Finite Group Databasebeta

← #13162005

Subgroups — #13162005

Order 1728.

912 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3#11723674C211yesmaximal, characteristic, complemented
4C2×#88811C211yesmaximal, characteristic, complemented
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722075C211yesmaximal, characteristic, complemented
7#11722072C211yesmaximal, characteristic, complemented
8#10591602—31nomaximal
9#10591599—31nomaximal
10#10591597—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C3×#3832—22no—
13C3×#3832—22no—
14C2×C3×C4⋊(C3⋊D3)—21no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3—21no—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C3×#3756—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#88811—22no—
26C2×#3832D311yescharacteristic
27C2×C3×#239—31no—
28C2×C3×#239—31no—
29C2×C3×#239—61no—
30C2×#3756D311yescharacteristic, complemented
31C2×C3×#216—31no—
32C2×C3×#216—31no—
33C2×C3×#216—61no—
34#63871—31no—
35#63857—31no—
36#63856—31no—
37C2×#3717—31no—
38C2×#3716—31no—
39C2×#3715—31no—
40#63662D311yescharacteristic, complemented
41C3×#835—31no—
42C3×#835—31no—
43C3×#835—61no—
44#63564—31no—
45#63563—31no—
46#63562—31no—
47#63557—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—31no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3D411yescharacteristic
52C33×Q8—21no—
53C33×Q8D412yes—
54C2×C33×C4—21no—
55C2×C33×C4C2311yescharacteristic
56C2×C3×C3⋊Dic3—21no—
57C2×C3×C3⋊Dic3—21no—
58C2×C3×C3⋊Dic3D411yescharacteristic
59C3×C4⋊(C3⋊D3)—24no—
60C3×C4⋊(C3⋊D3)—21no—
61C3×C4⋊(C3⋊D3)D412yes—
62C3×C4×C3⋊D3—24no—
63C3×C4×C3⋊D3—21no—
64C3×C4×C3⋊D3—21no—
65C3×#634—21no—
66C3×#634—21no—
67C3×C3⋊(C3⋊C8)—21no—
68C3×C3⋊(C3⋊C8)—21no—
69#5239—91no—
70#5187—91no—
71#4808—91no—
72C2×C32×Q8—21no—
73C2×C32×Q8—21no—
74C2×C32×Q8—21no—
75C2×C32×Q8—41no—
76C2×C32×Q8C2×D311yescharacteristic
77C2×C32×Q8C2×D311yescharacteristic
78C2×C32×Q8C2×D311yescharacteristic
79#3832—22no—
80#3832—22no—
81C2×C4⋊(C3⋊D3)—21no—
82C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
83C2×C4×C3⋊D3—21no—
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×#634C2×D311yescharacteristic
86C3×#239—62no—
87C3×#239—62no—
88C3×#239—64no—
89C3×#239—64no—
90C3×#239—62no—
91C3×#239—62no—
92C2×C3×D12—31no—
93C2×C3×D12—31no—
94C2×C3×D12—61no—
95C2×C3×D12—61no—
96C2×C3×D12—61no—
97C2×C3×D12—62no—
98C2×C3×C4×D3—31no—
99C2×C3×C4×D3—31no—
100C2×C3×C4×D3—61no—
101C2×C3×C4×D3—61no—
102C2×C3×C4×D3—61no—
103C2×C3×C4×D3—62no—
104C2×C3×Dic6—31no—
105C2×C3×Dic6—31no—
106C2×C3×Dic6—61no—
107#3756—22no—
108C4⋊(C3⋊Dic3)C2×D311yescharacteristic
109C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
