Finite Group Databasebeta

← #13161428

Subgroups — #13161428

Order 1728.

2372 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6312C211yesmaximal, characteristic
2C2×C3×#3757C211yesmaximal, characteristic, complemented
3C2×C32×#217C211yesmaximal, characteristic, complemented
4C2×#88807C211yesmaximal, characteristic, complemented
5C2×#88804C211yesmaximal, characteristic, complemented
6C2×#88804C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592658—31nomaximal
9#10591712—34nomaximal
10C24×C33C2211yescharacteristic
11C23×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×#6312—22no—
13C2×#6312—22no—
14C2×#6312—22no—
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20C3×#3757—22no—
21C32×#217—22no—
22#88807—22no—
23#88804—22no—
24#88804—22no—
25C22×#638—31no—
26C22×#638—34no—
27C22×#638—64no—
28C2×C32×C22⋊C4—31no—
29C2×#3757D311yescharacteristic, complemented
30C2×#3752—31no—
31C2×#3752—31no—
32C2×#3750—31no—
33C2×C3×#217—34no—
34C2×C3×#217D314yescomplemented
35C2×Dic3⋊Dic3—34no—
36C2×#3722—34no—
37C2×#3722—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⋊D3)D412yes—
55#6312—28no—
56C2×C3⋊(C3⋊Dic3)—21no—
57C2×C3⋊(C3⋊Dic3)—21no—
58C2×C3⋊(C3⋊Dic3)D412yes—
59C2×C3×C3⋊Dic3—21no—
60C2×C3×C3⋊Dic3—21no—
61C2×C3×C3⋊Dic3—21no—
62C2×C3×C3⋊Dic3—21no—
63C2×C3×C3⋊Dic3—21no—
64C2×C3×C3⋊Dic3D412yescomplemented
65C2×C32×Dic3—21no—
66C2×C32×Dic3—21no—
67C2×C32×Dic3—21no—
68C2×C32×Dic3—21no—
69C2×C32×Dic3—21no—
70C2×C32×Dic3—21no—
71#4961—91no—
72#4959—94no—
73C24×C32—24no—
74C24×C32C2×D311yescharacteristic
75C24×C32C2×D314yes—
76C23×C3⋊D3—31no—
77C23×C3⋊D3—34no—
78C23×C3⋊D3—64no—
79C22×C32×C4—31no—
80C22×C32×C4—31no—
81C2×#638—62no—
82C2×#638—68no—
83C2×#638—62no—
84C2×#638—68no—
85C2×#638—616no—
86C2×#638—616no—
87C2×#638—616no—
88C2×#638—62no—
89C2×#638—68no—
90C22×C3⋊Dic3—31no—
91C22×C3⋊Dic3—34no—
92C22×C3⋊Dic3—64no—
93C22×C3⋊Dic3C2×D311yescharacteristic
94C22×C3⋊Dic3C2×D311yescharacteristic
95C22×C3×Dic3—34no—
96C22×C3×Dic3—34no—
97C22×C3×Dic3C2×D314yes—
98C22×C3×Dic3C2×D314yes—
99C32×C22⋊C4—62no—
100#3757—22no—
101#3752—62no—
102#3752—62no—
103#3750—62no—
104C3×#217—28no—
105C3×#217—68no—
106Dic3⋊Dic3—68no—
107#3722—68no—
108#3722—68no—
109C22×C33—22no—
110C22×C33—22no—
111C22×C33—22no—
112C22×C33—41no—
113C22×C33—41no—
114C22×C33—41no—
115C22×C33—41no—
116C22×C33C2×D411yescharacteristic
117C22×C33C2×D411yescharacteristic
118C22×C33C2×D411yescharacteristic
119C22×C33C2×D411yescharacteristic
120C22×C33#4111yescharacteristic
121C22×C33#4111yescharacteristic
122C22×C33#4111yescharacteristic
123C2×C3⋊(C3⋊D3)—22no—
124C2×C3⋊(C3⋊D3)—22no—
125C2×C3⋊(C3⋊D3)—22no—
126C2×C3⋊(C3⋊D3)—41no—
127C2×C3⋊(C3⋊D3)—41no—
128C2×C3⋊(C3⋊D3)—41no—
