Finite Group Databasebeta

← C3×#10595030 #13174606

Subgroups — C3×#10595030 #13174606

Order 1728.

1264 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64138C211yesmaximal, characteristic, complemented
2C32×C4×#67C212yesmaximal, complemented
3C3×#63979C211yesmaximal, characteristic
4C3×#63975C211yesmaximal, characteristic, complemented
5C3×#63959C212yesmaximal, complemented
6C3×#63905C212yesmaximal, complemented
7C3×#63904C212yesmaximal, complemented
8C3×#63852C212yesmaximal, complemented
9C3×#63837C212yesmaximal, complemented
10C3×#5559—32nomaximal
11#10595030C311yesmaximal, characteristic, direct product factor
12C22×C33×C4C2211yescharacteristic
13C2×C3×#638C2211yescharacteristic
14C2×C32×#67C2212yes—
15C2×C32×C4×D3C2212yes—
16C2×C3×#627C2212yes—
17C2×C3×#626C2212yes—
18C2×C3×#625C2211yescharacteristic
19C2×C3×#624C2211yescharacteristic
20C3×#3757C2211yescharacteristic
21C3×#3752C2212yes—
22C3×#3750C2212yes—
23C32×#217C2212yes—
24C32×(C2×D3)⋊C4C2212yes—
25C32×Dic3⋊C4C2212yes—
26C32×C4×Dic3C2212yes—
27C3×Dic3⋊Dic3C2212yes—
28C3×#3722C2211yescharacteristic
29C3×(C2×D3)⋊Dic3C2212yes—
30C3×(C2×D3)⋊Dic3C2212yes—
31C3×Dic32C2212yes—
32C3×Dic32C2211yescharacteristic
33C32×C4×D4—32no—
34#64138C2×C311yescharacteristic, complemented
35C3×#940—32no—
36C3×#936—32no—
37C3×#929—32no—
38C3×#929—32no—
39C3×#929—61no—
40C3×C4×#67—22no—
41C3×C4×#67—32no—
42C3×C4×#67D312yescomplemented
43C3×C4×#67C2×C312yescomplemented
44C3×#893—32no—
45C3×#887—32no—
46C3×#885—32no—
47C3×#885—32no—
48C3×#883—32no—
49C3×#883—32no—
50C3×#875—32no—
51C3×C4×Dic6—32no—
52#63979C2×C311yescharacteristic
53#63975C2×C311yescharacteristic, complemented
54#63959C2×C312yescomplemented
55#63905C2×C312yescomplemented
56#63904C2×C312yescomplemented
57#63852C2×C312yescomplemented
58#63837C2×C312yescomplemented
59C23×C33C2311yescharacteristic
60C22×C3×C3⋊D3C2311yescharacteristic
61C22×C32×D3C2312yes—
62C2×C33×C4—41no—
63C2×C33×C4C2312yes—
64C3×#638—41no—
65C2×C3×C3⋊Dic3C2311yescharacteristic
66C2×C3×C3⋊Dic3C2312yes—
67C32×#67—24no—
68C2×C32×Dic3C2312yes—
69C2×C32×Dic3C2312yes—
70C2×C32×Dic3C2312yes—
71C32×C4×D3—42no—
72C3×#627—24no—
73C3×#626—24no—
74C3×#625—41no—
75C3×#624—41no—
76C3×#559—91no—
77#5559—32no—
78C2×C32×D4—32no—
79C22×C32×C4—21no—
80C22×C32×C4—22no—
81C22×C32×C4—32no—
82C22×C32×C4—41no—
83C22×C32×C4C2×D312yes—
84C22×C32×C4C22×C311yescharacteristic
85C2×#638C22×C311yescharacteristic
86C2×C3×#67—22no—
87C2×C3×#67—32no—
88C2×C3×#67—32no—
89C2×C3×#67—32no—
90C2×C3×#67—61no—
91C2×C3×#67C2×D312yes—
92C2×C3×#67C22×C312yes—
93C2×C3×D12—32no—
94C2×C3×C4×D3—22no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3C2×D312yes—
97C2×C3×C4×D3C22×C312yes—
98C2×C3×Dic6—32no—
99C2×C3×Dic6—32no—
100C2×#627C22×C312yes—