110C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
111C3×#216—64no—
112C3×#216—62no—
113C3×#216—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#3717—62no—
130#3716—62no—
131#3715—62no—
132C22×C33C2×D411yescharacteristic
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—22no—
135C2×C3×C3⋊D3—21no—
136C2×C3×C3⋊D3—21no—
137C2×C3×C3⋊D3—41no—
138C2×C3×C3⋊D3—41no—
139C33×C4—22no—
140C33×C4C2×D411yescharacteristic
141C33×C4C2×D411yescharacteristic
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—41no—
144C3×C3⋊Dic3—41no—
145C2×C3×#41—91no—
146C2×#239—31no—
147C2×#239—31no—
148C2×#239—61no—
149C2×C3×Q16—91no—
150#946—91no—
151C2×#216—31no—
152C2×#216—31no—
153C2×#216—61no—
154C2×#215—91no—
155C2×#214—91no—
156#924—91no—
157C2×#204—91no—
158#897—91no—
159C3×#103—91no—
160#835—31no—
161#835—31no—
162#835—61no—
163#832—91no—
164#830—91no—
165#821—91no—
166(C2×D3)⋊C8—91no—
167(C2×D3)⋊C8—91no—
168#809—91no—
169C22×C3⋊D3—21no—
170C22×C3⋊D3#6711yescharacteristic
171C22×C3×D3—31no—
172C22×C3×D3—31no—
173C22×C3×D3—61no—
174C22×C3×D3—62no—
175C22×C3×D3—61no—
176C22×C3×D3—61no—
177C32×Q8—21no—
178C32×Q8—22no—
179C32×Q8—21no—
180C32×Q8—21no—
181C32×Q8—22no—
182C32×Q8—22no—
183C32×Q8—22no—
184C32×Q8—42no—
185C32×Q8—41no—
186C32×Q8—41no—
187C32×Q8—42no—
188C32×Q8D1212yes—
189C32×Q8#6712yes—
190C32×Q8#6712yes—
191C2×C32×C4—21no—
192C2×C32×C4—22no—
193C2×C32×C4—21no—
194C2×C32×C4—21no—
195C2×C32×C4—21no—
196C2×C32×C4—21no—
197C2×C32×C4—21no—
198C2×C32×C4—41no—
199C2×C32×C4—41no—
200C2×C32×C4—41no—
201C2×C32×C4—42no—
202C2×C32×C4C22×D311yescharacteristic
203C2×C32×C4C22×D311yescharacteristic
204C2×C32×C4C22×D311yescharacteristic
205C2×C3⋊Dic3—21no—
206C2×C3⋊Dic3—21no—
207C2×C3⋊Dic3#6711yescharacteristic
208C4⋊(C3⋊D3)—24no—
209C4⋊(C3⋊D3)—21no—
210C4⋊(C3⋊D3)#6712yes—
211C4×C3⋊D3—24no—
212C4×C3⋊D3—21no—
213C4×C3⋊D3—21no—
214#634—21no—
215#634—21no—
216C2×C3×Dic3—31no—
217C2×C3×Dic3—31no—
218C2×C3×Dic3—61no—
219C2×C3×Dic3—61no—
220C2×C3×Dic3—61no—
221C2×C3×Dic3—61no—
222C2×C3×Dic3—61no—
223C2×C3×Dic3—121no—
224C2×C3×Dic3—121no—
225C3×D12—32no—
226C3×D12—32no—
227C3×D12—62no—
228C3×D12—68no—
229C3×D12—64no—
230C3×D12—64no—
231C3×D12—62no—
232C3×D12—61no—
233C3×D12—61no—
234C3×C4×D3—68no—
235C3×C4×D3—64no—
236C3×C4×D3—64no—
237C3×C4×D3—62no—
238C3×C4×D3—62no—
239C3×C4×D3—61no—
240C3×C4×D3—61no—
241C3×C4×D3—61no—
242C3×C4×D3—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#12011yescharacteristic
259C2×C33#13511yescharacteristic
260C2×C33#13711yescharacteristic
261C3×C3⋊D3—41no—
262C3×C3⋊D3—42no—
263C3×#41—182no—
264C3×#41—182no—
265C2×C3×Q8—21no—
266C2×C3×Q8—21no—
267C2×C3×Q8—21no—
268C2×C3×Q8—41no—
269C2×C3×Q8—91no—
