Finite Group Databasebeta

← #13178247

Subgroups — #13178247

Order 1728.

1126 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64094C212yesmaximal, complemented
2C3×#64086C211yesmaximal, characteristic, complemented
3#11723724C212yesmaximal, complemented
4#11723721C211yesmaximal, characteristic, complemented
5#11723717C211yesmaximal, characteristic
6C4×#6296C211yesmaximal, characteristic, complemented
7#11723688C212yesmaximal, complemented
8#11723685C211yesmaximal, characteristic, complemented
9#11723678C211yesmaximal, characteristic, complemented
10#11723676C212yesmaximal, complemented
11#11723672C211yesmaximal, characteristic, complemented
12#10594224—31nomaximal
13#10594207—32nomaximal
14C2×C3×#638C2212yes—
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C2×C3×#634C2212yes—
17C2×#6296C2211yescharacteristic
18C2×#6295C2211yescharacteristic
19C2×#6294C2212yes—
20C33×C22⋊C4C2211yescharacteristic
21C3×#3757C2211yescharacteristic
22C3×#3757C2212yes—
23C3×#3752C2212yes—
24C3×#3752C2212yes—
25C3×#3750C2211yescharacteristic
26C3×#3750C2211yescharacteristic
27C3×C4×C3⋊Dic3C2211yescharacteristic
28C3×C4×C3⋊Dic3C2211yescharacteristic
29C3×C4×C3⋊Dic3C2212yes—
30#88815C2212yes—
31#88815C2212yes—
32#88815C2211yescharacteristic
33#88815C2211yescharacteristic
34#88815C2212yes—
35#88814C2212yes—
36#88814C2212yes—
37#64094D312yescomplemented
38#64086D311yescharacteristic, complemented
39C3×#885—32no—
40C3×#885—32no—
41C3×#885—62no—
42C3×#877—32no—
43C3×#877—61no—
44#63982—31no—
45#63980—31no—
46#63978—32no—
47#63975—32no—
48#63969—32no—
49#63969—32no—
50#63964—32no—
51C4×#627—32no—
52C4×#627—31no—
53#63915—31no—
54#63902—31no—
55#63896—32no—
56#63888—32no—
57#63887—32no—
58#63880—32no—
59#63873—31no—
60#63868—32no—
61#63866—32no—
62#63862—32no—
63#63859—32no—
64#63852—32no—
65C23×C33C2311yescharacteristic
66C22×C3×C3⋊D3C2312yes—
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C3×#638—42no—
70C2×C3×C3⋊Dic3—22no—
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2312yes—
77C2×C3×C3⋊Dic3C2311yescharacteristic
78C3×#634—42no—
79#6296—22no—
80#6295—22no—
81#6294—42no—
82#5669—92no—
83#5534—91no—
84C2×#638C2×D312yes—
85C22×C3⋊Dic3C2×D311yescharacteristic
86C2×#634C2×D312yes—
87C2×C3×#67—32no—
88C2×C3×#67—32no—
89C2×C3×#67—62no—
90C22×C3×Dic3—32no—
91C22×C3×Dic3—61no—
92C2×C3×Dic6—32no—
93C2×C3×Dic6—32no—
94C2×C3×Dic6—62no—
95C2×#627—32no—
96C2×#627—31no—
97C2×#626—32no—
98C2×#625—31no—
99C2×#624—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102C32×C22⋊C4—21no—
103C32×C22⋊C4—22no—
104C32×C22⋊C4—41no—
105C32×C22⋊C4C2×D312yes—
106C32×C22⋊C4C2×D311yescharacteristic
107#3757C2×D311yescharacteristic
108#3757C2×D312yes—
