Finite Group Databasebeta

← #13155816

Subgroups — #13155816

Order 1728.

838 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3799C211yesmaximal, characteristic, complemented
2C2×#88789C211yesmaximal, characteristic, complemented
3C3×#63908C211yesmaximal, characteristic, complemented
4C2×C3×#3715C211yesmaximal, characteristic, complemented
5#11722075C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10591586—31nomaximal
9#10591521—31nomaximal
10#10591161—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×#626—21no—
16C2×C3×#625—21no—
17C2×C3×#624—21no—
18C3×#3799—22no—
19C3×#3799—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23#88789—22no—
24C3×(C2×D3)⋊Dic3—21no—
25C3×#3715—22no—
26C2×C3×#239—31no—
27C2×C3×#235—31no—
28C2×#3799D311yescharacteristic, complemented
29C2×#3753—31no—
30C2×C3×#215—31no—
31C3×#938—31no—
32C2×C3×#214—31no—
33C3×#930—31no—
34#63908D311yescharacteristic, complemented
35C2×#3715D311yescharacteristic, complemented
36C2×#3713—31no—
37C2×#3713—31no—
38C2×#3713—61no—
39#63662—31no—
40#63657—31no—
41#63565—31no—
42#63565—31no—
43#63565—61no—
44#63564—31no—
45#63563—31no—
46#63562—31no—
47(C2×D3)⋊(C3⋊C8)—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—61no—
50C22×C3×C3⋊D3—21no—
51C22×C32×D3—21no—
52C22×C32×D3D411yescharacteristic
53C2×C33×C4C2311yescharacteristic
54C2×C3×C3⋊Dic3—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)D412yescomplemented
57C2×C32×Dic3—21no—
58C2×C32×Dic3D411yescharacteristic
59C32×D12—21no—
60C32×D12—21no—
61C32×C4×D3—21no—
62C32×C4×D3—21no—
63C32×Dic6—21no—
64C32×Dic6D412yescomplemented
65C3×#626—24no—
66C3×#625—42no—
67C3×#624—24no—
68C3⋊(C3⋊(C3⋊C8))—21no—
69C3×C3⋊(C3⋊C8)—21no—
70#5238—91no—
71#5172—91no—
72#5037—91no—
73C2×C32×Q8—31no—
74C2×C32×D4—31no—
75C22×C32×C4—31no—
76C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
77C2×C3×#67—61no—
78C2×C3×#67—61no—
79C2×C3×#67—61no—
80C3×#239—62no—
81C3×#239—62no—
82C3×#235—62no—
83C3×#235—62no—
84C2×C3×D12—21no—
85C2×C3×D12—31no—
86C2×C3×D12—31no—
87C2×C3×D12—61no—
88C2×C3×D12—61no—
89C2×C3×D12C2×D311yescharacteristic
90C2×C3×D12C2×D311yescharacteristic
91C2×C3×C4×D3—21no—
92C2×C3×C4×D3—61no—
93C2×C3×C4×D3—61no—
94C2×C3×C4×D3C2×D311yescharacteristic
95C2×C3×C4×D3C2×D311yescharacteristic
96C2×C3×Dic6—21no—
97C2×C3×Dic6C2×D311yescharacteristic
98C2×C3×Dic6C2×D311yescharacteristic
99C2×#626—21no—
100C2×#625—21no—
101C2×#624—21no—
102#3799—22no—
103#3799—22no—
104#3753—62no—
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106C2×C3⋊(C3⋊C8)—31no—
107C2×C3⋊(C3⋊C8)—31no—
108C2×C3⋊(C3⋊C8)—31no—
109C2×C3⋊(C3⋊C8)—61no—
110C2×C3⋊(C3⋊C8)—61no—
111C2×C3⋊(C3⋊C8)—61no—
112C2×C3⋊(C3⋊C8)—121no—
113C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
114C3×#217—61no—
115C3×#215—62no—
116C3×#214—62no—
117C3×(C2×D3)⋊C4—61no—
118C3×C4⋊Dic3—31no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—61no—