101C2×#626C22×C312yes—
102C2×#625C22×C311yescharacteristic
103C2×#624C22×C311yescharacteristic
104C32×C4⋊C4—32no—
105C32×C22⋊C4—32no—
106C32×C22⋊C4—32no—
107C32×C42—32no—
108#3757C22×C311yescharacteristic
109#3752C22×C312yes—
110#3750C22×C312yes—
111C3×#217—22no—
112C3×#217—32no—
113C3×#217—32no—
114C3×#217—32no—
115C3×#217—61no—
116C3×#217C2×D312yes—
117C3×#217C22×C312yes—
118C3×(C2×D3)⋊C4—22no—
119C3×(C2×D3)⋊C4—32no—
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—32no—
122C3×(C2×D3)⋊C4—32no—
123C3×(C2×D3)⋊C4—32no—
124C3×(C2×D3)⋊C4—62no—
125C3×(C2×D3)⋊C4C2×D312yes—
126C3×(C2×D3)⋊C4C22×C312yes—
127C3×C4⋊Dic3—32no—
128C3×Dic3⋊C4—22no—
129C3×Dic3⋊C4—32no—
130C3×Dic3⋊C4—32no—
131C3×Dic3⋊C4—32no—
132C3×Dic3⋊C4—62no—
133C3×Dic3⋊C4C2×D312yes—
134C3×Dic3⋊C4C22×C312yes—
135C3×C4×Dic3—22no—
136C3×C4×Dic3—32no—
137C3×C4×Dic3—32no—
138C3×C4×Dic3—32no—
139C3×C4×Dic3C2×D312yes—
140C3×C4×Dic3C22×C312yes—
141Dic3⋊Dic3C22×C312yes—
142#3722C22×C311yescharacteristic
143(C2×D3)⋊Dic3C22×C312yes—
144(C2×D3)⋊Dic3C22×C312yes—
145Dic32C22×C311yescharacteristic
146Dic32C22×C312yes—
147C22×C33—22no—
148C22×C33—21no—
149C22×C33C2411yescharacteristic
150C2×C3×C3⋊D3—22no—
151C2×C3×C3⋊D3—21no—
152C2×C32×D3—22no—
153C2×C32×D3—22no—
154C2×C32×D3—22no—
155C33×C4—22no—
156C3×C3⋊Dic3—22no—
157C3×C3⋊Dic3—21no—
158C32×Dic3—22no—
159C32×Dic3—22no—
160C32×Dic3#4114yes—
161C3×#127—91no—
162C3×#124—92no—
163C3×#124—92no—
164C3×#123—91no—
165C3×#123—92no—
166C3×#123—91no—
167C3×C4×Q8—92no—
168C3×C4×D4—32no—
169C3×C4×D4—32no—
170C3×C4×D4—62no—
171C3×C4×D4—92no—
172#940—32no—
173#936—32no—
174#929—32no—
175#929—32no—
176#929—61no—
177C4×#67—32no—
178C4×#67C3×D312yescomplemented
179#893—32no—
180#887—32no—
181#885—32no—
182#885—32no—
183#883—32no—
184#883—32no—
185#875—32no—
186C4×Dic6—32no—
187C23×C32—21no—
188C23×C32—22no—
189C23×C32—32no—
190C23×C32—41no—
191C23×C32C22×D312yes—
192C23×C32C23×C311yescharacteristic
193C22×C3⋊D3C23×C311yescharacteristic
194C22×C3×D3—22no—
195C22×C3×D3—32no—
196C22×C3×D3—61no—
197C22×C3×D3C22×D312yes—
198C22×C3×D3C23×C312yes—
199C32×D4—64no—
200C2×C32×C4—22no—
201C2×C32×C4—22no—
202C2×C32×C4—22no—
203C2×C32×C4—32no—
204C2×C32×C4—32no—
205C2×C32×C4—32no—
206C2×C32×C4—42no—
207C2×C32×C4—42no—
208C2×C32×C4—41no—
209C2×C32×C4—44no—
210C2×C32×C4—42no—
211C2×C32×C4—44no—
212C2×C32×C4—122no—
213C2×C32×C4C22×D312yes—
214C2×C32×C4C22×D312yes—
215C2×C32×C4C23×C312yes—
216#638—41no—
217C2×C3⋊Dic3C23×C312yes—
218C2×C3⋊Dic3C23×C311yescharacteristic
219C3×#67—28no—
220C3×#67—24no—
221C3×#67—24no—
222C3×#67—64no—
223C3×#67—122no—
224C3×#67—122no—
225C3×#67—122no—
226C2×C3×Dic3—22no—
227C2×C3×Dic3—22no—
228C2×C3×Dic3—22no—
229C2×C3×Dic3—32no—
230C2×C3×Dic3—32no—
231C2×C3×Dic3—32no—
232C2×C3×Dic3—61no—
