Finite Group Databasebeta

← #13178274

Subgroups — #13178274

Order 1728.

1136 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64090C211yesmaximal, characteristic
2C3×#64086C212yesmaximal, complemented
3C2×#88815C211yesmaximal, characteristic, complemented
4#11723721C211yesmaximal, characteristic, complemented
5#11723720C212yesmaximal
6C4×#6296C211yesmaximal, characteristic, complemented
7#11723679C212yesmaximal, complemented
8#11723679C211yesmaximal, characteristic, complemented
9#11723676C212yesmaximal, complemented
10#11723673C211yesmaximal, characteristic, complemented
11#11723671C211yesmaximal, characteristic, complemented
12#10594046—31nomaximal
13#10594039—32nomaximal
14C2×C3×#638C2211yescharacteristic
15C22×C3×C3⋊Dic3C2212yes—
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×#6296C2211yescharacteristic
19C33×C22⋊C4C2211yescharacteristic
20C3×#3757C2212yes—
21C3×#3752C2211yescharacteristic
22C3×#3750C2212yes—
23C3×#3750C2212yes—
24C3×#3750C2211yescharacteristic
25C3×C4×C3⋊Dic3C2212yes—
26C3×C4×C3⋊Dic3C2212yes—
27C3×C4×C3⋊Dic3C2211yescharacteristic
28#88815C2211yescharacteristic
29#88815C2211yescharacteristic
30#88815C2211yescharacteristic
31#88815C2212yes—
32#88815C2212yes—
33#88815C2212yes—
34#88815C2212yes—
35#88814C2212yes—
36#88814C2212yes—
37#64090D311yescharacteristic
38#64086D312yescomplemented
39C3×#67⋊C4—32no—
40C3×#67⋊C4—61no—
41C3×#877—32no—
42C3×#877—32no—
43C3×#877—62no—
44#63982—31no—
45C2×#3724—31no—
46#63966—32no—
47C2×Dic3⋊Dic3—32no—
48#63964—32no—
49#63962—32no—
50#63961—32no—
51C4×#627—32no—
52C4×#627—31no—
53#63874—31no—
54#63870—32no—
55#63870—31no—
56#63868—32no—
57#63867—32no—
58#63867—32no—
59#63867—32no—
60#63865—31no—
61#63862—32no—
62#63859—32no—
63#63855—32no—
64#63851—32no—
65C23×C33C2311yescharacteristic
66C22×C3×C3⋊D3C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C3×#638—22no—
70C2×C3×C3⋊Dic3—24no—
71C2×C3×C3⋊Dic3—22no—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2312yes—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C2×C3×C3⋊Dic3C2311yescharacteristic
79C3×C4×C3⋊D3—22no—
80#6296—22no—
81#5541—91no—
82#5531—92no—
83C2×#638C2×D311yescharacteristic
84C22×C3⋊Dic3C2×D312yes—
85C22×C3⋊Dic3C2×D311yescharacteristic
86C2×C4×C3⋊D3C2×D311yescharacteristic
87C2×C3×#67—32no—
88C2×C3×#67—61no—
89C22×C3×Dic3—32no—
90C22×C3×Dic3—32no—
91C22×C3×Dic3—32no—
92C22×C3×Dic3—62no—
93C22×C3×Dic3—61no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—61no—
96C2×#627—32no—
97C2×#627—31no—
98C32×C22⋊C4—22no—
