Finite Group Databasebeta

← C3×#10596023 #13179288

Subgroups — C3×#10596023 #13179288

Order 1728.

2116 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic, complemented
2C22×C3×#634C211yesmaximal, characteristic, complemented
3C33×C4×D4C211yesmaximal, characteristic, complemented
4C3×#64148C212yesmaximal, complemented
5C3×#64139C211yesmaximal, characteristic, complemented
6C3×#64133C211yesmaximal, characteristic, complemented
7C3×#64105C212yesmaximal, complemented
8C3×#64094C212yesmaximal, complemented
9C3×#64090C212yesmaximal, complemented
10C3×#64084C211yesmaximal, characteristic, complemented
11C3×C4×#634C211yesmaximal, characteristic, complemented
12#10596023C311yesmaximal, characteristic, direct product factor
13C3×#5552—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C33×C4C2211yescharacteristic
16C22×C33×C4C2211yescharacteristic
17C2×C3×#638C2212yes—
18C2×C3×#638C2212yes—
19C22×C3×C3⋊Dic3C2212yes—
20C3×#3828—24no—
21C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
22C2×C3×C4×C3⋊D3C2212yes—
23C2×C3×#634—24no—
24C2×C3×#634C2211yescharacteristic
25C2×C3×#634C2212yes—
26C2×C3×#634C2211yescharacteristic
27C33×C4⋊C4C2211yescharacteristic
28C33×C22⋊C4C2211yescharacteristic
29C33×C22⋊C4C2211yescharacteristic
30C33×C42C2211yescharacteristic
31C3×#3757C2212yes—
32C3×#3752C2212yes—
33C3×#3752C2212yes—
34C3×#3752C2212yes—
35C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
36C3×#3750C2212yes—
37C3×#3750C2212yes—
38C3×C4×C3⋊Dic3C2212yes—
39C2×#3828C2×C311yescharacteristic, complemented
40C22×#634C2×C311yescharacteristic, complemented
41C2×C3×#235—34no—
42C22×C3×Dic6—34no—
43C32×C4×D4—24no—
44C32×C4×D4D314yescomplemented
45C32×C4×D4C2×C311yescharacteristic, complemented
46#64148C2×C312yescomplemented
47#64139C2×C311yescharacteristic, complemented
48#64133C2×C311yescharacteristic, complemented
49#64105C2×C312yescomplemented
50#64094C2×C312yescomplemented
51#64090C2×C312yescomplemented
52#64084C2×C311yescharacteristic, complemented
53C4×#634C2×C311yescharacteristic, complemented
54C3×#939—38no—
55C3×#930—34no—
56C3×#924—34no—
57C3×#896—38no—
58C3×#885—38no—
59C3×#67⋊C4—38no—
60C3×#875—34no—
61C3×C4×Dic6—34no—
62C23×C33C2311yescharacteristic
63C23×C33C2311yescharacteristic
64C22×C3×C3⋊D3C2312yes—
65C33×D4—22no—
66C2×C33×C4—22no—
67C2×C33×C4—22no—
68C2×C33×C4C2311yescharacteristic
69C2×C33×C4C2311yescharacteristic
70C2×C33×C4C2311yescharacteristic
71C2×C33×C4C2311yescharacteristic
72C2×C33×C4C2311yescharacteristic
73C3×#638—24no—
74C3×#638—24no—
75C2×C3×C3⋊Dic3—24no—
76C2×C3×C3⋊Dic3C2312yes—
77C2×C3×C3⋊Dic3C2312yes—
78C2×C3×C3⋊Dic3C2312yes—
79C3×C4⋊(C3⋊D3)—22no—
80C3×C4×C3⋊D3—24no—
81C3×#634—22no—
82C3×#634—24no—
83C3×#634D414yescomplemented
84C3×#548—91no—
85#5552—34no—
86C2×C32×D4—24no—
87C2×C32×D4C2×D314yes—
88C2×C32×D4C22×C311yescharacteristic
89C22×C32×C4—24no—
90C22×C32×C4—24no—