233C2×C3×Dic3—62no—
234C2×C3×Dic3C22×D312yes—
235C2×C3×Dic3C22×D312yes—
236C2×C3×Dic3C22×D312yes—
237C2×C3×Dic3C23×C312yes—
238C2×C3×Dic3C23×C312yes—
239C2×C3×Dic3C23×C312yes—
240C3×D12—64no—
241C3×C4×D3—42no—
242C3×C4×D3—42no—
243C3×C4×D3—44no—
244C3×C4×D3—122no—
245C3×Dic6—64no—
246C3×Dic6—64no—
247#627—24no—
248#626—24no—
249#625—41no—
250#624—41no—
251#559—91noSylow
252C2×C33—41no—
253C2×C33C2×#4112yes—
254C2×C33H(5,𝔽2)11yescharacteristic
255C3×C3⋊D3—41no—
256C32×D3—42no—
257C2×C3×Q8—92no—
258C2×C3×D4—32no—
259C2×C3×D4—32no—
260C2×C3×D4—62no—
261C2×C3×D4—92no—
262C2×C3×D4—91no—
263C2×C3×D4—91no—
264C22×C3×C4—21no—
265C22×C3×C4—22no—
266C22×C3×C4—32no—
267C22×C3×C4—32no—
268C22×C3×C4—41no—
269C22×C3×C4—62no—
270C22×C3×C4—91no—
271C22×C3×C4D3211yescharacteristic
272C22×C3×C4C2×C3×D312yes—
273C2×#67—32no—
274C2×#67—32no—
275C2×#67—32no—
276C2×#67—61no—
277C2×#67C2×C3×D312yes—
278C2×D12—32no—
279C2×C4×D3—32no—
280C2×C4×D3C2×C3×D312yes—
281C2×Dic6—32no—
282C2×Dic6—32no—
283C3×C4⋊C4—32no—
284C3×C4⋊C4—32no—
285C3×C4⋊C4—62no—
286C3×C4⋊C4—92no—
287C3×C4⋊C4—92no—
288C3×C22⋊C4—32no—
289C3×C22⋊C4—32no—
290C3×C22⋊C4—32no—
291C3×C22⋊C4—32no—
292C3×C22⋊C4—62no—
293C3×C22⋊C4—62no—
294C3×C22⋊C4—91no—
295C3×C22⋊C4—92no—
296C3×C22⋊C4—92no—
297C3×C22⋊C4—91no—
298C3×C22⋊C4—92no—
299C3×C42—32no—
300C3×C42—32no—
301C3×C42—62no—
302C3×C42—92no—
303C3×C42—91no—
304#217—32no—
305#217—32no—
306#217—32no—
307#217—61no—
308#217C2×C3×D312yes—
309(C2×D3)⋊C4—32no—
310(C2×D3)⋊C4—32no—
311(C2×D3)⋊C4—32no—
312(C2×D3)⋊C4—32no—
313(C2×D3)⋊C4—32no—
314(C2×D3)⋊C4—62no—
315(C2×D3)⋊C4C2×C3×D312yes—
316C4⋊Dic3—32no—
317Dic3⋊C4—32no—
318Dic3⋊C4—32no—
319Dic3⋊C4—32no—
320Dic3⋊C4—62no—
321Dic3⋊C4C2×C3×D312yes—
322C4×Dic3—32no—
323C4×Dic3—32no—
324C4×Dic3—32no—
325C4×Dic3C2×C3×D312yes—
326C22×C32—22no—
327C22×C32—21no—
328C22×C32—24no—
329C22×C32—22no—
330C22×C32—22no—
331C22×C32—22no—
332C22×C32—22no—
333C22×C32—21no—
334C22×C32—22no—
335C22×C32—41no—
336C22×C32—42no—
337C22×C32—42no—
338C22×C32—42no—
339C22×C32—42no—
340C22×C32—44no—
341C22×C32—62no—
342C22×C32—62no—
343C22×C32—62no—
344C22×C32C23×D312yes—
345C22×C32C24×C311yescharacteristic
346C2×C3⋊D3—22no—
347C2×C3⋊D3—21no—
348C2×C3×D3—24no—
349C2×C3×D3—22no—
350C2×C3×D3—22no—
351C2×C3×D3—22no—
352C2×C3×D3—22no—
353C2×C3×D3—22no—
354C2×C3×D3—22no—
355C2×C3×D3—42no—
356C2×C3×D3—42no—
357C2×C3×D3—62no—
358C2×C3×D3—62no—
359C2×C3×D3—62no—
360C2×C3×D3—121no—
361C2×C3×D3—122no—
362C32×C4—22no—
363C32×C4—24no—
364C32×C4—22no—
365C32×C4—22no—
366C32×C4—34no—
367C32×C4—42no—
368C32×C4—42no—
369C32×C4—44no—
370C32×C4—62no—
371C32×C4—62no—
372C3⋊Dic3—22no—
373C3⋊Dic3—21no—
374C3×Dic3—22no—
375C3×Dic3—22no—
376C3×Dic3—22no—
377C3×Dic3—22no—
378C3×Dic3—24no—
379C3×Dic3—42no—
