Finite Group Databasebeta

← #13161424

Subgroups — #13161424

Order 1728.

2076 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×#67C211yesmaximal, characteristic
2C2×#88864C211yesmaximal, characteristic, complemented
3C2×C3×#3757C211yesmaximal, characteristic, complemented
4C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6C2×(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
7C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592920—31nomaximal
9#10591707—34nomaximal
10C24×C33C2211yescharacteristic
11C23×C32×D3C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C2×C32×#67—22no—
17C2×C32×#67—22no—
18C2×C32×#67C414yescomplemented
19C22×C32×Dic3C2211yescharacteristic
20#88864—22no—
21C3×#3757—22no—
22Dic3⋊(C3⋊Dic3)—22no—
23(C2×D3)⋊(C3⋊Dic3)—22no—
24(C2×D3)⋊(C3⋊Dic3)—22no—
25Dic3×C3⋊Dic3—22no—
26C22×C32×D4—31no—
27C22×C3×#67D314yes—
28C2×#3757—31no—
29C2×#3757—31no—
30C2×#3757—31no—
31C2×#3757—34no—
32C2×#3757—64no—
33C2×#3757D311yescharacteristic, complemented
34C2×C4⋊(C3⋊Dic3)—31no—
35C2×C4×C3⋊Dic3—31no—
36C2×C3×#217—34no—
37C2×Dic3⋊Dic3—34no—
38C2×(C2×D3)⋊Dic3—34no—
39C2×(C2×D3)⋊Dic3—34no—
40C2×Dic32—34no—
41C23×C33—21no—
42C23×C33—21no—
43C23×C33—21no—
44C23×C33—21no—
45C23×C33—21no—
46C23×C33—21no—
47C23×C33C2×C412yes—
48C23×C33C2311yescharacteristic
49C22×C32×D3—21no—
50C22×C32×D3—21no—
51C22×C32×D3—21no—
52C22×C32×D3—21no—
53C22×C32×D3—21no—
54C22×C32×D3—21no—
55C22×C32×D3C2×C412yes—
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)D412yescomplemented
62C2×C3×C3⋊Dic3—21no—
63C2×C3×C3⋊Dic3—21no—
64C2×C3×C3⋊Dic3—21no—
65C2×C3×C3⋊Dic3—21no—
66C2×C3×C3⋊Dic3—21no—
67C2×C3×C3⋊Dic3D412yescomplemented
68C32×#67—28no—
69C2×C32×Dic3C2×C412yes—
70C2×C32×Dic3D412yes—
71C2×C32×Dic3D412yes—
72#5221—94no—
73#4957—91no—
74C24×C32—24no—
75C24×C32—31no—
76C24×C32C2×D311yescharacteristic
77C24×C32C2×D314yes—
78C23×C3×D3C2×D314yes—
79C2×C32×D4—34no—
80C2×C32×D4—62no—
81C2×C32×D4—62no—
82C22×C32×C4—31no—
83C22×C3⋊Dic3—31no—
84C22×C3⋊Dic3—34no—
85C22×C3⋊Dic3—31no—
86C22×C3⋊Dic3—34no—
87C22×C3⋊Dic3—64no—
88C22×C3⋊Dic3—64no—
89C22×C3⋊Dic3C2×D311yescharacteristic
90C22×C3⋊Dic3C2×D311yescharacteristic
91C2×C3×#67—28no—
92C2×C3×#67—28no—
93C2×C3×#67Dic3116yescomplemented
94C22×C3×Dic3—34no—
95C22×C3×Dic3—34no—
96C22×C3×Dic3C2×D314yes—
97#3757—22no—
98#3757—62no—
99#3757—62no—
100#3757—68no—
101#3757—616no—
102#3757—62no—
103C4⋊(C3⋊Dic3)—62no—
104C4×C3⋊Dic3—62no—
105C3×#217—68no—
106Dic3⋊Dic3—68no—
107(C2×D3)⋊Dic3—68no—
108(C2×D3)⋊Dic3—68no—
109Dic32—68no—
110C22×C33—22no—
111C22×C33—22no—
112C22×C33—22no—
113C22×C33—41no—
114C22×C33—41no—
115C22×C33—41no—
116C22×C33—41no—
117C22×C33C22×C411yescharacteristic
118C22×C33C2×D411yescharacteristic
119C22×C33C2×D411yescharacteristic
120C22×C33C2×D411yescharacteristic