129C2×C3⋊(C3⋊D3)—41no—
130C3⋊(C3⋊Dic3)—22no—
131C3×C3⋊Dic3—22no—
132C3×C3⋊Dic3—41no—
133C32×Dic3—41no—
134C32×Dic3—41no—
135C22×#67—94no—
136C22×#67—91no—
137C22×#67—184no—
138C2×C3×C22⋊C4—34no—
139C2×C3×C22⋊C4—91no—
140C2×#217—34no—
141C2×#217C3⋊D311yescharacteristic, complemented
142C2×(C2×D3)⋊C4—94no—
143C2×(C2×D3)⋊C4—94no—
144C2×(C2×D3)⋊C4—91no—
145C2×(C2×D3)⋊C4—91no—
146C2×C4⋊Dic3—91no—
147C2×Dic3⋊C4—94no—
148C3⋊#95—94no—
149C3⋊#95—91no—
150C23×C32—28no—
151C23×C32—28no—
152C23×C32—21no—
153C23×C32—24no—
154C23×C32—24no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32—21no—
159C23×C32—21no—
160C23×C32—28no—
161C23×C32—24no—
162C23×C32—24no—
163C23×C32—24no—
164C23×C32—24no—
165C23×C32—24no—
166C23×C32—44no—
167C23×C32—44no—
168C23×C32—44no—
169C23×C32—44no—
170C23×C32D1212yes—
171C23×C32#6718yes—
172C23×C32C22×D314yes—
173C23×C32C22×D311yescharacteristic
174C22×C3⋊D3—32no—
175C22×C3⋊D3—38no—
176C22×C3⋊D3—68no—
177C22×C3⋊D3—68no—
178C22×C3⋊D3—68no—
179C22×C3⋊D3—61no—
180C22×C3⋊D3—61no—
181C22×C3⋊D3—64no—
182C22×C3⋊D3—61no—
183C22×C3⋊D3—61no—
184C22×C3⋊D3—61no—
185C22×C3⋊D3—61no—
186C22×C3⋊D3—64no—
187C22×C3⋊D3—64no—
188C22×C3⋊D3—64no—
189C22×C3⋊D3—64no—
190C22×C3⋊D3—64no—
191C22×C3⋊D3—124no—
192C22×C3⋊D3—124no—
193C22×C3⋊D3—124no—
194C22×C3⋊D3—124no—
195C2×C32×C4—61no—
196C2×C32×C4—61no—
197C2×C32×C4—61no—
198C2×C32×C4—61no—
199C2×C32×C4—61no—
200C2×C32×C4—61no—
201#638—664no—
202#638—68no—
203#638—632no—
204C2×C3⋊Dic3—21no—
205C2×C3⋊Dic3—21no—
206C2×C3⋊Dic3—21no—
207C2×C3⋊Dic3—21no—
208C2×C3⋊Dic3—21no—
209C2×C3⋊Dic3—32no—
210C2×C3⋊Dic3—38no—
211C2×C3⋊Dic3—68no—
212C2×C3⋊Dic3—68no—
213C2×C3⋊Dic3—68no—
214C2×C3⋊Dic3—61no—
215C2×C3⋊Dic3—64no—
216C2×C3⋊Dic3—61no—
217C2×C3⋊Dic3—64no—
218C2×C3⋊Dic3#6712yescomplemented
219C2×C3×Dic3—24no—
220C2×C3×Dic3—24no—
221C2×C3×Dic3—24no—
222C2×C3×Dic3—24no—
223C2×C3×Dic3—24no—
224C2×C3×Dic3—24no—
225C2×C3×Dic3—38no—
226C2×C3×Dic3—64no—
227C2×C3×Dic3—64no—
228C2×C3×Dic3—64no—
229C2×C3×Dic3—64no—
230C2×C3×Dic3—64no—
231#394—271noSylow
232C2×C33—42no—
233C2×C33#12011yescharacteristic
234C2×C33#12111yescharacteristic
235C2×C33#12111yescharacteristic
236C2×C33#12111yescharacteristic
237C2×C33#12311yescharacteristic
238C2×C33#12311yescharacteristic
239C2×C33#12411yescharacteristic
240C3⋊(C3⋊D3)—42no—
241C24×C3—24no—
242C24×C3D3214yes—
243C24×C3C2×C3⋊D311yescharacteristic
244C23×D3—94no—
245C23×D3—91no—
246C23×D3—184no—
247C22×C3×C4—34no—
248C22×C3×C4—34no—
249C22×C3×C4—91no—
250C22×C3×C4—91no—
251C2×#67—188no—
252C2×#67—182no—
253C2×#67—188no—
254C2×#67—182no—
255C2×#67—1816no—
256C2×#67—1816no—
257C2×#67—1816no—
258C2×#67—188no—
259C2×#67—182no—
260C22×Dic3—34no—
261C22×Dic3—34no—