121C2×C3×C3⋊C8—31no—
122C2×C3×C3⋊C8—31no—
123C2×C3×C3⋊C8—61no—
124(C2×D3)⋊Dic3—21no—
125#3715—22no—
126#3713—62no—
127#3713—62no—
128#3713—64no—
129C22×C33C2×D411yescharacteristic
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—41no—
132C2×C3×C3⋊D3—41no—
133C2×C32×D3—21no—
134C2×C32×D3—21no—
135C2×C32×D3—21no—
136C2×C32×D3—41no—
137C2×C32×D3—41no—
138C2×C32×D3—41no—
139C33×C4C2×D411yescharacteristic
140C33×C4C2×D411yescharacteristic
141C3×C3⋊Dic3—41no—
142C32×Dic3—21no—
143C32×Dic3—22no—
144C2×C3×#41—91no—
145C2×#239—31no—
146C2×#235—31no—
147C2×C3×SD16—91no—
148C3×#121—91no—
149C2×#215—31no—
150#938—31no—
151C2×#214—31no—
152C2×#213—91no—
153C2×#213—91no—
154C2×#213—91no—
155C2×#213—181no—
156#930—31no—
157#835—91no—
158#835—91no—
159#835—181no—
160#832—91no—
161#830—91no—
162#830—91no—
163#830—181no—
164#821—91no—
165(C2×D3)⋊C8—91no—
166#810—91no—
167#809—91no—
168C23×C32—31no—
169C23×C32—61no—
170C22×C3⋊D3—21no—
171C22×C3×D3—21no—
172C22×C3×D3—21no—
173C22×C3×D3—21no—
174C22×C3×D3—41no—
175C22×C3×D3—61no—
176C22×C3×D3—61no—
177C22×C3×D3—121no—
178C22×C3×D3#6711yescharacteristic
179C22×C3×D3#6711yescharacteristic
180C32×Q8—32no—
181C32×Q8—61no—
182C32×D4—61no—
183C32×D4—61no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—31no—
188C2×C32×C4—41no—
189C2×C32×C4—61no—
190C2×C32×C4—61no—
191C2×C32×C4—61no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C3⋊Dic3—21no—
196C4⋊(C3⋊D3)—21no—
197C4⋊(C3⋊D3)#6712yescomplemented
198C3×#67—64no—
199C3×#67—122no—
200C3×#67—122no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—21no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—41no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—121no—
208C2×C3×Dic3#6711yescharacteristic
209C2×C3×Dic3#6711yescharacteristic
210C3×D12—22no—
211C3×D12—21no—
212C3×D12—21no—
213C3×D12—21no—
214C3×D12—21no—
215C3×D12—32no—
216C3×D12—32no—
217C3×D12—41no—
218C3×D12—62no—
219C3×D12—64no—
220C3×D12—61no—
221C3×D12—61no—
222C3×D12—121no—
223C3×C4×D3—21no—
224C3×C4×D3—21no—
225C3×C4×D3—21no—
226C3×C4×D3—21no—
227C3×C4×D3—41no—
228C3×C4×D3—41no—
229C3×C4×D3—64no—
230C3×C4×D3—64no—
231C3×Dic6—21no—
232C3×Dic6—21no—
233C3×Dic6—22no—
234C3×Dic6—41no—
235C3×Dic6#6712yescomplemented
236C3×Dic6#6712yescomplemented
237#626—24no—
238#625—42no—
239#624—24no—
240C3⋊(C3⋊C8)—21no—
241C3⋊(C3⋊C8)—61no—
242C3⋊(C3⋊C8)—62no—
243C3⋊(C3⋊C8)—61no—
244C3⋊(C3⋊C8)—61no—
245C3⋊(C3⋊C8)—121no—
246C3⋊(C3⋊C8)—121no—
247C3⋊(C3⋊C8)—122no—
248C3×C3⋊C8—61no—
249C3×C3⋊C8—61no—
250C3×C3⋊C8—121no—
251#449—271noSylow
252C2×C33#12011yescharacteristic
253C2×C33#13511yescharacteristic
254C2×C33AGL(1,ℤ/8)11yescharacteristic
255C3×C3⋊D3—42no—
256C32×D3—41no—
257C32×D3—81no—
258C3×#41—182no—
259C3×#41—182no—
260C2×C3×Q8—31no—
261C2×C3×Q8—31no—
262C2×C3×Q8—61no—
263C2×C3×D4—31no—
264C2×C3×D4—31no—
265C2×C3×D4—61no—
