Finite Group Databasebeta

← C2×#11723663 #13178515

Subgroups — C2×#11723663 #13178515

Order 1728.

1474 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C22×C3×#634C211yesmaximal, characteristic, complemented
3C2×C3×#3748C211yesmaximal, characteristic, complemented
4C2×#88812C212yesmaximal, complemented
5#11723663C218yesmaximal, direct product factor
6C2×#88811C212yesmaximal, complemented
7C2×#63833—31nomaximal
8C2×#63831—31nomaximal
9C2×#63827—31nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×C3×#638—21no—
12C22×C3×C3⋊Dic3—21no—
13C3×#3828—22no—
14C3×#3828C2214yes—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3—21no—
17C2×C3×#634—22no—
18C2×C3×#634—21no—
19C2×C3×#634C2212yes—
20C2×C3×#634C2211yescharacteristic
21C2×C3×#634C2214yescomplemented
22C3×#3748C2214yescomplemented
23C2×C3×C3⋊(C3⋊C8)C2212yes—
24#88812C2218yescomplemented
25#88811C2218yescomplemented
26C2×#3828D311yescharacteristic
27C22×#634D311yescharacteristic, complemented
28C2×C3×#235—31no—
29C2×C3×#235—31no—
30C2×C3×#235—61no—
31C22×C3×Dic6—31no—
32C22×C3×Dic6—31no—
33C22×C3×Dic6—61no—
34C2×#3748D311yescharacteristic, complemented
35C2×C3×#208—31no—
36C2×C3×#208—31no—
37C2×C3×#208—61no—
38C2×#3719—32no—
39C2×#3719—32no—
40C2×#3718—32no—
41#63833—38no—
42C2×#3717—32no—
43#63831—38no—
44C2×#3716—32no—
45#63827—38no—
46C2×#3715—32no—
47C23×C33D411yescharacteristic
48C22×C3×C3⋊D3—21no—
49C2×C33×C4D411yescharacteristic
50C2×C33×C4D412yes—
51C2×C33×C4C2311yescharacteristic
52C2×C33×C4C2312yes—
53C3×#638—24no—
54C2×C3×C3⋊Dic3—22no—
55C2×C3×C3⋊Dic3—24no—
56C2×C3×C3⋊Dic3—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)C2312yes—
59C3×C4×C3⋊D3—24no—
60C3×#634—22no—
61C3×#634—24no—
62C3×#634—21no—
63C3×#634C2314yes—
64C3×#634C2312yes—
65C3×C3⋊(C3⋊C8)C2314yes—
66C2×#951—91no—
67C2×#942—91no—
68C2×#909—91no—
69C22×C32×C4—21no—
70C22×C32×C4—21no—
71C22×C32×C4—21no—
72C22×C32×C4—41no—
73C22×C32×C4C2×D311yescharacteristic
74C22×C32×C4C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C2×#638—21no—
77C22×C3⋊Dic3—21no—
78#3828—22no—
79#3828C2×D314yes—
80C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
81C2×C4×C3⋊D3—21no—
82C2×#634—22no—
83C2×#634—21no—
84C2×#634C2×D312yes—
85C2×#634C2×D311yescharacteristic
86C2×#634C2×D314yescomplemented
87C2×C3×#67—62no—
88C2×C3×#67—61no—
89C2×C3×#67—61no—
90C22×C3×Dic3—62no—
91C22×C3×Dic3—61no—
92C22×C3×Dic3—61no—
93C3×#235—34no—
94C3×#235—34no—
95C3×#235—64no—
96C3×#235—64no—
97C3×#235—62no—
98C3×#235—62no—
99C2×C3×D12—31no—
100C2×C3×D12—31no—
101C2×C3×D12—61no—
102C2×C3×C4×D3—62no—
103C2×C3×C4×D3—61no—
104C2×C3×C4×D3—61no—
105C2×C3×Dic6—32no—
106C2×C3×Dic6—34no—
107C2×C3×Dic6—32no—
108C2×C3×Dic6—34no—
109C2×C3×Dic6—31no—
110C2×C3×Dic6—31no—
111C2×C3×Dic6—62no—
112C2×C3×Dic6—64no—
113C2×C3×Dic6—61no—
114C2×C3×Dic6—64no—
115C2×C3×Dic6—62no—
116C2×C3×Dic6—61no—
117C2×C3×Dic6—62no—
118C2×C3×Dic6—61no—
