Finite Group Databasebeta

← C3×#10595247 #13174746

Subgroups — C3×#10595247 #13174746

Order 1728.

1262 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×#64145C211yesmaximal, characteristic, complemented
2C32×#939C212yesmaximal, complemented
3C3×Dic3×#67C212yesmaximal, complemented
4C3×#63970C212yesmaximal, complemented
5C3×#63959C212yesmaximal, complemented
6C3×#63955C212yesmaximal, complemented
7C3×#63913C211yesmaximal, characteristic, complemented
8C3×#63878C212yesmaximal, complemented
9C3×#63840C211yesmaximal, characteristic, complemented
10C3×#5616—32nomaximal
11#10595247C311yesmaximal, characteristic, direct product factor
12C2×C33×D4C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×#67C2212yes—
15C2×C32×#67C2212yes—
16C22×C32×Dic3C2212yes—
17C2×C3×#627C2211yescharacteristic
18C2×C3×#625C2211yescharacteristic
19C2×C3×D3×Dic3C2212yes—
20C3×#3757C2212yes—
21C3×#3757C2212yes—
22C3×C4×C3⋊Dic3C2211yescharacteristic
23C32×#217C2212yes—
24C32×(C2×D3)⋊C4C2212yes—
25C32×Dic3⋊C4C2212yes—
26C3×Dic3⋊Dic3C2212yes—
27C3×Dic3⋊Dic3C2212yes—
28C3×(C2×D3)⋊Dic3C2212yes—
29C3×(C2×D3)⋊Dic3C2212yes—
30C3×Dic32C2211yescharacteristic
31C3×Dic32C2212yes—
32C3×Dic32C2211yescharacteristic
33C32×#121—32no—
34#64145C2×C311yescharacteristic, complemented
35C3×#939—22no—
36C3×#939—32no—
37C3×#939D312yescomplemented
38C3×#939C2×C312yescomplemented
39C3×#938—32no—
40C3×#936—32no—
41C3×#936—32no—
42C3×#936—61no—
43C3×D4×Dic3—32no—
44C3×C4×#67—32no—
45C3×#926—32no—
46C3×#892—32no—
47C3×#888—32no—
48C3×#885—32no—
49C3×#885—32no—
50C3×#877—32no—
51C3×#877—32no—
52Dic3×#67C2×C312yescomplemented
53#63970C2×C312yescomplemented
54#63959C2×C312yescomplemented
55#63955C2×C312yescomplemented
56#63913C2×C311yescharacteristic, complemented
57#63878C2×C312yescomplemented
58#63840C2×C311yescharacteristic, complemented
59C23×C33C2312yes—
60C22×C32×D3C2312yes—
61C33×D4—41no—
62C2×C33×C4C2311yescharacteristic
63C2×C3×C3⋊Dic3C2312yes—
64C2×C3×C3⋊Dic3C2311yescharacteristic
65C2×C3×C3⋊Dic3C2311yescharacteristic
66C3×#634—41no—
67C32×#67—24no—
68C32×#67—42no—
69C2×C32×Dic3—24no—
70C2×C32×Dic3C2312yes—
71C2×C32×Dic3C2312yes—
72C2×C32×Dic3C2312yes—
73C3×#627—41no—
74C3×#625—41no—
75C3×D3×Dic3—24no—
76C3×#555—91no—
77#5616—32no—
78C2×C32×D4—21no—
79C2×C32×D4—22no—
80C2×C32×D4—32no—
81C2×C32×D4—32no—
82C2×C32×D4—41no—
83C2×C32×D4C2×D312yes—
84C2×C32×D4C22×C311yescharacteristic
85C22×C32×C4—32no—
86C2×#634C22×C311yescharacteristic
87C2×C3×#67—22no—
88C2×C3×#67—22no—
89C2×C3×#67—32no—
90C2×C3×#67C2×D312yes—
91C2×C3×#67C2×D312yes—
92C2×C3×#67C22×C312yes—
93C2×C3×#67C22×C312yes—
94C22×C3×Dic3—22no—
95C22×C3×Dic3—32no—
96C22×C3×Dic3C2×D312yes—
97C22×C3×Dic3C22×C312yes—
98C2×C3×C4×D3—32no—
99C2×C3×Dic6—32no—