91C22×C32×C4C2×D314yes—
92C22×C32×C4C2×D314yes—
93C22×C32×C4C22×C311yescharacteristic
94C22×C32×C4C22×C311yescharacteristic
95C2×#638C22×C312yes—
96C2×#638C22×C312yes—
97C22×C3⋊Dic3C22×C312yes—
98#3828—24no—
99C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
100C2×C4×C3⋊D3C22×C312yes—
101C2×#634—24no—
102C2×#634C22×C311yescharacteristic
103C2×#634C22×C312yes—
104C2×#634C22×C311yescharacteristic
105C2×C3×#67—38no—
106C2×C3×#67—38no—
107C22×C3×Dic3—38no—
108C3×#235—616no—
109C2×C3×D12—34no—
110C2×C3×C4×D3—38no—
111C2×C3×Dic6—34no—
112C2×C3×Dic6—38no—
113C2×C3×Dic6—34no—
114C2×C3×Dic6—616no—
115C32×C4⋊C4—24no—
116C32×C4⋊C4C2×D314yes—
117C32×C4⋊C4C22×C311yescharacteristic
118C32×C22⋊C4—24no—
119C32×C22⋊C4—24no—
120C32×C22⋊C4C2×D314yes—
121C32×C22⋊C4C2×D314yes—
122C32×C22⋊C4C22×C311yescharacteristic
123C32×C22⋊C4C22×C311yescharacteristic
124C32×C42—24no—
125C32×C42C2×D314yes—
126C32×C42C22×C311yescharacteristic
127#3757C22×C312yes—
128#3752C22×C312yes—
129#3752C22×C312yes—
130#3752C22×C312yes—
131C4⋊(C3⋊Dic3)C22×C311yescharacteristic
132#3750C22×C312yes—
133#3750C22×C312yes—
134C4×C3⋊Dic3C22×C312yes—
135C3×#217—38no—
136C3×(C2×D3)⋊C4—38no—
137C3×(C2×D3)⋊C4—38no—
138C3×(C2×D3)⋊C4—38no—
139C3×C4⋊Dic3—34no—
140C3×Dic3⋊C4—38no—
141C3×Dic3⋊C4—38no—
142C3×C4×Dic3—38no—
143C22×C33—21no—
144C22×C33—21no—
145C22×C33—21no—
146C22×C33—21no—
147C22×C33—21no—
148C22×C33#4112yescomplemented
149C22×C33C2411yescharacteristic
150C2×C3×C3⋊D3—22no—
151C2×C3×C3⋊D3—22no—
152C2×C3×C3⋊D3—22no—
153C33×C4—21no—
154C33×C4—21no—
155C33×C4—21no—
156C33×C4—21no—
157C33×C4C2×D412yes—
158C3×C3⋊Dic3—22no—
159C3×C3⋊Dic3—22no—
160C3×C3⋊Dic3C2×D414yes—
161C2×C3×#41—91no—
162C22×C3×Q8—91no—
163C2×#235—34no—
164C22×Dic6—34no—
165C3×#124—92no—
166C3×#124—91no—
167C3×C22⋊Q8—91no—
168C3×C22⋊Q8—92no—
169C3×#121—92no—
170C3×#121—91no—
171C3×C4×Q8—91no—
172C3×C4×D4—24no—
173C3×C4×D4—92no—
174C3×C4×D4C3×D314yescomplemented
175C3×C4×D4C3⋊D311yescharacteristic, complemented
176#939—38no—
177#930—34no—
178#924—34no—
179#896—38no—
180#885—38no—
181#67⋊C4—38no—
182#875—34no—
183C4×Dic6—34no—
184C23×C32—24no—
185C23×C32—24no—
186C23×C32C22×D314yes—
187C23×C32C22×D314yes—
188C23×C32C23×C311yescharacteristic
189C23×C32C23×C311yescharacteristic
190C22×C3⋊D3C23×C312yes—
191C22×C3×D3—38no—
192C32×D4—216no—
193C32×D4—28no—
194C32×D4—22no—
195C2×C32×C4—22no—
196C2×C32×C4—22no—
197C2×C32×C4—24no—
198C2×C32×C4—24no—
199C2×C32×C4—24no—
200C2×C32×C4—24no—
201C2×C32×C4—24no—
202C2×C32×C4—28no—
203C2×C32×C4—28no—
204C2×C32×C4—48no—
205C2×C32×C4—48no—
206C2×C32×C4C22×D314yes—
207C2×C32×C4C22×D314yes—
208C2×C32×C4C22×D314yes—
209C2×C32×C4C22×D314yes—
210C2×C32×C4C22×D314yes—
211C2×C32×C4C23×C311yescharacteristic
212C2×C32×C4C23×C311yescharacteristic
213C2×C32×C4C23×C311yescharacteristic
214C2×C32×C4C23×C311yescharacteristic
215C2×C32×C4C23×C311yescharacteristic
