Finite Group Databasebeta

← C3×#10595051 #13174648

Subgroups — C3×#10595051 #13174648

Order 1728.

1122 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
1C3×#64135C211yesmaximal, characteristic
2C32×#926C211yesmaximal, characteristic
3C2×C32×C4⋊Dic3C211yesmaximal, characteristic
4C3×#63964C214yesmaximal
5C3×#63919C212yesmaximal, complemented
6C3×#63918C212yesmaximal, complemented
7C3×#63872C212yesmaximal, complemented
8C3×#63869C212yesmaximal, complemented
9C3×#5519—31nomaximal
10C3×#5500—31nomaximal
11#10595051C311yesmaximal, characteristic, direct product factor
12C22×C33×C4C2211yescharacteristic
13C22×C32×Dic3C2212yes—
14C2×C3×#627C2214yes—
15C3×#3757C2212yes—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4×C3⋊Dic3C2212yes—
19C32×#217C2212yes—
20C32×C4⋊Dic3C2212yes—
21C32×C4⋊Dic3C2212yes—
22C32×C4⋊Dic3C2211yescharacteristic
23C32×C4⋊Dic3C2211yescharacteristic
24C32×C4×Dic3C2212yes—
25C3×#3724C2214yes—
26C3×Dic3⋊Dic3C2214yes—
27C3×Dic3⋊Dic3C2214yes—
28C32×#117—31no—
29C2×C32×C4⋊C4—31no—
30#64135C2×C311yescharacteristic
31C3×#926—21no—
32C3×#926—31no—
33C3×#926—31no—
34C3×#926—61no—
35C3×#926D311yescharacteristic
36C3×#926C2×C311yescharacteristic
37C2×C3×C4⋊Dic3—21no—
38C2×C3×C4⋊Dic3D311yescharacteristic
39C2×C3×C4⋊Dic3C2×C311yescharacteristic
40C3×#924—34no—
41C3×#890—32no—
42C3×#890—32no—
43C3×#890—32no—
44C3×#888—32no—
45C3×#888—32no—
46C3×#888—32no—
47C3×#878—34no—
48C3×#870—32no—
49C3×#869—32no—
50#63964C2×C314yes—
51#63919C2×C312yescomplemented
52#63918C2×C312yescomplemented
53#63872C2×C312yescomplemented
54#63869C2×C312yescomplemented
55C23×C33C2311yescharacteristic
56C2×C33×C4Q814yescomplemented
57C2×C33×C4C2311yescharacteristic
58C2×C33×C4C2311yescharacteristic
59C2×C3×C3⋊Dic3C2312yes—
60C2×C3×C3⋊Dic3C2312yes—
61C2×C32×Dic3—42no—
62C2×C32×Dic3C2312yes—
63C2×C32×Dic3C2312yes—
64C2×C32×Dic3C2314yes—
65C3×#627—44no—
66C3×#544—91no—
67#5519—31no—
68#5500—31no—
69C22×C32×C4—21no—
70C22×C32×C4—21no—
71C22×C32×C4—21no—
72C22×C32×C4—32no—
73C22×C32×C4—41no—
74C22×C32×C4C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C22×C32×C4C22×C311yescharacteristic
77C22×C3×Dic3—22no—
78C22×C3×Dic3C2×D312yes—
79C22×C3×Dic3C22×C312yes—
80C2×C3×Dic6—34no—
81C2×C3×Dic6—34no—
82C2×#627C22×C314yes—
83C32×C4⋊C4—31no—
84C32×C4⋊C4—32no—
85C32×C4⋊C4—32no—
86C32×C4⋊C4—31no—
87C32×C22⋊C4—32no—
88C32×C42—32no—
89#3757C22×C312yes—
90C4⋊(C3⋊Dic3)C22×C311yescharacteristic
91C4⋊(C3⋊Dic3)C22×C311yescharacteristic
92C4×C3⋊Dic3C22×C312yes—
93C3×#217—22no—
94C3×#217—32no—
95C3×#217—32no—
96C3×#217—62no—
97C3×#217C2×D312yes—
98C3×#217C22×C312yes—
99C3×C4⋊Dic3—22no—
100C3×C4⋊Dic3—22no—
