Finite Group Databasebeta

← #13161422

Subgroups — #13161422

Order 1728.

1956 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic
2C2×#88864C211yesmaximal, characteristic, complemented
3C2×C32×#217C211yesmaximal, characteristic, complemented
4C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5C2×#88803C211yesmaximal, characteristic, complemented
6C2×#88803C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592660—31nomaximal
9#10591709—34nomaximal
10C24×C33C2211yescharacteristic
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×#638—22no—
15C2×C3×#638—22no—
16C2×C3×#638—22no—
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20#88864—22no—
21C32×#217—22no—
22Dic3⋊(C3⋊Dic3)—22no—
23#88803—22no—
24#88803—22no—
25C22×#638D311yescharacteristic
26C22×C3×#67—34no—
27C2×C32×C22⋊C4—31no—
28C2×#3757—31no—
29C2×#3757—34no—
30C2×#3757—64no—
31C2×#3752—31no—
32C2×#3752—31no—
33C2×C4⋊(C3⋊Dic3)—31no—
34C2×C3×#217D314yescomplemented
35C2×Dic3⋊Dic3—34no—
36C2×(C2×D3)⋊Dic3—34no—
37C2×(C2×D3)⋊Dic3—34no—
38#63658—31no—
39#63579—34no—
40C23×C33—21no—
41C23×C33—21no—
42C23×C33—21no—
43C23×C33—21no—
44C23×C33—21no—
45C23×C33—21no—
46C23×C33D412yes—
47C23×C33C2311yescharacteristic
48C22×C3×C3⋊D3—21no—
49C22×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3—21no—
52C22×C3×C3⋊D3—21no—
53C22×C3×C3⋊D3—21no—
54C22×C3×C3⋊D3D412yes—
55C2×C3⋊(C3⋊Dic3)—21no—
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—
61C3×#638—28no—
62C2×C3×C3⋊Dic3—21no—
63C2×C3×C3⋊Dic3—21no—
64C2×C3×C3⋊Dic3D412yes—
65C2×C32×Dic3—21no—
66C2×C32×Dic3—21no—
67C2×C32×Dic3—21no—
68C2×C32×Dic3—21no—
69C2×C32×Dic3—21no—
70C2×C32×Dic3D412yescomplemented
71#5222—91no—
72#4961—94no—
73C24×C32—24no—
74C24×C32C2×D314yes—
75C24×C32C2×D311yescharacteristic
76C23×C3⋊D3C2×D311yescharacteristic
77C23×C3×D3—34no—
78C22×C32×C4—31no—
79C22×C32×C4—31no—
80C2×#638—22no—
81C2×#638—22no—
82C2×#638—22no—
83C22×C3⋊Dic3—31no—
84C22×C3⋊Dic3—31no—
85C22×C3⋊Dic3—34no—
86C22×C3⋊Dic3—34no—
87C22×C3⋊Dic3—64no—
88C22×C3⋊Dic3—64no—
89C22×C3⋊Dic3C2×D311yescharacteristic
90C2×C3×#67—68no—
91C2×C3×#67—68no—
92C2×C3×#67—68no—
93C22×C3×Dic3—34no—
94C22×C3×Dic3C2×D314yes—
95C22×C3×Dic3C2×D314yes—
96C32×C22⋊C4—62no—
97#3757—616no—
98#3757—68no—
99#3757—62no—
100#3752—62no—
101#3752—62no—
102C4⋊(C3⋊Dic3)—62no—
103C3×#217—28no—
104Dic3⋊Dic3—68no—
105(C2×D3)⋊Dic3—68no—
106(C2×D3)⋊Dic3—68no—
107C22×C33—22no—
108C22×C33—22no—
109C22×C33—22no—
110C22×C33—41no—
111C22×C33—41no—
112C22×C33—41no—
113C22×C33—41no—
114C22×C33C2×D411yescharacteristic
115C22×C33C2×D411yescharacteristic
116C22×C33C2×D411yescharacteristic
117C22×C33C2×D411yescharacteristic
