Finite Group Databasebeta

← #13161457

Subgroups — #13161457

Order 1728.

2660 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6312C211yesmaximal, characteristic
2C22×#6289C211yesmaximal, characteristic, complemented
3C2×C3×#3757C211yesmaximal, characteristic, complemented
4C2×C32×#217C211yesmaximal, characteristic, complemented
5C2×#88804C211yesmaximal, characteristic, complemented
6#11722061C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592655—31nomaximal
9#10591722—34nomaximal
10C24×C33C2211yescharacteristic
11C23×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×#6312—22no—
13C2×#6312—22no—
14C2×#6312C414yescomplemented
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20C2×#6289—22no—
21C2×#6289—22no—
22C2×#6289—22no—
23C3×#3757—22no—
24C32×#217—22no—
25#88804—22no—
26C22×#638—31no—
27C22×#638—34no—
28C22×#638—64no—
29C22×C4×C3⋊D3—31no—
30C22×#624—34no—
31C2×C32×C22⋊C4—31no—
32C2×#3757D311yescharacteristic, complemented
33C2×#3752—31no—
34C2×C3×#217—34no—
35C2×C3×#217D314yescomplemented
36C2×#3722—34no—
37#63658—31no—
38#63658—31no—
39#63579—34no—
40#63579—34no—
41C23×C33—21no—
42C23×C33—21no—
43C23×C33—21no—
44C23×C33—21no—
45C23×C33—21no—
46C23×C33—21no—
47C23×C33C2×C412yescomplemented
48C23×C33C2311yescharacteristic
49C22×C3⋊(C3⋊D3)—21no—
50C22×C3⋊(C3⋊D3)—21no—
51C22×C3⋊(C3⋊D3)—21no—
52C22×C3⋊(C3⋊D3)—21no—
53C22×C3⋊(C3⋊D3)C2×C412yes—
54C22×C3⋊(C3⋊D3)D412yes—
55C22×C3⋊(C3⋊D3)D412yes—
56#6312—28no—
57C2×C3⋊(C3⋊Dic3)—21no—
58C2×C3⋊(C3⋊Dic3)—21no—
59C2×C3⋊(C3⋊Dic3)C2×C412yescomplemented
60C2×C3×C3⋊Dic3—21no—
61C2×C3×C3⋊Dic3—21no—
62C2×C3×C3⋊Dic3—21no—
63C2×C3×C3⋊Dic3—21no—
64C2×C3×C3⋊Dic3—21no—
65C2×C3×C3⋊Dic3D412yescomplemented
66C2×C32×Dic3—21no—
67C2×C32×Dic3—21no—
68C2×C32×Dic3—21no—
69C2×C32×Dic3—21no—
70C2×C32×Dic3—21no—
71C2×C32×Dic3D412yescomplemented
72#6289—28no—
73#4956—94no—
74#4956—91no—
75C24×C32—24no—
76C24×C32C2×D314yes—
77C24×C32C2×D311yescharacteristic
78C23×C3⋊D3—31no—
79C23×C3⋊D3—34no—
80C23×C3⋊D3—64no—
81C22×C32×C4—31no—
82C22×C32×C4—31no—
83C2×#638—34no—
84C2×#638—316no—
85C2×#638—62no—
86C2×#638—62no—
87C2×#638—68no—
88C2×#638—68no—
89C2×#638—616no—
90C2×#638—616no—
91C2×#638—616no—
92C22×C3⋊Dic3—31no—
93C22×C3⋊Dic3—34no—
94C22×C3⋊Dic3—64no—
95C22×C3⋊Dic3C2×D311yescharacteristic
96C22×C3⋊Dic3C2×D311yescharacteristic
97C2×C4×C3⋊D3—62no—
98C2×C4×C3⋊D3—62no—
99C2×C4×C3⋊D3—62no—
100C22×C3×Dic3—34no—
101C22×C3×Dic3—34no—
102C22×C3×Dic3C2×D314yes—
103C22×C3×Dic3C2×D314yes—
104C2×#624—68no—
105C2×#624—68no—
106C2×#624—68no—
107C32×C22⋊C4—62no—
108#3757—22no—
109#3752—62no—
110C3×#217—28no—
