Finite Group Databasebeta

← C3×#10592367 #13162708

Subgroups — C3×#10592367 #13162708

Order 1728.

1708 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3752C211yesmaximal, characteristic
3C2×C3×#3752C211yesmaximal, characteristic
4C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5C2×C3×#3750C211yesmaximal, characteristic, complemented
6C33×#95C211yesmaximal, characteristic, complemented
7C3×#63658C211yesmaximal, characteristic, complemented
8#10592367C311yesmaximal, characteristic, direct product factor
9C3×#4668—34nomaximal
10C23×C3×C3⋊D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C2×C3×C4×C3⋊D3—22no—
18C2×C3×C4×C3⋊D3—22no—
19C2×C3×C4×C3⋊D3—22no—
20C3×#3752—22no—
21C3×#3752C414yescomplemented
22C3×C4⋊(C3⋊Dic3)—22no—
23C3×#3750—22no—
24C22×C4×C3⋊D3C2×C311yescharacteristic
25C22×C3×C4×D3—34no—
26C2×#3752C2×C311yescharacteristic
27C2×#3752C2×C311yescharacteristic
28C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
29C2×#3750C2×C311yescharacteristic, complemented
30C2×C3×(C2×D3)⋊C4—34no—
31C2×C3×(C2×D3)⋊C4—34no—
32C2×C3×C4⋊Dic3—34no—
33C2×C3×Dic3⋊C4—34no—
34C32×#95—24no—
35C32×#95D314yescomplemented
36C32×#95C2×C311yescharacteristic, complemented
37#63658C2×C311yescharacteristic, complemented
38C3×C3⋊#95—34no—
39C23×C33C2311yescharacteristic
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3—21no—
44C22×C3×C3⋊D3C2×C412yes—
45C22×C3×C3⋊D3D412yes—
46C22×C3×C3⋊D3Q812yes—
47C2×C33×C4—21no—
48C2×C33×C4—21no—
49C2×C33×C4—21no—
50C2×C33×C4—21no—
51C2×C33×C4—21no—
52C2×C33×C4—21no—
53C2×C33×C4—21no—
54C2×C33×C4C2×C412yescomplemented
55C2×C33×C4D412yes—
56C2×C3×C3⋊Dic3—21no—
57C2×C3×C3⋊Dic3—21no—
58C2×C3×C3⋊Dic3—21no—
59C2×C3×C3⋊Dic3—21no—
60C2×C3×C3⋊Dic3—21no—
61C2×C3×C3⋊Dic3—21no—
62C2×C3×C3⋊Dic3—21no—
63C2×C3×C3⋊Dic3C2×C412yescomplemented
64C2×C3×C3⋊Dic3D412yes—
65C3×C4×C3⋊D3—28no—
66C3×#385—91no—
67#4668—34no—
68C23×C3⋊D3C22×C311yescharacteristic
69C23×C3×D3—34no—
70C22×C32×C4—24no—
71C22×C32×C4—24no—
72C22×C32×C4—24no—
73C22×C32×C4C2×D314yes—
74C22×C32×C4C2×D314yes—
75C22×C32×C4C2×D314yes—
76C22×C32×C4C22×C311yescharacteristic
77C22×C32×C4C22×C311yescharacteristic
78C22×C32×C4C22×C311yescharacteristic
79C22×C3⋊Dic3C22×C311yescharacteristic
80C22×C3⋊Dic3C22×C311yescharacteristic
81C22×C3⋊Dic3C22×C311yescharacteristic
82C2×C4×C3⋊D3—22no—
83C2×C4×C3⋊D3—22no—
84C2×C4×C3⋊D3—22no—
85C22×C3×Dic3—34no—
86C22×C3×Dic3—34no—
87C22×C3×Dic3—34no—
88C2×C3×C4×D3—68no—
89C2×C3×C4×D3—68no—
90C2×C3×C4×D3—68no—
91#3752—22no—
92#3752C3×C414yescomplemented
93C4⋊(C3⋊Dic3)—22no—
94#3750—22no—
95C3×(C2×D3)⋊C4—316no—
96C3×(C2×D3)⋊C4—68no—
97C3×C4⋊Dic3—68no—
98C3×Dic3⋊C4—68no—