109#3752C2×D312yes—
110#3752C2×D312yes—
111#3750C2×D311yescharacteristic
112#3750C2×D311yescharacteristic
113C4×C3⋊Dic3C2×D311yescharacteristic
114C4×C3⋊Dic3C2×D311yescharacteristic
115C4×C3⋊Dic3C2×D312yes—
116C3×#217—32no—
117C3×#217—32no—
118C3×#217—32no—
119C3×#217—61no—
120C3×#217—62no—
121C3×(C2×D3)⋊C4—32no—
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×(C2×D3)⋊C4—62no—
126C3×(C2×D3)⋊C4—62no—
127C3×Dic3⋊C4—32no—
128C3×Dic3⋊C4—32no—
129C3×Dic3⋊C4—61no—
130C3×Dic3⋊C4—61no—
131C3×C4×Dic3—32no—
132C3×C4×Dic3—32no—
133C3×C4×Dic3—32no—
134C3×C4×Dic3—32no—
135C3×C4×Dic3—61no—
136C3×C4×Dic3—61no—
137C3×C4×Dic3—62no—
138#3724—32no—
139#3724—32no—
140#3724—31no—
141#3724—31no—
142#3724—32no—
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—32no—
145Dic3⋊Dic3—32no—
146Dic3⋊Dic3—32no—
147Dic3⋊Dic3—32no—
148Dic3⋊Dic3—32no—
149Dic3⋊Dic3—32no—
150Dic3⋊Dic3—32no—
151#3722—32no—
152#3722—32no—
153(C2×D3)⋊Dic3—32no—
154(C2×D3)⋊Dic3—32no—
155(C2×D3)⋊Dic3—32no—
156(C2×D3)⋊Dic3—32no—
157C22×C33—21no—
158C22×C33—21no—
159C22×C33—21no—
160C22×C33C2411yescharacteristic
161C2×C3×C3⋊D3—22no—
162C2×C3×C3⋊D3—22no—
163C2×C3×C3⋊D3—22no—
164C33×C4—21no—
165C33×C4—21no—
166C3×C3⋊Dic3—21no—
167C3×C3⋊Dic3—22no—
168C3×C3⋊Dic3—22no—
169C3×C3⋊Dic3—21no—
170C3×C3⋊Dic3—22no—
171C3×C3⋊Dic3—21no—
172C3×C3⋊Dic3#4112yes—
173C3×#124—92no—
174C3×#117—91no—
175#946—92no—
176#944—92no—
177#939—92no—
178#935—92no—
179#930—91no—
180#924—91no—
181#898—92no—
182#896—92no—
183#885—32no—
184#885—32no—
185#885—62no—
186#885—92no—
187#883—92no—
188#883—92no—
189#67⋊C4—92no—
190#879—92no—
191#878—92no—
192#878—92no—
193#877—32no—
194#877—61no—
195#876—92no—
196#875—91no—
197C4×D12—91no—
198#869—91no—
199C4×Dic6—92no—
200C4×Dic6—91no—
201C23×C32—21no—
202C23×C32—22no—
203C23×C32—41no—
204C23×C32C22×D312yes—
205C23×C32C22×D311yescharacteristic
206C22×C3⋊D3C22×D312yes—
207C22×C3×D3—32no—
208C22×C3×D3—32no—
209C22×C3×D3—62no—
210C2×C32×C4—21no—
211C2×C32×C4—21no—
212C2×C32×C4—22no—
213C2×C32×C4—22no—
214C2×C32×C4—41no—
215C2×C32×C4—41no—
216C2×C32×C4C22×D312yes—
217C2×C32×C4C22×D311yescharacteristic
218C2×C32×C4C22×D312yes—
219C2×C32×C4C22×D311yescharacteristic
220#638—42no—
221C2×C3⋊Dic3—22no—
222C2×C3⋊Dic3C22×D311yescharacteristic
223C2×C3⋊Dic3C22×D311yescharacteristic
224C2×C3⋊Dic3C22×D312yes—
225C2×C3⋊Dic3C22×D312yes—
226C2×C3⋊Dic3C22×D312yes—
227C2×C3⋊Dic3C22×D311yescharacteristic
228C2×C3⋊Dic3C22×D311yescharacteristic
229#634—42no—
230C3×#67—122no—
231C3×#67—122no—
232C3×#67—124no—
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—62no—
245C2×C3×Dic3—61no—
246C2×C3×Dic3—61no—
247C2×C3×Dic3—61no—
248C2×C3×Dic3—62no—
249C2×C3×Dic3—62no—
