Finite Group Databasebeta

← C3×#10591318 #13158236

Subgroups — C3×#10591318 #13158236

Order 1728.

1522 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×D12C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C2×C32×C4×Dic3C211yesmaximal, characteristic, complemented
4C2×C3×(C2×D3)⋊Dic3C214yesmaximal, complemented
5C3×#5221—31nomaximal
6C3×#4939—31nomaximal
7#10591318C311yesmaximal, characteristic, direct product factor
8C23×C32×D3C2212yes—
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2212yes—
12C2×C32×D12—22no—
13C2×C32×D12—22no—
14C2×C32×D12C414yescomplemented
15C3×C4⋊(C3⋊Dic3)—22no—
16C32×C4×Dic3—22no—
17C3×(C2×D3)⋊Dic3—28no—
18C22×C32×D4—31no—
19C22×C3×D12—21no—
20C22×C3×D12D311yescharacteristic
21C22×C3×D12C2×C311yescharacteristic
22C2×C32×C42—31no—
23C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
24C2×C3×#217—34no—
25C2×C3×(C2×D3)⋊C4—34no—
26C2×C3×C4⋊Dic3—31no—
27C2×C3×C4⋊Dic3—31no—
28C2×C3×C4⋊Dic3—61no—
29C2×C3×C4×Dic3—21no—
30C2×C3×C4×Dic3D311yescharacteristic, complemented
31C2×C3×C4×Dic3C2×C311yescharacteristic, complemented
32C2×(C2×D3)⋊Dic3C2×C314yescomplemented
33C23×C33C2311yescharacteristic
34C22×C32×D3—22no—
35C22×C32×D3—22no—
36C22×C32×D3—24no—
37C22×C32×D3—24no—
38C22×C32×D3C2×C414yes—
39C2×C33×C4C2×C412yes—
40C2×C33×C4D412yes—
41C2×C33×C4D412yes—
42C2×C3×C3⋊Dic3—22no—
43C2×C3×C3⋊Dic3—24no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3D414yescomplemented
47C32×D12—28no—
48C3×#390—91no—
49#5221—31no—
50#4939—31no—
51C24×C32—32no—
52C23×C3×D3—22no—
53C23×C3×D3C2×D312yes—
54C23×C3×D3C22×C312yes—
55C2×C32×D4—34no—
56C2×C32×D4—62no—
57C2×C32×D4—62no—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C22×C32×C4—32no—
62C22×C32×C4—41no—
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C22×C311yescharacteristic
66C22×C3⋊Dic3C22×C312yes—
67C22×C3×Dic3—22no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—62no—
71C22×C3×Dic3C2×D312yes—
72C22×C3×Dic3C22×C312yes—
73C2×C3×D12—24no—
74C2×C3×D12—24no—
75C2×C3×D12—24no—
76C2×C3×D12—22no—
77C2×C3×D12—22no—
78C2×C3×D12—22no—
79C2×C3×D12—22no—
80C2×C3×D12Dic314yescomplemented
81C2×C3×D12C3×C414yescomplemented
82C32×C42—62no—
83C4⋊(C3⋊Dic3)—22no—
84C3×#217—68no—
85C3×(C2×D3)⋊C4—68no—
86C3×C4⋊Dic3—64no—
87C3×C4⋊Dic3—62no—
88C3×C4⋊Dic3—62no—
89C3×C4×Dic3—24no—
90C3×C4×Dic3—22no—
91C3×C4×Dic3—22no—
92(C2×D3)⋊Dic3—28no—
93C22×C33C22×C411yescharacteristic
94C22×C33C2×D411yescharacteristic
95C22×C33C2×D411yescharacteristic
96C22×C33C2×D412yes—
97C22×C33#4112yes—
98C2×C32×D3—24no—
99C2×C32×D3—28no—
100C2×C32×D3—44no—
101C2×C32×D3—42no—
102C2×C32×D3—42no—
103C33×C4C22⋊C414yescomplemented
104C3×C3⋊Dic3—42no—
105C32×Dic3—24no—
106C22×C3×D4—31no—
107C22×C3×D4—31no—
108C22×C3×D4—61no—