99C22×C33C22×C411yescharacteristic
100C22×C33C2×D411yescharacteristic
101C22×C33C2×D411yescharacteristic
102C22×C33C2×D411yescharacteristic
103C22×C33C2×Q811yescharacteristic
104C22×C33#4111yescharacteristic
105C22×C33#4111yescharacteristic
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3—22no—
109C2×C3×C3⋊D3—22no—
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3C4⋊C414yescomplemented
113C33×C4—22no—
114C33×C4—22no—
115C33×C4—41no—
116C3×C3⋊Dic3—22no—
117C3×C3⋊Dic3—22no—
118C3×C3⋊Dic3—41no—
119C23×C3×C4—91no—
120C22×C4×D3—34no—
121C2×C3×C4⋊C4—91no—
122C2×C3×C4⋊C4—91no—
123C2×C3×C22⋊C4—91no—
124C2×C3×C22⋊C4—91no—
125C2×(C2×D3)⋊C4—34no—
126C2×(C2×D3)⋊C4—34no—
127C2×C4⋊Dic3—34no—
128C2×Dic3⋊C4—34no—
129C3×#95—24no—
130C3×#95—91no—
131C3×#95C3×D314yescomplemented
132C3×#95C3⋊D311yescharacteristic, complemented
133C3⋊#95—34no—
134C23×C32—24no—
135C23×C32C22×D314yes—
136C23×C32C23×C311yescharacteristic
137C22×C3⋊D3—21no—
138C22×C3⋊D3—21no—
139C22×C3⋊D3—21no—
140C22×C3⋊D3—21no—
141C22×C3⋊D3C2×C3×C412yes—
142C22×C3⋊D3C3×D412yes—
143C22×C3⋊D3C3×Q812yes—
144C22×C3×D3—38no—
145C22×C3×D3—38no—
146C22×C3×D3—38no—
147C22×C3×D3—64no—
148C22×C3×D3—64no—
149C22×C3×D3—64no—
150C22×C3×D3—64no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—28no—
156C2×C32×C4—24no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—28no—
161C2×C32×C4—28no—
162C2×C32×C4—24no—
163C2×C32×C4—24no—
164C2×C32×C4—24no—
165C2×C32×C4—24no—
166C2×C32×C4—24no—
167C2×C32×C4—24no—
168C2×C32×C4—44no—
169C2×C32×C4—44no—
170C2×C32×C4—44no—
171C2×C32×C4—44no—
172C2×C32×C4—44no—
173C2×C32×C4—44no—
174C2×C32×C4C4×D318yescomplemented
175C2×C32×C4D1218yes—
176C2×C32×C4C2×C3×C412yescomplemented
177C2×C32×C4C3×D412yes—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3—21no—
180C2×C3⋊Dic3—21no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—21no—
183C2×C3⋊Dic3—21no—
184C2×C3⋊Dic3—21no—
185C2×C3⋊Dic3C2×C3×C412yescomplemented
186C2×C3⋊Dic3C3×D412yes—
187C4×C3⋊D3—28no—
188C2×C3×Dic3—38no—
189C2×C3×Dic3—38no—
190C2×C3×Dic3—64no—
191C2×C3×Dic3—64no—
192C2×C3×Dic3—64no—
193C2×C3×Dic3—64no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3—64no—
196C2×C3×Dic3—64no—
197C3×C4×D3—632no—
198#385—91noSylow
199C2×C33C2×C4⋊C411yescharacteristic
200C2×C33C4×D411yescharacteristic
201C2×C33C4×D411yescharacteristic
202C2×C33#12011yescharacteristic
203C2×C33C22⋊Q811yescharacteristic
204C2×C33C22⋊Q811yescharacteristic
205C2×C33#12311yescharacteristic
206C3×C3⋊D3—24no—
207C24×C3—91no—
208C23×D3—34no—
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—24no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—182no—
216C22×C3×C4—182no—