118C22×C33#4111yescharacteristic
119C22×C33#4111yescharacteristic
120C22×C33#4111yescharacteristic
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—41no—
125C2×C3×C3⋊D3—41no—
126C2×C3×C3⋊D3—41no—
127C2×C3×C3⋊D3—41no—
128C3⋊(C3⋊Dic3)—41no—
129C3⋊(C3⋊Dic3)—41no—
130C3×C3⋊Dic3—22no—
131C32×Dic3—22no—
132C32×Dic3—41no—
133C22×C3×D4—91no—
134C22×#67—34no—
135C2×C3×C22⋊C4—34no—
136C2×#217—94no—
137C2×#217—91no—
138C2×#217—91no—
139C2×#217—91no—
140C2×#217—184no—
141C2×#217C3⋊D311yescharacteristic, complemented
142C2×(C2×D3)⋊C4—94no—
143C2×(C2×D3)⋊C4—94no—
144C2×C4⋊Dic3—94no—
145C2×Dic3⋊C4—91no—
146C3⋊#95—94no—
147C3⋊#95—91no—
148C23×C32—28no—
149C23×C32—28no—
150C23×C32—21no—
151C23×C32—24no—
152C23×C32—24no—
153C23×C32—21no—
154C23×C32—21no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32—28no—
159C23×C32—24no—
160C23×C32—24no—
161C23×C32—24no—
162C23×C32—24no—
163C23×C32—24no—
164C23×C32—44no—
165C23×C32—44no—
166C23×C32—44no—
167C23×C32—44no—
168C23×C32D1218yes—
169C23×C32#6712yes—
170C23×C32C22×D314yes—
171C23×C32C22×D311yescharacteristic
172C22×C3⋊D3—21no—
173C22×C3⋊D3—21no—
174C22×C3⋊D3—21no—
175C22×C3⋊D3—21no—
176C22×C3⋊D3—21no—
177C22×C3⋊D3—21no—
178C22×C3⋊D3#6712yes—
179C22×C3×D3—38no—
180C22×C3×D3—64no—
181C22×C3×D3—64no—
182C22×C3×D3—64no—
183C22×C3×D3—64no—
184C22×C3×D3—64no—
185C22×C3×D3—64no—
186C2×C32×C4—32no—
187C2×C32×C4—61no—
188C2×C32×C4—61no—
189C2×C32×C4—61no—
190C2×C32×C4—61no—
191C2×C32×C4—61no—
192#638—28no—
193C2×C3⋊Dic3—21no—
194C2×C3⋊Dic3—21no—
195C2×C3⋊Dic3—68no—
196C2×C3⋊Dic3—68no—
197C2×C3⋊Dic3—64no—
198C2×C3⋊Dic3—64no—
199C2×C3⋊Dic3—61no—
200C2×C3⋊Dic3—61no—
201C2×C3⋊Dic3—61no—
202C2×C3⋊Dic3—61no—
203C2×C3⋊Dic3—61no—
204C2×C3⋊Dic3—61no—
205C2×C3⋊Dic3—64no—
206C2×C3⋊Dic3—64no—
207C2×C3⋊Dic3—64no—
208C2×C3⋊Dic3—64no—
209C2×C3⋊Dic3—124no—
210C2×C3⋊Dic3—124no—
211C2×C3⋊Dic3—124no—
212C2×C3⋊Dic3—124no—
213C2×C3⋊Dic3#6712yes—
214C3×#67—632no—
215C2×C3×Dic3—24no—
216C2×C3×Dic3—24no—
217C2×C3×Dic3—24no—
218C2×C3×Dic3—24no—
219C2×C3×Dic3—24no—
220C2×C3×Dic3—38no—
221C2×C3×Dic3—64no—
222C2×C3×Dic3—64no—
223C2×C3×Dic3#6718yescomplemented
224#394—271noSylow
225C2×C33—42no—
226C2×C33#12011yescharacteristic
227C2×C33#12111yescharacteristic
228C2×C33#12111yescharacteristic
229C2×C33#12111yescharacteristic
230C2×C33#12311yescharacteristic
231C2×C33#12311yescharacteristic
232C2×C33#12411yescharacteristic
233C3×C3⋊D3—42no—
234C24×C3—24no—
235C24×C3—91no—
236C24×C3D3214yes—
237C24×C3C2×C3⋊D311yescharacteristic
238C23×D3—34no—
239C2×C3×D4—182no—
240C2×C3×D4—182no—
241C2×C3×D4—182no—
242C22×C3×C4—34no—
243C22×C3×C4—34no—
244C22×C3×C4—91no—
245C2×#67—68no—
246C2×#67—68no—