270C2×C3×Q8D3211yescharacteristic
271C2×C3×Q8D3211yescharacteristic
272C2×C3×Q8D3211yescharacteristic
273C2×C3×D4—91no—
274C2×C3×D4—181no—
275C22×C3×C4—91no—
276C22×C3×C4—181no—
277#239—62no—
278#239—62no—
279#239—64no—
280#239—64no—
281#239—62no—
282#239—62no—
283C2×D12—31no—
284C2×D12—31no—
285C2×D12—61no—
286C2×D12—61no—
287C2×D12—61no—
288C2×D12—62no—
289C2×C4×D3—31no—
290C2×C4×D3—31no—
291C2×C4×D3—61no—
292C2×C4×D3—61no—
293C2×C4×D3—61no—
294C2×C4×D3—62no—
295C2×Dic6—31no—
296C2×Dic6—31no—
297C2×Dic6—61no—
298C3×Q16—182no—
299C2×C3×C8—91no—
300C2×C3×C8—91no—
301C3×C4⋊C4—91no—
302#217—181no—
303#216—64no—
304#216—62no—
305#216—62no—
306#215—182no—
307#214—182no—
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—
322#204—182no—
323C22×C32—21no—
324C22×C32—21no—
325C22×C32—21no—
326C22×C32—41no—
327C22×C32D3×D411yescharacteristic
328C22×C32D3×D411yescharacteristic
329C22×C32C2×#6711yescharacteristic
330C2×C3⋊D3—21no—
331C2×C3⋊D3—22no—
332C2×C3⋊D3—21no—
333C2×C3⋊D3—21no—
334C2×C3⋊D3—41no—
335C2×C3⋊D3—41no—
336C2×C3×D3—64no—
337C2×C3×D3—62no—
338C2×C3×D3—62no—
339C2×C3×D3—61no—
340C2×C3×D3—61no—
341C2×C3×D3—61no—
342C2×C3×D3—61no—
343C2×C3×D3—61no—
344C2×C3×D3—62no—
345C2×C3×D3—61no—
346C2×C3×D3—121no—
347C2×C3×D3—121no—
348C2×C3×D3—122no—
349C2×C3×D3—122no—
350C2×C3×D3—121no—
351C2×C3×D3—121no—
352C2×C3×D3—121no—
353C2×C3×D3—121no—
354C32×C4—22no—
355C32×C4—24no—
356C32×C4—22no—
357C32×C4—22no—
358C32×C4—21no—
359C32×C4—21no—
360C32×C4—21no—
361C32×C4—21no—
362C32×C4—21no—
363C32×C4—21no—
364C32×C4—41no—
365C32×C4—41no—
366C32×C4—42no—
367C32×C4—42no—
368C32×C4—44no—
369C32×C4C2×D1211yescharacteristic
370C32×C4D3×D411yescharacteristic
371C32×C4D3×D411yescharacteristic
372C32×C4C2×#6711yescharacteristic
373C32×C4C2×#6711yescharacteristic
374C32×C4C2×#6711yescharacteristic
375C3⋊Dic3—21no—
376C3⋊Dic3—41no—
377C3⋊Dic3—41no—
378C3×Dic3—62no—
379C3×Dic3—61no—
380C3×Dic3—61no—
381C3×Dic3—122no—
382C3×Dic3—121no—
383C3×Dic3—121no—
384C3×Dic3—121no—
385C3×Dic3—121no—
386C3×Dic3—241no—
387C2×#41—91no—
388C2×Q16—91no—
389C2×SD16—271no—
390C22⋊Q8—271no—
391#103—91no—
392#102—271no—
393C22⋊C8—271no—
394C33#45211yescharacteristic, complemented, Sylow
395C23×C3—91no—
396C23×C3—181no—
397C22×D3—31no—
398C22×D3—31no—
399C22×D3—61no—
400C22×D3—62no—
401C22×D3—61no—
402C22×D3—61no—
403C3×Q8—22no—
404C3×Q8—21no—
405C3×Q8—21no—
406C3×Q8—21no—
407C3×Q8—22no—
408C3×Q8—22no—
409C3×Q8—22no—
410C3×Q8—41no—
411C3×Q8—41no—
412C3×Q8—42no—
413C3×Q8—42no—
414C3×Q8—181no—
415C3×Q8—181no—
416C3×Q8#62512yes—
417C3×Q8#62612yes—
418C3×Q8#62612yes—
419C3×D4—92no—
420C3×D4—184no—
421C3×D4—181no—
422C2×C3×C4—22no—
