Finite Group Databasebeta

← C3×#10594401 #13173958

Subgroups — C3×#10594401 #13173958

Order 1728.

1036 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64106C211yesmaximal, characteristic
2C32×#889C212yesmaximal, complemented
3C3×#63918C212yesmaximal, complemented
4C3×#63895C212yesmaximal
5C3×#63883C211yesmaximal, characteristic
6C3×#63865C211yesmaximal, characteristic
7C3×#63845C212yesmaximal, complemented
8C3×#63840C212yesmaximal, complemented
9C3×#63836C212yesmaximal
10C3×#5689—32nomaximal
11#10594401C311yesmaximal, characteristic, direct product factor
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C3×#627C2212yes—
15C2×C3×#627C2211yescharacteristic
16C2×C3×#624C2211yescharacteristic
17C2×C3×#624C2211yescharacteristic
18C33×C4⋊C4C2211yescharacteristic
19C3×#3752C2212yes—
20C3×C4⋊(C3⋊Dic3)C2212yes—
21C32×C4⋊Dic3C2212yes—
22C32×Dic3⋊C4C2212yes—
23C32×Dic3⋊C4C2212yes—
24C32×Dic3⋊C4C2212yes—
25C32×Dic3⋊C4C2212yes—
26C32×C4×Dic3C2212yes—
27C3×#3724C2211yescharacteristic
28C3×Dic3⋊Dic3C2212yes—
29C3×Dic3⋊Dic3C2212yes—
30C3×Dic3⋊Dic3C2212yes—
31C3×#3722C2212yes—
32C3×Dic32C2212yes—
33C32×#125—32no—
34#64106C2×C311yescharacteristic
35C3×#897—32no—
36C3×#897—32no—
37C3×#897—61no—
38C3×#896—32no—
39C3×#896—32no—
40C3×#892—32no—
41C3×D3×C4⋊C4—32no—
42C3×#890—32no—
43C3×#889—22no—
44C3×#889—32no—
45C3×#889D312yescomplemented
46C3×#889C2×C312yescomplemented
47C3×#888—32no—
48C3×#888—32no—
49C3×#888—32no—
50C3×#872—32no—
51C3×#869—32no—
52#63918C2×C312yescomplemented
53#63895C2×C312yes—
54#63883C2×C311yescharacteristic
55#63865C2×C311yescharacteristic
56#63845C2×C312yescomplemented
57#63840C2×C312yescomplemented
58#63836C2×C312yes—
59C22×C3×C3⋊D3C2311yescharacteristic
60C2×C33×C4C2312yes—
61C2×C33×C4C2311yescharacteristic
62C2×C3×C3⋊Dic3C2311yescharacteristic
63C2×C3×C3⋊Dic3C2312yes—
64C3×C4×C3⋊D3—41no—
65C3×#634—41no—
66C2×C32×Dic3C2312yes—
67C2×C32×Dic3C2312yes—
68C2×C32×Dic3C2312yes—
69C2×C32×Dic3C2312yes—
70C3×#627—42no—
71C3×#627—41no—
72C3×#624—41no—
73C3×#624Q814yescomplemented
74C3×#544—91no—
75#5689—32no—
76C2×C4×C3⋊D3C22×C311yescharacteristic
77C2×#634C22×C311yescharacteristic
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—61no—
82C2×C3×Dic6—32no—
83C2×C3×Dic6—32no—
84C2×C3×Dic6—32no—
85C2×C3×Dic6—32no—
86C2×C3×Dic6—61no—
87C2×#627C22×C312yes—
88C2×#627C22×C311yescharacteristic
89C2×#624C22×C311yescharacteristic
90C2×#624C22×C311yescharacteristic
91C32×C4⋊C4—21no—
92C32×C4⋊C4—22no—
93C32×C4⋊C4—32no—