217C22×C3×C4—182no—
218C22×C3×C4C2×C3×D314yes—
219C22×C3×C4C2×C3×D314yes—
220C22×C3×C4C2×C3×D314yes—
221C22×C3×C4C2×C3⋊D311yescharacteristic
222C22×C3×C4C2×C3⋊D311yescharacteristic
223C22×C3×C4C2×C3⋊D311yescharacteristic
224C22×Dic3—34no—
225C22×Dic3—34no—
226C22×Dic3—34no—
227C2×C4×D3—68no—
228C2×C4×D3—68no—
229C2×C4×D3—68no—
230C3×C4⋊C4—182no—
231C3×C4⋊C4—182no—
232C3×C22⋊C4—94no—
233C3×C22⋊C4—182no—
234(C2×D3)⋊C4—316no—
235(C2×D3)⋊C4—68no—
236C4⋊Dic3—68no—
237Dic3⋊C4—68no—
238C22×C32—24no—
239C22×C32—24no—
240C22×C32—24no—
241C22×C32—24no—
242C22×C32—24no—
243C22×C32—24no—
244C22×C32—24no—
245C22×C32C2×C4×D314yes—
246C22×C32C2×D1214yes—
247C22×C32#23514yes—
248C22×C32D3×D414yes—
249C22×C32D3×D414yes—
250C22×C32#23714yes—
251C22×C32D3×Q814yes—
252C22×C32C22×C3×C411yescharacteristic
253C22×C32C2×C3×D411yescharacteristic
254C22×C32C2×C3×D411yescharacteristic
255C22×C32C2×C3×D411yescharacteristic
256C22×C32C2×C3×Q811yescharacteristic
257C22×C32C3×#4111yescharacteristic
258C22×C32C3×#4111yescharacteristic
259C2×C3⋊D3—22no—
260C2×C3⋊D3—22no—
261C2×C3⋊D3—22no—
262C2×C3⋊D3—22no—
263C2×C3⋊D3—22no—
264C2×C3⋊D3—22no—
265C2×C3⋊D3C3×C4⋊C414yescomplemented
266C2×C3×D3—316no—
267C2×C3×D3—68no—
268C2×C3×D3—68no—
269C2×C3×D3—68no—
270C2×C3×D3—68no—
271C2×C3×D3—68no—
272C2×C3×D3—68no—
273C32×C4—22no—
274C32×C4—22no—
275C32×C4—28no—
276C32×C4—28no—
277C32×C4—48no—
278C32×C4—48no—
279C32×C4—48no—
280C32×C4—44no—
281C32×C4—41no—
282C3⋊Dic3—22no—
283C3⋊Dic3—22no—
284C3⋊Dic3—41no—
285C3×Dic3—68no—
286C3×Dic3—68no—
287C3×Dic3—124no—
288C23×C4—91no—
289C2×C4⋊C4—91no—
290C2×C4⋊C4—91no—
291C2×C22⋊C4—91no—
292C2×C22⋊C4—91no—
293#95—91no—
294#95C3×C3⋊D311yescharacteristic, complemented
295C33#38511yescharacteristic, complemented, Sylow
296C23×C3—24no—
297C23×C3—92no—
298C23×C3—92no—
299C23×C3—92no—
300C23×C3—181no—
301C23×C3—181no—
302C23×C3—181no—
303C23×C3—181no—
304C23×C3C22×C3×D314yes—
305C23×C3C22×C3⋊D311yescharacteristic
306C22×D3—38no—
307C22×D3—38no—
308C22×D3—38no—
309C22×D3—64no—
310C22×D3—64no—
311C22×D3—64no—
312C22×D3—64no—
313C2×C3×C4—24no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—28no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—28no—
329C2×C3×C4—28no—
330C2×C3×C4—44no—
331C2×C3×C4—44no—
332C2×C3×C4—44no—
333C2×C3×C4—44no—
334C2×C3×C4—44no—
335C2×C3×C4—44no—
336C2×C3×C4—92no—
337C2×C3×C4—92no—
338C2×C3×C4—188no—
339C2×C3×C4—181no—
340C2×C3×C4—181no—
341C2×C3×C4—181no—
342C2×C3×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4—181no—
345C2×C3×C4—181no—
346C2×C3×C4C3×C4×D318yescomplemented
347C2×C3×C4C3×D1218yes—
348C2×C3×C4C4×C3⋊D312yescomplemented
349C2×C3×C4C4⋊(C3⋊D3)12yes—