266C2×C3×D4—91no—
267C2×C3×D4—181no—
268C2×C3×D4—181no—
269C22×C3×C4—31no—
270C22×C3×C4—31no—
271C22×C3×C4—61no—
272C22×C3×C4—181no—
273C2×#67—61no—
274C2×#67—61no—
275C2×#67—61no—
276#239—62no—
277#239—62no—
278#235—62no—
279#235—62no—
280C2×D12—31no—
281C2×D12—31no—
282C2×D12—61no—
283C2×D12—61no—
284C2×D12D3211yescharacteristic
285C2×C4×D3—61no—
286C2×C4×D3—61no—
287C2×C4×D3D3211yescharacteristic
288C2×Dic6D3211yescharacteristic
289C3×SD16—182no—
290C2×C3×C8—91no—
291C3×C4⋊C4—91no—
292C3×C22⋊C4—181no—
293#217—61no—
294#215—62no—
295#214—62no—
296#213—182no—
297#213—182no—
298#213—182no—
299#213—184no—
300(C2×D3)⋊C4—61no—
301C4⋊Dic3—31no—
302C4⋊Dic3—31no—
303C4⋊Dic3—61no—
304C2×C3⋊C8—31no—
305C2×C3⋊C8—31no—
306C2×C3⋊C8—61no—
307C2×C3⋊C8—91no—
308C2×C3⋊C8—91no—
309C2×C3⋊C8—91no—
310C2×C3⋊C8—181no—
311C2×C3⋊C8—181no—
312C2×C3⋊C8—181no—
313C2×C3⋊C8—361no—
314C22×C32—21no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32—41no—
318C22×C32—61no—
319C22×C32—61no—
320C22×C32—61no—
321C22×C32—121no—
322C22×C32—121no—
323C22×C32—121no—
324C22×C32D3×D411yescharacteristic
325C22×C32C2×#6711yescharacteristic
326C22×C32C2×#6711yescharacteristic
327C2×C3⋊D3—22no—
328C2×C3⋊D3—41no—
329C2×C3⋊D3—41no—
330C2×C3×D3—22no—
331C2×C3×D3—21no—
332C2×C3×D3—21no—
333C2×C3×D3—21no—
334C2×C3×D3—21no—
335C2×C3×D3—21no—
336C2×C3×D3—21no—
337C2×C3×D3—41no—
338C2×C3×D3—41no—
339C2×C3×D3—42no—
340C2×C3×D3—41no—
341C2×C3×D3—41no—
342C2×C3×D3—41no—
343C2×C3×D3—41no—
344C2×C3×D3—41no—
345C2×C3×D3—41no—
346C2×C3×D3—62no—
347C2×C3×D3—62no—
348C2×C3×D3—81no—
349C2×C3×D3—81no—
350C2×C3×D3—122no—
351C2×C3×D3—121no—
352C2×C3×D3—121no—
353C2×C3×D3—121no—
354C2×C3×D3—121no—
355C2×C3×D3—241no—
356C2×C3×D3—241no—
357C32×C4—21no—
358C32×C4—21no—
359C32×C4—21no—
360C32×C4—21no—
361C32×C4—21no—
362C32×C4—21no—
363C32×C4—41no—
364C32×C4—41no—
365C32×C4—61no—
366C32×C4—62no—
367C32×C4D3×D411yescharacteristic
368C32×C4D3×D411yescharacteristic
369C32×C4D3×D411yescharacteristic
370C32×C4C2×#6711yescharacteristic
371C32×C4C2×#6711yescharacteristic
372C32×C4C2×#6711yescharacteristic
373C3⋊Dic3—41no—
374C3×Dic3—22no—
375C3×Dic3—21no—
376C3×Dic3—21no—
377C3×Dic3—22no—
378C3×Dic3—42no—
379C3×Dic3—41no—
380C3×Dic3—121no—
381C3×Dic3—121no—
382C3×Dic3—241no—
383C2×#41—91no—
384C2×SD16—91no—
385C2×D8—271no—
386#121—91no—
387#103—271no—
388#102—271no—
389C22⋊C8—271no—
390C33#44911yescharacteristic, complemented, Sylow
391C23×C3—31no—
392C23×C3—31no—
393C23×C3—61no—
394C23×C3—61no—
395C23×C3—61no—
396C23×C3—121no—
397C23×C3—181no—
398C22×D3—21no—
399C22×D3—61no—
400C22×D3—61no—
401C22×D3—121no—
402C22×D3#62511yescharacteristic
403C3×Q8—32no—
404C3×Q8—32no—
405C3×Q8—62no—
406C3×Q8—61no—
407C3×Q8—61no—
408C3×Q8—121no—
409C3×D4—62no—
410C3×D4—61no—
411C3×D4—61no—
412C3×D4—61no—
413C3×D4—61no—
414C3×D4—92no—
415C3×D4—121no—
416C3×D4—184no—
417C3×D4—181no—
