Finite Group Databasebeta

← C3×#10596510 #13181029

Subgroups — C3×#10596510 #13181029

Order 1728.

1154 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64306C211yesmaximal, characteristic, complemented
2C2×C3×C4×D32C211yesmaximal, characteristic, complemented
3C3×#64286C211yesmaximal, characteristic, complemented
4C3×#64241C212yesmaximal, complemented
5C3×#64233C212yesmaximal, complemented
6C3×#64230C212yesmaximal, complemented
7C3×C4×GO+(4,𝔽2)C214yesmaximal, complemented
8C3×D32⋊Q8C212yesmaximal, complemented
9#10596510C311yesmaximal, characteristic, direct product factor
10C22×C3×D32C2211yescharacteristic
11C2×C3×GO+(4,𝔽2)C2214yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C32×#235—21no—
14C2×C32×C4×D3—21no—
15C3×#3811C2212yes—
16C2×C3×#624C2211yescharacteristic
17C3×#3805—21no—
18C2×C3×D3×Dic3—21no—
19C3×#3802C2211yescharacteristic, complemented
20C3×C4×D32—21no—
21C3×C4×D32—21no—
22C3×C4×D32C2212yes—
23C3×C4×D32C2212yes—
24C3×C4×D32C2212yescomplemented
25C3×#3798—21no—
26C3×#3797C2211yescharacteristic, complemented
27C3×#3796—21no—
28C3×C22⋊#157C2212yes—
29C3×C4⋊#157C2212yescomplemented
30C3×C4×#157C2212yescomplemented
31C3×#3773C2214yescomplemented
32C3×D32⋊C4C2214yes—
33C3×D32⋊C4C2214yescomplemented
34C3×D3×#41—61no—
35C22×C3×C4×D3—61no—
36#64306C2×C311yescharacteristic, complemented
37C2×C4×D32C2×C311yescharacteristic, complemented
38#64286C2×C311yescharacteristic, complemented
39#64241C2×C312yescomplemented
40#64233C2×C312yescomplemented
41#64230C2×C312yescomplemented
42C4×GO+(4,𝔽2)C2×C314yescomplemented
43D32⋊Q8C2×C312yescomplemented
44C22×C3×C3⋊D3C2311yescharacteristic
45C22×C32×D3—21no—
46C2×C3×D32—21no—
47C2×C3×D32—24no—
48C2×C3×D32C2312yes—
49C2×C3×D32C2312yes—
50C2×C3×#157C2314yes—
51C3×GO+(4,𝔽2)—28no—
52C2×C33×C4D411yescharacteristic, complemented
53C2×C3×C3⋊Dic3D411yescharacteristic, complemented
54C3×C4×C3⋊D3D411yescharacteristic, complemented
55C3×C4×C3⋊D3D411yescharacteristic, complemented
56C3×C4×C3⋊D3C2311yescharacteristic
57C3×C4×C3⋊D3C2311yescharacteristic
58C32×#67—22no—
59C2×C32×Dic3—21no—
60C32×D12—21no—
61C32×C4×D3—22no—
62C32×C4×D3—22no—
63C32×C4×D3—22no—
64C32×Dic6—21no—
65C3×#627—21no—
66C3×#626—22no—
67C3×#625—21no—
68C3×#624—21no—
69C3×#624C2312yes—
70C3×#624C2312yes—
71C3×D3×Dic3—22no—
72C3×D3×Dic3—22no—
73C3×D3×Dic3—22no—
74C3×#525—91nomaximal
75C23×C3×D3—61no—
76C22×D32C22×C311yescharacteristic
77C2×GO+(4,𝔽2)C22×C314yes—
78C32×#41—61no—
79C22×C32×C4—61no—
80C2×C4×C3⋊D3C22×C311yescharacteristic
81C22×C3×Dic3—61no—
82C3×#239—61no—
83C3×D3×Q8—61no—
84C3×#237—61no—
85C3×#237—62no—
86C3×D3×D4—61no—
87C3×D3×D4—62no—
88C3×#235—21no—
89C3×#235—21no—
90C3×#235—41no—
91C3×#235—62no—
92C2×C3×C4×D3—21no—
93C2×C3×C4×D3—21no—
94C2×C3×C4×D3—41no—
95C2×C3×C4×D3—62no—
96C2×C3×C4×D3—62no—
97C2×C3×C4×D3—62no—
98C2×C3×C4×D3—62no—
99C2×C3×C4×D3—62no—
100C2×C3×C4×D3—61no—
101C2×C3×C4×D3—61no—
