Finite Group Databasebeta

← #13161954

Subgroups — #13161954

Order 1728.

944 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2C2×C3×#3754C211yesmaximal, characteristic, complemented
3#11723674C211yesmaximal, characteristic, complemented
4C2×#88812C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6#11722076C211yesmaximal, characteristic, complemented
7#11722072C211yesmaximal, characteristic, complemented
8#10591466—31nomaximal
9#10591463—31nomaximal
10#10591461—31nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C22×C3×C3⋊Dic3—21no—
14C3×#3830—22no—
15C3×#3830—22no—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C3×#3754—22no—
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23#88815—21no—
24#88814—21no—
25#88812—22no—
26C2×#3830D311yescharacteristic
27C2×C3×#237—31no—
28C2×C3×#237—31no—
29C2×C3×#237—61no—
30C2×#3754D311yescharacteristic, complemented
31C2×C3×#214—31no—
32C2×C3×#214—31no—
33C2×C3×#214—61no—
34#63871—31no—
35#63857—31no—
36#63856—31no—
37C2×#3719—31no—
38C2×#3719—31no—
39C2×#3718—31no—
40#63659D311yescharacteristic, complemented
41C3×#832—31no—
42C3×#832—31no—
43C3×#832—61no—
44#63567—31no—
45#63566—31no—
46#63565—31no—
47#63557—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—31no—
50C23×C33—21no—
51C22×C3×C3⋊D3D411yescharacteristic
52C33×D4—21no—
53C33×D4D412yes—
54C2×C33×C4C2311yescharacteristic
55C3×#638—24no—
56C2×C3×C3⋊Dic3—24no—
57C2×C3×C3⋊Dic3—21no—
58C2×C3×C3⋊Dic3—21no—
59C2×C3×C3⋊Dic3—21no—
60C2×C3×C3⋊Dic3D411yescharacteristic
61C3×C4×C3⋊D3—21no—
62C3×C4×C3⋊D3—21no—
63C3×#634—21no—
64C3×#634—21no—
65C3×#634—21no—
66C3×#634D412yes—
67C3×C3⋊(C3⋊C8)—21no—
68C3×C3⋊(C3⋊C8)—21no—
69#5239—91no—
70#5173—91no—
71#4783—91no—
72C2×C32×D4—21no—
73C2×C32×D4—21no—
74C2×C32×D4—21no—
75C2×C32×D4—41no—
76C2×C32×D4C2×D311yescharacteristic
77C2×C32×D4C2×D311yescharacteristic
78C2×C32×D4C2×D311yescharacteristic
79C2×#638—21no—
80C22×C3⋊Dic3—21no—
81#3830—22no—
82#3830—22no—
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×#634C2×D311yescharacteristic
85C2×#634C2×D311yescharacteristic
86C2×C3×#67—61no—
87C2×C3×#67—61no—
88C2×C3×#67—62no—
89C22×C3×Dic3—61no—
90C22×C3×Dic3—61no—
91C22×C3×Dic3—62no—
92C3×#237—62no—
93C3×#237—62no—
94C3×#237—64no—
95C3×#237—64no—
96C3×#237—62no—
97C3×#237—62no—
98C2×C3×C4×D3—31no—
99C2×C3×C4×D3—31no—
100C2×C3×C4×D3—61no—
101C2×C3×Dic6—31no—
102C2×C3×Dic6—31no—
103C2×C3×Dic6—31no—
104C2×C3×Dic6—31no—
105C2×C3×Dic6—61no—
106C2×C3×Dic6—61no—
107#3754—22no—
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic
109C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
110C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
111C3×#214—64no—
112C3×#214—62no—
113C3×#214—62no—
114C3×C4⋊Dic3—31no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—61no—
117C2×C3×C3⋊C8—31no—
118C2×C3×C3⋊C8—31no—
119C2×C3×C3⋊C8—31no—