216#638—24no—
217#638—24no—
218C2×C3⋊Dic3—24no—
219C2×C3⋊Dic3C23×C312yes—
220C2×C3⋊Dic3C23×C312yes—
221C2×C3⋊Dic3C23×C312yes—
222C4⋊(C3⋊D3)—22no—
223C4×C3⋊D3—24no—
224#634—22no—
225#634—24no—
226#634C3×D414yescomplemented
227C3×#67—616no—
228C3×#67—616no—
229C2×C3×Dic3—38no—
230C2×C3×Dic3—38no—
231C2×C3×Dic3—38no—
232C2×C3×Dic3—616no—
233C3×D12—68no—
234C3×C4×D3—616no—
235C3×Dic6—316no—
236C3×Dic6—616no—
237C3×Dic6—68no—
238#548—91noSylow
239C2×C33—22no—
240C2×C33—41no—
241C2×C33C22×D411yescharacteristic
242C2×C33C2×#4111yescharacteristic
243C2×C332−1+411yescharacteristic
244C3×C3⋊D3—42no—
245C3×#41—184no—
246C2×C3×Q8—91no—
247C2×C3×Q8—92no—
248C2×C3×Q8—91no—
249C2×C3×Q8—184no—
250C2×C3×D4—24no—
251C2×C3×D4—92no—
252C2×C3×D4—92no—
253C2×C3×D4—91no—
254C2×C3×D4C2×C3×D314yes—
255C2×C3×D4C2×C3⋊D311yescharacteristic
256C22×C3×C4—24no—
257C22×C3×C4—24no—
258C22×C3×C4—92no—
259C22×C3×C4—92no—
260C22×C3×C4C2×C3×D314yes—
261C22×C3×C4C2×C3×D314yes—
262C22×C3×C4C2×C3⋊D311yescharacteristic
263C22×C3×C4C2×C3⋊D311yescharacteristic
264C2×#67—38no—
265C2×#67—38no—
266C22×Dic3—38no—
267#235—616no—
268C2×D12—34no—
269C2×C4×D3—38no—
270C2×Dic6—34no—
271C2×Dic6—34no—
272C2×Dic6—38no—
273C2×Dic6—616no—
274C3×C4⋊C4—24no—
275C3×C4⋊C4—92no—
276C3×C4⋊C4—92no—
277C3×C4⋊C4—91no—
278C3×C4⋊C4C2×C3×D314yes—
279C3×C4⋊C4C2×C3⋊D311yescharacteristic
280C3×C22⋊C4—24no—
281C3×C22⋊C4—24no—
282C3×C22⋊C4—92no—
283C3×C22⋊C4—92no—
284C3×C22⋊C4—92no—
285C3×C22⋊C4—92no—
286C3×C22⋊C4C2×C3×D314yes—
287C3×C22⋊C4C2×C3×D314yes—
288C3×C22⋊C4C2×C3⋊D311yescharacteristic
289C3×C22⋊C4C2×C3⋊D311yescharacteristic
290C3×C42—24no—
291C3×C42—92no—
292C3×C42C2×C3×D314yes—
293C3×C42C2×C3⋊D311yescharacteristic
294#217—38no—
295(C2×D3)⋊C4—38no—
296(C2×D3)⋊C4—38no—
297(C2×D3)⋊C4—38no—
298C4⋊Dic3—34no—
299Dic3⋊C4—38no—
300Dic3⋊C4—38no—
301C4×Dic3—38no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—28no—
305C22×C32—24no—
306C22×C32—21no—
307C22×C32—21no—
308C22×C32—21no—
309C22×C32—24no—
310C22×C32—28no—
311C22×C32—24no—
312C22×C32—24no—
313C22×C32—24no—
314C22×C32—24no—
315C22×C32—44no—
316C22×C32—44no—
317C22×C32—44no—
318C22×C32—44no—
319C22×C32#23518yescomplemented
320C22×C32C3×#4112yescomplemented
321C22×C32C23×D314yes—
322C22×C32C24×C311yescharacteristic
323C2×C3⋊D3—22no—
324C2×C3⋊D3—22no—
325C2×C3⋊D3—22no—
326C2×C3×D3—68no—
327C2×C3×D3—68no—
328C2×C3×D3—68no—
329C32×C4—28no—
330C32×C4—24no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—28no—
336C32×C4—24no—
337C32×C4—24no—
338C32×C4—24no—
339C32×C4—44no—
340C32×C4—44no—
341C32×C4—44no—
342C32×C4D3×D418yes—
343C32×C4C2×C3×D412yes—
344C3⋊Dic3—22no—
345C3⋊Dic3—22no—
346C3⋊Dic3C2×C3×D414yes—
347C3×Dic3—316no—
348C3×Dic3—68no—
349C3×Dic3—68no—
350C2×#41—91no—
351C22×Q8—91no—
352#124—92no—
353#124—91no—
354C22⋊Q8—91no—
355C22⋊Q8—92no—
356#121—92no—
357#121—91no—
358C4×Q8—91no—
359C4×D4—92no—