423C2×C3×C4—21no—
424C2×C3×C4—21no—
425C2×C3×C4—21no—
426C2×C3×C4—21no—
427C2×C3×C4—21no—
428C2×C3×C4—21no—
429C2×C3×C4—41no—
430C2×C3×C4—41no—
431C2×C3×C4—42no—
432C2×C3×C4—41no—
433C2×C3×C4—91no—
434C2×C3×C4—184no—
435C2×C3×C4—181no—
436C2×C3×C4—181no—
437C2×C3×C4—181no—
438C2×C3×C4—181no—
439C2×C3×C4C2×D3211yescharacteristic
440C2×C3×C4C2×D3211yescharacteristic
441C2×C3×C4C2×D3211yescharacteristic
442C2×Dic3—31no—
443C2×Dic3—31no—
444C2×Dic3—61no—
445C2×Dic3—61no—
446C2×Dic3—61no—
447C2×Dic3—61no—
448C2×Dic3—61no—
449C2×Dic3—121no—
450C2×Dic3—121no—
451D12—32no—
452D12—32no—
453D12—62no—
454D12—68no—
455D12—64no—
456D12—64no—
457D12—62no—
458D12—61no—
459D12—61no—
460C4×D3—68no—
461C4×D3—64no—
462C4×D3—64no—
463C4×D3—62no—
464C4×D3—62no—
465C4×D3—61no—
466C4×D3—61no—
467C4×D3—61no—
468C4×D3—61no—
469Dic6—62no—
470Dic6—61no—
471Dic6—61no—
472Dic6—61no—
473Dic6—61no—
474Dic6—121no—
475C3×C8—181no—
476C3×C8—181no—
477C3⋊C8—62no—
478C3⋊C8—61no—
479C3⋊C8—61no—
480C3⋊C8—61no—
481C3⋊C8—61no—
482C3⋊C8—121no—
483C2×C32—21no—
484C2×C32—21no—
485C2×C32—21no—
486C2×C32—21no—
487C2×C32—21no—
488C2×C32—21no—
489C2×C32—21no—
490C2×C32—21no—
491C2×C32—21no—
492C2×C32—41no—
493C2×C32—41no—
494C2×C32—41no—
495C2×C32#88211yescharacteristic
496C2×C32#91611yescharacteristic
497C2×C32#91811yescharacteristic
498C2×C32#91811yescharacteristic
499C2×C32#91911yescharacteristic
500C2×C32#93711yescharacteristic
501C2×C32#95011yescharacteristic
502C2×C32#95111yescharacteristic
503C2×C32#95311yescharacteristic
504C3⋊D3—41no—
505C3⋊D3—42no—
506C3×D3—122no—
507C3×D3—124no—
508C3×D3—121no—
509C3×D3—122no—
510C3×D3—121no—
511C3×D3—122no—
512#41—182no—
513#41—182no—
514C2×Q8—91no—
515C2×Q8#111311yescharacteristic
516C2×D4—91no—
517C2×D4—181no—
518C22×C4—91no—
519C22×C4—181no—
520Q16—182no—
521SD16—542no—
522C2×C8—91no—
523C2×C8—91no—
524C4⋊C4—91no—
525C4⋊C4—541no—
526C22⋊C4—541no—
527C22×C3—21no—
528C22×C3—21no—
529C22×C3—21no—
530C22×C3—41no—
531C22×C3—181no—
532C22×C3—181no—
533C22×C3—182no—
534C22×C3—181no—
535C22×C3—361no—
536C22×C3—361no—
537C22×C3D3×#6711yescharacteristic
538C22×C3D3×#6711yescharacteristic
539C22×C3#381111yescharacteristic
540C2×D3—64no—
541C2×D3—62no—
542C2×D3—61no—
543C2×D3—61no—
544C2×D3—61no—
545C2×D3—61no—
546C2×D3—62no—
547C2×D3—61no—
548C2×D3—62no—
549C2×D3—61no—
550C2×D3—121no—
551C2×D3—121no—
552C2×D3—122no—
553C2×D3—122no—
554C2×D3—121no—
555C2×D3—121no—
556C2×D3—121no—
557C2×D3—121no—
558C3×C4—24no—
559C3×C4—22no—
560C3×C4—22no—
561C3×C4—22no—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—21no—
566C3×C4—21no—
567C3×C4—21no—
568C3×C4—42no—
569C3×C4—42no—
570C3×C4—44no—
571C3×C4—41no—
572C3×C4—41no—