120C2×C3×C3⋊C8—31no—
121C2×C3×C3⋊C8—61no—
122C2×C3×C3⋊C8—61no—
123#3724—61no—
124Dic3⋊Dic3—61no—
125Dic3⋊Dic3—61no—
126#3722—61no—
127(C2×D3)⋊Dic3—61no—
128(C2×D3)⋊Dic3—61no—
129#3719—62no—
130#3719—62no—
131#3718—62no—
132C22×C33—22no—
133C22×C33—41no—
134C22×C33—41no—
135C22×C33C2×D411yescharacteristic
136C2×C3×C3⋊D3—21no—
137C2×C3×C3⋊D3—21no—
138C2×C3×C3⋊D3—21no—
139C33×C4C2×D411yescharacteristic
140C33×C4C2×D411yescharacteristic
141C3×C3⋊Dic3—22no—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—41no—
144C3×C3⋊Dic3—41no—
145C2×C3×#41—91no—
146C2×#237—31no—
147C2×#237—31no—
148C2×#237—61no—
149C2×C3×SD16—91no—
150#946—91no—
151C2×#216—91no—
152C2×#216—91no—
153C2×#214—31no—
154C2×#214—31no—
155C2×#214—61no—
156#924—91no—
157C2×Dic12—91no—
158#897—91no—
159C3×#102—91no—
160#835—91no—
161#832—31no—
162#832—31no—
163#832—61no—
164#830—91no—
165(C2×D3)⋊C8—91no—
166(C2×D3)⋊C8—91no—
167#816—91no—
168#810—91no—
169C23×C32—21no—
170C23×C32—22no—
171C23×C32—21no—
172C23×C32—21no—
173C23×C32—41no—
174C23×C32—41no—
175C23×C32—42no—
176C22×C3⋊D3#6711yescharacteristic
177C22×C3×D3—31no—
178C22×C3×D3—31no—
179C22×C3×D3—61no—
180C32×D4—21no—
181C32×D4—22no—
182C32×D4—21no—
183C32×D4—21no—
184C32×D4—22no—
185C32×D4—22no—
186C32×D4—22no—
187C32×D4—42no—
188C32×D4—41no—
189C32×D4—41no—
190C32×D4—42no—
191C32×D4D1212yes—
192C32×D4#6712yes—
193C32×D4#6712yes—
194C2×C32×C4—21no—
195C2×C32×C4—21no—
196C2×C32×C4—21no—
197C2×C32×C4—41no—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C22×D311yescharacteristic
201#638—24no—
202C2×C3⋊Dic3—24no—
203C2×C3⋊Dic3—21no—
204C2×C3⋊Dic3—21no—
205C2×C3⋊Dic3—21no—
206C2×C3⋊Dic3#6711yescharacteristic
207C4×C3⋊D3—21no—
208C4×C3⋊D3—21no—
209#634—21no—
210#634—21no—
211#634—21no—
212#634#6712yes—
213C3×#67—68no—
214C3×#67—64no—
215C3×#67—64no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—31no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—62no—
220C2×C3×Dic3—68no—
221C2×C3×Dic3—64no—
222C2×C3×Dic3—64no—
223C2×C3×Dic3—61no—
224C2×C3×Dic3—61no—
225C2×C3×Dic3—61no—
226C2×C3×Dic3—61no—
227C2×C3×Dic3—61no—
228C2×C3×Dic3—61no—
229C2×C3×Dic3—121no—
230C2×C3×Dic3—121no—
231C3×C4×D3—62no—
232C3×C4×D3—62no—
233C3×C4×D3—61no—
234C3×C4×D3—61no—
235C3×C4×D3—61no—
236C3×C4×D3—61no—
237C3×Dic6—32no—
238C3×Dic6—32no—
239C3×Dic6—62no—
240C3×Dic6—62no—
241C3×Dic6—61no—
242C3×Dic6—61no—
243C3×Dic6—62no—
244C3×Dic6—61no—
245C3×Dic6—61no—
246C3×Dic6—61no—
247C3×Dic6—61no—
248C3×Dic6—121no—
249C3⋊(C3⋊C8)—21no—
250C3⋊(C3⋊C8)—21no—
251C3×C3⋊C8—62no—
252C3×C3⋊C8—61no—
253C3×C3⋊C8—61no—
254C3×C3⋊C8—61no—
255C3×C3⋊C8—61no—
256C3×C3⋊C8—121no—
257#452—271noSylow
258C2×C33—42no—
259C2×C33#12011yescharacteristic
260C2×C33#13511yescharacteristic
261C2×C33#13711yescharacteristic
262C3×C3⋊D3—41no—
263C3×#41—182no—
264C3×#41—182no—
265C2×C3×Q8—91no—
266C2×C3×Q8—91no—