250C2×C3×Dic3—64no—
251C2×C3×Dic3—64no—
252C3×Dic6—122no—
253C3×Dic6—122no—
254C3×Dic6—124no—
255#627—64no—
256#627—62no—
257#626—64no—
258#625—62no—
259#624—122no—
260D3×Dic3—122no—
261D3×Dic3—122no—
262#542—271noSylow
263C2×C33—41no—
264C2×C33C2×#4111yescharacteristic
265C2×C33H(5,𝔽2)11yescharacteristic
266C2×C332−1+411yescharacteristic
267C3×C3⋊D3—42no—
268C2×C3×Q8—92no—
269C2×C3×D4—92no—
270C22×C3×C4—91no—
271C2×#67—32no—
272C2×#67—32no—
273C2×#67—62no—
274C2×#67—92no—
275C22×Dic3—32no—
276C22×Dic3—61no—
277C22×Dic3—92no—
278C2×D12—91no—
279C2×C4×D3—92no—
280C2×C4×D3—92no—
281C2×Dic6—32no—
282C2×Dic6—32no—
283C2×Dic6—62no—
284C2×Dic6—91no—
285C2×Dic6—92no—
286C3×C4⋊C4—91no—
287C3×C4⋊C4—91no—
288C3×C22⋊C4—21no—
289C3×C22⋊C4—22no—
290C3×C22⋊C4—41no—
291C3×C22⋊C4—92no—
292C3×C22⋊C4—91no—
293C3×C22⋊C4—92no—
294C3×C22⋊C4—92no—
295C3×C22⋊C4D3212yes—
296C3×C22⋊C4D3211yescharacteristic
297C3×C42—91no—
298C3×C42—92no—
299C3×C42—91no—
300#217—32no—
301#217—32no—
302#217—32no—
303#217—61no—
304#217—62no—
305#217—92no—
306#217—92no—
307(C2×D3)⋊C4—32no—
308(C2×D3)⋊C4—32no—
309(C2×D3)⋊C4—32no—
310(C2×D3)⋊C4—32no—
311(C2×D3)⋊C4—62no—
312(C2×D3)⋊C4—62no—
313(C2×D3)⋊C4—92no—
314(C2×D3)⋊C4—92no—
315(C2×D3)⋊C4—92no—
316(C2×D3)⋊C4—92no—
317C4⋊Dic3—91no—
318C4⋊Dic3—92no—
319C4⋊Dic3—91no—
320C4⋊Dic3—92no—
321C4⋊Dic3—92no—
322Dic3⋊C4—32no—
323Dic3⋊C4—32no—
324Dic3⋊C4—61no—
325Dic3⋊C4—61no—
326Dic3⋊C4—92no—
327Dic3⋊C4—92no—
328Dic3⋊C4—92no—
329Dic3⋊C4—92no—
330Dic3⋊C4—92no—
331Dic3⋊C4—92no—
332Dic3⋊C4—92no—
333Dic3⋊C4—92no—
334C4×Dic3—32no—
335C4×Dic3—32no—
336C4×Dic3—32no—
337C4×Dic3—32no—
338C4×Dic3—61no—
339C4×Dic3—61no—
340C4×Dic3—62no—
341C22×C32—21no—
342C22×C32—22no—
343C22×C32—22no—
344C22×C32—22no—
345C22×C32—21no—
346C22×C32—21no—
347C22×C32—22no—
348C22×C32—21no—
349C22×C32—22no—
350C22×C32—41no—
351C22×C32—42no—
352C22×C32—42no—
353C22×C32—41no—
354C22×C32—41no—
355C22×C32—42no—
356C22×C32—42no—
357C22×C32—81no—
358C22×C32—81no—
359C22×C32C23×D311yescharacteristic
360C22×C32C23×D312yes—
361C2×C3⋊D3—22no—
362C2×C3⋊D3—22no—
363C2×C3⋊D3—22no—
364C2×C3×D3—62no—
365C2×C3×D3—62no—
366C2×C3×D3—62no—
367C2×C3×D3—62no—
368C2×C3×D3—62no—
369C2×C3×D3—62no—
370C2×C3×D3—122no—
371C2×C3×D3—122no—
372C2×C3×D3—122no—
373C32×C4—22no—
374C32×C4—21no—
375C32×C4—22no—
376C32×C4—21no—
377C32×C4—42no—
378C32×C4—41no—
379C32×C4—42no—
380C32×C4—41no—
381C32×C4—81no—
382C32×C4—81no—
383C3⋊Dic3—21no—
384C3⋊Dic3—22no—
385C3⋊Dic3—22no—
386C3⋊Dic3—22no—
387C3⋊Dic3—21no—
388C3⋊Dic3—21no—
389C3⋊Dic3#23512yes—
390C3×Dic3—34no—
391C3×Dic3—62no—
392C3×Dic3—64no—
393C3×Dic3—62no—
394C3×Dic3—62no—
395C3×Dic3—62no—
396C3×Dic3—62no—
397C3×Dic3—62no—
398C3×Dic3—62no—