119C2×C3×Dic6—62no—
120#3748C2×D314yescomplemented
121C2×C3⋊(C3⋊C8)C2×D312yes—
122C3×#208—34no—
123C3×#208—34no—
124C3×#208—64no—
125C2×C3×C3⋊C8—32no—
126C2×C3×C3⋊C8—32no—
127C2×C3×C3⋊C8—62no—
128#3719—38no—
129#3719—38no—
130#3718—38no—
131#3717—38no—
132#3716—38no—
133#3715—38no—
134C22×C33—22no—
135C22×C33C2×D412yes—
136C22×C33C2×D411yescharacteristic
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—42no—
139C33×C4C2×D411yescharacteristic
140C33×C4C2×D412yes—
141C33×C4C2411yescharacteristic
142C3×C3⋊Dic3—24no—
143C3×C3⋊Dic3—22no—
144C2×C3×#41—91no—
145C22×C3×Q8—91no—
146C2×#235—31no—
147C2×#235—31no—
148C2×#235—61no—
149C22×Dic6—31no—
150C22×Dic6—31no—
151C22×Dic6—61no—
152C2×C3×M16—91no—
153#951—98no—
154C2×#216—92no—
155C2×#216—92no—
156#942—98no—
157C2×#215—92no—
158C2×#214—92no—
159C2×#208—31no—
160C2×#208—31no—
161C2×#208—61no—
162#909—98no—
163C2×Dic12—92no—
164C2×#204—92no—
165C23×C32—21no—
166C23×C32—21no—
167C23×C32—21no—
168C23×C32—41no—
169C23×C32D1211yescharacteristic
170C23×C32#6711yescharacteristic
171C23×C32#6711yescharacteristic
172C22×C3⋊D3—21no—
173C22×C3×D3—62no—
174C22×C3×D3—61no—
175C22×C3×D3—61no—
176C2×C32×C4—21no—
177C2×C32×C4—22no—
178C2×C32×C4—21no—
179C2×C32×C4—22no—
180C2×C32×C4—21no—
181C2×C32×C4—22no—
182C2×C32×C4—21no—
183C2×C32×C4—22no—
184C2×C32×C4—21no—
185C2×C32×C4—22no—
186C2×C32×C4—21no—
187C2×C32×C4—22no—
188C2×C32×C4—41no—
189C2×C32×C4—42no—
190C2×C32×C4—41no—
191C2×C32×C4—42no—
192C2×C32×C4D1211yescharacteristic
193C2×C32×C4D1212yes—
194C2×C32×C4#6711yescharacteristic
195C2×C32×C4#6712yes—
196C2×C32×C4#6711yescharacteristic
197C2×C32×C4#6712yes—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D312yes—
200C2×C32×C4C22×D311yescharacteristic
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C22×D312yes—
203C2×C32×C4C22×D312yes—
204#638—24no—
205C2×C3⋊Dic3—22no—
206C2×C3⋊Dic3—24no—
207C2×C3⋊Dic3—21no—
208C4⋊(C3⋊D3)—21no—
209C4⋊(C3⋊D3)C22×D312yes—
210C4×C3⋊D3—24no—
211#634—22no—
212#634—24no—
213#634—21no—
214#634C22×D314yes—
215#634C22×D312yes—
216C3×#67—68no—
217C3×#67—64no—
218C3×#67—64no—
219C2×C3×Dic3—64no—
220C2×C3×Dic3—68no—
221C2×C3×Dic3—62no—
222C2×C3×Dic3—64no—
223C2×C3×Dic3—62no—
224C2×C3×Dic3—61no—
225C2×C3×Dic3—64no—
226C2×C3×Dic3—62no—
227C2×C3×Dic3—61no—
228C3×D12—32no—
229C3×D12—32no—
230C3×D12—62no—
231C3×D12—62no—
232C3×D12—61no—
233C3×D12—61no—
234C3×C4×D3—68no—
235C3×C4×D3—64no—
236C3×C4×D3—64no—
237C3×Dic6—34no—
238C3×Dic6—34no—
239C3×Dic6—32no—
240C3×Dic6—32no—
241C3×Dic6—64no—
242C3×Dic6—62no—
243C3×Dic6—64no—
244C3×Dic6—68no—
245C3×Dic6—62no—
246C3×Dic6—62no—
247C3×Dic6—64no—
248C3×Dic6—61no—
249C3×Dic6—62no—
250C3×Dic6—64no—
251C3×Dic6—61no—
252C3⋊(C3⋊C8)C22×D314yes—
253C3×C3⋊C8—34no—
254C3×C3⋊C8—34no—
255C3×C3⋊C8—64no—
256C2×#137—271noSylow
257C2×C33—22no—
258C2×C33#13712yescomplemented
259C2×C33C22×D411yescharacteristic
260C3×C3⋊D3—42no—