100C2×C3×Dic6—32no—
101C2×C3×Dic6—61no—
102C2×#627C22×C311yescharacteristic
103C2×#625C22×C311yescharacteristic
104C2×D3×Dic3C22×C312yes—
105C32×C4⋊C4—32no—
106C32×C22⋊C4—32no—
107C32×C22⋊C4—32no—
108#3757C22×C312yes—
109#3757C22×C312yes—
110C4×C3⋊Dic3C22×C311yescharacteristic
111C3×#217—22no—
112C3×#217—32no—
113C3×#217—32no—
114C3×#217—32no—
115C3×#217—32no—
116C3×#217—32no—
117C3×#217—32no—
118C3×#217—62no—
119C3×#217—62no—
120C3×#217C2×D312yes—
121C3×#217C22×C312yes—
122C3×(C2×D3)⋊C4—22no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—32no—
125C3×(C2×D3)⋊C4C2×D312yes—
126C3×(C2×D3)⋊C4C22×C312yes—
127C3×C4⋊Dic3—32no—
128C3×C4⋊Dic3—32no—
129C3×Dic3⋊C4—22no—
130C3×Dic3⋊C4—32no—
131C3×Dic3⋊C4—32no—
132C3×Dic3⋊C4C2×D312yes—
133C3×Dic3⋊C4C22×C312yes—
134C3×C4×Dic3—32no—
135C3×C4×Dic3—32no—
136C3×C4×Dic3—32no—
137C3×C4×Dic3—32no—
138C3×C4×Dic3—32no—
139C3×C4×Dic3—61no—
140Dic3⋊Dic3C22×C312yes—
141Dic3⋊Dic3C22×C312yes—
142(C2×D3)⋊Dic3C22×C312yes—
143(C2×D3)⋊Dic3C22×C312yes—
144Dic32C22×C311yescharacteristic
145Dic32C22×C312yes—
146Dic32C22×C311yescharacteristic
147C22×C33—22no—
148C22×C33—22no—
149C22×C33—22no—
150C22×C33C2411yescharacteristic
151C2×C32×D3—22no—
152C2×C32×D3—22no—
153C2×C32×D3—22no—
154C33×C4—21no—
155C3×C3⋊Dic3—22no—
156C3×C3⋊Dic3—21no—
157C3×C3⋊Dic3—21no—
158C32×Dic3—22no—
159C32×Dic3—22no—
160C32×Dic3#4114yes—
161C3×C4⋊Q8—91no—
162C3×#124—91no—
163C3×#124—92no—
164C3×#124—91no—
165C3×#121—32no—
166C3×#121—32no—
167C3×#121—62no—
168C3×#121—92no—
169C3×C4×D4—92no—
170C3×#117—92no—
171C3×#117—92no—
172#939—32no—
173#939C3×D312yescomplemented
174#938—32no—
175#936—32no—
176#936—32no—
177#936—61no—
178D4×Dic3—32no—
179C4×#67—32no—
180#926—32no—
181#892—32no—
182#888—32no—
183#885—32no—
184#885—32no—
185#877—32no—
186#877—32no—
187C23×C32—22no—
188C23×C32—22no—
189C23×C32—22no—
190C23×C32—32no—
191C23×C32—42no—
192C23×C32C22×D312yes—
193C23×C32C22×D312yes—
194C23×C32C23×C312yes—
195C22×C3×D3—22no—
196C22×C3×D3C22×D312yes—
197C22×C3×D3C23×C312yes—
198C32×D4—42no—
199C32×D4—41no—
200C32×D4—42no—
201C32×D4—44no—
202C32×D4—44no—
203C32×D4—64no—
204C32×D4—122no—
205C2×C32×C4—21no—
206C2×C32×C4—22no—
207C2×C32×C4—32no—
208C2×C32×C4—32no—
209C2×C32×C4—32no—
210C2×C32×C4—41no—
211C2×C32×C4—64no—
212C2×C32×C4C22×D312yes—
213C2×C32×C4C23×C311yescharacteristic
214C2×C3⋊Dic3C23×C311yescharacteristic
215C2×C3⋊Dic3C23×C312yes—
216C2×C3⋊Dic3C23×C311yescharacteristic
217#634—41no—
218C3×#67—24no—
219C3×#67—24no—
220C3×#67—44no—
221C3×#67—42no—
222C3×#67—42no—
223C3×#67—44no—
224C3×#67—122no—
225C2×C3×Dic3—24no—
226C2×C3×Dic3—24no—
227C2×C3×Dic3—22no—
228C2×C3×Dic3—22no—
229C2×C3×Dic3—22no—
230C2×C3×Dic3—32no—
231C2×C3×Dic3—32no—
232C2×C3×Dic3—32no—
233C2×C3×Dic3—32no—