121C22×C33C2×D411yescharacteristic
122C22×C33#4111yescharacteristic
123C22×C33#4111yescharacteristic
124C2×C32×D3—22no—
125C2×C32×D3—22no—
126C2×C32×D3—22no—
127C2×C32×D3—41no—
128C2×C32×D3—41no—
129C2×C32×D3—41no—
130C2×C32×D3—41no—
131C3⋊(C3⋊Dic3)—22no—
132C3⋊(C3⋊Dic3)—41no—
133C3×C3⋊Dic3—22no—
134C3×C3⋊Dic3—41no—
135C32×Dic3C22⋊C414yescomplemented
136C22×C3×D4—34no—
137C22×#67C3⋊D311yescharacteristic
138C2×C3×C22⋊C4—91no—
139C2×#217—34no—
140C2×#217—94no—
141C2×#217—94no—
142C2×#217—94no—
143C2×#217—91no—
144C2×#217—184no—
145C2×(C2×D3)⋊C4—91no—
146C2×(C2×D3)⋊C4—91no—
147C2×C4⋊Dic3—94no—
148C2×Dic3⋊C4—91no—
149C2×C4×Dic3—94no—
150C2×C4×Dic3—91no—
151C23×C32—28no—
152C23×C32—28no—
153C23×C32—21no—
154C23×C32—24no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32—24no—
159C23×C32—21no—
160C23×C32—21no—
161C23×C32—28no—
162C23×C32—24no—
163C23×C32—24no—
164C23×C32—24no—
165C23×C32—24no—
166C23×C32—24no—
167C23×C32—32no—
168C23×C32—44no—
169C23×C32—44no—
170C23×C32—44no—
171C23×C32—44no—
172C23×C32—61no—
173C23×C32—61no—
174C23×C32—61no—
175C23×C32—61no—
176C23×C32—61no—
177C23×C32—61no—
178C23×C32C4×D312yes—
179C23×C32C2×Dic318yes—
180C23×C32C22×D314yes—
181C23×C32C22×D311yescharacteristic
182C22×C3×D3—24no—
183C22×C3×D3—24no—
184C22×C3×D3—24no—
185C22×C3×D3—24no—
186C22×C3×D3—24no—
187C22×C3×D3—24no—
188C22×C3×D3C2×Dic318yes—
189C32×D4—68no—
190C2×C32×C4—32no—
191C2×C32×C4—32no—
192C2×C32×C4—32no—
193C2×C3⋊Dic3—21no—
194C2×C3⋊Dic3—21no—
195C2×C3⋊Dic3—21no—
196C2×C3⋊Dic3—21no—
197C2×C3⋊Dic3—21no—
198C2×C3⋊Dic3—32no—
199C2×C3⋊Dic3—38no—
200C2×C3⋊Dic3—68no—
201C2×C3⋊Dic3—61no—
202C2×C3⋊Dic3—61no—
203C2×C3⋊Dic3—68no—
204C2×C3⋊Dic3—61no—
205C2×C3⋊Dic3—64no—
206C2×C3⋊Dic3—61no—
207C2×C3⋊Dic3—61no—
208C2×C3⋊Dic3—64no—
209C2×C3⋊Dic3—64no—
210C2×C3⋊Dic3—64no—
211C2×C3⋊Dic3—64no—
212C2×C3⋊Dic3—124no—
213C2×C3⋊Dic3—124no—
214C2×C3⋊Dic3—124no—
215C2×C3⋊Dic3—124no—
216C2×C3⋊Dic3#6712yescomplemented
217C3×#67—232no—
218C2×C3×Dic3—38no—
219C2×C3×Dic3—64no—
220C2×C3×Dic3—64no—
221C2×C3×Dic3—64no—
222C2×C3×Dic3—64no—
223C2×C3×Dic3—64no—
224C2×C3×Dic3C2×Dic318yes—
225C2×C3×Dic3#6718yes—
226C2×C3×Dic3#6718yes—
227#390—271noSylow
228C2×C33—42no—
229C2×C33C2×C22⋊C411yescharacteristic
230C2×C33C4×D411yescharacteristic
231C2×C33C4×D411yescharacteristic
232C2×C33#12111yescharacteristic
233C2×C33#12111yescharacteristic
234C2×C33#12411yescharacteristic
235C2×C33#12711yescharacteristic
236C32×D3—42no—
237C24×C3—24no—
238C24×C3—34no—
239C24×C3D3214yes—
240C24×C3C2×C3⋊D311yescharacteristic
241C23×D3C2×C3⋊D311yescharacteristic
242C2×C3×D4—316no—
243C2×C3×D4—68no—
244C2×C3×D4—68no—
245C22×C3×C4—34no—
246C22×C3×C4—91no—
247C22×C3×C4—91no—
248C2×#67—22no—
249C2×#67—22no—
250C2×#67C3⋊Dic314yescomplemented
251C22×Dic3—34no—
252C22×Dic3—34no—