99C32×C22⋊C4—21no—
100C32×C22⋊C4—41no—
101C32×C22⋊C4C2×D311yescharacteristic
102C32×C22⋊C4C2×D312yes—
103#3757C2×D312yes—
104#3752C2×D311yescharacteristic
105#3750C2×D312yes—
106#3750C2×D312yes—
107#3750C2×D311yescharacteristic
108C4×C3⋊Dic3C2×D312yes—
109C4×C3⋊Dic3C2×D312yes—
110C4×C3⋊Dic3C2×D311yescharacteristic
111C3×#217—32no—
112C3×#217—32no—
113C3×#217—62no—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—61no—
116C3×Dic3⋊C4—32no—
117C3×Dic3⋊C4—32no—
118C3×Dic3⋊C4—32no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—62no—
122C3×Dic3⋊C4—62no—
123C3×Dic3⋊C4—61no—
124C3×C4×Dic3—32no—
125C3×C4×Dic3—32no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—32no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—62no—
130C3×C4×Dic3—62no—
131C3×C4×Dic3—61no—
132#3724—32no—
133#3724—31no—
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—
150#3722—32no—
151#3722—32no—
152(C2×D3)⋊Dic3—32no—
153(C2×D3)⋊Dic3—32no—
154(C2×D3)⋊Dic3—32no—
155(C2×D3)⋊Dic3—32no—
156C22×C33—21no—
157C22×C33—21no—
158C22×C33#4112yes—
159C22×C33C2411yescharacteristic
160C2×C3×C3⋊D3—21no—
161C2×C3×C3⋊D3—21no—
162C2×C3×C3⋊D3—21no—
163C33×C4—21no—
164C33×C4—21no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3—21no—
168C3×C3⋊Dic3—22no—
169C3×C3⋊Dic3—21no—
170C3×C3⋊Dic3—22no—
171C3×C3⋊Dic3#4112yes—
172C3×C4×D4—91no—
173C3×#117—92no—
174#935—92no—
175#929—92no—
176#926—91no—
177C2×C4⋊Dic3—91no—
178#924—91no—
179#924—91no—
180C2×Dic3⋊C4—92no—
181#898—92no—
182#896—92no—
183#889—92no—
184#889—92no—
185#889—92no—
186#886—92no—
187#67⋊C4—32no—
188#67⋊C4—61no—
189#879—92no—
190#878—92no—
191#877—32no—
192#877—32no—
193#877—62no—
194#876—92no—
195#872—92no—
196#870—92no—
197#870—91no—
198C4×Dic6—92no—
199C4×Dic6—91no—
200C23×C32—22no—
201C23×C32—21no—
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—22no—
210C2×C32×C4—21no—
211C2×C32×C4—21no—
212C2×C32×C4—41no—
213C2×C32×C4—41no—
214C2×C32×C4C22×D311yescharacteristic
215C2×C32×C4C22×D311yescharacteristic
216C2×C32×C4C22×D312yes—
217C2×C32×C4C22×D312yes—
218#638—22no—
219C2×C3⋊Dic3—24no—
220C2×C3⋊Dic3—22no—
221C2×C3⋊Dic3C22×D312yes—
222C2×C3⋊Dic3C22×D312yes—
223C2×C3⋊Dic3C22×D312yes—
224C2×C3⋊Dic3C22×D312yes—
225C2×C3⋊Dic3C22×D311yescharacteristic
226C2×C3⋊Dic3C22×D311yescharacteristic
227C2×C3⋊Dic3C22×D311yescharacteristic
228C4×C3⋊D3—22no—
229C3×#67—64no—
230C3×#67—64no—
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—62no—
243C2×C3×Dic3—61no—
244C2×C3×Dic3—61no—
245C2×C3×Dic3—62no—
246C2×C3×Dic3—62no—
