Finite Group Databasebeta

← C2×#11722075 #13161808

Subgroups — C2×#11722075 #13161808

Order 1728.

1266 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4#11722075C214yesmaximal, direct product factor
5C2×#63564—31nomaximal
6C2×#63563—31nomaximal
7C2×#63562—31nomaximal
8C23×C3×C3⋊D3—21no—
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3—21no—
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)—22no—
13C2×C3×C4⋊(C3⋊D3)C414yescomplemented
14C2×C3×C4⋊(C3⋊D3)C2212yes—
15C3×C4⋊(C3⋊Dic3)—21no—
16C3×C4⋊(C3⋊Dic3)C2212yescomplemented
17C2×C3×C3⋊(C3⋊C8)—22no—
18C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
19C22×C4⋊(C3⋊D3)D311yescharacteristic
20C22×C3×D12—31no—
21C22×C3×D12—31no—
22C22×C3×D12—61no—
23C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
24C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
25C2×C3×C4⋊Dic3—31no—
26C2×C3×C4⋊Dic3—31no—
27C2×C3×C4⋊Dic3—61no—
28C22×C3×C3⋊C8—31no—
29C22×C3×C3⋊C8—31no—
30C22×C3×C3⋊C8—61no—
31#63564—34no—
32#63563—34no—
33#63562—34no—
34C23×C33D411yescharacteristic
35C22×C3×C3⋊D3—24no—
36C22×C3×C3⋊D3—22no—
37C22×C3×C3⋊D3—42no—
38C22×C3×C3⋊D3—42no—
39C2×C33×C4C2×C412yes—
40C2×C33×C4D411yescharacteristic
41C2×C33×C4D412yes—
42C2×C33×C4C2311yescharacteristic
43C2×C3×C3⋊Dic3—22no—
44C2×C3×C3⋊Dic3—42no—
45C3×C4⋊(C3⋊D3)—24no—
46C3×C4⋊(C3⋊D3)—22no—
47C3×C4⋊(C3⋊D3)C2×C414yescomplemented
48C3×C3⋊(C3⋊C8)—22no—
49C2×#832—91no—
50C2×#821—91no—
51C2×#809—91no—
52C23×C3⋊D3—21no—
53C23×C3×D3—62no—
54C23×C3×D3—61no—
55C23×C3×D3—61no—
56C22×C32×C4—21no—
57C22×C32×C4—21no—
58C22×C32×C4—21no—
59C22×C32×C4—41no—
60C22×C32×C4C2×D311yescharacteristic
61C22×C32×C4C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C22×C3⋊Dic3—21no—
64C2×C4⋊(C3⋊D3)—21no—
65C2×C4⋊(C3⋊D3)—22no—
66C2×C4⋊(C3⋊D3)Dic314yescomplemented
67C2×C4⋊(C3⋊D3)C2×D312yes—
68C22×C3×Dic3—62no—
69C22×C3×Dic3—61no—
70C22×C3×Dic3—61no—
71C2×C3×D12—34no—
72C2×C3×D12—34no—
73C2×C3×D12—32no—
74C2×C3×D12—32no—
75C2×C3×D12—64no—
76C2×C3×D12—62no—
77C2×C3×D12—62no—
78C2×C3×D12—64no—
79C2×C3×D12—61no—
80C2×C3×D12—62no—
81C2×C3×D12—61no—
82C2×C3×D12—62no—
83C4⋊(C3⋊Dic3)—21no—
84C4⋊(C3⋊Dic3)C2×D312yescomplemented
85C2×C3⋊(C3⋊C8)—22no—
86C2×C3⋊(C3⋊C8)C2×D312yescomplemented
87C3×C4⋊Dic3—32no—
88C3×C4⋊Dic3—32no—
89C3×C4⋊Dic3—62no—
90C3×C4⋊Dic3—62no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C2×C3×C3⋊C8—32no—
94C2×C3×C3⋊C8—32no—
95C2×C3×C3⋊C8—62no—
96C2×C3×C3⋊C8—64no—
97C2×C3×C3⋊C8—62no—
98C2×C3×C3⋊C8—62no—
99C22×C33C22⋊C412yes—
100C22×C33D812yes—
101C22×C33SD1612yes—
102C22×C33C2×D411yescharacteristic
103C2×C3×C3⋊D3—24no—
104C2×C3×C3⋊D3—48no—