261C3×#41—94no—
262C3×#41—182no—
263C2×C3×Q8—92no—
264C2×C3×Q8—94no—
265C2×C3×Q8—91no—
266C2×C3×Q8—181no—
267C2×C3×Q8—182no—
268C2×C3×D4—91no—
269C2×C3×D4—181no—
270C22×C3×C4—21no—
271C22×C3×C4—21no—
272C22×C3×C4—21no—
273C22×C3×C4—41no—
274C22×C3×C4—181no—
275C22×C3×C4—181no—
276C22×C3×C4D3211yescharacteristic
277C22×C3×C4D3211yescharacteristic
278C22×C3×C4D3211yescharacteristic
279C2×#67—62no—
280C2×#67—61no—
281C2×#67—61no—
282C22×Dic3—62no—
283C22×Dic3—61no—
284C22×Dic3—61no—
285#235—34no—
286#235—34no—
287#235—64no—
288#235—64no—
289#235—62no—
290#235—62no—
291C2×D12—31no—
292C2×D12—31no—
293C2×D12—61no—
294C2×C4×D3—62no—
295C2×C4×D3—61no—
296C2×C4×D3—61no—
297C2×Dic6—32no—
298C2×Dic6—31no—
299C2×Dic6—34no—
300C2×Dic6—32no—
301C2×Dic6—31no—
302C2×Dic6—34no—
303C2×Dic6—62no—
304C2×Dic6—64no—
305C2×Dic6—61no—
306C2×Dic6—64no—
307C2×Dic6—62no—
308C2×Dic6—61no—
309C2×Dic6—62no—
310C2×Dic6—61no—
311C2×Dic6—62no—
312C3×M16—94no—
313C2×C3×C8—92no—
314#216—98no—
315#216—98no—
316#215—98no—
317#214—98no—
318#208—34no—
319#208—34no—
320#208—64no—
321C2×C3⋊C8—32no—
322C2×C3⋊C8—32no—
323C2×C3⋊C8—62no—
324Dic12—98no—
325#204—98no—
326C22×C32—22no—
327C22×C32—24no—
328C22×C32—22no—
329C22×C32—22no—
330C22×C32—21no—
331C22×C32—22no—
332C22×C32—22no—
333C22×C32—21no—
334C22×C32—21no—
335C22×C32—22no—
336C22×C32—42no—
337C22×C32—41no—
338C22×C32—42no—
339C22×C32—42no—
340C22×C32—44no—
341C22×C32C2×D1212yes—
342C22×C32C2×D1211yescharacteristic
343C22×C32C2×#6712yes—
344C22×C32C2×#6712yes—
345C22×C32C2×#6711yescharacteristic
346C22×C32C2×#6711yescharacteristic
347C2×C3⋊D3—22no—
348C2×C3⋊D3—42no—
349C2×C3×D3—64no—
350C2×C3×D3—62no—
351C2×C3×D3—62no—
352C2×C3×D3—124no—
353C2×C3×D3—122no—
354C2×C3×D3—122no—
355C32×C4—21no—
356C32×C4—21no—
357C32×C4—21no—
358C32×C4—21no—
359C32×C4—22no—
360C32×C4—21no—
361C32×C4—22no—
362C32×C4—21no—
363C32×C4—22no—
364C32×C4—41no—
365C32×C4—42no—
366C32×C4—41no—
367C32×C4C2×D1211yescharacteristic
368C32×C4C2×D1212yes—
369C32×C4C2×#6711yescharacteristic
370C32×C4C2×#6712yes—
371C32×C4C2×#6711yescharacteristic
372C32×C4C2×#6712yes—
373C32×C4C23×D311yescharacteristic
374C32×C4C23×D311yescharacteristic
375C32×C4C23×D311yescharacteristic
376C3⋊Dic3—24no—
377C3⋊Dic3—22no—
378C3×Dic3—68no—
379C3×Dic3—64no—
380C3×Dic3—64no—
381C3×Dic3—62no—
382C3×Dic3—64no—
383C3×Dic3—62no—
384C2×#41—91no—
385C22×Q8—91no—
386#137—278no—
387C2×Q16—272no—
388C2×SD16—272no—
389C2×M16—91no—
390C33C2×#13711yescharacteristic, complemented, Sylow
391C23×C3—21no—
392C23×C3—21no—
393C23×C3—21no—
394C23×C3—41no—
395C23×C3—181no—
396C23×C3#62511yescharacteristic
397C23×C3#62611yescharacteristic
398C23×C3#62611yescharacteristic
399C22×D3—62no—
400C22×D3—61no—
401C22×D3—61no—
402C3×Q8—94no—
403C3×Q8—92no—
404C3×Q8—182no—
405C3×Q8—184no—
406C3×Q8—181no—
407C3×D4—92no—
408C3×D4—184no—
409C3×D4—181no—
410C2×C3×C4—22no—
411C2×C3×C4—21no—