101C3×C4⋊Dic3—21no—
102C3×C4⋊Dic3—21no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—34no—
108C3×C4⋊Dic3—34no—
109C3×C4⋊Dic3—61no—
110C3×C4⋊Dic3—61no—
111C3×C4⋊Dic3C2×D311yescharacteristic
112C3×C4⋊Dic3C2×D312yes—
113C3×C4⋊Dic3C2×D311yescharacteristic
114C3×C4⋊Dic3C2×D312yes—
115C3×C4⋊Dic3C22×C312yes—
116C3×C4⋊Dic3C22×C312yes—
117C3×C4⋊Dic3C22×C311yescharacteristic
118C3×C4⋊Dic3C22×C311yescharacteristic
119C3×Dic3⋊C4—34no—
120C3×Dic3⋊C4—34no—
121C3×Dic3⋊C4—34no—
122C3×Dic3⋊C4—34no—
123C3×C4×Dic3—22no—
124C3×C4×Dic3—32no—
125C3×C4×Dic3—32no—
126C3×C4×Dic3—62no—
127C3×C4×Dic3C2×D312yes—
128C3×C4×Dic3C22×C312yes—
129#3724C22×C314yes—
130Dic3⋊Dic3C22×C314yes—
131Dic3⋊Dic3C22×C314yes—
132C22×C33—21no—
133C22×C33—21no—
134C22×C33C2×Q812yes—
135C22×C33C2411yescharacteristic
136C33×C4C2×Q812yes—
137C33×C4C2×Q812yes—
138C3×C3⋊Dic3—22no—
139C3×C3⋊Dic3—22no—
140C32×Dic3—22no—
141C32×Dic3—24no—
142C32×Dic3—22no—
143C3×C4⋊Q8—92no—
144C3×C4⋊Q8—92no—
145C3×#125—92no—
146C3×#125—92no—
147C3×C22⋊Q8—94no—
148C3×#117—31no—
149C3×#117—31no—
150C3×#117—61no—
151C3×#117—91no—
152C2×C3×C4⋊C4—31no—
153C2×C3×C4⋊C4—31no—
154C2×C3×C4⋊C4—61no—
155#926—31no—
156#926—31no—
157#926—61no—
158#926C3×D311yescharacteristic
159C2×C4⋊Dic3C3×D311yescharacteristic
160#924—34no—
161#890—32no—
162#890—32no—
163#890—32no—
164#888—32no—
165#888—32no—
166#888—32no—
167#878—34no—
168#870—32no—
169#869—32no—
170C23×C32—21no—
171C23×C32—21no—
172C23×C32—21no—
173C23×C32—41no—
174C23×C32C22×D311yescharacteristic
175C23×C32C22×D311yescharacteristic
176C23×C32C23×C311yescharacteristic
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—24no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—32no—
187C2×C32×C4—34no—
188C2×C32×C4—32no—
189C2×C32×C4—44no—
190C2×C32×C4—41no—
191C2×C32×C4—41no—
192C2×C32×C4—122no—
193C2×C32×C4Dic614yescomplemented
194C2×C32×C4Dic614yescomplemented
195C2×C32×C4C3×Q814yescomplemented
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200C2×C32×C4C23×C311yescharacteristic
201C2×C32×C4C23×C311yescharacteristic
202C2×C3⋊Dic3C23×C312yes—
203C2×C3⋊Dic3C23×C312yes—
204C2×C3×Dic3—22no—
205C2×C3×Dic3—22no—
206C2×C3×Dic3—24no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—42no—
212C2×C3×Dic3—42no—
213C2×C3×Dic3—44no—
214C2×C3×Dic3—62no—
215C2×C3×Dic3—62no—
216C2×C3×Dic3C22×D312yes—
217C2×C3×Dic3C22×D312yes—
218C2×C3×Dic3C22×D314yes—
219C2×C3×Dic3C23×C312yes—
220C2×C3×Dic3C23×C314yes—
221C2×C3×Dic3C23×C312yes—
222C3×Dic6—124no—
223C3×Dic6—124no—
224#627—44no—
225#544—91noSylow
226C2×C33—22no—
227C2×C33C22×Q811yescharacteristic
228C2×C33H(5,𝔽2)11yescharacteristic
229C2×C332−1+411yescharacteristic
