Finite Group Databasebeta

← #13168697

Subgroups — #13168697

Order 1728.

1156 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723679C211yesmaximal, characteristic, complemented
2#11723415C212yesmaximal, complemented
3#11723390C211yesmaximal, characteristic, complemented
4C3×C4×#625C211yesmaximal, characteristic
5C3×D3×Dic3⋊C4C212yesmaximal, complemented
6#11723037C211yesmaximal, characteristic, complemented
7#11723018C211yesmaximal, characteristic, complemented
8#11722975C212yesmaximal, complemented
9#11722971C212yesmaximal, complemented
10Dic3×#625C211yesmaximal, characteristic
11Dic3×#625C211yesmaximal, characteristic
12#10594951—32nomaximal
13#10594531—31nomaximal
14C2×C32×C4×D3C2212yes—
15C2×#6293C2211yescharacteristic
16C2×#6293C2212yes—
17C2×C3×#625C2211yescharacteristic
18C2×C3×D3×Dic3C2212yes—
19C2×C3×D3×Dic3C2212yes—
20#88857C2211yescharacteristic
21C3×#3750C2212yes—
22C3×C4×C3⋊Dic3C2211yescharacteristic
23C32×Dic3⋊C4C2211yescharacteristic
24#88815C2211yescharacteristic
25#88815C2211yescharacteristic
26#88815C2212yes—
27#88807C2211yescharacteristic
28Dic3⋊(C3⋊Dic3)C2212yes—
29(C2×D3)⋊(C3⋊Dic3)C2212yes—
30(C2×D3)⋊(C3⋊Dic3)C2212yes—
31Dic3×C3⋊Dic3C2211yescharacteristic
32Dic3×C3⋊Dic3C2211yescharacteristic
33C3×#3724C2211yescharacteristic
34C3×#3724C2212yes—
35C3×Dic3⋊Dic3C2212yes—
36C3×(C2×D3)⋊Dic3C2212yes—
37#64133—32no—
38#64096—31no—
39C3×C4×#67—32no—
40C2×C3×Dic3⋊C4—32no—
41C3×D3×C4⋊C4—32no—
42Dic3×#67—32no—
43Dic3×#67—32no—
44#63964—32no—
45#63964—32no—
46#63919—31no—
47C4×#625—31no—
48C4×#625—31no—
49C4×#625D311yescharacteristic
50#63896—32no—
51#63893—32no—
52D3×Dic3⋊C4D312yescomplemented
53#63870—31no—
54#63870—31no—
55#63867—32no—
56#63857—32no—
57#63851—32no—
58#63851—32no—
59#63851—32no—
60#63851—62no—
61#63845—32no—
62#63840—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—
78#6293—42no—
79C3×#625Q814yescomplemented
80C3×D3×Dic3—24no—
81C3×D3×Dic3—42no—
82#5539—91no—
83#5511—92no—
84C22×C32×C4—32no—
85C2×#634—31no—
86C2×#634—32no—
87C2×C3×#67—32no—
88C22×C3×Dic3—32no—
89C22×C3×Dic3—32no—
90C2×C3×C4×D3—22no—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—32no—
93C2×C3×C4×D3C2×D312yes—
94C2×C3×C4×D3C2×D312yes—
95C2×#627—32no—
96C2×#627—32no—
97C2×#627—32no—
98C2×#627—61no—
99C2×#627—62no—
100C2×#625C2×D311yescharacteristic
101C2×D3×Dic3C2×D312yes—
102C2×D3×Dic3C2×D312yes—
103C32×C4⋊C4—31no—
104#3757—32no—
105#3757—32no—
106C4⋊(C3⋊Dic3)—32no—
107#3750—32no—
108#3750—31no—
109#3750—31no—
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×(C2×D3)⋊C4—32no—
119C3×C4⋊Dic3—32no—
120C3×Dic3⋊C4—21no—
121C3×Dic3⋊C4—32no—
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⋊C4C2×D312yes—
129C3×C4×Dic3—32no—
130C3×C4×Dic3—61no—
131#3724—32no—
132#3724—31no—
133#3724—31no—
134#3724C2×D312yes—
135#3724C2×D311yescharacteristic
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—61no—
145Dic3⋊Dic3C2×D312yes—