262C22×Dic3—94no—
263C22×Dic3—91no—
264C22×Dic3—184no—
265C22×Dic3C2×C3⋊D311yescharacteristic
266C22×Dic3C2×C3⋊D311yescharacteristic
267C3×C22⋊C4—68no—
268C3×C22⋊C4—182no—
269#217—22no—
270#217—68no—
271(C2×D3)⋊C4—182no—
272(C2×D3)⋊C4—182no—
273(C2×D3)⋊C4—188no—
274(C2×D3)⋊C4—188no—
275C4⋊Dic3—182no—
276Dic3⋊C4—188no—
277C22×C32—216no—
278C22×C32—22no—
279C22×C32—22no—
280C22×C32—22no—
281C22×C32—28no—
282C22×C32—24no—
283C22×C32—24no—
284C22×C32—24no—
285C22×C32—24no—
286C22×C32—24no—
287C22×C32—24no—
288C22×C32—24no—
289C22×C32—28no—
290C22×C32—28no—
291C22×C32—48no—
292C22×C32—48no—
293C22×C32—48no—
294C22×C32—48no—
295C22×C32—48no—
296C22×C32—48no—
297C22×C32—44no—
298C22×C32—41no—
299C22×C32—44no—
300C22×C32—44no—
301C22×C32—44no—
302C22×C32—41no—
303C22×C32—41no—
304C22×C32—41no—
305C22×C32C2×D1211yescharacteristic
306C22×C32#23514yes—
307C22×C32#23514yes—
308C22×C32#23511yescharacteristic
309C22×C32D3×D414yes—
310C22×C32D3×D414yes—
311C22×C32D3×D414yes—
312C22×C32D3×D411yescharacteristic
313C22×C32D3×D411yescharacteristic
314C22×C32#23714yes—
315C22×C32#23711yescharacteristic
316C22×C32#23711yescharacteristic
317C22×C32C2×#6714yes—
318C22×C32C2×#6711yescharacteristic
319C2×C3⋊D3—616no—
320C2×C3⋊D3—62no—
321C2×C3⋊D3—62no—
322C2×C3⋊D3—68no—
323C2×C3⋊D3—62no—
324C2×C3⋊D3—68no—
325C2×C3⋊D3—68no—
326C2×C3⋊D3—128no—
327C2×C3⋊D3—128no—
328C2×C3⋊D3—128no—
329C2×C3⋊D3—128no—
330C2×C3⋊D3—128no—
331C2×C3⋊D3—128no—
332C2×C3⋊D3—124no—
333C2×C3⋊D3—124no—
334C2×C3⋊D3—121no—
335C2×C3⋊D3—121no—
336C2×C3⋊D3—124no—
337C2×C3⋊D3—121no—
338C2×C3⋊D3—124no—
339C2×C3⋊D3—121no—
340C32×C4—121no—
341C32×C4—121no—
342C3⋊Dic3—22no—
343C3⋊Dic3—41no—
344C3⋊Dic3—616no—
345C3⋊Dic3—62no—
346C3⋊Dic3—68no—
347C3×Dic3—44no—
348C3×Dic3—44no—
349C3×Dic3—68no—
350C3×Dic3—124no—
351C22×D4—271no—
352C2×C4⋊C4—271no—
353C2×C22⋊C4—31no—
354C2×C22⋊C4—91no—
355C2×C22⋊C4—271no—
356C2×C22⋊C4—271no—
357#95—271no—
358C33#39411yescharacteristic, complemented, Sylow
359C23×C3—28no—
360C23×C3—28no—
361C23×C3—24no—
362C23×C3—24no—
363C23×C3—24no—
364C23×C3—24no—
365C23×C3—24no—
366C23×C3—24no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—28no—
374C23×C3—24no—
375C23×C3—44no—
376C23×C3—44no—
377C23×C3—44no—
378C23×C3—44no—
379C23×C3#62618yes—
380C23×C3#63812yes—
381C23×C3C2×D3214yes—
382C23×C3C22×C3⋊D311yescharacteristic
383C22×D3—98no—
384C22×D3—92no—
385C22×D3—188no—
386C22×D3—188no—
387C22×D3—188no—
388C22×D3—184no—
389C22×D3—184no—
390C22×D3—184no—
391C22×D3—184no—
392C22×D3—184no—
393C22×D3—184no—
394C22×D3—181no—
395C22×D3—181no—
396C22×D3—181no—
397C22×D3—181no—
398C22×D3—181no—
399C22×D3—181no—
400C22×D3—364no—
401C22×D3—364no—
402C22×D3—364no—
403C22×D3—364no—