350C2×Dic3—38no—
351C2×Dic3—38no—
352C2×Dic3—64no—
353C2×Dic3—64no—
354C2×Dic3—64no—
355C2×Dic3—64no—
356C2×Dic3—64no—
357C2×Dic3—64no—
358C2×Dic3—64no—
359C4×D3—632no—
360C2×C32—24no—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32—24no—
364C2×C32—24no—
365C2×C32—24no—
366C2×C32—24no—
367C2×C32C4×D1214yes—
368C2×C32#67⋊C414yes—
369C2×C32#88214yes—
370C2×C32#88314yes—
371C2×C32D3×C4⋊C414yes—
372C2×C32#89614yes—
373C2×C32#89714yes—
374C2×C32C2×C3×C4⋊C411yescharacteristic
375C2×C32C3×C4×D411yescharacteristic
376C2×C32C3×C4×D411yescharacteristic
377C2×C32C3×#12011yescharacteristic
378C2×C32C3×C22⋊Q811yescharacteristic
379C2×C32C3×C22⋊Q811yescharacteristic
380C2×C32C3×#12311yescharacteristic
381C3⋊D3—24no—
382C3×D3—616no—
383C24—91no—
384C22×C4—91no—
385C22×C4—91no—
386C22×C4—91no—
387C22×C4—182no—
388C22×C4—182no—
389C22×C4—182no—
390C22×C4C2×C3×C3⋊D311yescharacteristic
391C22×C4C2×C3×C3⋊D311yescharacteristic
392C22×C4C2×C3×C3⋊D311yescharacteristic
393C4⋊C4—182no—
394C4⋊C4—182no—
395C22⋊C4—94no—
396C22⋊C4—182no—
397C22×C3—24no—
398C22×C3—24no—
399C22×C3—24no—
400C22×C3—24no—
401C22×C3—24no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3—94no—
405C22×C3—182no—
406C22×C3—182no—
407C22×C3—182no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3—182no—
411C22×C3C2×C3×C4×D314yes—
412C22×C3C2×C3×D1214yes—
413C22×C3C3×#23514yes—
414C22×C3C3×D3×D414yes—
415C22×C3C3×D3×D414yes—
416C22×C3C3×#23714yes—
417C22×C3C3×D3×Q814yes—
418C22×C3C2×C4×C3⋊D311yescharacteristic
419C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
420C22×C3#382811yescharacteristic
421C22×C3D4×C3⋊D311yescharacteristic
422C22×C3D4×C3⋊D311yescharacteristic
423C22×C3#383011yescharacteristic
424C22×C3Q8×C3⋊D311yescharacteristic
425C2×D3—316no—
426C2×D3—68no—
427C2×D3—68no—
428C2×D3—68no—
429C2×D3—68no—
430C2×D3—68no—
431C2×D3—68no—
432C3×C4—28no—
433C3×C4—28no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—48no—
437C3×C4—48no—
438C3×C4—48no—
439C3×C4—44no—
440C3×C4—41no—
441C3×C4—182no—
442C3×C4—182no—
443C3×C4—361no—
444Dic3—68no—
445Dic3—68no—
446Dic3—124no—
447C32—24no—
448C32#466814yescomplemented
449C32C3×#38511yescharacteristic, complemented
450C23—92no—
451C23—92no—
452C23—92no—
453C23—181no—
454C23—181no—
455C23—181no—
456C23—181no—
457C23C22×C3×C3⋊D311yescharacteristic
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—92no—
466C2×C4—92no—
467C2×C4—188no—
468C2×C4—181no—
469C2×C4—181no—
470C2×C4—181no—
471C2×C4—181no—
472C2×C4—181no—
473C2×C4—181no—
474C2×C4—181no—
475C2×C4C3×C4×C3⋊D312yescomplemented
476C2×C4C3×C4⋊(C3⋊D3)12yes—
477C2×C3—24no—
478C2×C3—24no—
479C2×C3—24no—
480C2×C3—24no—
481C2×C3—24no—