412C2×C3×C4—22no—
413C2×C3×C4—21no—
414C2×C3×C4—21no—
415C2×C3×C4—22no—
416C2×C3×C4—21no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—21no—
420C2×C3×C4—21no—
421C2×C3×C4—22no—
422C2×C3×C4—42no—
423C2×C3×C4—41no—
424C2×C3×C4—42no—
425C2×C3×C4—41no—
426C2×C3×C4—181no—
427C2×C3×C4—184no—
428C2×C3×C4—182no—
429C2×C3×C4—184no—
430C2×C3×C4#62511yescharacteristic
431C2×C3×C4#62512yes—
432C2×C3×C4#62611yescharacteristic
433C2×C3×C4#62612yes—
434C2×C3×C4#62611yescharacteristic
435C2×C3×C4#62612yes—
436C2×C3×C4C2×D3211yescharacteristic
437C2×C3×C4C2×D3212yes—
438C2×C3×C4C2×D3211yescharacteristic
439C2×C3×C4C2×D3211yescharacteristic
440C2×C3×C4C2×D3212yes—
441C2×C3×C4C2×D3212yes—
442#67—68no—
443#67—64no—
444#67—64no—
445C2×Dic3—68no—
446C2×Dic3—64no—
447C2×Dic3—62no—
448C2×Dic3—64no—
449C2×Dic3—62no—
450C2×Dic3—61no—
451C2×Dic3—64no—
452C2×Dic3—62no—
453C2×Dic3—61no—
454D12—32no—
455D12—32no—
456D12—62no—
457D12—62no—
458D12—61no—
459D12—61no—
460C4×D3—68no—
461C4×D3—64no—
462C4×D3—64no—
463Dic6—34no—
464Dic6—32no—
465Dic6—34no—
466Dic6—32no—
467Dic6—64no—
468Dic6—62no—
469Dic6—64no—
470Dic6—68no—
471Dic6—62no—
472Dic6—62no—
473Dic6—64no—
474Dic6—61no—
475Dic6—64no—
476Dic6—62no—
477Dic6—61no—
478C3×C8—94no—
479C3⋊C8—34no—
480C3⋊C8—34no—
481C3⋊C8—64no—
482C2×C32—22no—
483C2×C32—24no—
484C2×C32—22no—
485C2×C32—22no—
486C2×C32—21no—
487C2×C32—21no—
488C2×C32—21no—
489C2×C32—22no—
490C2×C32—22no—
491C2×C32—22no—
492C2×C32—42no—
493C2×C32—41no—
494C2×C32—42no—
495C2×C32—42no—
496C2×C32—44no—
497C2×C32#90912yescomplemented
498C2×C32#94212yescomplemented
499C2×C32#95112yescomplemented
500C2×C32C22×D1211yescharacteristic
501C2×C32C22×#6711yescharacteristic
502C2×C32C22×#6711yescharacteristic
503C3⋊D3—42no—
504C3×D3—124no—
505C3×D3—122no—
506C3×D3—122no—
507#41—94no—
508#41—182no—
509C2×Q8—92no—
510C2×Q8—91no—
511C2×Q8—94no—
512C2×Q8—182no—
513C2×Q8—181no—
514C2×D4—91no—
515C2×D4—181no—
516C22×C4—181no—
517C22×C4—181no—
518C22×C4#111311yescharacteristic
519Q16—278no—
520SD16—278no—
521M16—94no—
522C2×C8—92no—
523C22×C3—24no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—22no—
530C22×C3—21no—
531C22×C3—21no—
532C22×C3—21no—
533C22×C3—42no—
534C22×C3—42no—
535C22×C3—44no—
536C22×C3—42no—
537C22×C3—41no—
538C22×C3—182no—
539C22×C3—362no—
540C22×C3C2×#62512yes—
541C22×C3C2×#62511yescharacteristic
542C22×C3C2×#62612yes—
543C22×C3C2×#62612yes—
544C22×C3C2×#62611yescharacteristic
545C22×C3C2×#62611yescharacteristic
546C2×D3—64no—
547C2×D3—62no—
548C2×D3—62no—
549C2×D3—124no—
550C2×D3—122no—
551C2×D3—122no—
552C3×C4—21no—
553C3×C4—21no—
554C3×C4—21no—
555C3×C4—22no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—22no—
559C3×C4—22no—
560C3×C4—21no—
561C3×C4—41no—
562C3×C4—42no—
563C3×C4—41no—
564C3×C4—182no—
565C3×C4—184no—
566C3×C4C2×#62511yescharacteristic
567C3×C4C2×#62512yes—
568C3×C4C2×#62612yes—
569C3×C4C2×#62611yescharacteristic