230C2×C3×Q8—94no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—21no—
234C22×C3×C4—32no—
235C22×C3×C4—32no—
236C22×C3×C4—41no—
237C22×C3×C4—62no—
238C22×C3×C4D3211yescharacteristic
239C22×C3×C4C2×C3×D311yescharacteristic
240C22×C3×C4C2×C3×D311yescharacteristic
241C22×Dic3C2×C3×D312yes—
242C2×Dic6—34no—
243C2×Dic6—34no—
244C3×C4⋊C4—32no—
245C3×C4⋊C4—32no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—31no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—32no—
250C3×C4⋊C4—32no—
251C3×C4⋊C4—31no—
252C3×C4⋊C4—62no—
253C3×C4⋊C4—62no—
254C3×C4⋊C4—61no—
255C3×C4⋊C4—61no—
256C3×C4⋊C4—94no—
257C3×C4⋊C4—94no—
258C3×C4⋊C4—94no—
259C3×C4⋊C4—91no—
260C3×C4⋊C4—91no—
261C3×C22⋊C4—32no—
262C3×C22⋊C4—32no—
263C3×C22⋊C4—62no—
264C3×C22⋊C4—92no—
265C3×C42—32no—
266C3×C42—32no—
267C3×C42—62no—
268C3×C42—92no—
269#217—32no—
270#217—32no—
271#217—62no—
272#217C2×C3×D312yes—
273C4⋊Dic3—31no—
274C4⋊Dic3—31no—
275C4⋊Dic3—31no—
276C4⋊Dic3—31no—
277C4⋊Dic3—34no—
278C4⋊Dic3—34no—
279C4⋊Dic3—61no—
280C4⋊Dic3—61no—
281C4⋊Dic3C2×C3×D312yes—
282C4⋊Dic3C2×C3×D311yescharacteristic
283C4⋊Dic3C2×C3×D312yes—
284C4⋊Dic3C2×C3×D311yescharacteristic
285Dic3⋊C4—34no—
286Dic3⋊C4—34no—
287Dic3⋊C4—34no—
288Dic3⋊C4—34no—
289C4×Dic3—32no—
290C4×Dic3—32no—
291C4×Dic3—62no—
292C4×Dic3C2×C3×D312yes—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—21no—
298C22×C32—21no—
299C22×C32—21no—
300C22×C32—21no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—22no—
305C22×C32—22no—
306C22×C32—22no—
307C22×C32—42no—
308C22×C32—41no—
309C22×C32—41no—
310C22×C32—41no—
311C22×C32—41no—
312C22×C32—41no—
313C22×C32—42no—
314C22×C32—42no—
315C22×C32C2×Dic612yes—
316C22×C32C2×Dic612yes—
317C22×C32C2×C3×Q812yes—
318C22×C32C23×D311yescharacteristic
319C22×C32C23×D311yescharacteristic
320C22×C32C24×C311yescharacteristic
321C32×C4—22no—
322C32×C4—22no—
323C32×C4—22no—
324C32×C4—22no—
325C32×C4—22no—
326C32×C4—22no—
327C32×C4—42no—
328C32×C4—42no—
329C32×C4—62no—
330C32×C4—62no—
331C32×C4—64no—
332C32×C4C2×Dic612yes—
333C32×C4C2×Dic612yes—
334C32×C4C2×Dic612yes—
335C32×C4C2×Dic612yes—
336C32×C4C2×C3×Q812yes—
337C32×C4C2×C3×Q812yes—
338C3⋊Dic3—22no—
339C3⋊Dic3—22no—
340C3×Dic3—24no—
341C3×Dic3—22no—
342C3×Dic3—22no—
343C3×Dic3—22no—
344C3×Dic3—24no—
345C3×Dic3—22no—
346C3×Dic3—42no—
347C3×Dic3—42no—
348C3×Dic3—44no—
349C3×Dic3—62no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C3×Dic3—62no—
353C3×Dic3—122no—
354C3×Dic3—122no—
355C4⋊Q8—92no—
356C4⋊Q8—92no—
357#125—92no—
358#125—92no—
359C22⋊Q8—94no—
360#117—31no—
361#117—91no—
362C2×C4⋊C4—31no—
363C33#54411yescharacteristic, complemented, Sylow