267C2×C3×D4—21no—
268C2×C3×D4—21no—
269C2×C3×D4—21no—
270C2×C3×D4—41no—
271C2×C3×D4—181no—
272C2×C3×D4D3211yescharacteristic
273C2×C3×D4D3211yescharacteristic
274C2×C3×D4D3211yescharacteristic
275C22×C3×C4—91no—
276C22×C3×C4—181no—
277C2×#67—61no—
278C2×#67—61no—
279C2×#67—62no—
280C22×Dic3—61no—
281C22×Dic3—61no—
282C22×Dic3—62no—
283#237—62no—
284#237—62no—
285#237—64no—
286#237—64no—
287#237—62no—
288#237—62no—
289C2×C4×D3—31no—
290C2×C4×D3—31no—
291C2×C4×D3—61no—
292C2×Dic6—31no—
293C2×Dic6—31no—
294C2×Dic6—31no—
295C2×Dic6—31no—
296C2×Dic6—61no—
297C2×Dic6—61no—
298C3×SD16—182no—
299C2×C3×C8—91no—
300C2×C3×C8—91no—
301C3×C4⋊C4—91no—
302#217—181no—
303#216—182no—
304#216—182no—
305#214—64no—
306#214—62no—
307#214—62no—
308(C2×D3)⋊C4—181no—
309(C2×D3)⋊C4—181no—
310C4⋊Dic3—31no—
311C4⋊Dic3—31no—
312C4⋊Dic3—61no—
313C4⋊Dic3—181no—
314Dic3⋊C4—181no—
315Dic3⋊C4—181no—
316C2×C3⋊C8—31no—
317C2×C3⋊C8—31no—
318C2×C3⋊C8—31no—
319C2×C3⋊C8—31no—
320C2×C3⋊C8—61no—
321C2×C3⋊C8—61no—
322Dic12—182no—
323C22×C32—22no—
324C22×C32—24no—
325C22×C32—22no—
326C22×C32—22no—
327C22×C32—21no—
328C22×C32—21no—
329C22×C32—21no—
330C22×C32—41no—
331C22×C32—42no—
332C22×C32—42no—
333C22×C32—44no—
334C22×C32—41no—
335C22×C32—41no—
336C22×C32—42no—
337C22×C32—42no—
338C22×C32—41no—
339C22×C32—41no—
340C22×C32—41no—
341C22×C32—41no—
342C22×C32—81no—
343C22×C32—81no—
344C22×C32—81no—
345C22×C32—81no—
346C22×C32—82no—
347C22×C32—82no—
348C22×C32D3×D411yescharacteristic
349C22×C32D3×D411yescharacteristic
350C22×C32C2×#6711yescharacteristic
351C2×C3⋊D3—21no—
352C2×C3⋊D3—21no—
353C2×C3⋊D3—21no—
354C2×C3×D3—62no—
355C2×C3×D3—61no—
356C2×C3×D3—61no—
357C2×C3×D3—61no—
358C2×C3×D3—61no—
359C2×C3×D3—61no—
360C2×C3×D3—61no—
361C2×C3×D3—121no—
362C2×C3×D3—121no—
363C32×C4—21no—
364C32×C4—21no—
365C32×C4—21no—
366C32×C4—21no—
367C32×C4—21no—
368C32×C4—21no—
369C32×C4—41no—
370C32×C4—41no—
371C32×C4C2×D1211yescharacteristic
372C32×C4D3×D411yescharacteristic
373C32×C4D3×D411yescharacteristic
374C32×C4C2×#6711yescharacteristic
375C32×C4C2×#6711yescharacteristic
376C32×C4C2×#6711yescharacteristic
377C3⋊Dic3—22no—
378C3⋊Dic3—21no—
379C3⋊Dic3—41no—
380C3⋊Dic3—41no—
381C3×Dic3—64no—
382C3×Dic3—62no—
383C3×Dic3—62no—
384C3×Dic3—61no—
385C3×Dic3—62no—
386C3×Dic3—61no—
387C3×Dic3—122no—
388C3×Dic3—121no—
389C3×Dic3—121no—
390C3×Dic3—121no—
391C3×Dic3—121no—
392C3×Dic3—241no—
393C2×#41—91no—
394C2×Q16—271no—
395C2×SD16—91no—
396C22⋊Q8—271no—
397#103—271no—
398#102—91no—
399C22⋊C8—271no—
400C33#45211yescharacteristic, complemented, Sylow
401C23×C3—22no—
402C23×C3—21no—
403C23×C3—21no—
404C23×C3—21no—
405C23×C3—41no—
406C23×C3—41no—
407C23×C3—42no—
408C23×C3—91no—
409C22×D3—31no—
410C22×D3—31no—
411C22×D3—61no—
412C3×Q8—92no—
413C3×Q8—181no—
414C3×Q8—181no—
415C3×Q8—181no—
416C3×D4—22no—
417C3×D4—21no—
418C3×D4—21no—