102C2×C3×C4×D3—61no—
103C2×C3×C4×D3—61no—
104#3811C22×C312yes—
105C2×#624C22×C311yescharacteristic
106#3805—21no—
107C2×D3×Dic3—21no—
108#3802C22×C311yescharacteristic, complemented
109C4×D32—21no—
110C4×D32—21no—
111C4×D32C22×C312yes—
112C4×D32C22×C312yes—
113C4×D32C22×C312yescomplemented
114#3798—21no—
115#3797C22×C311yescharacteristic, complemented
116#3796—21no—
117C22⋊#157C22×C312yes—
118C4⋊#157C22×C312yescomplemented
119C4×#157C22×C312yescomplemented
120#3773C22×C314yescomplemented
121D32⋊C4C22×C314yes—
122D32⋊C4C22×C314yescomplemented
123C22×C33C2×D411yescharacteristic
124C2×C3×C3⋊D3—22no—
125C2×C3×C3⋊D3C2×D411yescharacteristic
126C2×C3×C3⋊D3C2411yescharacteristic
127C2×C32×D3—24no—
128C2×C32×D3—22no—
129C2×C32×D3—22no—
130C3×D32—24no—
131C3×D32—22no—
132C3×D32—24no—
133C3×D32#4114yescomplemented
134C3×#157—24no—
135C33×C4C2×D411yescharacteristic
136C33×C4C2×D411yescharacteristic
137C3×C3⋊Dic3C2×D411yescharacteristic
138C3×C3⋊Dic3C2×D411yescharacteristic
139C32×Dic3—22no—
140C32×Dic3—22no—
141C2×C3×#41—91no—
142C23×C3×C4—91no—
143D3×#41—61no—
144C22×C4×D3—61no—
145C3×#123—92no—
146C3×C22⋊Q8—92no—
147C3×#121—92no—
148C3×#120—92no—
149C3×C4×D4—94no—
150C3×#117—91no—
151C23×C32—61no—
152C22×C3⋊D3C23×C311yescharacteristic
153C22×C3×D3—21no—
154C22×C3×D3—21no—
155C22×C3×D3—41no—
156C22×C3×D3—62no—
157C22×C3×D3—62no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—68no—
161C22×C3×D3—62no—
162C2×D32—21no—
163C2×D32—24no—
164C2×D32C23×C312yes—
165C2×D32C23×C312yes—
166C2×#157C23×C314yes—
167GO+(4,𝔽2)—28no—
168C32×Q8—61no—
169C32×D4—61no—
170C32×D4—62no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—41no—
174C2×C32×C4—41no—
175C2×C32×C4—62no—
176C2×C32×C4—62no—
177C2×C32×C4—62no—
178C2×C32×C4—61no—
179C2×C32×C4C3×D411yescharacteristic, complemented
180C2×C3⋊Dic3C3×D411yescharacteristic, complemented
181C4×C3⋊D3C3×D411yescharacteristic, complemented
182C4×C3⋊D3C3×D411yescharacteristic, complemented
183C4×C3⋊D3C23×C311yescharacteristic
184C4×C3⋊D3C23×C311yescharacteristic
185C3×#67—22no—
186C3×#67—22no—
187C3×#67—42no—
188C3×#67—62no—
189C3×#67—62no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—41no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—62no—
198C3×D12—21no—
199C3×D12—21no—
200C3×D12—41no—
201C3×D12—62no—
202C3×C4×D3—22no—
203C3×C4×D3—22no—
204C3×C4×D3—22no—
205C3×C4×D3—22no—
206C3×C4×D3—22no—
207C3×C4×D3—22no—
208C3×C4×D3—42no—
209C3×C4×D3—42no—
210C3×C4×D3—42no—
211C3×C4×D3—62no—
212C3×C4×D3—62no—
213C3×C4×D3—61no—
214C3×C4×D3—61no—
215C3×C4×D3—61no—
216C3×C4×D3—61no—
217C3×C4×D3—61no—
218C3×C4×D3—61no—
219C3×C4×D3—61no—
220C3×C4×D3—61no—
221C3×C4×D3—62no—
222C3×C4×D3—62no—
223C3×C4×D3—62no—
224C3×C4×D3—62no—
225C3×Dic6—21no—
226C3×Dic6—21no—
227C3×Dic6—41no—
228C3×Dic6—62no—
229#627—21no—
230#626—22no—
231#625—21no—
232#624—21no—
233#624C23×C312yes—
234#624C23×C312yes—
235D3×Dic3—22no—