380C3×Dic3—42no—
381C3×Dic3—62no—
382C3×Dic3—62no—
383C3×Dic3—62no—
384C3×Dic3—122no—
385C3×Dic3—121no—
386C3×Dic3#23514yes—
387C3×Dic3C3×#4114yes—
388#127—91no—
389#124—92no—
390#124—92no—
391#123—91no—
392#123—92no—
393#123—91no—
394C4×Q8—92no—
395C4×D4—32no—
396C4×D4—92no—
397C33#55911yescharacteristic, complemented, Sylow
398C23×C3—22no—
399C23×C3—21no—
400C23×C3—32no—
401C23×C3—32no—
402C23×C3—41no—
403C23×C3—62no—
404C23×C3—91no—
405C23×C3C2×D3211yescharacteristic
406C23×C3C22×C3×D312yes—
407C22×D3—32no—
408C22×D3—61no—
409C22×D3C22×C3×D312yes—
410C3×Q8—184no—
411C3×D4—64no—
412C3×D4—64no—
413C3×D4—68no—
414C3×D4—184no—
415C3×D4—361no—
416C3×D4—361no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—22no—
420C2×C3×C4—32no—
421C2×C3×C4—32no—
422C2×C3×C4—32no—
423C2×C3×C4—32no—
424C2×C3×C4—32no—
425C2×C3×C4—32no—
426C2×C3×C4—41no—
427C2×C3×C4—42no—
428C2×C3×C4—44no—
429C2×C3×C4—42no—
430C2×C3×C4—44no—
431C2×C3×C4—42no—
432C2×C3×C4—62no—
433C2×C3×C4—62no—
434C2×C3×C4—62no—
435C2×C3×C4—92no—
436C2×C3×C4—91no—
437C2×C3×C4—122no—
438C2×C3×C4—122no—
439C2×C3×C4—124no—
440C2×C3×C4—361no—
441C2×C3×C4C2×D3212yes—
442C2×C3×C4C22×C3×D312yes—
443C2×C3×C4C22×C3×D312yes—
444#67—24no—
445#67—64no—
446#67—122no—
447#67—122no—
448#67—122no—
449C2×Dic3—32no—
450C2×Dic3—32no—
451C2×Dic3—32no—
452C2×Dic3—61no—
453C2×Dic3—62no—
454C2×Dic3C22×C3×D312yes—
455C2×Dic3C22×C3×D312yes—
456C2×Dic3C22×C3×D312yes—
457D12—64no—
458C4×D3—42no—
459C4×D3—122no—
460Dic6—64no—
461Dic6—64no—
462C2×C32—22no—
463C2×C32—21no—
464C2×C32—22no—
465C2×C32—22no—
466C2×C32—22no—
467C2×C32—42no—
468C2×C32—41no—
469C2×C32—44no—
470C2×C32—42no—
471C2×C32—44no—
472C2×C32—41no—
473C2×C32—42no—
474C2×C32—122no—
475C2×C32C2×#23512yes—
476C2×C32#100412yes—
477C2×C32D3×#4112yes—
478C2×C32C2×C3×#4112yes—
479C2×C32C3×H(5,𝔽2)11yescharacteristic
480C3⋊D3—41no—
481C3×D3—44no—
482C3×D3—42no—
483C3×D3—42no—
484C3×D3—122no—
485C3×D3—241no—
486C2×Q8—92no—
487C2×D4—32no—
488C2×D4—92no—
489C2×D4—91no—
490C2×D4—91no—
491C22×C4—32no—
492C22×C4—91no—
493C22×C4C3×D3211yescharacteristic
494C4⋊C4—32no—
495C4⋊C4—92no—
496C4⋊C4—92no—
497C22⋊C4—32no—
498C22⋊C4—32no—
499C22⋊C4—91no—
500C22⋊C4—92no—
501C22⋊C4—92no—
502C22⋊C4—91no—
503C22⋊C4—92no—
504C42—32no—
505C42—92no—
506C42—91no—
507C22×C3—22no—
508C22×C3—24no—
509C22×C3—22no—
510C22×C3—21no—
511C22×C3—22no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—21no—
515C22×C3—22no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—44no—
521C22×C3—41no—
522C22×C3—64no—
523C22×C3—62no—
524C22×C3—62no—
525C22×C3—62no—
526C22×C3—62no—
527C22×C3—62no—
528C22×C3—62no—
529C22×C3—122no—
530C22×C3—122no—
531C22×C3—182no—
532C22×C3—181no—
533C22×C3C22×D3211yescharacteristic
534C22×C3C23×C3×D312yes—