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—41no—
368C23×C3C2×D3211yescharacteristic
369C23×C3C22×C3×D311yescharacteristic
370C23×C3C22×C3×D311yescharacteristic
371C3×Q8—364no—
372C2×C3×C4—24no—
373C2×C3×C4—24no—
374C2×C3×C4—24no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—32no—
382C2×C3×C4—32no—
383C2×C3×C4—34no—
384C2×C3×C4—32no—
385C2×C3×C4—34no—
386C2×C3×C4—32no—
387C2×C3×C4—44no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—62no—
391C2×C3×C4—62no—
392C2×C3×C4—64no—
393C2×C3×C4—92no—
394C2×C3×C4—92no—
395C2×C3×C4—122no—
396C2×C3×C4—122no—
397C2×C3×C4—124no—
398C2×C3×C4#62714yescomplemented
399C2×C3×C4C3×Dic614yescomplemented
400C2×C3×C4C3×Dic614yescomplemented
401C2×C3×C4C2×D3211yescharacteristic
402C2×C3×C4C2×D3211yescharacteristic
403C2×C3×C4C22×C3×D311yescharacteristic
404C2×C3×C4C22×C3×D311yescharacteristic
405C2×C3×C4C22×C3×D311yescharacteristic
406C2×C3×C4C22×C3×D311yescharacteristic
407C2×Dic3—32no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—32no—
411C2×Dic3—42no—
412C2×Dic3—62no—
413C2×Dic3—62no—
414C2×Dic3C22×C3×D312yes—
415C2×Dic3C22×C3×D312yes—
416C2×Dic3C22×C3×D314yes—
417Dic6—124no—
418Dic6—124no—
419C2×C32—22no—
420C2×C32—24no—
421C2×C32—22no—
422C2×C32—22no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—41no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—42no—
436C2×C32—42no—
437C2×C32—44no—
438C2×C32C22×Dic611yescharacteristic
439C2×C32C22×Dic611yescharacteristic
440C2×C32#100411yescharacteristic
441C2×C32#100711yescharacteristic
442C2×C32#100911yescharacteristic
443C2×C32#101011yescharacteristic
444C2×C32C22×C3×Q811yescharacteristic
445C2×C32C3×H(5,𝔽2)11yescharacteristic
446C2×C32C3×2−1+411yescharacteristic
447C2×Q8—94no—
448C22×C4—32no—
449C22×C4C3×D3211yescharacteristic
450C4⋊C4—32no—
451C4⋊C4—32no—
452C4⋊C4—31no—
453C4⋊C4—31no—
454C4⋊C4—94no—
455C4⋊C4—94no—
456C4⋊C4—94no—
457C4⋊C4—91no—
458C4⋊C4—91no—
459C22⋊C4—32no—
460C22⋊C4—92no—
461C42—32no—
462C42—92no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—41no—
484C22×C3—42no—
485C22×C3C2×#62712yes—
486C22×C3C2×C3×Dic612yes—
487C22×C3C2×C3×Dic612yes—
488C22×C3C22×D3211yescharacteristic
489C22×C3C23×C3×D311yescharacteristic
490C22×C3C23×C3×D311yescharacteristic
491C3×C4—22no—
492C3×C4—22no—
493C3×C4—22no—
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—22no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—62no—
500C3×C4—64no—
501C3×C4—62no—
502C3×C4—62no—
503C3×C4—62no—
504C3×C4—64no—
505C3×C4—124no—
506C3×C4—122no—
507C3×C4—122no—
508C3×C4—182no—
509C3×C4—182no—
510C3×C4C2×#62712yes—
511C3×C4C2×#62712yes—
512C3×C4C2×C3×Dic612yes—
513C3×C4C2×C3×Dic612yes—