94C32×C4⋊C4—32no—
95C32×C4⋊C4—32no—
96C32×C4⋊C4—32no—
97C32×C4⋊C4—32no—
98C32×C4⋊C4—41no—
99C32×C4⋊C4C2×D312yes—
100C32×C4⋊C4C22×C311yescharacteristic
101C32×C42—32no—
102#3752C22×C312yes—
103C4⋊(C3⋊Dic3)C22×C312yes—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—32no—
106C3×(C2×D3)⋊C4—32no—
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4—62no—
109C3×C4⋊Dic3—22no—
110C3×C4⋊Dic3—32no—
111C3×C4⋊Dic3—32no—
112C3×C4⋊Dic3—32no—
113C3×C4⋊Dic3—32no—
114C3×C4⋊Dic3—32no—
115C3×C4⋊Dic3—62no—
116C3×C4⋊Dic3C2×D312yes—
117C3×C4⋊Dic3C22×C312yes—
118C3×Dic3⋊C4—22no—
119C3×Dic3⋊C4—22no—
120C3×Dic3⋊C4—22no—
121C3×Dic3⋊C4—22no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—32no—
126C3×Dic3⋊C4C2×D312yes—
127C3×Dic3⋊C4C2×D312yes—
128C3×Dic3⋊C4C2×D312yes—
129C3×Dic3⋊C4C2×D312yes—
130C3×Dic3⋊C4C22×C312yes—
131C3×Dic3⋊C4C22×C312yes—
132C3×Dic3⋊C4C22×C312yes—
133C3×Dic3⋊C4C22×C312yes—
134C3×C4×Dic3—22no—
135C3×C4×Dic3—32no—
136C3×C4×Dic3—32no—
137C3×C4×Dic3C2×D312yes—
138C3×C4×Dic3C22×C312yes—
139#3724C22×C311yescharacteristic
140Dic3⋊Dic3C22×C312yes—
141Dic3⋊Dic3C22×C312yes—
142Dic3⋊Dic3C22×C312yes—
143#3722C22×C312yes—
144Dic32C22×C312yes—
145C22×C33C2411yescharacteristic
146C2×C3×C3⋊D3—21no—
147C2×C3×C3⋊D3—21no—
148C2×C3×C3⋊D3C2×Q812yes—
149C33×C4—21no—
150C33×C4—22no—
151C3×C3⋊Dic3—22no—
152C3×C3⋊Dic3—21no—
153C32×Dic3—22no—
154C32×Dic3—22no—
155C32×Dic3—22no—
156C32×Dic3C2×Q814yes—
157C3×C4⋊Q8—92no—
158C3×C4⋊Q8—92no—
159C3×#125—32no—
160C3×#125—32no—
161C3×#125—62no—
162C3×#125—92no—
163C3×C22⋊Q8—91no—
164C3×C22⋊Q8—92no—
165C3×C22⋊Q8—91no—
166C3×#117—92no—
167C2×C3×C4⋊C4—91no—
168#897—32no—
169#897—32no—
170#897—61no—
171#896—32no—
172#896—32no—
173#892—32no—
174D3×C4⋊C4—32no—
175#890—32no—
176#889—32no—
177#889C3×D312yescomplemented
178#888—32no—
179#888—32no—
180#888—32no—
181#872—32no—
182#869—32no—
183C22×C3⋊D3C23×C311yescharacteristic
184C22×C3×D3—32no—
185C22×C3×D3—61no—
186C2×C32×C4—21no—
187C2×C32×C4—22no—
188C2×C32×C4—22no—
189C2×C32×C4—22no—
190C2×C32×C4—22no—
191C2×C32×C4—32no—
192C2×C32×C4—32no—
193C2×C32×C4—32no—
194C2×C32×C4—32no—
195C2×C32×C4—41no—
196C2×C32×C4—42no—
197C2×C32×C4C22×D312yes—
198C2×C32×C4C22×D312yes—
199C2×C32×C4C22×D312yes—
200C2×C32×C4C23×C312yes—
201C2×C32×C4C23×C311yescharacteristic
202C2×C3⋊Dic3C23×C312yes—
203C2×C3⋊Dic3C23×C311yescharacteristic
204C4×C3⋊D3—41no—
205#634—41no—
206C2×C3×Dic3—22no—
207C2×C3×Dic3—22no—
208C2×C3×Dic3—22no—
209C2×C3×Dic3—22no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—61no—