360C4×D4C3×C3⋊D311yescharacteristic, complemented
361C33#54811yescharacteristic, complemented, Sylow
362C23×C3—24no—
363C23×C3—24no—
364C23×C3—92no—
365C23×C3C22×C3×D314yes—
366C23×C3C22×C3×D314yes—
367C23×C3C22×C3⋊D311yescharacteristic
368C23×C3C22×C3⋊D311yescharacteristic
369C22×D3—38no—
370C3×Q8—94no—
371C3×Q8—184no—
372C3×Q8—182no—
373C3×D4—216no—
374C3×D4—28no—
375C3×D4—22no—
376C3×D4—184no—
377C3×D4—184no—
378C3×D4—182no—
379C2×C3×C4—28no—
380C2×C3×C4—28no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—24no—
384C2×C3×C4—24no—
385C2×C3×C4—24no—
386C2×C3×C4—24no—
387C2×C3×C4—24no—
388C2×C3×C4—48no—
389C2×C3×C4—48no—
390C2×C3×C4—92no—
391C2×C3×C4—92no—
392C2×C3×C4—92no—
393C2×C3×C4—184no—
394C2×C3×C4—184no—
395C2×C3×C4C22×C3×D314yes—
396C2×C3×C4C22×C3×D314yes—
397C2×C3×C4C22×C3×D314yes—
398C2×C3×C4C22×C3×D314yes—
399C2×C3×C4C22×C3×D314yes—
400C2×C3×C4C22×C3⋊D311yescharacteristic
401C2×C3×C4C22×C3⋊D311yescharacteristic
402C2×C3×C4C22×C3⋊D311yescharacteristic
403C2×C3×C4C22×C3⋊D311yescharacteristic
404C2×C3×C4C22×C3⋊D311yescharacteristic
405#67—616no—
406#67—616no—
407C2×Dic3—38no—
408C2×Dic3—38no—
409C2×Dic3—38no—
410C2×Dic3—616no—
411D12—68no—
412C4×D3—616no—
413Dic6—316no—
414Dic6—616no—
415Dic6—68no—
416C2×C32—22no—
417C2×C32—24no—
418C2×C32—24no—
419C2×C32—24no—
420C2×C32—28no—
421C2×C32—48no—
422C2×C32—48no—
423C2×C32—44no—
424C2×C32—41no—
425C2×C32C2×#23514yes—
426C2×C32C2×D3×D414yes—
427C2×C32#101014yes—
428C2×C32C22×C3×D411yescharacteristic
429C2×C32C2×C3×#4111yescharacteristic
430C2×C32C3×2−1+411yescharacteristic
431C3⋊D3—42no—
432C3×D3—128no—
433#41—184no—
434C2×Q8—91no—
435C2×Q8—92no—
436C2×Q8—91no—
437C2×Q8—184no—
438C2×D4—92no—
439C2×D4—92no—
440C2×D4—91no—
441C2×D4C2×C3×C3⋊D311yescharacteristic
442C22×C4—92no—
443C22×C4—92no—
444C22×C4C2×C3×C3⋊D311yescharacteristic
445C22×C4C2×C3×C3⋊D311yescharacteristic
446C4⋊C4—92no—
447C4⋊C4—92no—
448C4⋊C4—91no—
449C4⋊C4C2×C3×C3⋊D311yescharacteristic
450C22⋊C4—92no—
451C22⋊C4—92no—
452C22⋊C4—92no—
453C22⋊C4—92no—
454C22⋊C4C2×C3×C3⋊D311yescharacteristic
455C22⋊C4C2×C3×C3⋊D311yescharacteristic
456C42—92no—
457C42C2×C3×C3⋊D311yescharacteristic
458C22×C3—24no—
459C22×C3—24no—
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—28no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—24no—
470C22×C3—28no—
471C22×C3—44no—
472C22×C3—44no—
473C22×C3—44no—
474C22×C3—44no—
475C22×C3—182no—
476C22×C3—182no—
477C22×C3—182no—
478C22×C3C3×#23518yescomplemented
479C22×C3#382812yescomplemented
480C22×C3C23×C3×D314yes—
481C22×C3C23×C3⋊D311yescharacteristic
482C2×D3—68no—
483C2×D3—68no—
484C2×D3—68no—
485C3×C4—28no—
486C3×C4—24no—
487C3×C4—24no—
488C3×C4—24no—
489C3×C4—24no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—28no—
495C3×C4—44no—
496C3×C4—44no—
497C3×C4—44no—
498C3×C4—94no—
499C3×C4—182no—
500C3×C4—182no—
501C3×C4C3×D3×D418yes—
502C3×C4D4×C3⋊D312yes—