236D3×Dic3—22no—
237D3×Dic3—22no—
238#525—91noSylow
239C2×C33—21no—
240C2×C33C22×D411yescharacteristic
241C3×C3⋊D3—21no—
242C3×C3⋊D3C2×#4112yes—
243C32×D3—24no—
244C32×D3—42no—
245C24×C3—91no—
246C23×D3—61no—
247C3×#41—61no—
248C3×#41—61no—
249C3×#41—121no—
250C3×#41—181no—
251C3×#41—181no—
252C3×#41—181no—
253C2×C3×Q8—91no—
254C2×C3×D4—92no—
255C2×C3×D4—91no—
256C2×C3×D4—94no—
257C22×C3×C4—61no—
258C22×C3×C4—61no—
259C22×C3×C4—92no—
260C22×C3×C4—92no—
261C22×C3×C4—92no—
262C22×C3×C4—91no—
263C22×C3×C4—91no—
264C22×C3×C4—121no—
265C22×C3×C4—181no—
266C22×C3×C4—181no—
267C22×C3×C4—181no—
268C22×Dic3—61no—
269#239—61no—
270D3×Q8—61no—
271#237—61no—
272#237—62no—
273D3×D4—61no—
274D3×D4—62no—
275#235—21no—
276#235—62no—
277C2×C4×D3—21no—
278C2×C4×D3—62no—
279C2×C4×D3—62no—
280C2×C4×D3—62no—
281C2×C4×D3—61no—
282C2×C4×D3—61no—
283C2×C4×D3—61no—
284C2×C4×D3—62no—
285C2×C4×D3—62no—
286C2×C4×D3—61no—
287C3×C4⋊C4—94no—
288C3×C4⋊C4—92no—
289C3×C22⋊C4—94no—
290C3×C22⋊C4—94no—
291C3×C22⋊C4—92no—
292C3×C42—92no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—41no—
296C22×C32—41no—
297C22×C32—62no—
298C22×C32—64no—
299C22×C32—62no—
300C22×C32C2×C3×D411yescharacteristic
301C2×C3⋊D3—22no—
302C2×C3⋊D3C2×C3×D411yescharacteristic
303C2×C3⋊D3C24×C311yescharacteristic
304C2×C3×D3—22no—
305C2×C3×D3—24no—
306C2×C3×D3—24no—
307C2×C3×D3—22no—
308C2×C3×D3—22no—
309C2×C3×D3—22no—
310C2×C3×D3—42no—
311C2×C3×D3—44no—
312C2×C3×D3—42no—
313C2×C3×D3—64no—
314C2×C3×D3—61no—
315C2×C3×D3—61no—
316C2×C3×D3—61no—
317C2×C3×D3—61no—
318C2×C3×D3—64no—
319C2×C3×D3—68no—
320C2×C3×D3—64no—
321C2×C3×D3—124no—
322C2×C3×D3—122no—
323D32—24no—
324D32—22no—
325D32—24no—
326D32C3×#4114yescomplemented
327#157—24no—
328C32×C4—21no—
329C32×C4—21no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—41no—
333C32×C4—41no—
334C32×C4—41no—
335C32×C4—41no—
336C32×C4—62no—
337C32×C4—62no—
338C32×C4C2×C3×D411yescharacteristic
339C32×C4C2×C3×D411yescharacteristic
340C3⋊Dic3C2×C3×D411yescharacteristic
341C3⋊Dic3C2×C3×D411yescharacteristic
342C3×Dic3—22no—
343C3×Dic3—22no—
344C3×Dic3—22no—
345C3×Dic3—22no—
346C3×Dic3—42no—
347C3×Dic3—42no—
348C3×Dic3—61no—
349C3×Dic3—61no—
350C3×Dic3—61no—
351C3×Dic3—61no—
352C2×#41—91no—
353C23×C4—91no—
354#123—92no—
355C22⋊Q8—92no—
356#121—92no—
357#120—92no—
358C4×D4—94no—
359#117—91no—
360C33#52511yescharacteristic, complemented, Sylow
361C23×C3—61no—
362C23×C3—61no—
363C23×C3—92no—
364C23×C3—92no—
365C23×C3—91no—
366C23×C3—121no—
367C23×C3—181no—
368C23×C3—184no—
369C22×D3—21no—
370C22×D3—62no—
371C22×D3—62no—
372C22×D3—61no—
373C22×D3—61no—
374C22×D3—62no—
375C22×D3—68no—
376C3×Q8—61no—
377C3×Q8—61no—
378C3×Q8—121no—
379C3×Q8—181no—
380C3×D4—61no—
381C3×D4—62no—
382C3×D4—62no—
383C3×D4—61no—
384C3×D4—121no—
385C3×D4—122no—
386C3×D4—182no—
387C3×D4—181no—
388C3×D4—188no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—62no—
394C2×C3×C4—62no—