514C3×C4C2×C3×Dic612yes—
515C3×C4C2×C3×Dic612yes—
516Dic3—24no—
517Dic3—22no—
518Dic3—22no—
519Dic3—62no—
520Dic3—62no—
521Dic3—62no—
522Dic3—62no—
523Dic3—122no—
524Dic3—122no—
525C32—21no—
526C32—21no—
527C32—21no—
528C32—41no—
529C32#550011yescharacteristic, complemented
530C32#551911yescharacteristic, complemented
531C32C3×#54411yescharacteristic, complemented
532C23C2×C3×D3211yescharacteristic
533Q8—364no—
534C2×C4—32no—
535C2×C4—32no—
536C2×C4—34no—
537C2×C4—92no—
538C2×C4—92no—
539C2×C4—122no—
540C2×C4C3×#62714yescomplemented
541C2×C4C2×C3×D3211yescharacteristic
542C2×C4C2×C3×D3211yescharacteristic
543C2×C3—24no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—22no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—42no—
557C2×C3—42no—
558C2×C3—44no—
559C2×C3—41no—
560C2×C3—41no—
561C2×C3—41no—
562C2×C3#6429711yescharacteristic
563C2×C3#6430711yescharacteristic
564C2×C3C22×#62711yescharacteristic
565C2×C3C22×C3×Dic611yescharacteristic
566C2×C3C22×C3×Dic611yescharacteristic
567C2×C3C3×#100411yescharacteristic
568C2×C3C3×#100711yescharacteristic
569C2×C3C3×#100911yescharacteristic
570C2×C3C3×#101011yescharacteristic
571C22—21no—
572C22—21no—
573C22C2×C3×#62712yes—
574C22C22×C3×D3211yescharacteristic
575C4—62no—
576C4—64no—
577C4—62no—
578C4—182no—
579C4—182no—
580C4C2×C3×#62712yes—
581C4C2×C3×#62712yes—
582C3—21no—
583C3—21no—
584C3—21no—
585C3—41no—
586C3#1059505111yescharacteristic, direct product factor
587C3C3×#550011yescharacteristic, complemented
588C3C3×#551911yescharacteristic, complemented
589C2—22no—
590C2C3×#6429711yescharacteristic
591C2C3×#6430711yescharacteristic
592C2C22×C3×#62711yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C3×#544
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
5. C3×#63919C2semidirect
6. C3×#63918C2semidirect
7. C3×#63872C2semidirect
8. C3×#63869C2semidirect
11. #10595051C3direct (N × H)
51. #63919C2×C3semidirect
52. #63918C2×C3semidirect
53. #63872C2×C3semidirect
54. #63869C2×C3semidirect
56. C2×C33×C4Q8semidirect
193. C2×C32×C4Dic6semidirect
194. C2×C32×C4Dic6semidirect
195. C2×C32×C4C3×Q8semidirect
363. C33#544semidirect
398. C2×C3×C4#627semidirect
399. C2×C3×C4C3×Dic6semidirect
400. C2×C3×C4C3×Dic6semidirect
529. C32#5500semidirect
530. C32#5519semidirect
531. C32C3×#544semidirect
540. C2×C4C3×#627semidirect
586. C3#10595051direct (N × H)
587. C3C3×#5500semidirect
588. C3C3×#5519semidirect

Maximal subgroups

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

subgroupindex
C3×#641352
C32×#9262
C2×C32×C4⋊Dic32
C3×#639642
C3×#639192
C3×#639182
C3×#638722
C3×#638692
C3×#55193
C3×#55003
#105950513