419C3×D4—21no—
420C3×D4—22no—
421C3×D4—22no—
422C3×D4—22no—
423C3×D4—41no—
424C3×D4—41no—
425C3×D4—42no—
426C3×D4—42no—
427C3×D4—184no—
428C3×D4#62512yes—
429C3×D4#62612yes—
430C3×D4#62612yes—
431C2×C3×C4—21no—
432C2×C3×C4—21no—
433C2×C3×C4—21no—
434C2×C3×C4—41no—
435C2×C3×C4—91no—
436C2×C3×C4—184no—
437C2×C3×C4—181no—
438C2×C3×C4—181no—
439C2×C3×C4—181no—
440C2×C3×C4—181no—
441C2×C3×C4—181no—
442C2×C3×C4C2×D3211yescharacteristic
443C2×C3×C4C2×D3211yescharacteristic
444C2×C3×C4C2×D3211yescharacteristic
445#67—68no—
446#67—64no—
447#67—64no—
448C2×Dic3—31no—
449C2×Dic3—31no—
450C2×Dic3—61no—
451C2×Dic3—62no—
452C2×Dic3—68no—
453C2×Dic3—64no—
454C2×Dic3—64no—
455C2×Dic3—61no—
456C2×Dic3—61no—
457C2×Dic3—61no—
458C2×Dic3—61no—
459C2×Dic3—61no—
460C2×Dic3—61no—
461C2×Dic3—121no—
462C2×Dic3—121no—
463C4×D3—62no—
464C4×D3—62no—
465C4×D3—61no—
466C4×D3—61no—
467C4×D3—61no—
468C4×D3—61no—
469Dic6—32no—
470Dic6—32no—
471Dic6—62no—
472Dic6—62no—
473Dic6—61no—
474Dic6—61no—
475Dic6—62no—
476Dic6—61no—
477Dic6—61no—
478Dic6—61no—
479Dic6—61no—
480Dic6—121no—
481C3×C8—181no—
482C3×C8—181no—
483C3⋊C8—62no—
484C3⋊C8—61no—
485C3⋊C8—61no—
486C3⋊C8—61no—
487C3⋊C8—61no—
488C3⋊C8—121no—
489C2×C32—21no—
490C2×C32—21no—
491C2×C32—21no—
492C2×C32—21no—
493C2×C32—21no—
494C2×C32—21no—
495C2×C32—21no—
496C2×C32—21no—
497C2×C32—21no—
498C2×C32—41no—
499C2×C32—41no—
500C2×C32—41no—
501C2×C32—42no—
502C2×C32—44no—
503C2×C32—42no—
504C2×C32—42no—
505C2×C32—82no—
506C2×C32—82no—
507C2×C32—84no—
508C2×C32#88211yescharacteristic
509C2×C32#91211yescharacteristic
510C2×C32#91511yescharacteristic
511C2×C32#91511yescharacteristic
512C2×C32#91611yescharacteristic
513C2×C32#93711yescharacteristic
514C2×C32#95011yescharacteristic
515C2×C32#95111yescharacteristic
516C2×C32#95311yescharacteristic
517C3⋊D3—41no—
518C3×D3—122no—
519C3×D3—121no—
520C3×D3—121no—
521#41—182no—
522#41—182no—
523C2×Q8—91no—
524C2×Q8—91no—
525C2×D4—181no—
526C2×D4#111311yescharacteristic
527C22×C4—91no—
528C22×C4—181no—
529Q16—542no—
530SD16—182no—
531C2×C8—91no—
532C2×C8—91no—
533C4⋊C4—91no—
534C4⋊C4—541no—
535C22⋊C4—541no—
536C22×C3—24no—
537C22×C3—22no—
538C22×C3—22no—
539C22×C3—22no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—42no—
544C22×C3—42no—
545C22×C3—44no—
546C22×C3—42no—
547C22×C3—42no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3—41no—
552C22×C3—41no—
553C22×C3—41no—
554C22×C3—41no—
555C22×C3—81no—
556C22×C3—81no—
557C22×C3—81no—
558C22×C3—81no—
559C22×C3—82no—
560C22×C3—82no—
561C22×C3—181no—
562C22×C3—181no—
563C22×C3—181no—
564C22×C3D3×#6711yescharacteristic
565C22×C3D3×#6711yescharacteristic
566C22×C3#381111yescharacteristic
567C2×D3—62no—
568C2×D3—61no—
569C2×D3—61no—
570C2×D3—61no—
571C2×D3—61no—
572C2×D3—61no—
573C2×D3—61no—
574C2×D3—121no—
575C2×D3—121no—
576C3×C4—21no—
577C3×C4—21no—
578C3×C4—21no—
