×C4—62no—
535C3×C4—62no—
536C3×C4—62no—
537C3×C4—62no—
538C3×C4—62no—
539C3×C4—81no—
540C3×C4—92no—
541C3×C4—122no—
542C3×C4—121no—
543C3×C4—181no—
544C3×C4—182no—
545C3×C4—182no—
546Dic3—21no—
547Dic3—21no—
548Dic3—22no—
549Dic3—34no—
550Dic3—62no—
551Dic3—62no—
552Dic3—64no—
553Dic3—62no—
554Dic3—62no—
555Dic3—62no—
556Dic3—62no—
557Dic3—62no—
558Dic3—94no—
559Dic3—92no—
560Dic3—122no—
561Dic3—182no—
562Dic3—184no—
563Dic3—181no—
564Dic3—182no—
565Dic3—362no—
566Dic3—362no—
567Dic3—361no—
568Dic3—721no—
569C32—22no—
570C32—21no—
571C32—41no—
572C32#553911yescharacteristic, complemented
573C32#563612yescomplemented
574C23—32no—
575Q8—362no—
576Q8—542no—
577D4—94no—
578C2×C4—31no—
579C2×C4—31no—
580C2×C4—32no—
581C2×C4—91no—
582C2×C4—91no—
583C2×C4—92no—
584C2×C4—92no—
585C2×C4—122no—
586C2×C4—271no—
587C2×C4—271no—
588C2×C4—544no—
589C2×C4—1082no—
590C2×C4D3311yescharacteristic
591C2×C3—21no—
592C2×C3—22no—
593C2×C3—21no—
594C2×C3—22no—
595C2×C3—22no—
596C2×C3—21no—
597C2×C3—41no—
598C2×C3—41no—
599C2×C3—41no—
600C2×C3—68no—
601C2×C3—64no—
602C2×C3—64no—
603C2×C3C2×D3×Dic612yes—
604C2×C3#6430612yes—
605C2×C3#6430611yescharacteristic
606C2×C3#6431311yescharacteristic
607C2×C3D32×Q811yescharacteristic
608C2×C3#6433012yes—
609D3—24no—
610C22—34no—
611C22—62no—
612C22—62no—
613C22C2×D3311yescharacteristic
614C4—21no—
615C4—61no—
616C4—62no—
617C4—61no—
618C4—92no—
619C4—182no—
620C4—181no—
621C4—182no—
622C4—272no—
623C4—541no—
624C3—22no—
625C3—21no—
626C3—41no—
627C3#1059453411yescharacteristic, complemented
628C3#1059495912yescomplemented
629C2—64no—
630C2D32×Dic611yescharacteristic
631C2#1172439811yescharacteristic
632C2#1172441411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×C3×#625C22
hypercenterC2×C4D33
hypocenterC33#554
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. D3×#3750C2semidirect
3. #11723390C2semidirect
5. C3×#63851C2semidirect
6. #11723032C2semidirect
7. #11723015C2semidirect
8. #11722980C2semidirect
9. #11722976C2semidirect
60. #63851D3semidirect
81. C3×#625Q8semidirect
255. #625Dic6semidirect
407. C33#554semidirect
572. C32#5539semidirect
573. C32#5636semidirect
627. C3#10594534semidirect
628. C3#10594959semidirect

Maximal subgroups

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

subgroupindex
#117236792
D3×#37502
#117233902
C3×C4×#6252
C3×#638512
#117230322
#117230152
#117229802
#117229762
#117229252
#105949593
#105945343