109C22×D12C3×D311yescharacteristic
110C2×C3×C4⋊C4—91no—
111C2×C3×C22⋊C4—94no—
112C2×C3×C42—31no—
113C2×C3×C42—31no—
114C2×C3×C42—61no—
115C2×#217—34no—
116C2×(C2×D3)⋊C4—34no—
117C2×C4⋊Dic3—31no—
118C2×C4⋊Dic3—31no—
119C2×C4⋊Dic3—61no—
120C2×C4×Dic3C3×D311yescharacteristic, complemented
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—21no—
124C23×C32—34no—
125C23×C32—41no—
126C23×C32—62no—
127C23×C32—62no—
128C23×C32—64no—
129C23×C32—64no—
130C23×C32C22×D311yescharacteristic
131C23×C32C22×D311yescharacteristic
132C23×C32C23×C311yescharacteristic
133C22×C3×D3—24no—
134C22×C3×D3—24no—
135C22×C3×D3—24no—
136C22×C3×D3—24no—
137C22×C3×D3—22no—
138C22×C3×D3—22no—
139C22×C3×D3—24no—
140C22×C3×D3—22no—
141C22×C3×D3—22no—
142C22×C3×D3—24no—
143C22×C3×D3—24no—
144C22×C3×D3—44no—
145C22×C3×D3—44no—
146C22×C3×D3C2×Dic314yes—
147C22×C3×D3C2×C3×C414yes—
148C32×D4—68no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—34no—
159C2×C32×C4—42no—
160C2×C32×C4—42no—
161C2×C32×C4—42no—
162C2×C32×C4—62no—
163C2×C32×C4—62no—
164C2×C32×C4C4×D312yes—
165C2×C32×C4D1212yes—
166C2×C32×C4C2×Dic312yes—
167C2×C32×C4#6712yes—
168C2×C32×C4#6712yes—
169C2×C32×C4#6712yes—
170C2×C32×C4C2×C3×C412yes—
171C2×C32×C4C3×D412yes—
172C2×C32×C4C3×D412yes—
173C2×C3⋊Dic3—22no—
174C2×C3⋊Dic3—24no—
175C2×C3×Dic3—24no—
176C2×C3×Dic3—24no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—64no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3—64no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—124no—
188C2×C3×Dic3D1214yescomplemented
189C2×C3×Dic3C3×D414yescomplemented
190C3×D12—216no—
191C3×D12—28no—
192C3×D12—28no—
193#390—91noSylow
194C2×C33C2×C22⋊C411yescharacteristic
195C2×C33C4×D412yes—
196C2×C33#12112yes—
197C2×C33#12411yescharacteristic
198C2×C33#12711yescharacteristic
199C32×D3—44no—
200C24×C3—32no—
201C24×C3—32no—
202C24×C3—62no—
203C23×D3C2×C3×D312yes—
204C2×C3×D4—34no—
205C2×C3×D4—34no—
206C2×C3×D4—64no—
207C2×C3×D4—64no—
208C2×C3×D4—64no—
209C2×C3×D4—62no—
210C2×C3×D4—62no—
211C2×C3×D4—62no—
212C2×C3×D4—62no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—32no—
217C22×C3×C4—32no—
218C22×C3×C4—41no—
219C22×C3×C4—62no—
220C22×C3×C4—92no—
221C22×C3×C4D3211yescharacteristic
222C22×C3×C4C2×C3×D311yescharacteristic
223C22×C3×C4C2×C3×D311yescharacteristic
224C22×Dic3—32no—
225C22×Dic3—32no—
226C22×Dic3—62no—
227C22×Dic3C2×C3×D312yes—
228C2×D12—22no—
229C2×D12—22no—
230C2×D12C3×Dic314yescomplemented
231C3×C4⋊C4—182no—
232C3×C22⋊C4—188no—
233C3×C42—64no—
234C3×C42—62no—
235C3×C42—62no—
236#217—68no—
237(C2×D3)⋊C4—68no—
238C4⋊Dic3—64no—
239C4⋊Dic3—62no—
240C4⋊Dic3—62no—
241C4×Dic3—22no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—21no—