395C2×C3×C4—62no—
396C2×C3×C4—61no—
397C2×C3×C4—61no—
398C2×C3×C4—62no—
399C2×C3×C4—62no—
400C2×C3×C4—62no—
401C2×C3×C4—92no—
402C2×C3×C4—92no—
403C2×C3×C4—91no—
404C2×C3×C4—91no—
405C2×C3×C4—91no—
406C2×C3×C4—91no—
407C2×C3×C4—91no—
408C2×C3×C4—94no—
409C2×C3×C4—122no—
410C2×C3×C4—122no—
411C2×C3×C4—122no—
412C2×C3×C4—121no—
413C2×C3×C4—181no—
414C2×C3×C4—182no—
415C2×C3×C4—182no—
416C2×C3×C4—182no—
417C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
418#67—22no—
419#67—62no—
420#67—62no—
421C2×Dic3—21no—
422C2×Dic3—61no—
423C2×Dic3—61no—
424C2×Dic3—62no—
425C2×Dic3—62no—
426C2×Dic3—62no—
427D12—21no—
428D12—62no—
429C4×D3—22no—
430C4×D3—22no—
431C4×D3—22no—
432C4×D3—62no—
433C4×D3—62no—
434C4×D3—61no—
435C4×D3—61no—
436C4×D3—61no—
437C4×D3—61no—
438C4×D3—61no—
439C4×D3—61no—
440C4×D3—61no—
441C4×D3—61no—
442C4×D3—62no—
443C4×D3—62no—
444C4×D3—62no—
445C4×D3—62no—
446Dic6—21no—
447Dic6—62no—
448C2×C32—22no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—41no—
453C2×C32—41no—
454C2×C32—41no—
455C2×C32—42no—
456C2×C32—64no—
457C2×C32—81no—
458C2×C32—122no—
459C2×C32C22×C3×D411yescharacteristic
460C3⋊D3—21no—
461C3⋊D3C2×C3×#4112yes—
462C3×D3—24no—
463C3×D3—24no—
464C3×D3—44no—
465C3×D3—42no—
466C3×D3—42no—
467C3×D3—62no—
468C3×D3—62no—
469C3×D3—62no—
470C3×D3—82no—
471C3×D3—121no—
472C24—91no—
473#41—61no—
474#41—181no—
475#41—181no—
476#41—181no—
477C2×Q8—91no—
478C2×D4—92no—
479C2×D4—91no—
480C2×D4—94no—
481C22×C4—61no—
482C22×C4—92no—
483C22×C4—92no—
484C22×C4—92no—
485C22×C4—91no—
486C22×C4—91no—
487C22×C4—181no—
488C22×C4—181no—
489C22×C4—181no—
490C4⋊C4—92no—
491C4⋊C4—94no—
492C22⋊C4—92no—
493C22⋊C4—94no—
494C22⋊C4—94no—
495C42—92no—
496C22×C3—21no—
497C22×C3—21no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—62no—
501C22×C3—64no—
502C22×C3—64no—
503C22×C3—62no—
504C22×C3—62no—
505C22×C3—62no—
506C22×C3—94no—
507C22×C3—91no—
508C22×C3—91no—
509C22×C3—122no—
510C22×C3—124no—
511C22×C3—122no—
512C22×C3—182no—
513C22×C3—182no—
514C22×C3—184no—
515C22×C3—184no—
516C22×C3C2×GO+(4,𝔽2)11yescharacteristic
517C2×D3—22no—
518C2×D3—24no—
519C2×D3—22no—
520C2×D3—64no—
521C2×D3—61no—
522C2×D3—61no—
523C2×D3—61no—
524C2×D3—61no—
525C2×D3—64no—
526C2×D3—68no—
527C2×D3—64no—
528C2×D3—124no—
529C2×D3—122no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—41no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—41no—
538C3×C4—62no—
539C3×C4—62no—
540C3×C4—62no—
541C3×C4—62no—
542C3×C4—91no—
543C3×C4—91no—
544C3×C4—122no—
545C3×C4—122no—
546C3×C4—184no—
547C3×C4C2×GO+(4,𝔽2)11yescharacteristic
548C3×C4C2×GO+(4,𝔽2)11yescharacteristic
549Dic3—22no—
550Dic3—22no—
551Dic3—61no—
552Dic3—61no—
553Dic3—61no—
554Dic3—61no—
555C32—21no—
556C32—21no—
557C32—41no—
558C32—41no—
559C32C3×#52511yescharacteristic, complemented
560C23—61no—
561C23—92no—
562C23—92no—
563C23—91no—
564C23—181no—
565C23—184no—
566Q8—61no—