214C2×C3×Dic3—62no—
215C2×C3×Dic3C22×D312yes—
216C2×C3×Dic3C22×D312yes—
217C2×C3×Dic3C22×D312yes—
218C2×C3×Dic3C22×D312yes—
219C2×C3×Dic3C23×C312yes—
220C2×C3×Dic3C23×C312yes—
221C2×C3×Dic3C23×C312yes—
222C2×C3×Dic3C23×C312yes—
223C3×C4×D3—38no—
224C3×C4×D3—122no—
225C3×C4×D3—122no—
226C3×C4×D3—122no—
227C3×Dic6—122no—
228C3×Dic6—122no—
229C3×Dic6—122no—
230C3×Dic6—122no—
231C3×Dic6—122no—
232#627—42no—
233#627—41no—
234#624—41no—
235#624C3×Q814yescomplemented
236#544—91noSylow
237C2×C33C22×Q811yescharacteristic
238C2×C33H(5,𝔽2)11yescharacteristic
239C2×C332−1+411yescharacteristic
240C3×C3⋊D3—22no—
241C2×C3×Q8—91no—
242C2×C3×Q8—92no—
243C2×C3×Q8—91no—
244C22×C3×C4—91no—
245C22×C3×C4—91no—
246C22×C3×C4—91no—
247C2×C4×D3—32no—
248C2×C4×D3—32no—
249C2×C4×D3—32no—
250C2×C4×D3—61no—
251C2×Dic6—32no—
252C2×Dic6—32no—
253C2×Dic6—32no—
254C2×Dic6—32no—
255C2×Dic6—61no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—32no—
259C3×C4⋊C4—32no—
260C3×C4⋊C4—32no—
261C3×C4⋊C4—32no—
262C3×C4⋊C4—32no—
263C3×C4⋊C4—32no—
264C3×C4⋊C4—32no—
265C3×C4⋊C4—32no—
266C3×C4⋊C4—32no—
267C3×C4⋊C4—32no—
268C3×C4⋊C4—41no—
269C3×C4⋊C4—62no—
270C3×C4⋊C4—62no—
271C3×C4⋊C4—62no—
272C3×C4⋊C4—62no—
273C3×C4⋊C4—62no—
274C3×C4⋊C4—91no—
275C3×C4⋊C4—92no—
276C3×C4⋊C4—92no—
277C3×C4⋊C4—92no—
278C3×C4⋊C4—92no—
279C3×C4⋊C4D3211yescharacteristic
280C3×C4⋊C4C2×C3×D312yes—
281C3×C22⋊C4—92no—
282C3×C22⋊C4—92no—
283C3×C42—32no—
284C3×C42—32no—
285C3×C42—62no—
286C3×C42—92no—
287(C2×D3)⋊C4—32no—
288(C2×D3)⋊C4—32no—
289(C2×D3)⋊C4—32no—
290(C2×D3)⋊C4—32no—
291(C2×D3)⋊C4—62no—
292C4⋊Dic3—32no—
293C4⋊Dic3—32no—
294C4⋊Dic3—32no—
295C4⋊Dic3—32no—
296C4⋊Dic3—32no—
297C4⋊Dic3—62no—
298C4⋊Dic3C2×C3×D312yes—
299Dic3⋊C4—32no—
300Dic3⋊C4—32no—
301Dic3⋊C4—32no—
302Dic3⋊C4—32no—
303Dic3⋊C4C2×C3×D312yes—
304Dic3⋊C4C2×C3×D312yes—
305Dic3⋊C4C2×C3×D312yes—
306Dic3⋊C4C2×C3×D312yes—
307C4×Dic3—32no—
308C4×Dic3—32no—
309C4×Dic3C2×C3×D312yes—
310C22×C32—21no—
311C22×C32—22no—
312C22×C32—41no—
313C22×C32C23×D312yes—
314C22×C32C24×C311yescharacteristic
315C2×C3⋊D3—21no—
316C2×C3⋊D3—21no—
317C2×C3⋊D3C2×C3×Q812yes—
318C2×C3×D3—34no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—121no—
323C2×C3×D3—121no—
324C32×C4—21no—
325C32×C4—22no—
326C32×C4—22no—
327C32×C4—22no—
328C32×C4—22no—
329C32×C4—22no—
330C32×C4—34no—
331C32×C4—42no—
332C32×C4—42no—
333C32×C4—42no—
334C32×C4—42no—
335C32×C4—42no—
336C32×C4—62no—
337C32×C4—62no—
338C32×C4—62no—