234C2×C3×Dic3—44no—
235C2×C3×Dic3—62no—
236C2×C3×Dic3—61no—
237C2×C3×Dic3—61no—
238C2×C3×Dic3—64no—
239C2×C3×Dic3C22×D312yes—
240C2×C3×Dic3C22×D312yes—
241C2×C3×Dic3C22×D312yes—
242C2×C3×Dic3C23×C312yes—
243C2×C3×Dic3C23×C312yes—
244C2×C3×Dic3C23×C312yes—
245C3×C4×D3—64no—
246C3×Dic6—122no—
247C3×Dic6—122no—
248C3×Dic6—122no—
249#627—41no—
250#625—41no—
251D3×Dic3—24no—
252#555—91noSylow
253C2×C33—42no—
254C2×C33C2×#4112yes—
255C2×C33H(5,𝔽2)11yescharacteristic
256C32×D3—42no—
257C2×C3×Q8—91no—
258C2×C3×Q8—91no—
259C2×C3×D4—21no—
260C2×C3×D4—22no—
261C2×C3×D4—32no—
262C2×C3×D4—32no—
263C2×C3×D4—32no—
264C2×C3×D4—32no—
265C2×C3×D4—41no—
266C2×C3×D4—62no—
267C2×C3×D4—62no—
268C2×C3×D4—91no—
269C2×C3×D4D3211yescharacteristic
270C2×C3×D4C2×C3×D312yes—
271C22×C3×C4—32no—
272C22×C3×C4—32no—
273C22×C3×C4—62no—
274C22×C3×C4—92no—
275C2×#67—32no—
276C2×#67C2×C3×D312yes—
277C2×#67C2×C3×D312yes—
278C22×Dic3—32no—
279C22×Dic3C2×C3×D312yes—
280C2×C4×D3—32no—
281C2×Dic6—32no—
282C2×Dic6—32no—
283C2×Dic6—61no—
284C3×C4⋊C4—32no—
285C3×C4⋊C4—32no—
286C3×C4⋊C4—62no—
287C3×C4⋊C4—92no—
288C3×C4⋊C4—92no—
289C3×C22⋊C4—32no—
290C3×C22⋊C4—32no—
291C3×C22⋊C4—32no—
292C3×C22⋊C4—32no—
293C3×C22⋊C4—62no—
294C3×C22⋊C4—62no—
295C3×C22⋊C4—92no—
296C3×C22⋊C4—92no—
297C3×C22⋊C4—92no—
298C3×C22⋊C4—92no—
299C3×C42—91no—
300C3×C42—92no—
301C3×C42—91no—
302C3×C42—91no—
303#217—32no—
304#217—32no—
305#217—32no—
306#217—32no—
307#217—32no—
308#217—32no—
309#217—62no—
310#217—62no—
311#217C2×C3×D312yes—
312(C2×D3)⋊C4—32no—
313(C2×D3)⋊C4—32no—
314(C2×D3)⋊C4C2×C3×D312yes—
315C4⋊Dic3—32no—
316C4⋊Dic3—32no—
317Dic3⋊C4—32no—
318Dic3⋊C4—32no—
319Dic3⋊C4C2×C3×D312yes—
320C4×Dic3—32no—
321C4×Dic3—32no—
322C4×Dic3—32no—
323C4×Dic3—32no—
324C4×Dic3—32no—
325C4×Dic3—61no—
326C22×C32—22no—
327C22×C32—24no—
328C22×C32—22no—
329C22×C32—22no—
330C22×C32—22no—
331C22×C32—22no—
332C22×C32—22no—
333C22×C32—22no—
334C22×C32—22no—
335C22×C32—22no—
336C22×C32—21no—
337C22×C32—22no—
338C22×C32—41no—
339C22×C32—42no—
340C22×C32—42no—
341C22×C32—44no—
342C22×C32—42no—
343C22×C32—42no—
344C22×C32—42no—
345C22×C32—42no—
346C22×C32—42no—
347C22×C32—42no—
348C22×C32—62no—
349C22×C32—62no—
350C22×C32—62no—
351C22×C32—82no—
352C22×C32—82no—
353C22×C32C23×D312yes—
354C22×C32C24×C311yescharacteristic
355C2×C3×D3—22no—
356C2×C3×D3—22no—
357C2×C3×D3—22no—
358C2×C3×D3—22no—
359C2×C3×D3—22no—
360C2×C3×D3—22no—
361C2×C3×D3—42no—
362C2×C3×D3—42no—
363C2×C3×D3—42no—
364C32×C4—21no—
365C32×C4—22no—
366C32×C4—22no—
367C32×C4—34no—
368C32×C4—42no—
369C32×C4—42no—
370C32×C4—62no—
371C32×C4—62no—
372C3⋊Dic3—22no—
373C3⋊Dic3—21no—
374C3⋊Dic3—21no—
375C3×Dic3—22no—
376C3×Dic3—22no—
377C3×Dic3—22no—
378C3×Dic3—22no—
379C3×Dic3—24no—
380C3×Dic3—42no—
381C3×Dic3—42no—