570C3×C4C2×#62611yescharacteristic
571C3×C4C2×#62612yes—
572C3×C4C22×D3211yescharacteristic
573C3×C4C22×D3211yescharacteristic
574C3×C4C22×D3211yescharacteristic
575Dic3—68no—
576Dic3—64no—
577Dic3—64no—
578Dic3—62no—
579Dic3—64no—
580Dic3—62no—
581C32—21no—
582C32—21no—
583C32—21no—
584C32—41no—
585C32C2×#90911yescharacteristic, complemented
586C32C2×#94211yescharacteristic, complemented
587C32C2×#95111yescharacteristic, complemented
588C23—181no—
589C23#629511yescharacteristic
590Q8—94no—
591Q8—92no—
592Q8—184no—
593Q8—182no—
594Q8—181no—
595D4—92no—
596D4—184no—
597D4—181no—
598C2×C4—184no—
599C2×C4—181no—
600C2×C4—182no—
601C2×C4—184no—
602C2×C4#629512yes—
603C2×C4#629511yescharacteristic
604C2×C4C2×#111312yes—
605C2×C4C2×#111311yescharacteristic
606C8—94no—
607C2×C3—24no—
608C2×C3—22no—
609C2×C3—22no—
610C2×C3—22no—
611C2×C3—21no—
612C2×C3—21no—
613C2×C3—21no—
614C2×C3—22no—
615C2×C3—22no—
616C2×C3—22no—
617C2×C3—42no—
618C2×C3—42no—
619C2×C3—44no—
620C2×C3—41no—
621C2×C3—42no—
622C2×C3—362no—
623C2×C3#6382712yescomplemented
624C2×C3#6383112yescomplemented
625C2×C3#6383312yescomplemented
626C2×C3C22×#62511yescharacteristic
627C2×C3C22×#62611yescharacteristic
628C2×C3C22×#62611yescharacteristic
629D3—124no—
630D3—122no—
631D3—122no—
632C22—22no—
633C22—182no—
634C22—362no—
635C22C2×#629512yes—
636C22C2×#629511yescharacteristic
637C4—182no—
638C4—184no—
639C4C2×#629512yes—
640C4C2×#629511yescharacteristic
641C4C22×#111311yescharacteristic
642C3—21no—
643C3—21no—
644C3—21no—
645C3—41no—
646C3C2×#6382711yescharacteristic, complemented
647C3C2×#6383111yescharacteristic, complemented
648C3C2×#6383311yescharacteristic, complemented
649C2—22no—
650C2—362no—
651C2#1172366312yesdirect product factor
652C2C22×#629511yescharacteristic
subgroupG/H
centerC22C2×#6295
commutator, cosocleC33×C4C24
FrattiniC4C22×#1113
FittingC22×C33×C4C22
socleC22×C33C2×D4
prime radicalC2×C3×#3828C2
hypercenterC22×C4#1113
hypocenterC33C2×#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×#634C2semidirect
3. C2×C3×#3748C2semidirect
4. C2×#88812C2semidirect
5. #11723663C2direct (N × H)
6. C2×#88811C2semidirect
21. C2×C3×#634C22semidirect
22. C3×#3748C22semidirect
24. #88812C22semidirect
25. #88811C22semidirect
27. C22×#634D3semidirect
34. C2×#3748D3semidirect
86. C2×#634C2×D3semidirect
120. #3748C2×D3semidirect
258. C2×C33#137semidirect
390. C33C2×#137semidirect
497. C2×C32#909semidirect
498. C2×C32#942semidirect
499. C2×C32#951semidirect
585. C32C2×#909semidirect
586. C32C2×#942semidirect
587. C32C2×#951semidirect
623. C2×C3#63827semidirect
624. C2×C3#63831semidirect
625. C2×C3#63833semidirect
646. C3C2×#63827semidirect
647. C3C2×#63831semidirect
648. C3C2×#63833semidirect
651. C2#11723663direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C22×C3×#6342
C2×C3×#37482
C2×#888122
#117236632
C2×#888112
C2×#638333
C2×#638313
C2×#638273