339C32×C4—82no—
340C3⋊Dic3—22no—
341C3⋊Dic3—21no—
342C3×Dic3—22no—
343C3×Dic3—22no—
344C3×Dic3—22no—
345C3×Dic3—22no—
346C3×Dic3—22no—
347C3×Dic3—22no—
348C3×Dic3—24no—
349C3×Dic3—42no—
350C3×Dic3—42no—
351C3×Dic3—42no—
352C3×Dic3—62no—
353C3×Dic3—62no—
354C3×Dic3—62no—
355C3×Dic3—122no—
356C3×Dic3—121no—
357C3×Dic3D3×Q814yes—
358C3×Dic3C2×C3×Q814yes—
359C4⋊Q8—92no—
360C4⋊Q8—92no—
361#125—32no—
362#125—92no—
363C22⋊Q8—91no—
364C22⋊Q8—92no—
365C22⋊Q8—91no—
366#117—92no—
367C2×C4⋊C4—91no—
368C33#54411yescharacteristic, complemented, Sylow
369C23×C3—91no—
370C22×D3—32no—
371C22×D3—61no—
372C3×Q8—361no—
373C3×Q8—362no—
374C3×Q8—361no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—32no—
381C2×C3×C4—32no—
382C2×C3×C4—32no—
383C2×C3×C4—32no—
384C2×C3×C4—32no—
385C2×C3×C4—32no—
386C2×C3×C4—32no—
387C2×C3×C4—32no—
388C2×C3×C4—41no—
389C2×C3×C4—42no—
390C2×C3×C4—62no—
391C2×C3×C4—62no—
392C2×C3×C4—62no—
393C2×C3×C4—62no—
394C2×C3×C4—94no—
395C2×C3×C4—92no—
396C2×C3×C4—91no—
397C2×C3×C4—361no—
398C2×C3×C4—361no—
399C2×C3×C4C2×D3212yes—
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C22×C3×D312yes—
402C2×C3×C4C22×C3×D312yes—
403C2×C3×C4C22×C3×D312yes—
404C2×Dic3—32no—
405C2×Dic3—32no—
406C2×Dic3—32no—
407C2×Dic3—61no—
408C2×Dic3—62no—
409C2×Dic3C22×C3×D312yes—
410C2×Dic3C22×C3×D312yes—
411C2×Dic3C22×C3×D312yes—
412C2×Dic3C22×C3×D312yes—
413C4×D3—38no—
414C4×D3—122no—
415C4×D3—122no—
416C4×D3—122no—
417Dic6—122no—
418Dic6—122no—
419Dic6—122no—
420Dic6—122no—
421Dic6—122no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—22no—
426C2×C32—22no—
427C2×C32—22no—
428C2×C32—41no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32C2×D3×Q812yes—
432C2×C32#100912yes—
433C2×C32#101012yes—
434C2×C32C22×C3×Q811yescharacteristic
435C2×C32C3×H(5,𝔽2)11yescharacteristic
436C2×C32C3×2−1+411yescharacteristic
437C3⋊D3—22no—
438C3×D3—64no—
439C3×D3—122no—
440C2×Q8—91no—
441C2×Q8—92no—
442C2×Q8—91no—
443C22×C4—91no—
444C22×C4—91no—
445C22×C4—91no—
446C4⋊C4—32no—
447C4⋊C4—32no—
448C4⋊C4—32no—
449C4⋊C4—32no—
450C4⋊C4—32no—
451C4⋊C4—92no—
452C4⋊C4—92no—
453C4⋊C4—91no—
454C4⋊C4—92no—
455C4⋊C4—92no—
456C4⋊C4C3×D3211yescharacteristic
457C22⋊C4—92no—
458C22⋊C4—92no—
459C42—32no—
460C42—92no—
461C22×C3—21no—
462C22×C3—22no—
463C22×C3—41no—
464C22×C3—92no—
465C22×C3—181no—
466C22×C3—181no—
467C22×C3C22×D3211yescharacteristic
468C22×C3C23×C3×D312yes—
469C2×D3—34no—
470C2×D3—62no—
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—121no—
474C2×D3—121no—