382C3×Dic3—62no—
383C3×Dic3—62no—
384C3×Dic3—62no—
385C3×Dic3—62no—
386C3×Dic3—121no—
387C3×Dic3—121no—
388C3×Dic3—122no—
389C3×Dic3#23714yes—
390C3×Dic3C3×#4114yes—
391C4⋊Q8—91no—
392#124—91no—
393#124—92no—
394#124—91no—
395#121—32no—
396#121—92no—
397C4×D4—92no—
398#117—92no—
399#117—92no—
400C33#55511yescharacteristic, complemented, Sylow
401C23×C3—22no—
402C23×C3—22no—
403C23×C3—22no—
404C23×C3—32no—
405C23×C3—32no—
406C23×C3—42no—
407C23×C3—62no—
408C23×C3C2×D3212yes—
409C23×C3C22×C3×D312yes—
410C23×C3C22×C3×D312yes—
411C22×D3C22×C3×D312yes—
412C3×Q8—361no—
413C3×Q8—361no—
414C3×D4—41no—
415C3×D4—42no—
416C3×D4—42no—
417C3×D4—44no—
418C3×D4—44no—
419C3×D4—64no—
420C3×D4—64no—
421C3×D4—124no—
422C3×D4—122no—
423C3×D4—122no—
424C3×D4—124no—
425C3×D4—361no—
426C2×C3×C4—21no—
427C2×C3×C4—22no—
428C2×C3×C4—32no—
429C2×C3×C4—32no—
430C2×C3×C4—32no—
431C2×C3×C4—32no—
432C2×C3×C4—32no—
433C2×C3×C4—32no—
434C2×C3×C4—41no—
435C2×C3×C4—62no—
436C2×C3×C4—62no—
437C2×C3×C4—62no—
438C2×C3×C4—64no—
439C2×C3×C4—64no—
440C2×C3×C4—92no—
441C2×C3×C4—91no—
442C2×C3×C4—91no—
443C2×C3×C4—124no—
444C2×C3×C4—184no—
445C2×C3×C4C2×D3211yescharacteristic
446C2×C3×C4C22×C3×D312yes—
447#67—24no—
448#67—42no—
449#67—122no—
450C2×Dic3—24no—
451C2×Dic3—32no—
452C2×Dic3—32no—
453C2×Dic3—32no—
454C2×Dic3—32no—
455C2×Dic3—62no—
456C2×Dic3—61no—
457C2×Dic3—61no—
458C2×Dic3—64no—
459C2×Dic3C22×C3×D312yes—
460C2×Dic3C22×C3×D312yes—
461C2×Dic3C22×C3×D312yes—
462C4×D3—64no—
463Dic6—122no—
464Dic6—122no—
465Dic6—122no—
466C2×C32—22no—
467C2×C32—21no—
468C2×C32—22no—
469C2×C32—22no—
470C2×C32—22no—
471C2×C32—42no—
472C2×C32—41no—
473C2×C32—42no—
474C2×C32—44no—
475C2×C32—42no—
476C2×C32—42no—
477C2×C32—82no—
478C2×C32—82no—
479C2×C32—84no—
480C2×C32—122no—
481C2×C32C2×#23712yes—
482C2×C32#100412yes—
483C2×C32D3×#4112yes—
484C2×C32C2×C3×#4112yes—
485C2×C32C3×H(5,𝔽2)11yescharacteristic
486C3×D3—42no—
487C3×D3—42no—
488C3×D3—82no—
489C2×Q8—91no—
490C2×Q8—91no—
491C2×D4—32no—
492C2×D4—32no—
493C2×D4—91no—
494C2×D4C3×D3211yescharacteristic
495C22×C4—32no—
496C22×C4—92no—
497C4⋊C4—32no—
498C4⋊C4—92no—
499C4⋊C4—92no—
500C22⋊C4—32no—
501C22⋊C4—32no—
502C22⋊C4—92no—
503C22⋊C4—92no—
504C22⋊C4—92no—
505C22⋊C4—92no—
506C42—91no—
507C42—92no—
508C42—91no—
509C42—91no—
510C22×C3—24no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—22no—
515C22×C3—22no—
516C22×C3—22no—
517C22×C3—22no—
518C22×C3—22no—
519C22×C3—22no—
520C22×C3—21no—
521C22×C3—22no—
522C22×C3—42no—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3—44no—
531C22×C3—41no—
532C22×C3—62no—
533C22×C3—62no—
534C22×C3—62no—
535C22×C3—62no—
536C22×C3—62no—
537C22×C3—62no—
538C22×C3—82no—
539C22×C3—82no—
540C22×C3—122no—
541C22×C3—122no—
542C22×C3—122no—