535C2×D3—22no—
536C2×D3—22no—
537C2×D3—22no—
538C2×D3—62no—
539C2×D3—62no—
540C2×D3—62no—
541C2×D3—121no—
542C2×D3—122no—
543C3×C4—24no—
544C3×C4—22no—
545C3×C4—22no—
546C3×C4—22no—
547C3×C4—34no—
548C3×C4—34no—
549C3×C4—42no—
550C3×C4—42no—
551C3×C4—44no—
552C3×C4—64no—
553C3×C4—62no—
554C3×C4—62no—
555C3×C4—62no—
556C3×C4—62no—
557C3×C4—122no—
558C3×C4—122no—
559C3×C4—182no—
560C3×C4—181no—
561Dic3—22no—
562Dic3—22no—
563Dic3—62no—
564Dic3—62no—
565Dic3—62no—
566Dic3—121no—
567Dic3—122no—
568Dic3C3×#23514yes—
569C32—22no—
570C32—21no—
571C32—41no—
572C32#555912yescomplemented
573C32C3×#55911yescharacteristic, complemented
574C23—32no—
575C23—91no—
576C23C2×C3×D3211yescharacteristic
577Q8—184no—
578D4—64no—
579D4—184no—
580D4—361no—
581D4—361no—
582C2×C4—32no—
583C2×C4—32no—
584C2×C4—32no—
585C2×C4—41no—
586C2×C4—92no—
587C2×C4—91no—
588C2×C4—122no—
589C2×C4—361no—
590C2×C4C2×C3×D3212yes—
591C2×C3—21no—
592C2×C3—22no—
593C2×C3—22no—
594C2×C3—22no—
595C2×C3—22no—
596C2×C3—42no—
597C2×C3—44no—
598C2×C3—44no—
599C2×C3—42no—
600C2×C3—41no—
601C2×C3—41no—
602C2×C3—42no—
603C2×C3—124no—
604C2×C3—122no—
605C2×C3—122no—
606C2×C3—361no—
607C2×C3D3×#23512yes—
608C2×C3#6433011yescharacteristic
609C2×C3C2×C3×#23512yes—
610C2×C3C3×#100412yes—
611C2×C3C3×D3×#4112yes—
612D3—42no—
613D3—122no—
614D3—241no—
615C22—21no—
616C22—22no—
617C22—62no—
618C22—62no—
619C22—62no—
620C22—181no—
621C22—182no—
622C22C22×C3×D3211yescharacteristic
623C4—22no—
624C4—34no—
625C4—62no—
626C4—62no—
627C4—181no—
628C4—182no—
629C3—21no—
630C3—22no—
631C3—41no—
632C3#1059503011yescharacteristic, direct product factor
633C3C3×#555912yescomplemented
634C2—41no—
635C2—122no—
636C2—361no—
637C2C3×D3×#23512yes—
638C2C3×#6433011yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
prime radicalC3×#63979C2
hypercenterC22×C3×C4D32
hypocenterC32C3×#559
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#64138C2semidirect
2. C32×C4×#67C2semidirect
4. C3×#63975C2semidirect
5. C3×#63959C2semidirect
6. C3×#63905C2semidirect
7. C3×#63904C2semidirect
8. C3×#63852C2semidirect
9. C3×#63837C2semidirect
11. #10595030C3direct (N × H)
34. #64138C2×C3semidirect
42. C3×C4×#67D3semidirect
43. C3×C4×#67C2×C3semidirect
53. #63975C2×C3semidirect
54. #63959C2×C3semidirect
55. #63905C2×C3semidirect
56. #63904C2×C3semidirect
57. #63852C2×C3semidirect
58. #63837C2×C3semidirect
178. C4×#67C3×D3semidirect
397. C33#559semidirect
572. C32#5559semidirect
573. C32C3×#559semidirect
632. C3#10595030direct (N × H)
633. C3C3×#5559semidirect

Maximal subgroups

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

subgroupindex
C3×#641382
C32×C4×#672
C3×#639792
C3×#639752
C3×#639592
C3×#639052
C3×#639042
C3×#638522
C3×#638372
C3×#55593
#105950303