Finite Group Databasebeta

← #13155815

Subgroups — #13155815

Order 1728.

860 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×#3796C211yesmaximal, characteristic, complemented
2#11723522C211yesmaximal, characteristic, complemented
3C2×C3×#3715C211yesmaximal, characteristic, complemented
4C2×C3×#3713C211yesmaximal, characteristic, complemented
5#11722072C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7#11722032C211yesmaximal, characteristic, complemented
8#10591586—31nomaximal
9#10591520—31nomaximal
10#10591450—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6290—21no—
16C2×C3×#626—21no—
17C2×C3×D3×Dic3—21no—
18C3×#3796—22no—
19C3×#3796—22no—
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23(C2×D3)⋊(C3⋊Dic3)—21no—
24C3×#3715—22no—
25C3×#3713—22no—
26C2×C3×#239—31no—
27C2×C3×#237—31no—
28C2×#3796D311yescharacteristic, complemented
29#64147—31no—
30C2×C3×#215—31no—
31C2×C3×#214—31no—
32C2×C3×#213—31no—
33C2×C3×#213—31no—
34#63908—31no—
35#63908—31no—
36#63908—61no—
37C2×#3715D311yescharacteristic, complemented
38C2×#3713D311yescharacteristic, complemented
39#63662—31no—
40#63659—31no—
41#63566—31no—
42#63566—31no—
43#63566—61no—
44#63563—31no—
45#63563—31no—
46#63563—61no—
47#63557—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—31no—
50C22×C3×C3⋊D3D411yescharacteristic
51C22×C32×D3—21no—
52C22×C32×D3—21no—
53C2×C33×C4C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)—21no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C3×C4×C3⋊D3—21no—
57C3×C4×C3⋊D3—21no—
58C2×C32×Dic3—21no—
59C32×D12—21no—
60C32×D12—21no—
61C32×D12—21no—
62C32×D12D412yescomplemented
63C32×Dic6—21no—
64C32×Dic6D412yescomplemented
65#6290—42no—
66C3×#626—24no—
67C3×D3×Dic3—24no—
68C3×C3⋊(C3⋊C8)—21no—
69C3×C3⋊(C3⋊C8)—21no—
70#5238—91no—
71#5172—91no—
72#5158—91no—
73C2×C32×Q8—31no—
74C2×C32×D4—31no—
75C2×C32×D4—31no—
76C2×#638—61no—
77C2×C4×C3⋊D3C2×D311yescharacteristic
78C2×C3×#67—61no—
79C22×C3×Dic3—61no—
80C3×#239—62no—
81C3×#239—62no—
82C3×#237—62no—
83C3×#237—62no—
84C2×C3×D12—21no—
85C2×C3×D12—21no—
86C2×C3×D12—61no—
87C2×C3×D12C2×D311yescharacteristic
88C2×C3×D12C2×D311yescharacteristic
89C2×C3×D12C2×D311yescharacteristic
90C2×C3×D12C2×D311yescharacteristic
91C2×C3×C4×D3—31no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—61no—
94C2×C3×C4×D3—61no—
95C2×C3×Dic6—21no—
96C2×C3×Dic6C2×D311yescharacteristic
97C2×C3×Dic6C2×D311yescharacteristic
98C2×#626—21no—
99C2×#625—61no—
100C2×#625—61no—
101C2×#625—62no—
102C2×D3×Dic3—21no—
103#3796—22no—
104#3796—22no—
105#3757—61no—
106C4⋊(C3⋊Dic3)—31no—
107C4⋊(C3⋊Dic3)—31no—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—61no—
110C4⋊(C3⋊Dic3)—61no—
111C4⋊(C3⋊Dic3)—61no—
112C4⋊(C3⋊Dic3)—121no—
113C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
114C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
115C3×#215—62no—
116C3×#214—62no—
117C3×#213—62no—
118C3×#213—62no—