399C3×Dic3—62no—
400C3×Dic3—62no—
401C3×Dic3—62no—
402C3×Dic3—62no—
403C3×Dic3—122no—
404C3×Dic3—122no—
405C3×Dic3—121no—
406C3×Dic3—121no—
407C4⋊Q8—271no—
408#126—272no—
409#124—92no—
410#124—271no—
411#123—272no—
412C22⋊Q8—271no—
413C22⋊Q8—272no—
414#121—271no—
415C4×Q8—271no—
416C4×D4—271no—
417#117—91no—
418C33#54211yescharacteristic, complemented, Sylow
419C23×C3—21no—
420C23×C3—22no—
421C23×C3—41no—
422C23×C3—92no—
423C23×C3C2×D3212yes—
424C23×C3C2×D3211yescharacteristic
425C22×D3—32no—
426C22×D3—32no—
427C22×D3—62no—
428C3×Q8—362no—
429C3×D4—362no—
430C2×C3×C4—21no—
431C2×C3×C4—21no—
432C2×C3×C4—22no—
433C2×C3×C4—22no—
434C2×C3×C4—41no—
435C2×C3×C4—41no—
436C2×C3×C4—91no—
437C2×C3×C4—92no—
438C2×C3×C4—92no—
439C2×C3×C4—91no—
440C2×C3×C4—91no—
441C2×C3×C4—92no—
442C2×C3×C4—91no—
443C2×C3×C4—182no—
444C2×C3×C4C2×D3212yes—
445C2×C3×C4C2×D3211yescharacteristic
446C2×C3×C4C2×D3212yes—
447C2×C3×C4C2×D3211yescharacteristic
448#67—122no—
449#67—122no—
450#67—124no—
451#67—184no—
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—62no—
464C2×Dic3—61no—
465C2×Dic3—61no—
466C2×Dic3—61no—
467C2×Dic3—62no—
468C2×Dic3—62no—
469C2×Dic3—64no—
470C2×Dic3—64no—
471C2×Dic3—362no—
472D12—182no—
473C4×D3—362no—
474C4×D3—362no—
475Dic6—122no—
476Dic6—122no—
477Dic6—124no—
478Dic6—184no—
479Dic6—182no—
480C2×C32—22no—
481C2×C32—21no—
482C2×C32—21no—
483C2×C32—22no—
484C2×C32—21no—
485C2×C32—22no—
486C2×C32—41no—
487C2×C32—41no—
488C2×C32—41no—
489C2×C32—41no—
490C2×C32—42no—
491C2×C32—42no—
492C2×C32—82no—
493C2×C32—82no—
494C2×C32C2×#23511yescharacteristic
495C2×C32#100412yes—
496C2×C32D3×#4112yes—
497C2×C32#100911yescharacteristic
498C2×C32#101012yes—
499C2×C32#101011yescharacteristic
500C3⋊D3—42no—
501C3×D3—122no—
502C3×D3—122no—
503C3×D3—242no—
504C2×Q8—92no—
505C2×Q8—271no—
506C2×D4—92no—
507C2×D4—271no—
508C22×C4—91no—
509C22×C4—272no—
510C4⋊C4—91no—
511C4⋊C4—91no—
512C4⋊C4—271no—
513C4⋊C4—271no—
514C4⋊C4—272no—
515C4⋊C4—272no—
516C4⋊C4—272no—
517C22⋊C4—91no—
518C22⋊C4—92no—
519C22⋊C4—92no—
520C22⋊C4—92no—
521C22⋊C4—272no—
522C22⋊C4—272no—
523C22⋊C4#111311yescharacteristic
524C42—91no—
525C42—91no—
526C42—92no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—21no—
530C22×C3—22no—
531C22×C3—21no—
532C22×C3—21no—
533C22×C3—22no—
534C22×C3—21no—
535C22×C3—22no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—42no—
539C22×C3—42no—
540C22×C3—42no—
541C22×C3—42no—
542C22×C3—41no—
543C22×C3—81no—
544C22×C3—81no—
545C22×C3—182no—
546C22×C3—182no—
547C22×C3—182no—
548C22×C3C22×D3212yes—
549C22×C3C22×D3211yescharacteristic
550C2×D3—62no—
551C2×D3—62no—
552C2×D3—62no—
553C2×D3—62no—
554C2×D3—62no—
555C2×D3—62no—
556C2×D3—122no—