253C22×Dic3—94no—
254C22×Dic3—91no—
255C22×Dic3—94no—
256C22×Dic3—91no—
257C22×Dic3—184no—
258C22×Dic3—184no—
259C22×Dic3C2×C3⋊D311yescharacteristic
260C3×C22⋊C4—182no—
261#217—68no—
262#217—188no—
263#217—188no—
264#217—188no—
265#217—182no—
266#217—1816no—
267(C2×D3)⋊C4—182no—
268(C2×D3)⋊C4—182no—
269C4⋊Dic3—188no—
270Dic3⋊C4—182no—
271C4×Dic3—188no—
272C4×Dic3—182no—
273C22×C32—216no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—22no—
277C22×C32—28no—
278C22×C32—24no—
279C22×C32—24no—
280C22×C32—24no—
281C22×C32—24no—
282C22×C32—24no—
283C22×C32—24no—
284C22×C32—24no—
285C22×C32—28no—
286C22×C32—28no—
287C22×C32—48no—
288C22×C32—48no—
289C22×C32—48no—
290C22×C32—48no—
291C22×C32—48no—
292C22×C32—48no—
293C22×C32—41no—
294C22×C32—41no—
295C22×C32—44no—
296C22×C32—44no—
297C22×C32—44no—
298C22×C32—41no—
299C22×C32—44no—
300C22×C32—41no—
301C22×C32—62no—
302C22×C32—62no—
303C22×C32—62no—
304C22×C32—121no—
305C22×C32—121no—
306C22×C32—121no—
307C22×C32—121no—
308C22×C32C2×C4×D311yescharacteristic
309C22×C32#23511yescharacteristic
310C22×C32D3×D414yes—
311C22×C32D3×D414yes—
312C22×C32D3×D411yescharacteristic
313C22×C32D3×D411yescharacteristic
314C22×C32D3×D411yescharacteristic
315C22×C32#23714yes—
316C22×C32#23714yes—
317C22×C32#23711yescharacteristic
318C22×C32C22×Dic314yes—
319C22×C32C2×#6714yes—
320C22×C32C2×#6714yes—
321C22×C32C2×#6711yescharacteristic
322C2×C3×D3—28no—
323C2×C3×D3—28no—
324C2×C3×D3—28no—
325C2×C3×D3—44no—
326C2×C3×D3—44no—
327C2×C3×D3—44no—
328C2×C3×D3—44no—
329C32×C4—34no—
330C3⋊Dic3—22no—
331C3⋊Dic3—41no—
332C3⋊Dic3—62no—
333C3⋊Dic3—68no—
334C3⋊Dic3—128no—
335C3⋊Dic3—128no—
336C3⋊Dic3—124no—
337C3⋊Dic3—121no—
338C3×Dic3—68no—
339C3×Dic3—124no—
340C3×Dic3#217116yescomplemented
341C22×D4—31no—
342C2×C4⋊C4—271no—
343C2×C22⋊C4—91no—
344C2×C22⋊C4—271no—
345C2×C22⋊C4—271no—
346C2×C22⋊C4—271no—
347C2×C42—271no—
348C33#39011yescharacteristic, complemented, Sylow
349C23×C3—28no—
350C23×C3—28no—
351C23×C3—24no—
352C23×C3—24no—
353C23×C3—24no—
354C23×C3—24no—
355C23×C3—24no—
356C23×C3—24no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—28no—
364C23×C3—24no—
365C23×C3—38no—
366C23×C3—44no—
367C23×C3—44no—
368C23×C3—44no—
369C23×C3—44no—
370C23×C3—64no—
371C23×C3—64no—
372C23×C3—64no—
373C23×C3—64no—
374C23×C3—64no—
375C23×C3—64no—
376C23×C3D3×Dic318yes—
377C23×C3C2×C3⋊Dic312yes—
378C23×C3C2×D3214yes—
379C23×C3C22×C3⋊D311yescharacteristic
380C22×D3—21no—
381C22×D3—21no—
382C22×D3—21no—
383C22×D3—21no—
384C22×D3—21no—
385C22×D3—21no—
386C22×D3C2×C3⋊Dic312yes—
387C3×D4—632no—
388C2×C3×C4—38no—
389C2×C3×C4—38no—
390C2×C3×C4—38no—
391C2×C3×C4—92no—
392C2×C3×C4—181no—
393C2×C3×C4—181no—
394C2×C3×C4—181no—
395C2×C3×C4—181no—
396C2×C3×C4—181no—
397#67—28no—
398C2×Dic3—38no—
399C2×Dic3—64no—