105C2×C3×C3⋊D3—42no—
106C2×C3×C3⋊D3—42no—
107C33×C4C22⋊C412yes—
108C33×C4C22×C411yescharacteristic
109C33×C4C2×D411yescharacteristic
110C3×C3⋊Dic3—42no—
111C22×C3×D4—91no—
112C22×D12—31no—
113C22×D12—31no—
114C22×D12—61no—
115C22×C3×C8—91no—
116C2×C3×C4⋊C4—91no—
117C2×C4⋊Dic3—31no—
118C2×C4⋊Dic3—31no—
119C2×C4⋊Dic3—61no—
120C22×C3⋊C8—31no—
121C22×C3⋊C8—31no—
122C22×C3⋊C8—61no—
123#832—94no—
124#821—94no—
125#809—94no—
126C23×C32—21no—
127C23×C32—21no—
128C23×C32—21no—
129C23×C32—41no—
130C23×C32D1211yescharacteristic
131C23×C32#6711yescharacteristic
132C23×C32#6711yescharacteristic
133C22×C3⋊D3—24no—
134C22×C3⋊D3—22no—
135C22×C3⋊D3—42no—
136C22×C3⋊D3—42no—
137C22×C3×D3—68no—
138C22×C3×D3—64no—
139C22×C3×D3—64no—
140C22×C3×D3—64no—
141C22×C3×D3—62no—
142C22×C3×D3—62no—
143C22×C3×D3—124no—
144C22×C3×D3—124no—
145C22×C3×D3—122no—
146C22×C3×D3—122no—
147C22×C3×D3—122no—
148C22×C3×D3—122no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—21no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—22no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—42no—
162C2×C32×C4—42no—
163C2×C32×C4—41no—
164C2×C32×C4—41no—
165C2×C32×C4C4×D312yes—
166C2×C32×C4C4×D312yes—
167C2×C32×C4D1211yescharacteristic
168C2×C32×C4D1212yes—
169C2×C32×C4C2×Dic312yes—
170C2×C32×C4#6711yescharacteristic
171C2×C32×C4#6712yes—
172C2×C32×C4#6711yescharacteristic
173C2×C32×C4#6712yes—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C3⋊Dic3—22no—
178C2×C3⋊Dic3—42no—
179C4⋊(C3⋊D3)—24no—
180C4⋊(C3⋊D3)—22no—
181C4⋊(C3⋊D3)C2×Dic314yescomplemented
182C2×C3×Dic3—64no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—62no—
185C2×C3×Dic3—124no—
186C2×C3×Dic3—122no—
187C2×C3×Dic3—122no—
188C3×D12—34no—
189C3×D12—34no—
190C3×D12—64no—
191C3×D12—68no—
192C3×D12—64no—
193C3×D12—64no—
194C3×D12—64no—
195C3×D12—62no—
196C3×D12—62no—
197C3⋊(C3⋊C8)—22no—
198C3×C3⋊C8—64no—
199C3×C3⋊C8—62no—
200C3×C3⋊C8—62no—
201C2×#102—271noSylow
202C2×C33#10214yescomplemented
203C2×C33C2×C22⋊C411yescharacteristic
204C2×C33C2×D811yescharacteristic
205C2×C33C2×SD1611yescharacteristic
206C3×C3⋊D3—44no—
207C24×C3—181no—
208C23×D3—62no—
209C23×D3—61no—
210C23×D3—61no—
211C2×C3×D4—94no—
212C2×C3×D4—92no—
213C2×C3×D4—181no—
214C2×C3×D4—182no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—41no—
219C22×C3×C4—181no—
220C22×C3×C4D3211yescharacteristic
221C22×C3×C4D3211yescharacteristic
222C22×C3×C4D3211yescharacteristic
223C22×Dic3—62no—
224C22×Dic3—61no—
225C22×Dic3—61no—
226C2×D12—34no—
227C2×D12—34no—
228C2×D12—32no—
229C2×D12—32no—
230C2×D12—64no—
231C2×D12—62no—
232C2×D12—64no—
233C2×D12—62no—