111C3×#217—68no—
112#3722—68no—
113C22×C33—22no—
114C22×C33—22no—
115C22×C33—22no—
116C22×C33—41no—
117C22×C33—41no—
118C22×C33—41no—
119C22×C33—41no—
120C22×C33C22×C411yescharacteristic
121C22×C33C2×D411yescharacteristic
122C22×C33C2×D411yescharacteristic
123C22×C33C2×D411yescharacteristic
124C22×C33C2×D411yescharacteristic
125C22×C33#4111yescharacteristic
126C22×C33#4111yescharacteristic
127C2×C3⋊(C3⋊D3)—22no—
128C2×C3⋊(C3⋊D3)—22no—
129C2×C3⋊(C3⋊D3)—22no—
130C2×C3⋊(C3⋊D3)—22no—
131C2×C3⋊(C3⋊D3)—22no—
132C2×C3⋊(C3⋊D3)—22no—
133C2×C3⋊(C3⋊D3)C22⋊C414yescomplemented
134C3⋊(C3⋊Dic3)—22no—
135C3×C3⋊Dic3—22no—
136C3×C3⋊Dic3—41no—
137C32×Dic3—22no—
138C32×Dic3—41no—
139C22×#67—94no—
140C22×#67—91no—
141C22×#67—184no—
142C22×C4×D3—94no—
143C22×C4×D3—91no—
144C2×C3×C22⋊C4—34no—
145C2×C3×C22⋊C4—91no—
146C2×#217—34no—
147C2×#217C3⋊D311yescharacteristic, complemented
148C2×(C2×D3)⋊C4—94no—
149C2×(C2×D3)⋊C4—91no—
150C3⋊#95—94no—
151C3⋊#95—94no—
152C3⋊#95—91no—
153C3⋊#95—91no—
154C23×C32—28no—
155C23×C32—28no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32—24no—
159C23×C32—24no—
160C23×C32—21no—
161C23×C32—21no—
162C23×C32—21no—
163C23×C32—21no—
164C23×C32—28no—
165C23×C32—24no—
166C23×C32—24no—
167C23×C32—24no—
168C23×C32—24no—
169C23×C32—24no—
170C23×C32—44no—
171C23×C32—44no—
172C23×C32—44no—
173C23×C32—44no—
174C23×C32C4×D318yescomplemented
175C23×C32C4×D312yescomplemented
176C23×C32C22×D314yes—
177C23×C32C22×D311yescharacteristic
178C22×C3⋊D3—32no—
179C22×C3⋊D3—32no—
180C22×C3⋊D3—32no—
181C22×C3⋊D3—38no—
182C22×C3⋊D3—38no—
183C22×C3⋊D3—38no—
184C22×C3⋊D3—68no—
185C22×C3⋊D3—68no—
186C22×C3⋊D3—68no—
187C22×C3⋊D3—61no—
188C22×C3⋊D3—61no—
189C22×C3⋊D3—61no—
190C22×C3⋊D3—61no—
191C22×C3⋊D3—64no—
192C22×C3⋊D3—64no—
193C22×C3⋊D3—64no—
194C22×C3⋊D3—64no—
195C22×C3⋊D3—124no—
196C22×C3⋊D3—124no—
197C22×C3⋊D3—124no—
198C22×C3⋊D3—124no—
199C2×C32×C4—32no—
200C2×C32×C4—61no—
201C2×C32×C4—61no—
202C2×C32×C4—61no—
203C2×C32×C4—61no—
204C2×C32×C4—61no—
205#638—68no—
206#638—632no—
207#638—664no—
208C2×C3⋊Dic3—21no—
209C2×C3⋊Dic3—21no—
210C2×C3⋊Dic3—21no—
211C2×C3⋊Dic3—21no—
212C2×C3⋊Dic3—21no—
213C2×C3⋊Dic3—32no—
214C2×C3⋊Dic3—38no—
215C2×C3⋊Dic3—68no—
216C2×C3⋊Dic3—68no—
217C2×C3⋊Dic3—68no—
218C2×C3⋊Dic3—64no—
219C2×C3⋊Dic3—64no—
220C2×C3⋊Dic3—61no—
221C2×C3⋊Dic3—61no—
222C2×C3⋊Dic3#6712yescomplemented
223C4×C3⋊D3—68no—
224C2×C3×Dic3—24no—
225C2×C3×Dic3—24no—
226C2×C3×Dic3—24no—
227C2×C3×Dic3—24no—
228C2×C3×Dic3—24no—
229C2×C3×Dic3—38no—
230C2×C3×Dic3—64no—
231C2×C3×Dic3—64no—
232C2×C3×Dic3—64no—
233C2×C3×Dic3—64no—
234C2×C3×Dic3—64no—