482C2×C3—24no—
483C2×C3—24no—
484C2×C3—184no—
485C2×C3C3×C4×D1214yes—
486C2×C3C3×#67⋊C414yes—
487C2×C3C3×#88214yes—
488C2×C3C3×#88314yes—
489C2×C3C3×D3×C4⋊C414yes—
490C2×C3C3×#89614yes—
491C2×C3C3×#89714yes—
492C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
493C2×C3#6409011yescharacteristic
494C2×C3#6409111yescharacteristic
495C2×C3#6409211yescharacteristic
496C2×C3C4⋊C4×C3⋊D311yescharacteristic
497C2×C3#6410511yescharacteristic
498C2×C3#6410611yescharacteristic
499D3—616no—
500C22—94no—
501C22—182no—
502C22—182no—
503C22—182no—
504C22—182no—
505C22—182no—
506C22—182no—
507C22C2×C3×C4×C3⋊D311yescharacteristic
508C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
509C22C3×#382811yescharacteristic
510C22C3×D4×C3⋊D311yescharacteristic
511C22C3×D4×C3⋊D311yescharacteristic
512C22C3×#383011yescharacteristic
513C22C3×Q8×C3⋊D311yescharacteristic
514C4—22no—
515C4—22no—
516C4—41no—
517C4—182no—
518C4—182no—
519C4—361no—
520C3—24no—
521C3C3×#466814yescomplemented
522C3#1059236711yescharacteristic, direct product factor
523C2—184no—
524C2C3×C4×C4⋊(C3⋊D3)11yescharacteristic
525C2C3×#6409011yescharacteristic
526C2C3×#6409111yescharacteristic
527C2C3×#6409211yescharacteristic
528C2C3×C4⋊C4×C3⋊D311yescharacteristic
529C2C3×#6410511yescharacteristic
530C2C3×#6410611yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC33×#95C2
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC23×C32C23×C3
hypercenterC3×#95C3⋊D3
hypocenterC32C3×#385
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C3×#95

Complements

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

NH ≅ G/Ntype
4. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
5. C2×C3×#3750C2semidirect
6. C33×#95C2semidirect
7. C3×#63658C2semidirect
8. #10592367C3direct (N × H)
21. C3×#3752C4semidirect
28. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
29. C2×#3750C2×C3semidirect
35. C32×#95D3semidirect
36. C32×#95C2×C3semidirect
37. #63658C2×C3semidirect
54. C2×C33×C4C2×C4semidirect
63. C2×C3×C3⋊Dic3C2×C4semidirect
92. #3752C3×C4semidirect
112. C2×C3×C3⋊D3C4⋊C4semidirect
131. C3×#95C3×D3semidirect
132. C3×#95C3⋊D3semidirect
174. C2×C32×C4C4×D3semidirect
176. C2×C32×C4C2×C3×C4semidirect
185. C2×C3⋊Dic3C2×C3×C4semidirect
265. C2×C3⋊D3C3×C4⋊C4semidirect
294. #95C3×C3⋊D3semidirect
295. C33#385semidirect
346. C2×C3×C4C3×C4×D3semidirect
348. C2×C3×C4C4×C3⋊D3semidirect
448. C32#4668semidirect
449. C32C3×#385semidirect
475. C2×C4C3×C4×C3⋊D3semidirect
521. C3C3×#4668semidirect
522. C3#10592367direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×C3×#37522
C2×C3×C4⋊(C3⋊Dic3)2
C2×C3×#37502
C33×#952
C3×#636582
#105923673
C3×#46683