119C2×C3×C3⋊C8—31no—
120C2×C3×C3⋊C8—31no—
121C2×C3×C3⋊C8—31no—
122C2×C3×C3⋊C8—31no—
123C2×C3×C3⋊C8—61no—
124C2×C3×C3⋊C8—61no—
125(C2×D3)⋊Dic3—61no—
126(C2×D3)⋊Dic3—61no—
127(C2×D3)⋊Dic3—62no—
128#3715—22no—
129#3713—22no—
130C22×C33C2×D411yescharacteristic
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—21no—
133C2×C3×C3⋊D3—21no—
134C2×C32×D3—22no—
135C2×C32×D3—41no—
136C2×C32×D3—41no—
137C2×C32×D3—41no—
138C2×C32×D3—41no—
139C2×C32×D3—41no—
140C33×C4C2×D411yescharacteristic
141C33×C4C2×D411yescharacteristic
142C3⋊(C3⋊Dic3)—41no—
143C3×C3⋊Dic3—21no—
144C32×Dic3—22no—
145C2×C3×#41—91no—
146C2×#239—31no—
147C2×#237—31no—
148C2×C3×SD16—91no—
149C2×C3×D8—91no—
150C2×#215—31no—
151#938—91no—
152#938—91no—
153#938—181no—
154C2×#214—31no—
155C2×#213—31no—
156C2×#213—31no—
157#930—91no—
158#835—91no—
159#835—91no—
160#835—181no—
161#832—91no—
162#832—91no—
163#832—181no—
164#830—91no—
165#821—91no—
166(C2×D3)⋊C8—91no—
167(C2×D3)⋊C8—91no—
168#816—91no—
169C23×C32—61no—
170C23×C32—61no—
171C22×C3⋊D3#6711yescharacteristic
172C22×C3×D3—21no—
173C22×C3×D3—21no—
174C22×C3×D3—22no—
175C22×C3×D3—21no—
176C22×C3×D3—21no—
177C22×C3×D3—31no—
178C22×C3×D3—31no—
179C22×C3×D3—41no—
180C22×C3×D3—61no—
181C32×Q8—32no—
182C32×Q8—61no—
183C32×D4—32no—
184C32×D4—61no—
185C32×D4—61no—
186C32×D4—61no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—41no—
191C2×C32×C4—61no—
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195#638—122no—
196C2×C3⋊Dic3—61no—
197C2×C3⋊Dic3—62no—
198C2×C3⋊Dic3—61no—
199C2×C3⋊Dic3—61no—
200C2×C3⋊Dic3—121no—
201C2×C3⋊Dic3—121no—
202C2×C3⋊Dic3—122no—
203C2×C3⋊Dic3#6711yescharacteristic
204C4×C3⋊D3—21no—
205C4×C3⋊D3—21no—
206C3×#67—64no—
207C2×C3×Dic3—21no—
208C2×C3×Dic3—21no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—31no—
211C2×C3×Dic3—41no—
212C2×C3×Dic3—61no—
213C2×C3×Dic3—64no—
214C3×D12—21no—
215C3×D12—21no—
216C3×D12—21no—
217C3×D12—21no—
218C3×D12—21no—
219C3×D12—21no—
220C3×D12—22no—
221C3×D12—41no—
222C3×D12—41no—
223C3×D12—41no—
224C3×D12—64no—
225C3×D12#6712yescomplemented
226C3×D12#6712yescomplemented
227C3×C4×D3—64no—
228C3×C4×D3—61no—
229C3×C4×D3—61no—
230C3×C4×D3—61no—
231C3×C4×D3—61no—
232C3×C4×D3—121no—
233C3×C4×D3—121no—
234C3×Dic6—21no—
235C3×Dic6—21no—
236C3×Dic6—22no—
237C3×Dic6—41no—
238C3×Dic6#6712yescomplemented
239C3×Dic6#6712yescomplemented
240#626—24no—
241#625—124no—
242#625—122no—
243#625—122no—
244D3×Dic3—24no—
245C3⋊(C3⋊C8)—21no—
246C3⋊(C3⋊C8)—21no—
247C3×C3⋊C8—61no—
248C3×C3⋊C8—61no—
249C3×C3⋊C8—61no—
250C3×C3⋊C8—61no—
251C3×C3⋊C8—121no—
252C3×C3⋊C8—121no—
253#449—271noSylow
254C2×C33#12011yescharacteristic
255C2×C33#13511yescharacteristic
256C2×C33AGL(1,ℤ/8)11yescharacteristic
257C3×C3⋊D3—41no—
258C32×D3—42no—
259C32×D3—81no—
260C3×#41—182no—
261C3×#41—182no—
262C2×C3×Q8—31no—
263C2×C3×Q8—31no—
264C2×C3×Q8—61no—
265C2×C3×D4—31no—
266C2×C3×D4—31no—
267C2×C3×D4—31no—
268C2×C3×D4—31no—