146(C2×D3)⋊Dic3—32no—
147(C2×D3)⋊Dic3—32no—
148(C2×D3)⋊Dic3—32no—
149(C2×D3)⋊Dic3—32no—
150(C2×D3)⋊Dic3—62no—
151(C2×D3)⋊Dic3—62no—
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)—21no—
164C3×C3⋊Dic3—21no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3C2×Q812yes—
168C32×Dic3—22no—
169C32×Dic3—21no—
170C32×Dic3#4112yes—
171C3×C4×D4—91no—
172C2×C3×C4⋊C4—92no—
173D4×Dic3—91no—
174D4×Dic3—91no—
175C4×#67—32no—
176C4×#67—92no—
177C4×#67—92no—
178#924—92no—
179#924—92no—
180#924—92no—
181#924—92no—
182#924—182no—
183C2×Dic3⋊C4—32no—
184#897—92no—
185#896—92no—
186#896—92no—
187D3×C4⋊C4—32no—
188#890—91no—
189#889—92no—
190#889—92no—
191#888—91no—
192#888—92no—
193#887—92no—
194#887—181no—
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—32no—
214C2×C3⋊Dic3—31no—
215C2×C3⋊Dic3—31no—
216C2×C3⋊Dic3—61no—
217C2×C3⋊Dic3—61no—
218C2×C3⋊Dic3—62no—
219C2×C3⋊Dic3—62no—
220C2×C3⋊Dic3—121no—
221C2×C3⋊Dic3—121no—
222C2×C3⋊Dic3C22×D311yescharacteristic
223C2×C3⋊Dic3C22×D312yes—
224C2×C3⋊Dic3C22×D312yes—
225C2×C3⋊Dic3C22×D311yescharacteristic
226#634—62no—
227#634—122no—
228C3×#67—38no—
229C2×C3×Dic3—21no—
230C2×C3×Dic3—22no—
231C2×C3×Dic3—21no—
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—61no—
239C2×C3×Dic3—61no—
240C2×C3×Dic3—62no—
241C2×C3×Dic3—62no—
242C2×C3×Dic3—64no—
243C2×C3×Dic3—122no—
244C2×C3×Dic3C22×D312yes—
245C2×C3×Dic3C22×D312yes—
246C2×C3×Dic3C22×D312yes—
247C2×C3×Dic3C22×D312yes—
248C3×C4×D3—42no—
249C3×C4×D3—42no—
250C3×C4×D3—44no—
251C3×C4×D3—64no—
252C3×C4×D3—122no—
253#627—64no—
254#627—64no—
255#627—122no—
256#627—122no—
257#627—124no—
258#625Dic614yescomplemented
259D3×Dic3—24no—
260D3×Dic3—42no—
261#554—271noSylow
262C2×C33C22×Q811yescharacteristic
263C2×C33C2×#4111yescharacteristic
264C2×C33H(5,𝔽2)11yescharacteristic
265C32×D3—24no—
266C2×C3×D4—91no—
267C22×C3×C4—32no—
268C22×C3×C4—32no—
269C22×C3×C4—62no—
270C22×C3×C4—92no—
271C22×C3×C4—92no—
272C2×#67—32no—
273C22×Dic3—32no—
274C22×Dic3—32no—
275C2×C4×D3—32no—
276C2×C4×D3—32no—
277C2×C4×D3D3212yes—
278C2×Dic6—92no—
279C2×Dic6—91no—
280C2×Dic6—92no—
281C2×Dic6—92no—
282C2×Dic6—92no—
283C2×Dic6—181no—
284C2×Dic6—182no—
285C3×C4⋊C4—32no—
286C3×C4⋊C4—61no—
287C3×C4⋊C4—91no—
288C3×C4⋊C4—92no—
289C3×C4⋊C4—92no—
290C3×C4⋊C4—92no—
291C3×C22⋊C4—92no—
292C3×C42—91no—
293#217—32no—
294#217—92no—
295#217—92no—
296#217—92no—
297#217—92no—
298#217—182no—
299#217—182no—
300(C2×D3)⋊C4—32no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—92no—
303C4⋊Dic3—32no—
304C4⋊Dic3—91no—
305C4⋊Dic3—91no—
306C4⋊Dic3—92no—
307C4⋊Dic3—92no—
308C4⋊Dic3—92no—
309C4⋊Dic3—91no—
310C4⋊Dic3—182no—
311Dic3⋊C4—32no—
312Dic3⋊C4—32no—
313Dic3⋊C4—32no—
314Dic3⋊C4—32no—
315Dic3⋊C4—32no—
316Dic3⋊C4—32no—
317Dic3⋊C4—62no—
318Dic3⋊C4—92no—
319Dic3⋊C4—92no—
320Dic3⋊C4—92no—
321Dic3⋊C4—92no—
322Dic3⋊C4—92no—
323Dic3⋊C4—91no—
324Dic3⋊C4—92no—
325Dic3⋊C4—92no—
326Dic3⋊C4—181no—
327Dic3⋊C4—181no—
328Dic3⋊C4—182no—