235C2×C3×Dic3#6718yescomplemented
236#624—632no—
237#386—271noSylow
238C2×C33—42no—
239C2×C33C2×C22⋊C411yescharacteristic
240C2×C33C4×D411yescharacteristic
241C2×C33C4×D411yescharacteristic
242C2×C33#12011yescharacteristic
243C2×C33#12111yescharacteristic
244C2×C33#12111yescharacteristic
245C2×C33#12311yescharacteristic
246C3⋊(C3⋊D3)—24no—
247C24×C3—24no—
248C24×C3D3214yes—
249C24×C3C2×C3⋊D311yescharacteristic
250C23×D3—94no—
251C23×D3—91no—
252C23×D3—184no—
253C22×C3×C4—34no—
254C22×C3×C4—34no—
255C22×C3×C4—91no—
256C22×C3×C4—91no—
257C2×#67—916no—
258C2×#67—94no—
259C2×#67—1816no—
260C2×#67—188no—
261C2×#67—188no—
262C2×#67—182no—
263C2×#67—182no—
264C2×#67—1816no—
265C2×#67—1816no—
266C22×Dic3—34no—
267C22×Dic3—34no—
268C22×Dic3—94no—
269C22×Dic3—91no—
270C22×Dic3—184no—
271C22×Dic3C2×C3⋊D311yescharacteristic
272C22×Dic3C2×C3⋊D311yescharacteristic
273C2×C4×D3—182no—
274C2×C4×D3—182no—
275C2×C4×D3—188no—
276C2×C4×D3—188no—
277C2×C4×D3—188no—
278C2×C4×D3—182no—
279C3×C22⋊C4—68no—
280C3×C22⋊C4—182no—
281#217—22no—
282#217—68no—
283(C2×D3)⋊C4—182no—
284(C2×D3)⋊C4—188no—
285C22×C32—216no—
286C22×C32—28no—
287C22×C32—22no—
288C22×C32—22no—
289C22×C32—22no—
290C22×C32—24no—
291C22×C32—24no—
292C22×C32—24no—
293C22×C32—24no—
294C22×C32—24no—
295C22×C32—24no—
296C22×C32—24no—
297C22×C32—28no—
298C22×C32—28no—
299C22×C32—48no—
300C22×C32—48no—
301C22×C32—48no—
302C22×C32—48no—
303C22×C32—48no—
304C22×C32—48no—
305C22×C32—44no—
306C22×C32—41no—
307C22×C32—44no—
308C22×C32—41no—
309C22×C32—44no—
310C22×C32—41no—
311C22×C32—44no—
312C22×C32—41no—
313C22×C32C2×C4×D314yes—
314C22×C32C2×C4×D311yescharacteristic
315C22×C32#23514yes—
316C22×C32#23511yescharacteristic
317C22×C32D3×D414yes—
318C22×C32D3×D414yes—
319C22×C32D3×D414yes—
320C22×C32D3×D411yescharacteristic
321C22×C32D3×D411yescharacteristic
322C22×C32D3×D411yescharacteristic
323C22×C32#23714yes—
324C22×C32#23711yescharacteristic
325C22×C32C2×#6714yes—
326C22×C32C2×#6711yescharacteristic
327C2×C3⋊D3—34no—
328C2×C3⋊D3—316no—
329C2×C3⋊D3—616no—
330C2×C3⋊D3—62no—
331C2×C3⋊D3—62no—
332C2×C3⋊D3—62no—
333C2×C3⋊D3—62no—
334C2×C3⋊D3—62no—
335C2×C3⋊D3—62no—
336C2×C3⋊D3—68no—
337C2×C3⋊D3—68no—
338C2×C3⋊D3—68no—
339C2×C3⋊D3—68no—
340C2×C3⋊D3—68no—
341C2×C3⋊D3—68no—
342C2×C3⋊D3—128no—
343C2×C3⋊D3—128no—
344C2×C3⋊D3—128no—
345C2×C3⋊D3—128no—
346C2×C3⋊D3—128no—
347C2×C3⋊D3—128no—
348C32×C4—62no—
349C32×C4—121no—
350C3⋊Dic3—22no—
351C3⋊Dic3—41no—
352C3⋊Dic3—616no—
353C3⋊Dic3—68no—
354C3⋊Dic3—62no—
355C3×Dic3—28no—
356C3×Dic3—44no—
357C3×Dic3—68no—
358C3×Dic3—124no—
359C22×D4—271no—
360C23×C4—271no—
361C2×C22⋊C4—31no—