247C2×C3×Dic3—62no—
248C2×C3×Dic3—61no—
249C2×C3×Dic3—64no—
250C2×C3×Dic3—68no—
251C2×C3×Dic3—64no—
252C2×C3×Dic3—64no—
253C2×C3×Dic3—122no—
254C3×C4×D3—64no—
255C3×C4×D3—122no—
256#627—64no—
257#627—62no—
258#552—271noSylow
259C2×C33—22no—
260C2×C33C2×#4111yescharacteristic
261C2×C33C2×#4111yescharacteristic
262C2×C332−1+411yescharacteristic
263C3×C3⋊D3—41no—
264C2×C3×D4—91no—
265C22×C3×C4—92no—
266C22×C3×C4—91no—
267C22×C3×C4—91no—
268C2×#67—32no—
269C2×#67—61no—
270C22×Dic3—32no—
271C22×Dic3—32no—
272C22×Dic3—32no—
273C22×Dic3—62no—
274C22×Dic3—61no—
275C2×C4×D3—32no—
276C2×C4×D3—61no—
277C2×Dic6—92no—
278C2×Dic6—91no—
279C3×C4⋊C4—91no—
280C3×C4⋊C4—92no—
281C3×C4⋊C4—92no—
282C3×C22⋊C4—22no—
283C3×C22⋊C4—21no—
284C3×C22⋊C4—41no—
285C3×C22⋊C4—92no—
286C3×C22⋊C4—91no—
287C3×C22⋊C4D3211yescharacteristic
288C3×C22⋊C4D3212yes—
289C3×C42—91no—
290C3×C42—92no—
291C3×C42—92no—
292#217—32no—
293#217—32no—
294#217—62no—
295#217—92no—
296#217—92no—
297(C2×D3)⋊C4—32no—
298(C2×D3)⋊C4—61no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—92no—
303C4⋊Dic3—92no—
304C4⋊Dic3—91no—
305C4⋊Dic3—92no—
306C4⋊Dic3—92no—
307C4⋊Dic3—91no—
308C4⋊Dic3—91no—
309C4⋊Dic3—92no—
310Dic3⋊C4—32no—
311Dic3⋊C4—32no—
312Dic3⋊C4—32no—
313Dic3⋊C4—32no—
314Dic3⋊C4—32no—
315Dic3⋊C4—62no—
316Dic3⋊C4—62no—
317Dic3⋊C4—61no—
318Dic3⋊C4—92no—
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—
328Dic3⋊C4—92no—
329C4×Dic3—32no—
330C4×Dic3—32no—
331C4×Dic3—32no—
332C4×Dic3—32no—
333C4×Dic3—32no—
334C4×Dic3—62no—
335C4×Dic3—62no—
336C4×Dic3—61no—
337C22×C32—21no—
338C22×C32—22no—
339C22×C32—22no—
340C22×C32—21no—
341C22×C32—22no—
342C22×C32—21no—
343C22×C32—24no—
344C22×C32—22no—
345C22×C32—41no—
346C22×C32—42no—
347C22×C32—41no—
348C22×C32—42no—
349C22×C32—41no—
350C22×C32—42no—
351C22×C32—81no—
352C22×C32—81no—
353C22×C32#23512yes—
354C22×C32#23514yes—
355C22×C32C23×D311yescharacteristic
356C22×C32C23×D312yes—
357C2×C3⋊D3—21no—
358C2×C3⋊D3—21no—
359C2×C3⋊D3—21no—
360C2×C3×D3—62no—
361C2×C3×D3—62no—
362C2×C3×D3—62no—
363C2×C3×D3—62no—
364C2×C3×D3—121no—
365C2×C3×D3—121no—
366C32×C4—21no—
367C32×C4—22no—
368C32×C4—22no—
369C32×C4—21no—
370C32×C4—41no—
371C32×C4—42no—
372C32×C4—41no—
373C32×C4—42no—
374C32×C4—81no—
375C32×C4—81no—
376C3⋊Dic3—22no—
377C3⋊Dic3—22no—
378C3⋊Dic3—21no—
379C3⋊Dic3—21no—
380C3⋊Dic3—22no—
381C3⋊Dic3—22no—
382C3⋊Dic3#23712yes—
383C3×Dic3—34no—
384C3×Dic3—62no—
385C3×Dic3—64no—
386C3×Dic3—64no—
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—62no—
396C3×Dic3—62no—
397C3×Dic3—62no—