404C2×C3×C4—64no—
405C2×C3×C4—64no—
406C2×C3×C4—64no—
407C2×C3×C4—64no—
408C2×C3×C4—64no—
409C2×C3×C4—64no—
410C2×C3×C4—92no—
411C2×C3×C4—181no—
412C2×C3×C4—181no—
413C2×C3×C4—181no—
414C2×C3×C4—181no—
415C2×C3×C4—181no—
416#67—1864no—
417#67—188no—
418#67—1832no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—38no—
426C2×Dic3—64no—
427C2×Dic3—64no—
428C2×Dic3—64no—
429C2×Dic3—64no—
430C2×Dic3—64no—
431C2×Dic3—98no—
432C2×Dic3—92no—
433C2×Dic3—188no—
434C2×Dic3—188no—
435C2×Dic3—188no—
436C2×Dic3—184no—
437C2×Dic3—184no—
438C2×Dic3—181no—
439C2×Dic3—181no—
440C2×C32—24no—
441C2×C32—24no—
442C2×C32—24no—
443C2×C32—24no—
444C2×C32—24no—
445C2×C32—24no—
446C2×C32—24no—
447C2×C32—416no—
448C2×C32—48no—
449C2×C32—42no—
450C2×C32#88211yescharacteristic
451C2×C32#88314yes—
452C2×C32#88311yescharacteristic
453C2×C32#88414yes—
454C2×C32#88414yes—
455C2×C32#88514yes—
456C2×C32#88511yescharacteristic
457C2×C32#88611yescharacteristic
458C2×C32#92914yes—
459C2×C32#93011yescharacteristic
460C2×C32#93714yes—
461C2×C32#93811yescharacteristic
462C2×C32#93914yes—
463C2×C32#93911yescharacteristic
464C3⋊D3—1216no—
465C3⋊D3—128no—
466C3⋊D3—122no—
467C24—271no—
468C24D3×C3⋊D311yescharacteristic
469C2×D4—542no—
470C2×D4—542no—
471C2×D4—542no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—91no—
475C22×C4—91no—
476C22×C4—271no—
477C4⋊C4—542no—
478C22⋊C4—62no—
479C22⋊C4—182no—
480C22⋊C4—542no—
481C22⋊C4—542no—
482C22×C3—216no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—28no—
487C22×C3—28no—
488C22×C3—28no—
489C22×C3—24no—
490C22×C3—24no—
491C22×C3—24no—
492C22×C3—24no—
493C22×C3—24no—
494C22×C3—24no—
495C22×C3—24no—
496C22×C3—48no—
497C22×C3—48no—
498C22×C3—48no—
499C22×C3—48no—
500C22×C3—48no—
501C22×C3—48no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—44no—
507C22×C3—44no—
508C22×C3—44no—
509C22×C3—44no—
510C22×C3#380414yes—
511C22×C3#380414yes—
512C22×C3#380414yes—
513C22×C3C2×#62614yes—
514C22×C3D3×#6714yes—
515C22×C3#381114yes—
516C22×C3#381114yes—
517C22×C3#382811yescharacteristic
518C22×C3#382811yescharacteristic
519C22×C3D4×C3⋊D311yescharacteristic
520C22×C3D4×C3⋊D311yescharacteristic
521C22×C3D4×C3⋊D311yescharacteristic
522C22×C3#383011yescharacteristic
523C22×C3C2×#63811yescharacteristic
524C2×D3—1816no—
525C2×D3—182no—
526C2×D3—182no—
527C2×D3—182no—
528C2×D3—188no—
529C2×D3—188no—
530C2×D3—188no—
531C2×D3—368no—
532C2×D3—368no—
533C2×D3—368no—
534C2×D3—368no—
535C2×D3—368no—
536C2×D3—368no—
537C2×D3—361no—
538C2×D3—361no—
539C2×D3—361no—
540C2×D3—361no—
541C2×D3—364no—
542C2×D3—364no—
543C2×D3—364no—
544C2×D3—364no—
545C3×C4—124no—
546C3×C4—124no—
547C3×C4—182no—
548C3×C4—361no—
549Dic3—41no—
550Dic3—41no—
551Dic3—68no—