567Q8—181no—
568D4—62no—
569D4—61no—
570D4—182no—
571D4—181no—
572D4—188no—
573C2×C4—61no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—92no—
578C2×C4—92no—
579C2×C4—91no—
580C2×C4—91no—
581C2×C4—91no—
582C2×C4—91no—
583C2×C4—91no—
584C2×C4—94no—
585C2×C4—181no—
586C2×C4—182no—
587C2×C4—182no—
588C2×C4—182no—
589C2×C4C3×GO+(4,𝔽2)11yescharacteristic, complemented
590C2×C3—22no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—41no—
595C2×C3—42no—
596C2×C3—41no—
597C2×C3—41no—
598C2×C3—64no—
599C2×C3—64no—
600C2×C3—81no—
601C2×C3—92no—
602C2×C3—124no—
603C2×C3—122no—
604C2×C3—122no—
605C2×C3—181no—
606C2×C3—242no—
607C2×C3C22×GO+(4,𝔽2)11yescharacteristic
608D3—24no—
609D3—42no—
610D3—62no—
611D3—62no—
612D3—62no—
613D3—121no—
614C22—64no—
615C22—62no—
616C22—62no—
617C22—94no—
618C22—91no—
619C22—91no—
620C22—182no—
621C22—182no—
622C22—184no—
623C22—184no—
624C22C2×C3×GO+(4,𝔽2)11yescharacteristic
625C4—62no—
626C4—62no—
627C4—91no—
628C4—91no—
629C4—184no—
630C4C2×C3×GO+(4,𝔽2)11yescharacteristic
631C4C2×C3×GO+(4,𝔽2)11yescharacteristic
632C3—21no—
633C3—21no—
634C3—41no—
635C3—41no—
636C3#1059651011yescharacteristic, direct product factor
637C2—21no—
638C2—64no—
639C2—92no—
640C2—122no—
641C2—181no—
642C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC3×C4C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC2×C33C22×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#525
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#64306C2semidirect
2. C2×C3×C4×D32C2semidirect
3. C3×#64286C2semidirect
4. C3×#64241C2semidirect
5. C3×#64233C2semidirect
6. C3×#64230C2semidirect
7. C3×C4×GO+(4,𝔽2)C2semidirect
8. C3×D32⋊Q8C2semidirect
9. #10596510C3direct (N × H)
19. C3×#3802C22semidirect
24. C3×C4×D32C22semidirect
26. C3×#3797C22semidirect
29. C3×C4⋊#157C22semidirect
30. C3×C4×#157C22semidirect
31. C3×#3773C22semidirect
33. C3×D32⋊C4C22semidirect
36. #64306C2×C3semidirect
37. C2×C4×D32C2×C3semidirect
38. #64286C2×C3semidirect
39. #64241C2×C3semidirect
40. #64233C2×C3semidirect
41. #64230C2×C3semidirect
42. C4×GO+(4,𝔽2)C2×C3semidirect
43. D32⋊Q8C2×C3semidirect
52. C2×C33×C4D4semidirect
53. C2×C3×C3⋊Dic3D4semidirect
54. C3×C4×C3⋊D3D4semidirect
55. C3×C4×C3⋊D3D4semidirect
108. #3802C22×C3semidirect
113. C4×D32C22×C3semidirect
115. #3797C22×C3semidirect
118. C4⋊#157C22×C3semidirect
119. C4×#157C22×C3semidirect
120. #3773C22×C3semidirect
122. D32⋊C4C22×C3semidirect
133. C3×D32#41semidirect
179. C2×C32×C4C3×D4semidirect
180. C2×C3⋊Dic3C3×D4semidirect
181. C4×C3⋊D3C3×D4semidirect
182. C4×C3⋊D3C3×D4semidirect
326. D32C3×#41semidirect
360. C33#525semidirect
417. C2×C3×C4GO+(4,𝔽2)semidirect
559. C32C3×#525semidirect
589. C2×C4C3×GO+(4,𝔽2)semidirect
636. C3#10596510direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643062
C2×C3×C4×D322
C3×#642862
C3×#642412
C3×#642332
C3×#642302
C3×C4×GO+(4,𝔽2)2
C3×D32⋊Q82
#105965103
C3×#5259