398C3×Dic3—122no—
399C3×Dic3—122no—
400C3×Dic3—121no—
401#126—272no—
402#125—272no—
403#125—271no—
404#123—272no—
405C22⋊Q8—271no—
406C22⋊Q8—271no—
407C4×Q8—271no—
408C4×D4—91no—
409#117—92no—
410#117—271no—
411C2×C4⋊C4—271no—
412C33#55211yescharacteristic, complemented, Sylow
413C23×C3—22no—
414C23×C3—21no—
415C23×C3—41no—
416C23×C3—91no—
417C23×C3C2×D3211yescharacteristic
418C23×C3C2×D3212yes—
419C22×D3—32no—
420C22×D3—61no—
421C3×D4—182no—
422C2×C3×C4—22no—
423C2×C3×C4—22no—
424C2×C3×C4—21no—
425C2×C3×C4—21no—
426C2×C3×C4—41no—
427C2×C3×C4—41no—
428C2×C3×C4—91no—
429C2×C3×C4—91no—
430C2×C3×C4—92no—
431C2×C3×C4—92no—
432C2×C3×C4—91no—
433C2×C3×C4—92no—
434C2×C3×C4—92no—
435C2×C3×C4—182no—
436C2×C3×C4—182no—
437C2×C3×C4—184no—
438C2×C3×C4C2×D3211yescharacteristic
439C2×C3×C4C2×D3211yescharacteristic
440C2×C3×C4C2×D3212yes—
441C2×C3×C4C2×D3212yes—
442#67—64no—
443#67—64no—
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—32no—
454C2×Dic3—32no—
455C2×Dic3—62no—
456C2×Dic3—61no—
457C2×Dic3—61no—
458C2×Dic3—62no—
459C2×Dic3—62no—
460C2×Dic3—62no—
461C2×Dic3—61no—
462C2×Dic3—64no—
463C2×Dic3—68no—
464C2×Dic3—64no—
465C2×Dic3—64no—
466C2×Dic3—122no—
467C4×D3—64no—
468C4×D3—122no—
469Dic6—184no—
470Dic6—182no—
471C2×C32—24no—
472C2×C32—22no—
473C2×C32—22no—
474C2×C32—21no—
475C2×C32—22no—
476C2×C32—21no—
477C2×C32—22no—
478C2×C32—21no—
479C2×C32—41no—
480C2×C32—41no—
481C2×C32—41no—
482C2×C32—44no—
483C2×C32—42no—
484C2×C32—82no—
485C2×C32C2×#23512yes—
486C2×C32C2×#23511yescharacteristic
487C2×C32C2×#23711yescharacteristic
488C2×C32D3×#4112yes—
489C2×C32#101012yes—
490C2×C32#101011yescharacteristic
491C3⋊D3—41no—
492C3×D3—122no—
493C3×D3—122no—
494C2×Q8—271no—
495C2×D4—91no—
496C22×C4—92no—
497C22×C4—91no—
498C22×C4—91no—
499C4⋊C4—92no—
500C4⋊C4—92no—
501C4⋊C4—91no—
502C4⋊C4—271no—
503C4⋊C4—271no—
504C4⋊C4—272no—
505C4⋊C4—271no—
506C4⋊C4—272no—
507C4⋊C4—272no—
508C4⋊C4—272no—
509C22⋊C4—92no—
510C22⋊C4—91no—
511C22⋊C4—272no—
512C22⋊C4—272no—
513C22⋊C4#111311yescharacteristic
514C42—92no—
515C42—92no—
516C42—91no—
517C22×C3—22no—
518C22×C3—21no—
519C22×C3—22no—
520C22×C3—21no—
521C22×C3—22no—
522C22×C3—21no—
523C22×C3—24no—
524C22×C3—22no—
525C22×C3—41no—
526C22×C3—42no—
527C22×C3—41no—
528C22×C3—42no—
529C22×C3—41no—
530C22×C3—42no—
531C22×C3—81no—
532C22×C3—81no—
533C22×C3—181no—
534C22×C3—181no—
535C22×C3—181no—
536C22×C3#379812yes—
537C22×C3#379814yes—
538C22×C3C22×D3212yes—
539C22×C3C22×D3211yescharacteristic
540C2×D3—62no—
541C2×D3—62no—