573C3×C4—181no—
574C3×C4—361no—
575C3×C4—361no—
576C3×C4C2×#62511yescharacteristic
577C3×C4C2×#62611yescharacteristic
578C3×C4C2×#62611yescharacteristic
579C3×C4D3×#6711yescharacteristic
580C3×C4D3×#6711yescharacteristic
581C3×C4#381111yescharacteristic
582Dic3—62no—
583Dic3—61no—
584Dic3—61no—
585Dic3—122no—
586Dic3—121no—
587Dic3—121no—
588Dic3—121no—
589Dic3—121no—
590Dic3—241no—
591C32—21no—
592C32—21no—
593C32—21no—
594C32—41no—
595C32#480811yescharacteristic, complemented
596C32#518711yescharacteristic, complemented
597C32#523911yescharacteristic, complemented
598C23—91no—
599C23—181no—
600Q8—21no—
601Q8—181no—
602Q8—181no—
603Q8#629512yes—
604D4—92no—
605D4—184no—
606D4—181no—
607C2×C4—21no—
608C2×C4—91no—
609C2×C4—184no—
610C2×C4—181no—
611C2×C4—181no—
612C2×C4—181no—
613C2×C4—181no—
614C2×C4C2×#111311yescharacteristic
615C8—181no—
616C8—181no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—21no—
621C2×C3—21no—
622C2×C3—21no—
623C2×C3—21no—
624C2×C3—21no—
625C2×C3—21no—
626C2×C3—41no—
627C2×C3—41no—
628C2×C3—41no—
629C2×C3—361no—
630C2×C3—362no—
631C2×C3#6394311yescharacteristic
632C2×C3#6394311yescharacteristic
633C2×C3#6394511yescharacteristic
634C2×C3#6394711yescharacteristic
635C2×C3#6394911yescharacteristic
636C2×C3#6395011yescharacteristic
637C2×C3#6397611yescharacteristic
638C2×C3#6397711yescharacteristic
639C2×C3#6398111yescharacteristic
640D3—122no—
641D3—124no—
642D3—121no—
643D3—122no—
644D3—121no—
645D3—122no—
646C22—181no—
647C22—181no—
648C22—182no—
649C22—181no—
650C22—361no—
651C22—361no—
652C22#8904911yescharacteristic
653C4—22no—
654C4—181no—
655C4—361no—
656C4—361no—
657C4C2×#629511yescharacteristic
658C4#8904911yescharacteristic
659C3—21no—
660C3—21no—
661C3—21no—
662C3—41no—
663C3#1059159711yescharacteristic, complemented
664C3#1059159911yescharacteristic, complemented
665C3#1059160211yescharacteristic, complemented
666C2—361no—
667C2—362no—
668C2#1172371111yescharacteristic
669C2#1172371311yescharacteristic
670C2#1172372711yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×#3832C2
hypercenterC2×Q8#1113
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3756C2semidirect
3. #11723674C2semidirect
4. C2×#88811C2semidirect
5. C3×#63662C2semidirect
6. #11722075C2semidirect
7. #11722072C2semidirect
30. C2×#3756D3semidirect
40. #63662D3semidirect
394. C33#452semidirect
595. C32#4808semidirect
596. C32#5187semidirect
597. C32#5239semidirect
663. C3#10591597semidirect
664. C3#10591599semidirect
665. C3#10591602semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C3×#37562
#117236742
C2×#888112
C3×#636622
#117220752
#117220722
#105916023
#105915993
#105915973