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—22no—
249C22×C32—21no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—22no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—41no—
258C22×C32—41no—
259C22×C32—41no—
260C22×C32—42no—
261C22×C32—42no—
262C22×C32—64no—
263C22×C32—68no—
264C22×C32—122no—
265C22×C32—122no—
266C22×C32—124no—
267C22×C32C2×C4×D311yescharacteristic
268C22×C32C2×D1212yes—
269C22×C32C2×D1211yescharacteristic
270C22×C32#23512yes—
271C22×C32D3×D412yes—
272C22×C32#23712yes—
273C22×C32C22×Dic311yescharacteristic
274C22×C32C2×#6711yescharacteristic
275C22×C32C2×#6711yescharacteristic
276C22×C32C2×#6711yescharacteristic
277C22×C32C22×C3×C411yescharacteristic
278C22×C32C2×C3×D412yes—
279C22×C32C2×C3×D411yescharacteristic
280C22×C32C2×C3×D411yescharacteristic
281C22×C32C3×#4112yes—
282C2×C3×D3—28no—
283C2×C3×D3—28no—
284C2×C3×D3—24no—
285C2×C3×D3—24no—
286C2×C3×D3—28no—
287C2×C3×D3—48no—
288C2×C3×D3—44no—
289C2×C3×D3—48no—
290C2×C3×D3—44no—
291C2×C3×D3—42no—
292C2×C3×D3—44no—
293C2×C3×D3—42no—
294C2×C3×D3—44no—
295C2×C3×D3—42no—
296C2×C3×D3—42no—
297C32×C4—24no—
298C32×C4—24no—
299C32×C4—24no—
300C32×C4—44no—
301C32×C4—64no—
302C32×C4(C2×D3)⋊C414yescomplemented
303C32×C4#21714yescomplemented
304C32×C4C3×C22⋊C414yescomplemented
305C3⋊Dic3—42no—
306C3×Dic3—28no—
307C3×Dic3—24no—
308C3×Dic3—24no—
309C3×Dic3—124no—
310C3×Dic3—122no—
311C3×Dic3—122no—
312C22×D4—31no—
313C2×C4⋊C4—91no—
314C2×C22⋊C4—94no—
315C2×C42—31no—
316C33#39011yescharacteristic, complemented, Sylow
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—21no—
320C23×C3—34no—
321C23×C3—34no—
322C23×C3—41no—
323C23×C3—64no—
324C23×C3—64no—
325C23×C3—64no—
326C23×C3—64no—
327C23×C3—64no—
328C23×C3—62no—
329C23×C3—62no—
330C23×C3—64no—
331C23×C3—62no—
332C23×C3—64no—
333C23×C3—62no—
334C23×C3—124no—
335C23×C3—124no—
336C23×C3C2×D3211yescharacteristic
337C23×C3C22×C3×D311yescharacteristic
338C23×C3C22×C3×D311yescharacteristic
339C22×D3—24no—
340C22×D3—24no—
341C22×D3—22no—
342C22×D3—22no—
343C22×D3C2×C3×Dic314yes—
344C3×D4—616no—
345C3×D4—68no—
346C3×D4—68no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—34no—
357C2×C3×C4—34no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—64no—
362C2×C3×C4—64no—
363C2×C3×C4—64no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—62no—
367C2×C3×C4—62no—
368C2×C3×C4—184no—
369C2×C3×C4—182no—
370C2×C3×C4D3×Dic312yes—
371C2×C3×C4#62512yes—
372C2×C3×C4#62612yes—
373C2×C3×C4C3×C4×D312yes—
374C2×C3×C4C3×D1212yes—
375C2×C3×C4C2×C3×Dic312yes—
376C2×C3×C4C3×#6712yes—
377C2×C3×C4C3×#6712yes—
378C2×C3×C4C3×#6712yes—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—64no—
382C2×Dic3—62no—
383C2×Dic3—64no—
384C2×Dic3—64no—
385C2×Dic3—62no—
386C2×Dic3—124no—