247C2×#67—68no—
248C22×Dic3—34no—
249C22×Dic3—94no—
250C22×Dic3—94no—
251C22×Dic3—91no—
252C22×Dic3—91no—
253C22×Dic3—184no—
254C22×Dic3—184no—
255C22×Dic3C2×C3⋊D311yescharacteristic
256C22×Dic3C2×C3⋊D311yescharacteristic
257C3×C22⋊C4—68no—
258#217—22no—
259#217—182no—
260#217—1816no—
261#217—188no—
262#217—182no—
263#217—182no—
264(C2×D3)⋊C4—188no—
265(C2×D3)⋊C4—188no—
266C4⋊Dic3—188no—
267Dic3⋊C4—182no—
268C22×C32—216no—
269C22×C32—28no—
270C22×C32—22no—
271C22×C32—22no—
272C22×C32—22no—
273C22×C32—24no—
274C22×C32—24no—
275C22×C32—24no—
276C22×C32—24no—
277C22×C32—24no—
278C22×C32—24no—
279C22×C32—24no—
280C22×C32—28no—
281C22×C32—28no—
282C22×C32—48no—
283C22×C32—48no—
284C22×C32—48no—
285C22×C32—48no—
286C22×C32—48no—
287C22×C32—48no—
288C22×C32—44no—
289C22×C32—44no—
290C22×C32—41no—
291C22×C32—41no—
292C22×C32—44no—
293C22×C32—41no—
294C22×C32—44no—
295C22×C32—41no—
296C22×C32C2×D1214yes—
297C22×C32#23514yes—
298C22×C32D3×D414yes—
299C22×C32D3×D414yes—
300C22×C32D3×D411yescharacteristic
301C22×C32#23714yes—
302C22×C32#23714yes—
303C22×C32#23711yescharacteristic
304C22×C32#23711yescharacteristic
305C22×C32#23711yescharacteristic
306C22×C32C2×#6714yes—
307C22×C32C2×#6711yescharacteristic
308C22×C32C2×#6711yescharacteristic
309C22×C32C2×#6711yescharacteristic
310C2×C3⋊D3—22no—
311C2×C3⋊D3—22no—
312C2×C3⋊D3—22no—
313C2×C3⋊D3—41no—
314C2×C3⋊D3—41no—
315C2×C3⋊D3—41no—
316C2×C3⋊D3—41no—
317C2×C3×D3—68no—
318C2×C3×D3—68no—
319C2×C3×D3—68no—
320C2×C3×D3—124no—
321C2×C3×D3—124no—
322C2×C3×D3—124no—
323C2×C3×D3—124no—
324C32×C4—62no—
325C32×C4—121no—
326C3⋊Dic3—22no—
327C3⋊Dic3—128no—
328C3⋊Dic3—128no—
329C3⋊Dic3—124no—
330C3⋊Dic3—124no—
331C3⋊Dic3—121no—
332C3⋊Dic3—121no—
333C3×Dic3—28no—
334C3×Dic3—44no—
335C3×Dic3—68no—
336C22×D4—91no—
337C2×C4⋊C4—271no—
338C2×C22⋊C4—31no—
339C2×C22⋊C4—271no—
340C2×C22⋊C4—271no—
341C2×C22⋊C4—271no—
342#95—271no—
343C33#39411yescharacteristic, complemented, Sylow
344C23×C3—28no—
345C23×C3—28no—
346C23×C3—24no—
347C23×C3—24no—
348C23×C3—24no—
349C23×C3—24no—
350C23×C3—24no—
351C23×C3—24no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—28no—
359C23×C3—24no—
360C23×C3—44no—
361C23×C3—44no—
362C23×C3—44no—
363C23×C3—44no—
364C23×C3—92no—
365C23×C3—181no—
366C23×C3—181no—
367C23×C3—181no—
368C23×C3—181no—
369C23×C3—181no—
370C23×C3—181no—
371C23×C3#62618yes—
372C23×C3C4⋊(C3⋊D3)12yes—
373C23×C3C2×D3214yes—
374C23×C3C22×C3⋊D311yescharacteristic
375C22×D3—38no—
376C22×D3—64no—
377C22×D3—64no—
378C22×D3—64no—
379C22×D3—64no—
380C22×D3—64no—
381C22×D3—64no—
382C3×D4—188no—
383C2×C3×C4—38no—
384C2×C3×C4—64no—
385C2×C3×C4—64no—
386C2×C3×C4—64no—
387C2×C3×C4—64no—