475C3×C4—22no—
476C3×C4—21no—
477C3×C4—22no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—34no—
482C3×C4—34no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—64no—
489C3×C4—62no—
490C3×C4—62no—
491C3×C4—62no—
492C3×C4—62no—
493C3×C4—62no—
494C3×C4—62no—
495C3×C4—82no—
496C3×C4—122no—
497C3×C4—122no—
498C3×C4—122no—
499C3×C4—181no—
500C3×C4—182no—
501Dic3—22no—
502Dic3—22no—
503Dic3—22no—
504Dic3—62no—
505Dic3—62no—
506Dic3—62no—
507Dic3—122no—
508Dic3—121no—
509Dic3C3×D3×Q814yes—
510C32—21no—
511C32—22no—
512C32—41no—
513C32#568912yescomplemented
514C32C3×#54411yescharacteristic, complemented
515C23—91no—
516Q8—361no—
517Q8—362no—
518Q8—361no—
519C2×C4—32no—
520C2×C4—32no—
521C2×C4—32no—
522C2×C4—32no—
523C2×C4—94no—
524C2×C4—92no—
525C2×C4—91no—
526C2×C4—361no—
527C2×C4—361no—
528C2×C4C2×C3×D3212yes—
529C2×C4C2×C3×D3211yescharacteristic
530C2×C3—22no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—21no—
536C2×C3—41no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—182no—
540C2×C3#6430811yescharacteristic
541C2×C3#6430911yescharacteristic
542C2×C3D32×Q811yescharacteristic
543C2×C3C2×C3×D3×Q812yes—
544C2×C3C3×#100912yes—
545C2×C3C3×#101012yes—
546D3—64no—
547D3—122no—
548C22—92no—
549C22—181no—
550C22—181no—
551C22C22×C3×D3211yescharacteristic
552C4—21no—
553C4—22no—
554C4—34no—
555C4—62no—
556C4—62no—
557C4—62no—
558C4—181no—
559C4—182no—
560C3—21no—
561C3—22no—
562C3—41no—
563C3#1059440111yescharacteristic, direct product factor
564C3C3×#568912yescomplemented
565C2—182no—
566C2C3×#6430811yescharacteristic
567C2C3×#6430911yescharacteristic
568C2C3×D32×Q811yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C24
prime radicalC22×C3×C3⋊D3C23
hypercenterC3×C4⋊C4D32
hypocenterC32C3×#544
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C32×#889C2semidirect
3. C3×#63918C2semidirect
7. C3×#63845C2semidirect
8. C3×#63840C2semidirect
11. #10594401C3direct (N × H)
45. C3×#889D3semidirect
46. C3×#889C2×C3semidirect
52. #63918C2×C3semidirect
56. #63845C2×C3semidirect
57. #63840C2×C3semidirect
73. C3×#624Q8semidirect
177. #889C3×D3semidirect
235. #624C3×Q8semidirect
368. C33#544semidirect
513. C32#5689semidirect
514. C32C3×#544semidirect
563. C3#10594401direct (N × H)
564. C3C3×#5689semidirect

Maximal subgroups

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

subgroupindex
C3×#641062
C32×#8892
C3×#639182
C3×#638952
C3×#638832
C3×#638652
C3×#638452
C3×#638402
C3×#638362
C3×#56893
#105944013