329Dic3⋊C4—361no—
330Dic3⋊C4D3211yescharacteristic
331C4×Dic3—32no—
332C4×Dic3—61no—
333C4×Dic3—92no—
334C4×Dic3—92no—
335C4×Dic3—91no—
336C4×Dic3—91no—
337C4×Dic3—181no—
338C4×Dic3—181no—
339C22×C32—21no—
340C22×C32—22no—
341C22×C32—34no—
342C22×C32—41no—
343C22×C32—62no—
344C22×C32—62no—
345C22×C32C23×D311yescharacteristic
346C22×C32C23×D312yes—
347C2×C3×D3—22no—
348C2×C3×D3—22no—
349C2×C3×D3—22no—
350C2×C3×D3—22no—
351C2×C3×D3—24no—
352C2×C3×D3—42no—
353C2×C3×D3—42no—
354C2×C3×D3C2×Dic614yes—
355C2×C3×D3D3×Q814yes—
356C32×C4—21no—
357C32×C4—22no—
358C32×C4—32no—
359C32×C4—42no—
360C32×C4—41no—
361C32×C4—62no—
362C32×C4—61no—
363C32×C4—81no—
364C3⋊Dic3—21no—
365C3⋊Dic3—22no—
366C3⋊Dic3—22no—
367C3⋊Dic3—61no—
368C3⋊Dic3—62no—
369C3⋊Dic3—62no—
370C3⋊Dic3—61no—
371C3⋊Dic3—62no—
372C3⋊Dic3—122no—
373C3⋊Dic3—122no—
374C3⋊Dic3—122no—
375C3⋊Dic3—121no—
376C3⋊Dic3—241no—
377C3⋊Dic3C2×Dic612yes—
378C3×Dic3—24no—
379C3×Dic3—22no—
380C3×Dic3—22no—
381C3×Dic3—22no—
382C3×Dic3—22no—
383C3×Dic3—34no—
384C3×Dic3—41no—
385C3×Dic3—62no—
386C3×Dic3—62no—
387C3×Dic3—62no—
388C3×Dic3—62no—
389C3×Dic3—62no—
390C3×Dic3—62no—
391C3×Dic3—62no—
392C3×Dic3—122no—
393C3×Dic3—122no—
394C3×Dic3#23514yes—
395C4⋊Q8—271no—
396#125—271no—
397#125—271no—
398C22⋊Q8—272no—
399C22⋊Q8—272no—
400C22⋊Q8—272no—
401C4×Q8—271no—
402C4×D4—91no—
403C4×D4—271no—
404C4×D4—271no—
405C2×C4⋊C4—92no—
406C33#55411yescharacteristic, complemented, Sylow
407C23×C3—32no—
408C23×C3—32no—
409C23×C3—62no—
410C22×D3C2×D3212yes—
411C3×D4—94no—
412C2×C3×C4—21no—
413C2×C3×C4—22no—
414C2×C3×C4—32no—
415C2×C3×C4—32no—
416C2×C3×C4—32no—
417C2×C3×C4—32no—
418C2×C3×C4—41no—
419C2×C3×C4—61no—
420C2×C3×C4—62no—
421C2×C3×C4—61no—
422C2×C3×C4—91no—
423C2×C3×C4—92no—
424C2×C3×C4—91no—
425C2×C3×C4—92no—
426C2×C3×C4—122no—
427C2×C3×C4—122no—
428C2×C3×C4—124no—
429C2×C3×C4—184no—
430C2×C3×C4—362no—
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—64no—
445C2×Dic3—92no—
446C2×Dic3—92no—
447C2×Dic3—91no—
448C2×Dic3—91no—
449C2×Dic3—122no—
450C2×Dic3—181no—
451C2×Dic3—181no—
452C2×Dic3—182no—
453C2×Dic3—182no—
454C2×Dic3—361no—
455C2×Dic3—361no—
456C2×Dic3C2×D3211yescharacteristic
457C2×Dic3C2×D3212yes—
458C2×Dic3C2×D3211yescharacteristic
459C4×D3—42no—
460C4×D3—64no—
461C4×D3—122no—
462Dic6—184no—
463Dic6—184no—
464Dic6—182no—
465Dic6—362no—
466Dic6—362no—
467Dic6—362no—
468Dic6—364no—
469C2×C32—22no—
470C2×C32—21no—
471C2×C32—22no—
472C2×C32—21no—
473C2×C32—21no—
474C2×C32—22no—
475C2×C32—41no—
476C2×C32—41no—
477C2×C32—41no—
478C2×C32—64no—
479C2×C32C22×Dic611yescharacteristic
480C2×C32C2×#23512yes—
481C2×C32#100412yes—
482C2×C32#100411yescharacteristic
483C2×C32C2×D3×Q812yes—
484C2×C32D3×#4111yescharacteristic
485C3×D3—24no—
486C3×D3—24no—
487C3×D3—44no—
488C2×Q8—271no—
489C2×Q8—272no—
490C2×D4—91no—
491C22×C4—32no—
492C22×C4—92no—
493C22×C4—92no—
494C4⋊C4—31no—
495C4⋊C4—92no—
496C4⋊C4—92no—
497C4⋊C4—91no—