552Dic3—124no—
553Dic3—1816no—
554Dic3—182no—
555Dic3—188no—
556C32—24no—
557C32#495914yescomplemented
558C32#496111yescharacteristic, complemented
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—272no—
566C23—541no—
567C23—541no—
568C23—541no—
569C23—541no—
570C23—541no—
571C23—541no—
572C23#629112yes—
573C23C2×D3×C3⋊D311yescharacteristic
574D4—548no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—92no—
582C2×C4—181no—
583C2×C4—181no—
584C2×C4—181no—
585C2×C4—181no—
586C2×C4—181no—
587C2×C4—272no—
588C2×C4—541no—
589C2×C4—541no—
590C2×C3—24no—
591C2×C3—24no—
592C2×C3—24no—
593C2×C3—24no—
594C2×C3—24no—
595C2×C3—24no—
596C2×C3—24no—
597C2×C3—416no—
598C2×C3—42no—
599C2×C3—48no—
600C2×C3#6395314yes—
601C2×C3#6395814yes—
602C2×C3#6396614yes—
603C2×C3#6397114yes—
604C2×C3#6397514yes—
605C2×C3#6397814yes—
606C2×C3#6398114yes—
607C2×C3#6409211yescharacteristic
608C2×C3#6409311yescharacteristic
609C2×C3#6409311yescharacteristic
610C2×C3#6409411yescharacteristic
611C2×C3#6413811yescharacteristic
612C2×C3#6414611yescharacteristic
613C2×C3#6414811yescharacteristic
614D3—3616no—
615D3—362no—
616D3—368no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—542no—
625C22—542no—
626C22—542no—
627C22—1081no—
628C22—1081no—
629C22—1081no—
630C22—1081no—
631C22#8902811yescharacteristic
632C22#8903111yescharacteristic
633C22#8903111yescharacteristic
634C22C2×#629111yescharacteristic
635C22C3⋊D3×#6711yescharacteristic
636C22#8903911yescharacteristic
637C22#8903911yescharacteristic
638C4—121no—
639C4—121no—
640C4—182no—
641C4—361no—
642C4—542no—
643C3—24no—
644C3#1059171214yescomplemented
645C3#1059265811yescharacteristic, complemented
646C2—42no—
647C2—1082no—
648C2#1172359211yescharacteristic
649C2#1172361211yescharacteristic
650C2#1172362311yescharacteristic
651C2#1172362411yescharacteristic
652C2#1172362811yescharacteristic
653C2#1172363511yescharacteristic
654C2#1172363811yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutator, socle, cosocleC23×C33C23
FittingC24×C33C22
prime radicalC22×#6312C2
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×C3×#3757C2semidirect
3. C2×C32×#217C2semidirect
4. C2×#88807C2semidirect
5. C2×#88804C2semidirect
6. C2×#88804C2semidirect
7. #11722061C2semidirect
29. C2×#3757D3semidirect
34. C2×C3×#217D3semidirect
64. C2×C3×C3⋊Dic3D4semidirect
141. C2×#217C3⋊D3semidirect
218. C2×C3⋊Dic3#67semidirect
358. C33#394semidirect
557. C32#4959semidirect
558. C32#4961semidirect
644. C3#10591712semidirect
645. C3#10592658semidirect

Maximal subgroups

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

subgroupindex
C22×#63122
C2×C3×#37572
C2×C32×#2172
C2×#888072
C2×#888042
#117220612
#105926583
#105917123