269C2×C3×D4—61no—
270C2×C3×D4—61no—
271C2×C3×D4—181no—
272C22×C3×C4—91no—
273C22×C3×C4—181no—
274C2×#67—61no—
275C2×#67—181no—
276C2×#67—181no—
277C2×#67—181no—
278C2×#67—182no—
279C22×Dic3—61no—
280#239—62no—
281#239—62no—
282#237—62no—
283#237—62no—
284C2×D12—61no—
285C2×D12D3211yescharacteristic
286C2×D12D3211yescharacteristic
287C2×C4×D3—31no—
288C2×C4×D3—31no—
289C2×C4×D3—61no—
290C2×C4×D3—61no—
291C2×Dic6D3211yescharacteristic
292C3×SD16—182no—
293C3×D8—182no—
294C2×C3×C8—91no—
295C2×C3×C8—91no—
296#217—181no—
297#217—181no—
298#217—182no—
299#215—62no—
300#214—62no—
301#213—62no—
302#213—62no—
303(C2×D3)⋊C4—181no—
304C4⋊Dic3—91no—
305C4⋊Dic3—91no—
306C4⋊Dic3—91no—
307C4⋊Dic3—181no—
308C4⋊Dic3—181no—
309C4⋊Dic3—181no—
310C4⋊Dic3—361no—
311C2×C3⋊C8—31no—
312C2×C3⋊C8—31no—
313C2×C3⋊C8—31no—
314C2×C3⋊C8—31no—
315C2×C3⋊C8—61no—
316C2×C3⋊C8—61no—
317C22×C32—21no—
318C22×C32—21no—
319C22×C32—21no—
320C22×C32—41no—
321C22×C32—62no—
322C22×C32—121no—
323C22×C32—121no—
324C22×C32—121no—
325C22×C32—121no—
326C22×C32—121no—
327C22×C32D3×D411yescharacteristic
328C22×C32D3×D411yescharacteristic
329C22×C32C2×#6711yescharacteristic
330C2×C3⋊D3—21no—
331C2×C3⋊D3—21no—
332C2×C3⋊D3—21no—
333C2×C3×D3—22no—
334C2×C3×D3—22no—
335C2×C3×D3—42no—
336C2×C3×D3—41no—
337C2×C3×D3—41no—
338C2×C3×D3—41no—
339C2×C3×D3—41no—
340C2×C3×D3—42no—
341C2×C3×D3—42no—
342C2×C3×D3—42no—
343C2×C3×D3—41no—
344C2×C3×D3—41no—
345C2×C3×D3—41no—
346C2×C3×D3—41no—
347C2×C3×D3—41no—
348C2×C3×D3—41no—
349C2×C3×D3—62no—
350C2×C3×D3—61no—
351C2×C3×D3—61no—
352C2×C3×D3—61no—
353C2×C3×D3—61no—
354C2×C3×D3—61no—
355C2×C3×D3—61no—
356C2×C3×D3—81no—
357C2×C3×D3—81no—
358C2×C3×D3—121no—
359C2×C3×D3—121no—
360C32×C4—21no—
361C32×C4—21no—
362C32×C4—21no—
363C32×C4—21no—
364C32×C4—21no—
365C32×C4—21no—
366C32×C4—41no—
367C32×C4—41no—
368C32×C4—62no—
369C32×C4D3×D411yescharacteristic
370C32×C4D3×D411yescharacteristic
371C32×C4C2×#6711yescharacteristic
372C32×C4C2×#6711yescharacteristic
373C32×C4C2×#6711yescharacteristic
374C32×C4C2×#6711yescharacteristic
375C3⋊Dic3—21no—
376C3⋊Dic3—121no—
377C3⋊Dic3—122no—
378C3⋊Dic3—121no—
379C3⋊Dic3—121no—
380C3⋊Dic3—241no—
381C3⋊Dic3—241no—
382C3⋊Dic3—242no—
383C3×Dic3—22no—
384C3×Dic3—22no—
385C3×Dic3—42no—
386C3×Dic3—61no—
387C3×Dic3—61no—
388C3×Dic3—121no—
389C2×#41—91no—
390C2×SD16—91no—
391C2×D8—91no—
392#121—271no—
393#103—271no—
394#102—271no—
395C22⋊C8—271no—
396C33#44911yescharacteristic, complemented, Sylow
397C23×C3—62no—
398C23×C3—61no—
399C23×C3—61no—
400C23×C3—61no—
401C23×C3—61no—
402C23×C3—91no—
403C23×C3—121no—
404C22×D3—21no—
405C22×D3—21no—
406C22×D3—31no—
407C22×D3—31no—
408C22×D3—61no—
409C3×Q8—32no—
410C3×Q8—32no—
411C3×Q8—62no—
412C3×Q8—61no—
413C3×Q8—61no—
414C3×Q8—121no—
415C3×D4—32no—
416C3×D4—32no—
417C3×D4—62no—
418C3×D4—61no—
419C3×D4—61no—
420C3×D4—61no—
421C3×D4—61no—
422C3×D4—61no—
423C3×D4—61no—
424C3×D4—121no—
425C3×D4—121no—