234C2×D12—61no—
235C2×D12—62no—
236C2×D12—61no—
237C2×D12—62no—
238C2×C3×C8—92no—
239C2×C3×C8—182no—
240C3×C4⋊C4—92no—
241C3×C4⋊C4—181no—
242C4⋊Dic3—32no—
243C4⋊Dic3—32no—
244C4⋊Dic3—62no—
245C4⋊Dic3—62no—
246C4⋊Dic3—61no—
247C4⋊Dic3—61no—
248C2×C3⋊C8—32no—
249C2×C3⋊C8—32no—
250C2×C3⋊C8—62no—
251C2×C3⋊C8—64no—
252C2×C3⋊C8—62no—
253C2×C3⋊C8—62no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32—42no—
267C22×C32—42no—
268C22×C32—42no—
269C22×C32—41no—
270C22×C32#20412yes—
271C22×C32D2412yes—
272C22×C32(C2×D3)⋊C412yes—
273C22×C32(C2×D3)⋊C412yes—
274C22×C32#21312yes—
275C22×C32#21312yes—
276C22×C32#21412yes—
277C22×C32#21512yes—
278C22×C32#21712yes—
279C22×C32C2×D1211yescharacteristic
280C22×C32C2×#6711yescharacteristic
281C22×C32C2×#6711yescharacteristic
282C2×C3⋊D3—24no—
283C2×C3⋊D3—48no—
284C2×C3⋊D3—42no—
285C2×C3⋊D3—42no—
286C2×C3×D3—68no—
287C2×C3×D3—64no—
288C2×C3×D3—64no—
289C2×C3×D3—1216no—
290C2×C3×D3—124no—
291C2×C3×D3—124no—
292C2×C3×D3—128no—
293C2×C3×D3—128no—
294C2×C3×D3—122no—
295C2×C3×D3—122no—
296C2×C3×D3—122no—
297C2×C3×D3—122no—
298C32×C4—22no—
299C32×C4—22no—
300C32×C4—22no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—42no—
308C32×C4—41no—
309C32×C4—41no—
310C32×C4(C2×D3)⋊C412yes—
311C32×C4(C2×D3)⋊C412yes—
312C32×C4#21712yes—
313C32×C4C2×C4×D311yescharacteristic
314C32×C4C2×C4×D311yescharacteristic
315C32×C4C2×D1211yescharacteristic
316C32×C4C22×Dic311yescharacteristic
317C32×C4C2×#6711yescharacteristic
318C32×C4C2×#6711yescharacteristic
319C3⋊Dic3—42no—
320C3×Dic3—124no—
321C3×Dic3—122no—
322C3×Dic3—122no—
323C22×D4—91no—
324C22×C8—91no—
325C2×C4⋊C4—91no—
326#102—274no—
327C33C2×#10211yescharacteristic, complemented, Sylow
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—41no—
332C23×C3—184no—
333C23×C3—182no—
334C23×C3—362no—
335C23×C3—362no—
336C23×C3#62511yescharacteristic
337C23×C3#62611yescharacteristic
338C23×C3#62611yescharacteristic
339C22×D3—64no—
340C22×D3—68no—
341C22×D3—64no—
342C22×D3—64no—
343C22×D3—62no—
344C22×D3—62no—
345C22×D3—124no—
346C22×D3—124no—
347C22×D3—122no—
348C22×D3—122no—
349C22×D3—122no—
350C22×D3—122no—
351C3×D4—94no—
352C3×D4—184no—
353C3×D4—182no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—182no—
371C2×C3×C4—362no—
372C2×C3×C4D3×Dic312yes—
373C2×C3×C4D3×Dic312yes—
374C2×C3×C4#62412yes—
375C2×C3×C4#62511yescharacteristic
376C2×C3×C4#62512yes—
377C2×C3×C4#62611yescharacteristic
378C2×C3×C4#62612yes—
379C2×C3×C4#62611yescharacteristic
380C2×C3×C4#62612yes—
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384C2×Dic3—64no—
385C2×Dic3—62no—
386C2×Dic3—62no—