400C2×Dic3—64no—
401C2×Dic3—64no—
402C2×Dic3—64no—
403C2×Dic3—64no—
404C2×Dic3—98no—
405C2×Dic3—92no—
406C2×Dic3—188no—
407C2×Dic3—184no—
408C2×Dic3—184no—
409C2×Dic3—181no—
410C2×Dic3—181no—
411C2×Dic3—188no—
412C2×Dic3—184no—
413C2×Dic3—184no—
414C2×Dic3—184no—
415C2×Dic3—181no—
416C2×Dic3—181no—
417C2×Dic3—181no—
418C2×Dic3—364no—
419C2×Dic3—364no—
420C2×Dic3—364no—
421C2×Dic3—364no—
422C2×Dic3C2×C3⋊Dic312yes—
423C2×Dic3#63812yes—
424C2×Dic3#63812yes—
425C2×C32—24no—
426C2×C32—24no—
427C2×C32—24no—
428C2×C32—24no—
429C2×C32—24no—
430C2×C32—24no—
431C2×C32—24no—
432C2×C32—416no—
433C2×C32—42no—
434C2×C32—48no—
435C2×C32—122no—
436C2×C32D3×C22⋊C411yescharacteristic
437C2×C32#67⋊C411yescharacteristic
438C2×C32#88411yescharacteristic
439C2×C32#88511yescharacteristic
440C2×C32C4×#6711yescharacteristic
441C2×C32D4×Dic314yes—
442C2×C32D4×Dic314yes—
443C2×C32#93614yes—
444C2×C32#93814yes—
445C2×C32#93814yes—
446C2×C32#93911yescharacteristic
447C2×C32#94014yes—
448C2×C32#94011yescharacteristic
449C2×C32C2×#21714yes—
450C3×D3—48no—
451C24—31no—
452C24D3×C3⋊D311yescharacteristic
453C2×D4—34no—
454C2×D4—62no—
455C2×D4—62no—
456C22×C4—31no—
457C22×C4—91no—
458C22×C4—91no—
459C22×C4—271no—
460C22×C4—271no—
461C4⋊C4—542no—
462C22⋊C4—182no—
463C22⋊C4—542no—
464C22⋊C4—542no—
465C22⋊C4—542no—
466C42—542no—
467C22×C3—216no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—28no—
472C22×C3—28no—
473C22×C3—28no—
474C22×C3—24no—
475C22×C3—24no—
476C22×C3—24no—
477C22×C3—24no—
478C22×C3—24no—
479C22×C3—24no—
480C22×C3—24no—
481C22×C3—48no—
482C22×C3—48no—
483C22×C3—48no—
484C22×C3—48no—
485C22×C3—48no—
486C22×C3—48no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—44no—
492C22×C3—44no—
493C22×C3—44no—
494C22×C3—44no—
495C22×C3—68no—
496C22×C3—68no—
497C22×C3—68no—
498C22×C3—124no—
499C22×C3—124no—
500C22×C3—124no—
501C22×C3—124no—
502C22×C3C2×D3×Dic314yes—
503C22×C3#380414yes—
504C22×C3#380514yes—
505C22×C3D3×#6714yes—
506C22×C3D3×#6714yes—
507C22×C3D3×#6714yes—
508C22×C3#381114yes—
509C22×C3D4×C3⋊D311yescharacteristic
510C22×C3D4×C3⋊D311yescharacteristic
511C22×C3#383011yescharacteristic
512C22×C3#383011yescharacteristic
513C22×C3C22×C3⋊Dic311yescharacteristic
514C22×C3C2×#63811yescharacteristic
515C22×C3C2×#63811yescharacteristic
516C2×D3—22no—
517C2×D3—22no—
518C2×D3—22no—
519C2×D3—41no—
520C2×D3—41no—
521C2×D3—41no—
522C2×D3—41no—
523C3×C4—316no—
524C3×C4—182no—
525C3×C4—361no—
526Dic3—68no—
527Dic3—124no—
528Dic3—182no—
529Dic3—188no—
530Dic3—368no—
531Dic3—368no—
532Dic3—361no—
533Dic3—364no—
534Dic3#375714yescomplemented
535C32—24no—
536C32#495711yescharacteristic, complemented
537C32#522114yescomplemented
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—32no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23—61no—