426C3×D4—121no—
427C3×D4—184no—
428C2×C3×C4—21no—
429C2×C3×C4—21no—
430C2×C3×C4—21no—
431C2×C3×C4—41no—
432C2×C3×C4—61no—
433C2×C3×C4—61no—
434C2×C3×C4—91no—
435C2×C3×C4—121no—
436C2×C3×C4—184no—
437C2×C3×C4—181no—
438C2×C3×C4—181no—
439C2×C3×C4C2×D3211yescharacteristic
440C2×C3×C4C2×D3211yescharacteristic
441C2×C3×C4C2×D3211yescharacteristic
442#67—64no—
443#67—362no—
444#67—364no—
445#67—362no—
446#67—362no—
447C2×Dic3—21no—
448C2×Dic3—31no—
449C2×Dic3—31no—
450C2×Dic3—61no—
451C2×Dic3—64no—
452C2×Dic3—182no—
453C2×Dic3—181no—
454C2×Dic3—181no—
455C2×Dic3—181no—
456C2×Dic3—361no—
457C2×Dic3—361no—
458C2×Dic3—362no—
459D12—21no—
460D12—21no—
461D12—21no—
462D12—64no—
463D12#62512yescomplemented
464C4×D3—64no—
465C4×D3—61no—
466C4×D3—61no—
467C4×D3—61no—
468C4×D3—61no—
469C4×D3—121no—
470C4×D3—121no—
471Dic6—21no—
472Dic6#62512yescomplemented
473C3×C8—181no—
474C3×C8—181no—
475C3⋊C8—61no—
476C3⋊C8—61no—
477C3⋊C8—61no—
478C3⋊C8—61no—
479C3⋊C8—121no—
480C3⋊C8—121no—
481C2×C32—21no—
482C2×C32—21no—
483C2×C32—21no—
484C2×C32—21no—
485C2×C32—21no—
486C2×C32—21no—
487C2×C32—21no—
488C2×C32—21no—
489C2×C32—21no—
490C2×C32—41no—
491C2×C32—41no—
492C2×C32—41no—
493C2×C32—122no—
494C2×C32—241no—
495C2×C32#91111yescharacteristic
496C2×C32#91211yescharacteristic
497C2×C32#91411yescharacteristic
498C2×C32#91611yescharacteristic
499C2×C32#93711yescharacteristic
500C2×C32#93711yescharacteristic
501C2×C32#94911yescharacteristic
502C2×C32#95011yescharacteristic
503C2×C32#95311yescharacteristic
504C3⋊D3—41no—
505C3×D3—42no—
506C3×D3—42no—
507C3×D3—82no—
508C3×D3—82no—
509C3×D3—81no—
510C3×D3—81no—
511C3×D3—122no—
512C3×D3—121no—
513C3×D3—121no—
514#41—182no—
515#41—182no—
516C2×Q8—31no—
517C2×D4—31no—
518C2×D4—31no—
519C2×D4—181no—
520C2×D4—541no—
521C22×C4—91no—
522C22×C4—181no—
523SD16—182no—
524D8—182no—
525C2×C8—91no—
526C2×C8—91no—
527C4⋊C4—271no—
528C22⋊C4—541no—
529C22×C3—21no—
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—41no—
533C22×C3—62no—
534C22×C3—62no—
535C22×C3—122no—
536C22×C3—122no—
537C22×C3—122no—
538C22×C3—122no—
539C22×C3—121no—
540C22×C3—121no—
541C22×C3—121no—
542C22×C3—121no—
543C22×C3—121no—
544C22×C3—121no—
545C22×C3—121no—
546C22×C3—121no—
547C22×C3—121no—
548C22×C3—121no—
549C22×C3—181no—
550C22×C3—181no—
551C22×C3—181no—
552C22×C3—241no—
553C22×C3—241no—
554C22×C3D3×#6711yescharacteristic
555C22×C3D3×#6711yescharacteristic
556C22×C3#381111yescharacteristic
557C2×D3—22no—
558C2×D3—41no—
559C2×D3—41no—
560C2×D3—41no—
561C2×D3—41no—
562C2×D3—41no—
563C2×D3—62no—
564C2×D3—61no—
565C2×D3—61no—
566C2×D3—61no—
567C2×D3—61no—
568C2×D3—61no—
569C2×D3—61no—
570C2×D3—121no—
571C2×D3—121no—
572C3×C4—21no—
573C3×C4—21no—
574C3×C4—21no—
575C3×C4—21no—
576C3×C4—21no—
577C3×C4—21no—
578C3×C4—41no—
579C3×C4—41no—
580C3×C4—62no—
581C3×C4—62no—
582C3×C4—122no—
583C3×C4—181no—
584C3×C4C2×#62511yescharacteristic
585C3×C4C2×#62511yescharacteristic