387C2×Dic3C3×D1214yescomplemented
388D12—28no—
389C2×C32—21no—
390C2×C32—22no—
391C2×C32—22no—
392C2×C32—21no—
393C2×C32—22no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—22no—
398C2×C32—22no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—22no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—41no—
405C2×C32—42no—
406C2×C32—42no—
407C2×C32—41no—
408C2×C32—41no—
409C2×C32—124no—
410C2×C32C4×D1212yes—
411C2×C32#87411yescharacteristic
412C2×C32#87511yescharacteristic
413C2×C32C2×(C2×D3)⋊C411yescharacteristic
414C2×C32#93012yes—
415C2×C32D4×Dic312yes—
416C2×C32#93611yescharacteristic
417C2×C32#93812yes—
418C2×C32#94011yescharacteristic
419C2×C32C2×#21711yescharacteristic
420C2×C32C2×C3×C22⋊C411yescharacteristic
421C2×C32C3×C4×D412yes—
422C2×C32C3×#12112yes—
423C2×C32C3×#12411yescharacteristic
424C2×C32C3×#12711yescharacteristic
425C3×D3—48no—
426C3×D3—44no—
427C3×D3—44no—
428C24—32no—
429C2×D4—34no—
430C2×D4—62no—
431C2×D4—62no—
432C22×C4—32no—
433C22×C4—92no—
434C22×C4C3×D3211yescharacteristic
435C4⋊C4—182no—
436C22⋊C4—188no—
437C42—62no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—68no—
459C22×C3—64no—
460C22×C3—68no—
461C22×C3—64no—
462C22×C3—68no—
463C22×C3—128no—
464C22×C3—124no—
465C22×C3—128no—
466C22×C3—124no—
467C22×C3—124no—
468C22×C3—122no—
469C22×C3—124no—
470C22×C3—122no—
471C22×C3—122no—
472C22×C3—122no—
473C22×C3#379512yes—
474C22×C3D3×D1212yes—
475C22×C3C2×D3×Dic311yescharacteristic
476C22×C3C2×#62511yescharacteristic
477C22×C3C2×#62611yescharacteristic
478C22×C3C2×C3×C4×D311yescharacteristic
479C22×C3C2×C3×D1212yes—
480C22×C3C2×C3×D1211yescharacteristic
481C22×C3C3×#23512yes—
482C22×C3C3×D3×D412yes—
483C22×C3C3×#23712yes—
484C22×C3C22×C3×Dic311yescharacteristic
485C22×C3C2×C3×#6711yescharacteristic
486C22×C3C2×C3×#6711yescharacteristic
487C22×C3C2×C3×#6711yescharacteristic
488C2×D3—28no—
489C2×D3—24no—
490C2×D3—42no—
491C2×D3—44no—
492C2×D3—42no—
493C3×C4—24no—
494C3×C4—24no—
495C3×C4—24no—
496C3×C4—44no—
497C3×C4—68no—
498C3×C4—64no—
499C3×C4—64no—
500C3×C4—362no—
501C3×C4(C2×D3)⋊Dic314yescomplemented
502C3×C4C3×(C2×D3)⋊C414yescomplemented
503C3×C4C3×#21714yescomplemented
504Dic3—24no—
505Dic3—124no—
506Dic3—122no—
507Dic3—122no—
508C32—21no—
509C32—21no—
510C32—21no—
511C32—41no—
512C32#493911yescharacteristic, complemented
513C32#522111yescharacteristic, complemented
514C32C3×#39011yescharacteristic, complemented
515C23—34no—
516C23—64no—
517C23—64no—
518C23—62no—
519C23—62no—
520C23C2×C3×D3211yescharacteristic
521D4—68no—
522C2×C4—34no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—184no—
526C2×C4—182no—
527C2×C4C3×D3×Dic312yes—
528C2×C4C3×#62512yes—
529C2×C4C3×#62612yes—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—22no—