388C2×C3×C4—64no—
389C2×C3×C4—92no—
390C2×C3×C4—181no—
391C2×C3×C4—181no—
392#67—632no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—38no—
399C2×Dic3—64no—
400C2×Dic3—64no—
401C2×Dic3—188no—
402C2×Dic3—188no—
403C2×Dic3—184no—
404C2×Dic3—184no—
405C2×Dic3—184no—
406C2×Dic3—184no—
407C2×Dic3—184no—
408C2×Dic3—184no—
409C2×Dic3—181no—
410C2×Dic3—181no—
411C2×Dic3—181no—
412C2×Dic3—181no—
413C2×Dic3—181no—
414C2×Dic3—181no—
415C2×Dic3—364no—
416C2×Dic3—364no—
417C2×Dic3—364no—
418C2×Dic3—364no—
419C2×Dic3#63812yescomplemented
420C2×C32—24no—
421C2×C32—24no—
422C2×C32—24no—
423C2×C32—24no—
424C2×C32—24no—
425C2×C32—24no—
426C2×C32—24no—
427C2×C32—416no—
428C2×C32—48no—
429C2×C32—42no—
430C2×C32#88214yes—
431C2×C32#88314yes—
432C2×C32#88514yes—
433C2×C32#88614yes—
434C2×C32#93014yes—
435C2×C32#93511yescharacteristic
436C2×C32#93511yescharacteristic
437C2×C32#93611yescharacteristic
438C2×C32#93814yes—
439C2×C32#93811yescharacteristic
440C2×C32#93914yes—
441C2×C32#93911yescharacteristic
442C2×C32#93911yescharacteristic
443C2×C32#95311yescharacteristic
444C3⋊D3—42no—
445C3×D3—128no—
446C24—91no—
447C24D3×C3⋊D311yescharacteristic
448C2×D4—182no—
449C2×D4—182no—
450C2×D4—182no—
451C22×C4—31no—
452C22×C4—31no—
453C22×C4—91no—
454C22×C4—271no—
455C22×C4—271no—
456C4⋊C4—542no—
457C22⋊C4—62no—
458C22⋊C4—542no—
459C22⋊C4—542no—
460C22⋊C4—542no—
461C22×C3—216no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—28no—
466C22×C3—28no—
467C22×C3—28no—
468C22×C3—24no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—24no—
472C22×C3—24no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—48no—
476C22×C3—48no—
477C22×C3—48no—
478C22×C3—48no—
479C22×C3—48no—
480C22×C3—48no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—44no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3—182no—
490C22×C3—182no—
491C22×C3—182no—
492C22×C3—361no—
493C22×C3—361no—
494C22×C3—361no—
495C22×C3—361no—
496C22×C3#380414yes—
497C22×C3#380514yes—
498C22×C3#380514yes—
499C22×C3C2×#62614yes—
500C22×C3D3×#6714yes—
501C22×C3D3×#6714yes—
502C22×C3D3×#6714yes—
503C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
504C22×C3#382811yescharacteristic
505C22×C3D4×C3⋊D311yescharacteristic
506C22×C3D4×C3⋊D311yescharacteristic
507C22×C3#383011yescharacteristic
508C22×C3#383011yescharacteristic
509C22×C3C2×#63811yescharacteristic
510C2×D3—68no—
511C2×D3—68no—
512C2×D3—68no—
513C2×D3—124no—
514C2×D3—124no—
515C2×D3—124no—
516C2×D3—124no—
517C3×C4—68no—
518C3×C4—124no—
519C3×C4—182no—
520Dic3—22no—
521Dic3—41no—
522Dic3—68no—
523Dic3—368no—
524Dic3—368no—
525Dic3—361no—
526Dic3—361no—
527Dic3—364no—
528Dic3—364no—