542C2×D3—62no—
543C2×D3—62no—
544C2×D3—121no—
545C2×D3—121no—
546C3×C4—22no—
547C3×C4—21no—
548C3×C4—22no—
549C3×C4—21no—
550C3×C4—41no—
551C3×C4—42no—
552C3×C4—41no—
553C3×C4—42no—
554C3×C4—81no—
555C3×C4—81no—
556C3×C4—92no—
557C3×C4—182no—
558C3×C4—182no—
559C3×C4—181no—
560C3×C4—182no—
561C3×C4—182no—
562C3×C4—181no—
563Dic3—34no—
564Dic3—62no—
565Dic3—64no—
566Dic3—64no—
567Dic3—62no—
568Dic3—62no—
569Dic3—62no—
570Dic3—62no—
571Dic3—62no—
572Dic3—62no—
573Dic3—62no—
574Dic3—62no—
575Dic3—62no—
576Dic3—62no—
577Dic3—62no—
578Dic3—121no—
579Dic3—122no—
580Dic3—122no—
581C32—22no—
582C32—21no—
583C32—41no—
584C32#553112yescomplemented
585C32#554111yescharacteristic, complemented
586C23—91no—
587C23C2×#111311yescharacteristic
588Q8—542no—
589D4—182no—
590C2×C4—91no—
591C2×C4—92no—
592C2×C4—92no—
593C2×C4—92no—
594C2×C4—91no—
595C2×C4—91no—
596C2×C4—92no—
597C2×C4—182no—
598C2×C4—182no—
599C2×C4—184no—
600C2×C4C2×#111311yescharacteristic
601C2×C4C2×#111311yescharacteristic
602C2×C3—24no—
603C2×C3—22no—
604C2×C3—22no—
605C2×C3—21no—
606C2×C3—22no—
607C2×C3—21no—
608C2×C3—21no—
609C2×C3—22no—
610C2×C3—42no—
611C2×C3—44no—
612C2×C3—41no—
613C2×C3—41no—
614C2×C3—41no—
615C2×C3—82no—
616C2×C3—361no—
617C2×C3C2×#379812yes—
618C2×C3C2×#379811yescharacteristic
619C2×C3#6430912yes—
620C2×C3#6430911yescharacteristic
621C2×C3D3×#23712yes—
622C2×C3#6431211yescharacteristic
623D3—122no—
624D3—122no—
625C22—21no—
626C22—21no—
627C22—181no—
628C22—181no—
629C22—181no—
630C22#8904212yes—
631C22C22×#111311yescharacteristic
632C4—21no—
633C4—21no—
634C4—92no—
635C4—182no—
636C4—182no—
637C4—181no—
638C4—182no—
639C4—181no—
640C4—182no—
641C3—21no—
642C3—22no—
643C3—41no—
644C3#1059403912yescomplemented
645C3#1059404611yescharacteristic, complemented
646C2—22no—
647C2—361no—
648C2C2×#8904211yescharacteristic
649C2#1172462011yescharacteristic
650C2#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×#64086C2semidirect
3. C2×#88815C2semidirect
4. #11723721C2semidirect
6. C4×#6296C2semidirect
7. #11723679C2semidirect
8. #11723679C2semidirect
9. #11723676C2semidirect
10. #11723673C2semidirect
11. #11723671C2semidirect
38. #64086D3semidirect
412. C33#552semidirect
584. C32#5531semidirect
585. C32#5541semidirect
644. C3#10594039semidirect
645. C3#10594046semidirect

Maximal subgroups

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

subgroupindex
C3×#640902
C3×#640862
C2×#888152
#117237212
#117237202
C4×#62962
#117236792
#117236762
#117236732
#117236712
#105940463
#105940393