586C3×C4D3×#6711yescharacteristic
587C3×C4D3×#6711yescharacteristic
588C3×C4D3×#6711yescharacteristic
589C3×C4D3×#6711yescharacteristic
590Dic3—22no—
591Dic3—61no—
592Dic3—61no—
593Dic3—121no—
594Dic3—362no—
595Dic3—361no—
596Dic3—361no—
597Dic3—361no—
598Dic3—721no—
599Dic3—721no—
600Dic3—722no—
601C32—21no—
602C32—21no—
603C32—21no—
604C32—41no—
605C32#515811yescharacteristic, complemented
606C32#517211yescharacteristic, complemented
607C32#523811yescharacteristic, complemented
608C23—61no—
609C23—61no—
610C23—91no—
611Q8—32no—
612Q8—61no—
613D4—32no—
614D4—61no—
615D4—61no—
616D4—61no—
617D4—184no—
618D4—1082no—
619C2×C4—61no—
620C2×C4—91no—
621C2×C4—184no—
622C2×C4—181no—
623C2×C4—181no—
624C2×C4—541no—
625C2×C4D3311yescharacteristic
626C8—181no—
627C8—181no—
628C2×C3—21no—
629C2×C3—21no—
630C2×C3—21no—
631C2×C3—21no—
632C2×C3—21no—
633C2×C3—21no—
634C2×C3—21no—
635C2×C3—21no—
636C2×C3—21no—
637C2×C3—41no—
638C2×C3—41no—
639C2×C3—41no—
640C2×C3—122no—
641C2×C3—122no—
642C2×C3—242no—
643C2×C3—242no—
644C2×C3—241no—
645C2×C3—241no—
646C2×C3—361no—
647C2×C3#6392511yescharacteristic
648C2×C3#6393311yescharacteristic
649C2×C3#6393611yescharacteristic
650C2×C3#6393711yescharacteristic
651C2×C3#6393911yescharacteristic
652C2×C3#6394711yescharacteristic
653C2×C3#6397611yescharacteristic
654C2×C3#6397611yescharacteristic
655C2×C3#6398311yescharacteristic
656D3—42no—
657D3—81no—
658D3—122no—
659D3—121no—
660D3—121no—
661C22—62no—
662C22—121no—
663C22—121no—
664C22—121no—
665C22—121no—
666C22—121no—
667C22—181no—
668C22—181no—
669C22—181no—
670C22#8895211yescharacteristic
671C4—62no—
672C4—181no—
673C4—1081no—
674C4D3×#62511yescharacteristic
675C4D3×#62511yescharacteristic
676C3—21no—
677C3—21no—
678C3—21no—
679C3—41no—
680C3#1059145011yescharacteristic, complemented
681C3#1059152011yescharacteristic, complemented
682C3#1059158611yescharacteristic, complemented
683C2—122no—
684C2—241no—
685C2—361no—
686C2#1172285711yescharacteristic
687C2#1172287411yescharacteristic
688C2#1172294611yescharacteristic
subgroupG/H
centerC22#88952
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×#3796C2semidirect
2. #11723522C2semidirect
3. C2×C3×#3715C2semidirect
4. C2×C3×#3713C2semidirect
5. #11722072C2semidirect
6. #11722038C2semidirect
7. #11722032C2semidirect
28. C2×#3796D3semidirect
37. C2×#3715D3semidirect
38. C2×#3713D3semidirect
62. C32×D12D4semidirect
64. C32×Dic6D4semidirect
225. C3×D12#67semidirect
226. C3×D12#67semidirect
238. C3×Dic6#67semidirect
239. C3×Dic6#67semidirect
396. C33#449semidirect
463. D12#625semidirect
472. Dic6#625semidirect
605. C32#5158semidirect
606. C32#5172semidirect
607. C32#5238semidirect
680. C3#10591450semidirect
681. C3#10591520semidirect
682. C3#10591586semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37962
#117235222
C2×C3×#37152
C2×C3×#37132
#117220722
#117220382
#117220322
#105915863
#105915203
#105914503