543C22×C3C22×D3211yescharacteristic
544C22×C3C23×C3×D312yes—
545C2×D3—22no—
546C2×D3—22no—
547C2×D3—22no—
548C3×C4—22no—
549C3×C4—21no—
550C3×C4—22no—
551C3×C4—34no—
552C3×C4—34no—
553C3×C4—42no—
554C3×C4—42no—
555C3×C4—64no—
556C3×C4—62no—
557C3×C4—62no—
558C3×C4—62no—
559C3×C4—62no—
560C3×C4—122no—
561C3×C4—122no—
562C3×C4—182no—
563C3×C4—181no—
564C3×C4—181no—
565Dic3—22no—
566Dic3—22no—
567Dic3—62no—
568Dic3—62no—
569Dic3—62no—
570Dic3—62no—
571Dic3—121no—
572Dic3—121no—
573Dic3—122no—
574Dic3C3×#23714yes—
575C32—22no—
576C32—21no—
577C32—41no—
578C32#561612yescomplemented
579C32C3×#55511yescharacteristic, complemented
580C23—32no—
581C23C2×C3×D3212yes—
582Q8—361no—
583Q8—361no—
584D4—41no—
585D4—64no—
586D4—122no—
587D4—361no—
588C2×C4—32no—
589C2×C4—32no—
590C2×C4—32no—
591C2×C4—64no—
592C2×C4—92no—
593C2×C4—91no—
594C2×C4—91no—
595C2×C4—184no—
596C2×C4C2×C3×D3211yescharacteristic
597C2×C3—21no—
598C2×C3—22no—
599C2×C3—22no—
600C2×C3—22no—
601C2×C3—22no—
602C2×C3—44no—
603C2×C3—42no—
604C2×C3—42no—
605C2×C3—42no—
606C2×C3—41no—
607C2×C3—42no—
608C2×C3—82no—
609C2×C3—82no—
610C2×C3—84no—
611C2×C3—122no—
612C2×C3—122no—
613C2×C3—242no—
614C2×C3D3×#23712yes—
615C2×C3#6433011yescharacteristic
616C2×C3C2×C3×#23712yes—
617C2×C3C3×#100412yes—
618C2×C3C3×D3×#4112yes—
619D3—42no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—62no—
624C22—62no—
625C22—62no—
626C22C22×C3×D3211yescharacteristic
627C4—21no—
628C4—34no—
629C4—62no—
630C4—62no—
631C4—181no—
632C4—181no—
633C4—182no—
634C3—22no—
635C3—21no—
636C3—41no—
637C3#1059524711yescharacteristic, direct product factor
638C3C3×#561612yescomplemented
639C2—42no—
640C2—122no—
641C2C3×D3×#23712yes—
642C2C3×#6433011yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C24
hypercenterC2×C3×D4D32
hypocenterC32C3×#555
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64145C2semidirect
2. C32×#939C2semidirect
3. C3×Dic3×#67C2semidirect
4. C3×#63970C2semidirect
5. C3×#63959C2semidirect
6. C3×#63955C2semidirect
7. C3×#63913C2semidirect
8. C3×#63878C2semidirect
9. C3×#63840C2semidirect
11. #10595247C3direct (N × H)
34. #64145C2×C3semidirect
37. C3×#939D3semidirect
38. C3×#939C2×C3semidirect
52. Dic3×#67C2×C3semidirect
53. #63970C2×C3semidirect
54. #63959C2×C3semidirect
55. #63955C2×C3semidirect
56. #63913C2×C3semidirect
57. #63878C2×C3semidirect
58. #63840C2×C3semidirect
173. #939C3×D3semidirect
400. C33#555semidirect
578. C32#5616semidirect
579. C32C3×#555semidirect
637. C3#10595247direct (N × H)
638. C3C3×#5616semidirect

Maximal subgroups

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

subgroupindex
C3×#641452
C32×#9392
C3×Dic3×#672
C3×#639702
C3×#639592
C3×#639552
C3×#639132
C3×#638782
C3×#638402
C3×#56163
#105952473