537C2×C3—22no—
538C2×C3—22no—
539C2×C3—22no—
540C2×C3—22no—
541C2×C3—22no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—41no—
546C2×C3—41no—
547C2×C3—42no—
548C2×C3—42no—
549C2×C3—41no—
550C2×C3—128no—
551C2×C3—124no—
552C2×C3—124no—
553C2×C3#6386011yescharacteristic
554C2×C3Dic3×D1212yes—
555C2×C3#6390812yes—
556C2×C3#6391111yescharacteristic
557C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
558C2×C3C3×C4×D1212yes—
559C2×C3C3×#87411yescharacteristic
560C2×C3C3×#87511yescharacteristic
561C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
562C2×C3C3×#93012yes—
563C2×C3C3×D4×Dic312yes—
564C2×C3C3×#93611yescharacteristic
565C2×C3C3×#93812yes—
566C2×C3C3×#94011yescharacteristic
567C2×C3C2×C3×#21711yescharacteristic
568D3—44no—
569C22—64no—
570C22—68no—
571C22—122no—
572C22—124no—
573C22—122no—
574C22C3×#379512yes—
575C22C3×D3×D1212yes—
576C22C2×C3×D3×Dic311yescharacteristic
577C22C2×C3×#62511yescharacteristic
578C22C2×C3×#62611yescharacteristic
579C4—64no—
580C4—362no—
581C4C3×(C2×D3)⋊Dic314yescomplemented
582C3—21no—
583C3—21no—
584C3—21no—
585C3—41no—
586C3#1059131811yescharacteristic, direct product factor
587C3C3×#493911yescharacteristic, complemented
588C3C3×#522111yescharacteristic, complemented
589C2—124no—
590C2C3×#6386011yescharacteristic
591C2C3×Dic3×D1212yes—
592C2C3×#6390812yes—
593C2C3×#6391111yescharacteristic
594C2C2×C3×(C2×D3)⋊Dic311yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC22×C32×D12C2
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#390
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
3. C2×C32×C4×Dic3C2semidirect
4. C2×C3×(C2×D3)⋊Dic3C2semidirect
7. #10591318C3direct (N × H)
14. C2×C32×D12C4semidirect
23. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
30. C2×C3×C4×Dic3D3semidirect
31. C2×C3×C4×Dic3C2×C3semidirect
32. C2×(C2×D3)⋊Dic3C2×C3semidirect
46. C2×C32×Dic3D4semidirect
80. C2×C3×D12Dic3semidirect
81. C2×C3×D12C3×C4semidirect
103. C33×C4C22⋊C4semidirect
120. C2×C4×Dic3C3×D3semidirect
188. C2×C3×Dic3D12semidirect
189. C2×C3×Dic3C3×D4semidirect
230. C2×D12C3×Dic3semidirect
302. C32×C4(C2×D3)⋊C4semidirect
303. C32×C4#217semidirect
304. C32×C4C3×C22⋊C4semidirect
316. C33#390semidirect
387. C2×Dic3C3×D12semidirect
501. C3×C4(C2×D3)⋊Dic3semidirect
502. C3×C4C3×(C2×D3)⋊C4semidirect
503. C3×C4C3×#217semidirect
512. C32#4939semidirect
513. C32#5221semidirect
514. C32C3×#390semidirect
581. C4C3×(C2×D3)⋊Dic3semidirect
586. C3#10591318direct (N × H)
587. C3C3×#4939semidirect
588. C3C3×#5221semidirect

Maximal subgroups

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

subgroupindex
C22×C32×D122
C2×C3×C4⋊(C3⋊Dic3)2
C2×C32×C4×Dic32
C2×C3×(C2×D3)⋊Dic32
C3×#52213
C3×#49393
#105913183