418C3×D4—362no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—21no—
422C2×C3×C4—31no—
423C2×C3×C4—31no—
424C2×C3×C4—41no—
425C2×C3×C4—61no—
426C2×C3×C4—61no—
427C2×C3×C4—61no—
428C2×C3×C4—61no—
429C2×C3×C4—61no—
430C2×C3×C4—61no—
431C2×C3×C4—61no—
432C2×C3×C4—121no—
433C2×C3×C4—121no—
434C2×C3×C4—121no—
435C2×C3×C4—184no—
436C2×C3×C4—181no—
437C2×C3×C4C2×D3211yescharacteristic
438C2×C3×C4C2×D3211yescharacteristic
439C2×C3×C4C2×D3211yescharacteristic
440#67—64no—
441#67—122no—
442#67—122no—
443C2×Dic3—21no—
444C2×Dic3—61no—
445C2×Dic3—61no—
446C2×Dic3—121no—
447C2×Dic3#62511yescharacteristic
448D12—21no—
449D12—21no—
450D12—32no—
451D12—32no—
452D12—62no—
453D12—64no—
454D12—61no—
455D12—61no—
456D12—121no—
457C4×D3—21no—
458C4×D3—21no—
459C4×D3—64no—
460C4×D3—64no—
461Dic6—21no—
462Dic6#62512yescomplemented
463C3×C8—181no—
464C3⋊C8—61no—
465C3⋊C8—61no—
466C3⋊C8—121no—
467C3⋊C8—182no—
468C3⋊C8—181no—
469C3⋊C8—181no—
470C3⋊C8—181no—
471C3⋊C8—361no—
472C3⋊C8—361no—
473C3⋊C8—362no—
474C2×C32—21no—
475C2×C32—21no—
476C2×C32—21no—
477C2×C32—21no—
478C2×C32—21no—
479C2×C32—21no—
480C2×C32—21no—
481C2×C32—21no—
482C2×C32—21no—
483C2×C32—41no—
484C2×C32—41no—
485C2×C32—41no—
486C2×C32—121no—
487C2×C32—241no—
488C2×C32#91411yescharacteristic
489C2×C32#91611yescharacteristic
490C2×C32#93211yescharacteristic
491C2×C32#93711yescharacteristic
492C2×C32#93711yescharacteristic
493C2×C32#94911yescharacteristic
494C2×C32#95011yescharacteristic
495C2×C32#95011yescharacteristic
496C2×C32#95311yescharacteristic
497C3⋊D3—42no—
498C3×D3—42no—
499C3×D3—41no—
500C3×D3—41no—
501C3×D3—82no—
502C3×D3—81no—
503C3×D3—81no—
504C3×D3—122no—
505C3×D3—122no—
506C3×D3—242no—
507#41—182no—
508#41—182no—
509C2×Q8—31no—
510C2×D4—31no—
511C2×D4—91no—
512C2×D4—181no—
513C2×D4—181no—
514C22×C4—31no—
515C22×C4—181no—
516SD16—182no—
517D8—542no—
518C2×C8—91no—
519C2×C8—271no—
520C4⋊C4—91no—
521C22⋊C4—181no—
522C22×C3—21no—
523C22×C3—21no—
524C22×C3—21no—
525C22×C3—41no—
526C22×C3—62no—
527C22×C3—61no—
528C22×C3—61no—
529C22×C3—61no—
530C22×C3—61no—
531C22×C3—61no—
532C22×C3—61no—
533C22×C3—121no—
534C22×C3—121no—
535C22×C3—122no—
536C22×C3—121no—
537C22×C3—121no—
538C22×C3—121no—
539C22×C3—121no—
540C22×C3—121no—
541C22×C3—121no—
542C22×C3—182no—
543C22×C3—241no—
544C22×C3—241no—
545C22×C3—361no—
546C22×C3—361no—
547C22×C3C2×#62511yescharacteristic
548C22×C3D3×#6711yescharacteristic
549C22×C3D3×#6711yescharacteristic
550C2×D3—21no—
551C2×D3—21no—
552C2×D3—21no—
553C2×D3—41no—
554C2×D3—41no—
555C2×D3—41no—
556C2×D3—62no—
557C2×D3—62no—
558C2×D3—122no—
559C2×D3—121no—
560C2×D3—121no—
561C2×D3—121no—
562C2×D3—121no—
563C2×D3—241no—
564C2×D3—241no—
565C3×C4—21no—
566C3×C4—21no—
567C3×C4—21no—
568C3×C4—21no—
569C3×C4—21no—
570C3×C4—21no—
571C3×C4—41no—
572C3×C4—41no—
573C3×C4—61no—
574C3×C4—62no—
575C3×C4—61no—
576C3×C4—62no—