387C2×Dic3—124no—
388C2×Dic3—122no—
389C2×Dic3—122no—
390D12—34no—
391D12—34no—
392D12—64no—
393D12—64no—
394D12—68no—
395D12—64no—
396D12—64no—
397D12—62no—
398D12—62no—
399C3×C8—182no—
400C3⋊C8—64no—
401C3⋊C8—62no—
402C3⋊C8—62no—
403C2×C32—24no—
404C2×C32—24no—
405C2×C32—24no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—44no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32#80914yescomplemented
420C2×C32#82114yescomplemented
421C2×C32#83214yescomplemented
422C2×C32C2×#20411yescharacteristic
423C2×C32C2×D2411yescharacteristic
424C2×C32C2×(C2×D3)⋊C411yescharacteristic
425C2×C32C2×(C2×D3)⋊C411yescharacteristic
426C2×C32C2×#21311yescharacteristic
427C2×C32C2×#21311yescharacteristic
428C2×C32C2×#21411yescharacteristic
429C2×C32C2×#21511yescharacteristic
430C2×C32C2×#21711yescharacteristic
431C3⋊D3—44no—
432C3×D3—128no—
433C3×D3—124no—
434C3×D3—124no—
435C24—181no—
436C2×D4—94no—
437C2×D4—92no—
438C2×D4—182no—
439C2×D4—181no—
440C22×C4—181no—
441C22×C4#111311yescharacteristic
442C2×C8—92no—
443C2×C8—182no—
444C4⋊C4—92no—
445C4⋊C4—181no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—41no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—42no—
462C22×C3—184no—
463C22×C3—368no—
464C22×C3—362no—
465C22×C3—362no—
466C22×C3#371312yes—
467C22×C3#371412yes—
468C22×C3#371412yes—
469C22×C3#371512yes—
470C22×C3#371612yes—
471C22×C3#371712yes—
472C22×C3(C2×D3)⋊Dic312yes—
473C22×C3(C2×D3)⋊Dic312yes—
474C22×C3#372212yes—
475C22×C3C2×#62511yescharacteristic
476C22×C3C2×#62611yescharacteristic
477C22×C3C2×#62611yescharacteristic
478C2×D3—68no—
479C2×D3—64no—
480C2×D3—64no—
481C2×D3—1216no—
482C2×D3—124no—
483C2×D3—124no—
484C2×D3—128no—
485C2×D3—128no—
486C2×D3—122no—
487C2×D3—122no—
488C2×D3—122no—
489C2×D3—122no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—22no—
497C3×C4—22no—
498C3×C4—22no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—42no—
502C3×C4—362no—
503C3×C4(C2×D3)⋊Dic312yes—
504C3×C4(C2×D3)⋊Dic312yes—
505C3×C4#372212yes—
506C3×C4C2×D3×Dic311yescharacteristic
507C3×C4C2×D3×Dic311yescharacteristic
508C3×C4C2×#62411yescharacteristic
509C3×C4C2×#62511yescharacteristic
510C3×C4C2×#62611yescharacteristic
511C3×C4C2×#62611yescharacteristic
512Dic3—124no—
513Dic3—122no—
514Dic3—122no—
515C32—21no—
516C32—21no—
517C32—21no—
518C32—41no—
519C32C2×#80911yescharacteristic, complemented
520C32C2×#82111yescharacteristic, complemented
521C32C2×#83211yescharacteristic, complemented
522C23—182no—
523C23—184no—
524C23—362no—
525C23—362no—
526C23#629511yescharacteristic
527D4—94no—
528D4—182no—
529D4—184no—
530C2×C4—182no—