503Dic3—316no—
504Dic3—68no—
505Dic3—68no—
506C32—24no—
507C32#555214yescomplemented
508C32C3×#54811yescharacteristic, complemented
509C23—92no—
510C23C22×C3×C3⋊D311yescharacteristic
511C23C22×C3×C3⋊D311yescharacteristic
512Q8—94no—
513Q8—184no—
514Q8—182no—
515D4—22no—
516D4—184no—
517D4—184no—
518D4—182no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—92no—
522C2×C4—92no—
523C2×C4—92no—
524C2×C4—184no—
525C2×C4—184no—
526C2×C4C22×C3×C3⋊D311yescharacteristic
527C2×C4C22×C3×C3⋊D311yescharacteristic
528C2×C4C22×C3×C3⋊D311yescharacteristic
529C2×C4C22×C3×C3⋊D311yescharacteristic
530C2×C4C22×C3×C3⋊D311yescharacteristic
531C2×C3—28no—
532C2×C3—22no—
533C2×C3—24no—
534C2×C3—24no—
535C2×C3—24no—
536C2×C3—48no—
537C2×C3—48no—
538C2×C3—44no—
539C2×C3—41no—
540C2×C3—362no—
541C2×C3C2×C3×#23514yes—
542C2×C3C2×C3×D3×D414yes—
543C2×C3C3×#101014yes—
544C2×C3C2×#382811yescharacteristic
545C2×C3C2×D4×C3⋊D311yescharacteristic
546C2×C3#6436711yescharacteristic
547D3—128no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—182no—
554C22—182no—
555C22—182no—
556C22C3×#382812yescomplemented
557C22C23×C3×C3⋊D311yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—94no—
563C4—182no—
564C4—182no—
565C4C3×D4×C3⋊D312yes—
566C3—24no—
567C3C3×#555214yescomplemented
568C3#1059602311yescharacteristic, direct product factor
569C2—22no—
570C2—41no—
571C2—362no—
572C2C2×C3×#382811yescharacteristic
573C2C2×C3×D4×C3⋊D311yescharacteristic
574C2C3×#6436711yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C24
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#548
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3828C2semidirect
2. C22×C3×#634C2semidirect
3. C33×C4×D4C2semidirect
4. C3×#64148C2semidirect
5. C3×#64139C2semidirect
6. C3×#64133C2semidirect
7. C3×#64105C2semidirect
8. C3×#64094C2semidirect
9. C3×#64090C2semidirect
10. C3×#64084C2semidirect
11. C3×C4×#634C2semidirect
12. #10596023C3direct (N × H)
39. C2×#3828C2×C3semidirect
40. C22×#634C2×C3semidirect
44. C32×C4×D4D3semidirect
45. C32×C4×D4C2×C3semidirect
46. #64148C2×C3semidirect
47. #64139C2×C3semidirect
48. #64133C2×C3semidirect
49. #64105C2×C3semidirect
50. #64094C2×C3semidirect
51. #64090C2×C3semidirect
52. #64084C2×C3semidirect
53. C4×#634C2×C3semidirect
83. C3×#634D4semidirect
148. C22×C33#41semidirect
174. C3×C4×D4C3×D3semidirect
175. C3×C4×D4C3⋊D3semidirect
226. #634C3×D4semidirect
319. C22×C32#235semidirect
320. C22×C32C3×#41semidirect
360. C4×D4C3×C3⋊D3semidirect
361. C33#548semidirect
478. C22×C3C3×#235semidirect
479. C22×C3#3828semidirect
507. C32#5552semidirect
508. C32C3×#548semidirect
556. C22C3×#3828semidirect
567. C3C3×#5552semidirect
568. C3#10596023direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C22×C3×#6342
C33×C4×D42
C3×#641482
C3×#641392
C3×#641332
C3×#641052
C3×#640942
C3×#640902
C3×#640842
C3×C4×#6342
#105960233
C3×#55523