577C3×C4—121no—
578C3×C4—122no—
579C3×C4—361no—
580C3×C4C2×#62511yescharacteristic
581C3×C4D3×#6711yescharacteristic
582C3×C4D3×#6711yescharacteristic
583C3×C4D3×#6711yescharacteristic
584C3×C4D3×#6711yescharacteristic
585C3×C4#381111yescharacteristic
586Dic3—22no—
587Dic3—21no—
588Dic3—121no—
589Dic3—121no—
590Dic3—241no—
591C32—21no—
592C32—21no—
593C32—21no—
594C32—41no—
595C32#503711yescharacteristic, complemented
596C32#517211yescharacteristic, complemented
597C32#523811yescharacteristic, complemented
598C23—31no—
599C23—61no—
600C23—181no—
601Q8—32no—
602Q8—61no—
603D4—61no—
604D4—61no—
605D4—92no—
606D4—184no—
607D4—181no—
608D4—362no—
609C2×C4—31no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—184no—
614C2×C4—181no—
615C2×C4D3311yescharacteristic
616C8—181no—
617C8—541no—
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—21no—
627C2×C3—41no—
628C2×C3—41no—
629C2×C3—41no—
630C2×C3—122no—
631C2×C3—121no—
632C2×C3—121no—
633C2×C3—242no—
634C2×C3—241no—
635C2×C3—241no—
636C2×C3—362no—
637C2×C3#6382211yescharacteristic
638C2×C3#6382311yescharacteristic
639C2×C3#6393411yescharacteristic
640C2×C3#6393511yescharacteristic
641C2×C3#6393911yescharacteristic
642C2×C3#6394711yescharacteristic
643C2×C3#6397611yescharacteristic
644C2×C3#6397611yescharacteristic
645C2×C3#6398311yescharacteristic
646D3—41no—
647D3—81no—
648D3—122no—
649D3—122no—
650D3—242no—
651C22—61no—
652C22—61no—
653C22—61no—
654C22—121no—
655C22—121no—
656C22—121no—
657C22—182no—
658C22—361no—
659C22—361no—
660C22D3×#62511yescharacteristic
661C4—61no—
662C4—62no—
663C4—361no—
664C4D3×#62511yescharacteristic
665C4#8895211yescharacteristic
666C3—21no—
667C3—21no—
668C3—21no—
669C3—41no—
670C3#1059116111yescharacteristic, complemented
671C3#1059152111yescharacteristic, complemented
672C3#1059158611yescharacteristic, complemented
673C2—121no—
674C2—241no—
675C2—362no—
676C2#1172285111yescharacteristic
677C2#1172286711yescharacteristic
678C2#1172294611yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#449
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3799C2semidirect
2. C2×#88789C2semidirect
3. C3×#63908C2semidirect
4. C2×C3×#3715C2semidirect
5. #11722075C2semidirect
6. #11722037C2semidirect
7. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
28. C2×#3799D3semidirect
34. #63908D3semidirect
35. C2×#3715D3semidirect
56. C3×C4⋊(C3⋊D3)D4semidirect
64. C32×Dic6D4semidirect
197. C4⋊(C3⋊D3)#67semidirect
235. C3×Dic6#67semidirect
236. C3×Dic6#67semidirect
390. C33#449semidirect
462. Dic6#625semidirect
595. C32#5037semidirect
596. C32#5172semidirect
597. C32#5238semidirect
670. C3#10591161semidirect
671. C3#10591521semidirect
672. C3#10591586semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37992
C2×#887892
C3×#639082
C2×C3×#37152
#117220752
#117220372
(C2×D3)⋊(C3⋊(C3⋊C8))2
#105915863
#105915213
#105911613