557C2×D3—122no—
558C2×D3—122no—
559C3×C4—22no—
560C3×C4—21no—
561C3×C4—21no—
562C3×C4—22no—
563C3×C4—42no—
564C3×C4—41no—
565C3×C4—42no—
566C3×C4—41no—
567C3×C4—81no—
568C3×C4—81no—
569C3×C4—92no—
570C3×C4—182no—
571C3×C4—181no—
572C3×C4—181no—
573C3×C4—181no—
574C3×C4—182no—
575C3×C4—182no—
576Dic3—34no—
577Dic3—62no—
578Dic3—64no—
579Dic3—62no—
580Dic3—62no—
581Dic3—62no—
582Dic3—62no—
583Dic3—62no—
584Dic3—62no—
585Dic3—62no—
586Dic3—62no—
587Dic3—62no—
588Dic3—62no—
589Dic3—122no—
590Dic3—122no—
591Dic3—121no—
592Dic3—121no—
593C32—22no—
594C32—21no—
595C32—41no—
596C32#553411yescharacteristic, complemented
597C32#566912yescomplemented
598C23—92no—
599C23C2×#111311yescharacteristic
600Q8—362no—
601Q8—542no—
602D4—362no—
603D4—542no—
604C2×C4—91no—
605C2×C4—92no—
606C2×C4—91no—
607C2×C4—91no—
608C2×C4—92no—
609C2×C4—92no—
610C2×C4—91no—
611C2×C4—182no—
612C2×C4—1082no—
613C2×C4C2×#111311yescharacteristic
614C2×C4C2×#111311yescharacteristic
615C2×C3—21no—
616C2×C3—22no—
617C2×C3—21no—
618C2×C3—22no—
619C2×C3—22no—
620C2×C3—21no—
621C2×C3—42no—
622C2×C3—42no—
623C2×C3—41no—
624C2×C3—41no—
625C2×C3—41no—
626C2×C3—41no—
627C2×C3—82no—
628C2×C3—82no—
629C2×C3—362no—
630C2×C3D3×#23512yes—
631C2×C3#6430611yescharacteristic
632C2×C3#6430912yes—
633C2×C3#6430911yescharacteristic
634C2×C3#6431311yescharacteristic
635C2×C3#6431412yes—
636D3—122no—
637D3—122no—
638D3—242no—
639C22—21no—
640C22—21no—
641C22—21no—
642C22—182no—
643C22—182no—
644C22—182no—
645C22C22×#111311yescharacteristic
646C4—21no—
647C4—21no—
648C4—92no—
649C4—181no—
650C4—182no—
651C4—181no—
652C4—181no—
653C4—182no—
654C4—182no—
655C3—22no—
656C3—21no—
657C3—41no—
658C3#1059420712yescomplemented
659C3#1059422411yescharacteristic, complemented
660C2—41no—
661C2—362no—
662C2#1172461811yescharacteristic
663C2#1172462011yescharacteristic
664C2#1172462311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radical#11723717C2
hypercenterC22⋊C4#1113
hypocenterC33#542
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#64094C2semidirect
2. C3×#64086C2semidirect
3. #11723724C2semidirect
4. #11723721C2semidirect
6. C4×#6296C2semidirect
7. #11723688C2semidirect
8. #11723685C2semidirect
9. #11723678C2semidirect
10. #11723676C2semidirect
11. #11723672C2semidirect
37. #64094D3semidirect
38. #64086D3semidirect
418. C33#542semidirect
596. C32#5534semidirect
597. C32#5669semidirect
658. C3#10594207semidirect
659. C3#10594224semidirect

Maximal subgroups

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

subgroupindex
C3×#640942
C3×#640862
#117237242
#117237212
#117237172
C4×#62962
#117236882
#117236852
#117236782
#117236762
#117236722
#105942243
#105942073