531C2×C4—362no—
532C2×C4#629412yes—
533C2×C4#629512yes—
534C2×C4#629511yescharacteristic
535C2×C4C2×#111311yescharacteristic
536C8—182no—
537C2×C3—24no—
538C2×C3—24no—
539C2×C3—24no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—44no—
550C2×C3—41no—
551C2×C3—41no—
552C2×C3—41no—
553C2×C3—364no—
554C2×C3#6356214yescomplemented
555C2×C3#6356314yescomplemented
556C2×C3#6356414yescomplemented
557C2×C3C2×#371311yescharacteristic
558C2×C3C2×#371411yescharacteristic
559C2×C3C2×#371411yescharacteristic
560C2×C3C2×#371511yescharacteristic
561C2×C3C2×#371611yescharacteristic
562C2×C3C2×#371711yescharacteristic
563C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
564C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
565C2×C3C2×#372211yescharacteristic
566D3—128no—
567D3—124no—
568D3—124no—
569C22—184no—
570C22—368no—
571C22—362no—
572C22—362no—
573C22#8881012yes—
574C22#8881112yes—
575C22#8881412yes—
576C22C2×#629511yescharacteristic
577C4—362no—
578C4#8881412yes—
579C4C2×#629411yescharacteristic
580C4C2×#629511yescharacteristic
581C3—21no—
582C3—21no—
583C3—21no—
584C3—41no—
585C3C2×#6356211yescharacteristic, complemented
586C3C2×#6356311yescharacteristic, complemented
587C3C2×#6356411yescharacteristic, complemented
588C2—364no—
589C2#1172207514yesdirect product factor
590C2C2×#8881011yescharacteristic
591C2C2×#8881111yescharacteristic
592C2C2×#8881411yescharacteristic
subgroupG/H
centerC23#6295
commutatorC33×C4C22×C4
FrattiniC2×C4C2×#1113
FittingC22×C33×C4C22
socleC23×C33D4
prime radicalC22×C3×C4⋊(C3⋊D3)C2
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33C2×#102
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
3. C22×C3×C3⋊(C3⋊C8)C2semidirect
4. #11722075C2direct (N × H)
13. C2×C3×C4⋊(C3⋊D3)C4semidirect
16. C3×C4⋊(C3⋊Dic3)C22semidirect
18. C2×C3×C3⋊(C3⋊C8)C22semidirect
23. C2×C4⋊(C3⋊Dic3)D3semidirect
24. C22×C3⋊(C3⋊C8)D3semidirect
47. C3×C4⋊(C3⋊D3)C2×C4semidirect
66. C2×C4⋊(C3⋊D3)Dic3semidirect
84. C4⋊(C3⋊Dic3)C2×D3semidirect
86. C2×C3⋊(C3⋊C8)C2×D3semidirect
181. C4⋊(C3⋊D3)C2×Dic3semidirect
202. C2×C33#102semidirect
327. C33C2×#102semidirect
419. C2×C32#809semidirect
420. C2×C32#821semidirect
421. C2×C32#832semidirect
519. C32C2×#809semidirect
520. C32C2×#821semidirect
521. C32C2×#832semidirect
554. C2×C3#63562semidirect
555. C2×C3#63563semidirect
556. C2×C3#63564semidirect
585. C3C2×#63562semidirect
586. C3C2×#63563semidirect
587. C3C2×#63564semidirect
589. C2#11722075direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊(C3⋊D3)2
C2×C3×C4⋊(C3⋊Dic3)2
C22×C3×C3⋊(C3⋊C8)2
#117220752
C2×#635643
C2×#635633
C2×#635623