529C32—24no—
530C32#496114yescomplemented
531C32#522211yescharacteristic, complemented
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—92no—
539C23—181no—
540C23—181no—
541C23—181no—
542C23—181no—
543C23—181no—
544C23—181no—
545C23#629212yes—
546C23C2×D3×C3⋊D311yescharacteristic
547D4—188no—
548C2×C4—32no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—92no—
555C2×C4—181no—
556C2×C4—181no—
557C2×C4—541no—
558C2×C4—541no—
559C2×C4—541no—
560C2×C4—541no—
561C2×C4—541no—
562C2×C4—541no—
563C2×C3—24no—
564C2×C3—24no—
565C2×C3—24no—
566C2×C3—24no—
567C2×C3—24no—
568C2×C3—24no—
569C2×C3—24no—
570C2×C3—416no—
571C2×C3—42no—
572C2×C3—48no—
573C2×C3—362no—
574C2×C3#6395914yes—
575C2×C3#6396214yes—
576C2×C3#6396914yes—
577C2×C3#6397014yes—
578C2×C3#6397014yes—
579C2×C3#6397714yes—
580C2×C3#6397814yes—
581C2×C3#6409111yescharacteristic
582C2×C3#6409211yescharacteristic
583C2×C3#6409411yescharacteristic
584C2×C3#6409511yescharacteristic
585C2×C3#6413911yescharacteristic
586C2×C3#6414711yescharacteristic
587C2×C3#6414811yescharacteristic
588D3—128no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—182no—
597C22—182no—
598C22—182no—
599C22—361no—
600C22—361no—
601C22—361no—
602C22—361no—
603C22#8902911yescharacteristic
604C22#8902911yescharacteristic
605C22#8903111yescharacteristic
606C22C2×#629211yescharacteristic
607C22D3×#63811yescharacteristic
608C22C3⋊D3×#6711yescharacteristic
609C22C3⋊D3×#6711yescharacteristic
610C4—62no—
611C4—121no—
612C4—182no—
613C4—1081no—
614C4—1081no—
615C3—24no—
616C3#1059170914yescomplemented
617C3#1059266011yescharacteristic, complemented
618C2—42no—
619C2—362no—
620C2#1172359111yescharacteristic
621C2#1172360911yescharacteristic
622C2#1172361511yescharacteristic
623C2#1172362111yescharacteristic
624C2#1172363111yescharacteristic
625C2#1172363211yescharacteristic
626C2#1172363611yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutator, socle, cosocleC23×C33C23
FittingC24×C33C22
prime radicalC22×C3×#638C2
hypercenterC24D3×C3⋊D3
hypocenterC33#394
LCSG ▷ C23×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#88864C2semidirect
3. C2×C32×#217C2semidirect
4. C2×Dic3⋊(C3⋊Dic3)C2semidirect
5. C2×#88803C2semidirect
6. C2×#88803C2semidirect
7. #11722061C2semidirect
34. C2×C3×#217D3semidirect
70. C2×C32×Dic3D4semidirect
141. C2×#217C3⋊D3semidirect
223. C2×C3×Dic3#67semidirect
343. C33#394semidirect
419. C2×Dic3#638semidirect
530. C32#4961semidirect
531. C32#5222semidirect
616. C3#10591709semidirect
617. C3#10592660semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C2×#888642
C2×C32×#2172
C2×Dic3⋊(C3⋊Dic3)2
C2×#888032
#117220612
#105926603
#105917093