579C3×C4—21no—
580C3×C4—21no—
581C3×C4—21no—
582C3×C4—41no—
583C3×C4—41no—
584C3×C4—181no—
585C3×C4—182no—
586C3×C4—361no—
587C3×C4—361no—
588C3×C4C2×#62511yescharacteristic
589C3×C4C2×#62611yescharacteristic
590C3×C4C2×#62611yescharacteristic
591C3×C4D3×#6711yescharacteristic
592C3×C4D3×#6711yescharacteristic
593C3×C4#381111yescharacteristic
594Dic3—64no—
595Dic3—62no—
596Dic3—62no—
597Dic3—61no—
598Dic3—62no—
599Dic3—61no—
600Dic3—122no—
601Dic3—121no—
602Dic3—121no—
603Dic3—121no—
604Dic3—121no—
605Dic3—241no—
606C32—21no—
607C32—21no—
608C32—21no—
609C32—41no—
610C32#478311yescharacteristic, complemented
611C32#517311yescharacteristic, complemented
612C32#523911yescharacteristic, complemented
613C23—21no—
614C23—91no—
615Q8—92no—
616Q8—181no—
617Q8—181no—
618Q8—181no—
619D4—21no—
620D4—184no—
621D4#629512yes—
622C2×C4—91no—
623C2×C4—184no—
624C2×C4—181no—
625C2×C4—181no—
626C2×C4—181no—
627C2×C4—181no—
628C2×C4—181no—
629C2×C4C2×#111311yescharacteristic
630C8—181no—
631C8—181no—
632C2×C3—21no—
633C2×C3—21no—
634C2×C3—21no—
635C2×C3—21no—
636C2×C3—21no—
637C2×C3—21no—
638C2×C3—21no—
639C2×C3—21no—
640C2×C3—21no—
641C2×C3—44no—
642C2×C3—42no—
643C2×C3—42no—
644C2×C3—42no—
645C2×C3—41no—
646C2×C3—41no—
647C2×C3—41no—
648C2×C3—82no—
649C2×C3—82no—
650C2×C3—84no—
651C2×C3—361no—
652C2×C3#6392911yescharacteristic
653C2×C3#6392911yescharacteristic
654C2×C3#6393111yescharacteristic
655C2×C3#6393311yescharacteristic
656C2×C3#6393511yescharacteristic
657C2×C3#6393611yescharacteristic
658C2×C3#6397611yescharacteristic
659C2×C3#6397711yescharacteristic
660C2×C3#6398111yescharacteristic
661D3—122no—
662D3—121no—
663D3—121no—
664C22—22no—
665C22—41no—
666C22—41no—
667C22—181no—
668C22—181no—
669C22—181no—
670C22#8904911yescharacteristic
671C4—181no—
672C4—182no—
673C4—361no—
674C4—361no—
675C4C2×#629511yescharacteristic
676C4#8904911yescharacteristic
677C3—21no—
678C3—21no—
679C3—21no—
680C3—41no—
681C3#1059146111yescharacteristic, complemented
682C3#1059146311yescharacteristic, complemented
683C3#1059146611yescharacteristic, complemented
684C2—42no—
685C2—361no—
686C2#1172370511yescharacteristic
687C2#1172370711yescharacteristic
688C2#1172372711yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3830C2
hypercenterC2×D4#1113
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3754C2semidirect
3. #11723674C2semidirect
4. C2×#88812C2semidirect
5. C3×#63659C2semidirect
6. #11722076C2semidirect
7. #11722072C2semidirect
30. C2×#3754D3semidirect
40. #63659D3semidirect
400. C33#452semidirect
610. C32#4783semidirect
611. C32#5173semidirect
612. C32#5239semidirect
681. C3#10591461semidirect
682. C3#10591463semidirect
683. C3#10591466semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C3×#37542
#117236742
C2×#888122
C3×#636592
#117220762
#117220722
#105914663
#105914633
#105914613