549C23—61no—
550C23—61no—
551C23D3×C3⋊Dic312yes—
552C23C2×D3×C3⋊D311yescharacteristic
553D4—68no—
554C2×C4—32no—
555C2×C4—32no—
556C2×C4—32no—
557C2×C4—92no—
558C2×C4—181no—
559C2×C4—181no—
560C2×C4—181no—
561C2×C4—181no—
562C2×C4—181no—
563C2×C4—272no—
564C2×C4—541no—
565C2×C4—541no—
566C2×C4—541no—
567C2×C4—541no—
568C2×C4—541no—
569C2×C3—24no—
570C2×C3—24no—
571C2×C3—24no—
572C2×C3—24no—
573C2×C3—24no—
574C2×C3—24no—
575C2×C3—24no—
576C2×C3—416no—
577C2×C3—42no—
578C2×C3—48no—
579C2×C3—128no—
580C2×C3#6395814yes—
581C2×C3#6395914yes—
582C2×C3D3×#21714yes—
583C2×C3#6397014yes—
584C2×C3Dic3×#6714yes—
585C2×C3#67⋊Dic314yes—
586C2×C3#6397914yes—
587C2×C3D4×C3⋊Dic311yescharacteristic
588C2×C3D4×C3⋊Dic311yescharacteristic
589C2×C3#6414511yescharacteristic
590C2×C3#6414711yescharacteristic
591C2×C3#6414711yescharacteristic
592C2×C3#6414911yescharacteristic
593C2×C3C2×#375711yescharacteristic
594D3—42no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—62no—
603C22—62no—
604C22—62no—
605C22—121no—
606C22—121no—
607C22—121no—
608C22—121no—
609C22C2×D3×C3⋊Dic311yescharacteristic
610C22#8902811yescharacteristic
611C22#8902911yescharacteristic
612C22D3×#63811yescharacteristic
613C22D3×#63811yescharacteristic
614C22C3⋊D3×#6711yescharacteristic
615C22#8903911yescharacteristic
616C4—34no—
617C4—182no—
618C4—361no—
619C4—542no—
620C4—1081no—
621C3—24no—
622C3#1059170714yescomplemented
623C3#1059292011yescharacteristic, complemented
624C2—42no—
625C2—122no—
626C2#1172359911yescharacteristic
627C2#1172360011yescharacteristic
628C2D3×#375711yescharacteristic
629C2#67×C3⋊Dic311yescharacteristic
630C2#67⋊(C3⋊Dic3)11yescharacteristic
631C2#1172363111yescharacteristic
632C2#1172363311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
FittingC24×C33C22
socle, cosocleC23×C33C23
prime radicalC22×C32×#67C2
hypercenterC24D3×C3⋊D3
hypocenterC33#390
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#88864C2semidirect
3. C2×C3×#3757C2semidirect
4. C2×Dic3⋊(C3⋊Dic3)C2semidirect
5. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
6. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
7. C2×Dic3×C3⋊Dic3C2semidirect
18. C2×C32×#67C4semidirect
33. C2×#3757D3semidirect
61. C2×C3⋊(C3⋊Dic3)D4semidirect
67. C2×C3×C3⋊Dic3D4semidirect
93. C2×C3×#67Dic3semidirect
135. C32×Dic3C22⋊C4semidirect
216. C2×C3⋊Dic3#67semidirect
250. C2×#67C3⋊Dic3semidirect
340. C3×Dic3#217semidirect
348. C33#390semidirect
534. Dic3#3757semidirect
536. C32#4957semidirect
537. C32#5221semidirect
622. C3#10591707semidirect
623. C3#10592920semidirect

Maximal subgroups

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

subgroupindex
C22×C32×#672
C2×#888642
C2×C3×#37572
C2×Dic3⋊(C3⋊Dic3)2
C2×(C2×D3)⋊(C3⋊Dic3)2
C2×Dic3×C3⋊Dic32
#105929203
#105917073