498C4⋊C4—92no—
499C4⋊C4—271no—
500C4⋊C4—272no—
501C4⋊C4—271no—
502C4⋊C4—271no—
503C4⋊C4—272no—
504C4⋊C4—271no—
505C22⋊C4—92no—
506C22⋊C4—272no—
507C22⋊C4—272no—
508C42—91no—
509C42—271no—
510C42—271no—
511C22×C3—21no—
512C22×C3—22no—
513C22×C3—34no—
514C22×C3—34no—
515C22×C3—41no—
516C22×C3—64no—
517C22×C3—62no—
518C22×C3—62no—
519C22×C3—62no—
520C22×C3—62no—
521C22×C3—122no—
522C22×C3—122no—
523C22×C3C22×D3212yes—
524C22×C3C22×D3211yescharacteristic
525C2×D3—22no—
526C2×D3—22no—
527C2×D3D3×Dic614yes—
528C3×C4—21no—
529C3×C4—22no—
530C3×C4—34no—
531C3×C4—42no—
532C3×C4—41no—
533C3×C4—62no—
534C3×C4—64no—
535C3×C4—62no—
536C3×C4—62no—
537C3×C4—62no—
538C3×C4—81no—
539C3×C4—92no—
540C3×C4—121no—
541C3×C4—181no—
542C3×C4—182no—
543C3×C4—182no—
544Dic3—21no—
545Dic3—22no—
546Dic3—34no—
547Dic3—62no—
548Dic3—62no—
549Dic3—62no—
550Dic3—62no—
551Dic3—62no—
552Dic3—62no—
553Dic3—62no—
554Dic3—122no—
555Dic3—122no—
556Dic3—182no—
557Dic3—181no—
558Dic3—182no—
559Dic3—181no—
560Dic3—182no—
561Dic3—362no—
562Dic3—362no—
563Dic3—361no—
564Dic3—362no—
565Dic3—721no—
566Dic3#379812yes—
567C32—22no—
568C32—21no—
569C32—41no—
570C32#551112yescomplemented
571C32#553911yescharacteristic, complemented
572C23—32no—
573Q8—542no—
574Q8—1082no—
575D4—94no—
576C2×C4—31no—
577C2×C4—31no—
578C2×C4—32no—
579C2×C4—91no—
580C2×C4—91no—
581C2×C4—92no—
582C2×C4—92no—
583C2×C4—122no—
584C2×C4—184no—
585C2×C4—271no—
586C2×C4—271no—
587C2×C4—362no—
588C2×C4D3311yescharacteristic
589C2×C3—21no—
590C2×C3—22no—
591C2×C3—21no—
592C2×C3—22no—
593C2×C3—22no—
594C2×C3—21no—
595C2×C3—41no—
596C2×C3—41no—
597C2×C3—41no—
598C2×C3—64no—
599C2×C3—64no—
600C2×C3—124no—
601C2×C3C2×D3×Dic612yes—
602C2×C3C2×#379811yescharacteristic
603C2×C3D3×#23512yes—
604C2×C3#6431311yescharacteristic
605C2×C3D32×Q811yescharacteristic
606C2×C3#6433012yes—
607D3—24no—
608C22—34no—
609C22—62no—
610C22—62no—
611C22C2×D3311yescharacteristic
612C4—21no—
613C4—32no—
614C4—62no—
615C4—61no—
616C4—92no—
617C4—182no—
618C4—181no—
619C4—182no—
620C4—541no—
621C4—541no—
622C3—22no—
623C3—21no—
624C3—41no—
625C3#1059453111yescharacteristic, complemented
626C3#1059495112yescomplemented
627C2—64no—
628C2D32×Dic611yescharacteristic
629C2D3×#379811yescharacteristic
630C2#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. #11723415C2semidirect
3. #11723390C2semidirect
5. C3×D3×Dic3⋊C4C2semidirect
6. #11723037C2semidirect
7. #11723018C2semidirect
8. #11722975C2semidirect
9. #11722971C2semidirect
52. D3×Dic3⋊C4D3semidirect
79. C3×#625Q8semidirect
258. #625Dic6semidirect
406. C33#554semidirect
570. C32#5511semidirect
571. C32#5539semidirect
625. C3#10594531semidirect
626. C3#10594951semidirect

Maximal subgroups

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

subgroupindex
#117236792
#117234152
#117233902
C3×C4×#6252
C3×D3×Dic3⋊C42
#117230372
#117230182
#117229752
#117229712
Dic3×#6252
#105949513
#105945313