362C2×C22⋊C4—91no—
363C2×C22⋊C4—271no—
364#95—271no—
365#95—271no—
366C33#38611yescharacteristic, complemented, Sylow
367C23×C3—28no—
368C23×C3—28no—
369C23×C3—24no—
370C23×C3—24no—
371C23×C3—24no—
372C23×C3—24no—
373C23×C3—24no—
374C23×C3—24no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—28no—
382C23×C3—24no—
383C23×C3—44no—
384C23×C3—44no—
385C23×C3—44no—
386C23×C3—44no—
387C23×C3#62418yescomplemented
388C23×C3C4×C3⋊D312yescomplemented
389C23×C3C2×D3214yes—
390C23×C3C22×C3⋊D311yescharacteristic
391C22×D3—98no—
392C22×D3—98no—
393C22×D3—98no—
394C22×D3—92no—
395C22×D3—92no—
396C22×D3—92no—
397C22×D3—188no—
398C22×D3—188no—
399C22×D3—188no—
400C22×D3—184no—
401C22×D3—184no—
402C22×D3—184no—
403C22×D3—184no—
404C22×D3—181no—
405C22×D3—181no—
406C22×D3—181no—
407C22×D3—181no—
408C22×D3—364no—
409C22×D3—364no—
410C22×D3—364no—
411C22×D3—364no—
412C2×C3×C4—38no—
413C2×C3×C4—64no—
414C2×C3×C4—64no—
415C2×C3×C4—64no—
416C2×C3×C4—64no—
417C2×C3×C4—64no—
418C2×C3×C4—92no—
419C2×C3×C4—181no—
420C2×C3×C4—181no—
421C2×C3×C4—181no—
422C2×C3×C4—181no—
423C2×C3×C4—181no—
424#67—188no—
425#67—1832no—
426#67—1864no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—38no—
433C2×Dic3—64no—
434C2×Dic3—64no—
435C2×Dic3—64no—
436C2×Dic3—64no—
437C2×Dic3—64no—
438C2×Dic3—98no—
439C2×Dic3—92no—
440C2×Dic3—188no—
441C2×Dic3—188no—
442C2×Dic3—188no—
443C2×Dic3—184no—
444C2×Dic3—184no—
445C2×Dic3—181no—
446C2×Dic3—181no—
447C2×Dic3#63812yescomplemented
448C4×D3—188no—
449C4×D3—1832no—
450C2×C32—24no—
451C2×C32—24no—
452C2×C32—24no—
453C2×C32—24no—
454C2×C32—24no—
455C2×C32—24no—
456C2×C32—24no—
457C2×C32—416no—
458C2×C32—48no—
459C2×C32—42no—
460C2×C32D3×C22⋊C414yes—
461C2×C32D3×C22⋊C411yescharacteristic
462C2×C32#67⋊C414yes—
463C2×C32#67⋊C411yescharacteristic
464C2×C32#88314yes—
465C2×C32#88311yescharacteristic
466C2×C32#88414yes—
467C2×C32#88411yescharacteristic
468C2×C32C4×#6714yes—
469C2×C32C4×#6711yescharacteristic
470C2×C32#93714yes—
471C2×C32#93711yescharacteristic
472C2×C32#93814yes—
473C2×C32#93811yescharacteristic
474C3⋊D3—64no—
475C3⋊D3—616no—
476C3⋊D3—1216no—
477C24—271no—
478C24D3×C3⋊D311yescharacteristic
479C2×D4—274no—
480C2×D4—542no—
481C2×D4—542no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—91no—
485C22×C4—91no—
486C22×C4—271no—
487C22×C4—542no—
488C22×C4—542no—
489C22×C4—542no—
490C22⋊C4—62no—
491C22⋊C4—182no—
492C22⋊C4—542no—
493C22×C3—216no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—28no—
498C22×C3—28no—
499C22×C3—28no—
500C22×C3—24no—
501C22×C3—24no—
502C22×C3—24no—
503C22×C3—24no—
504C22×C3—24no—
505C22×C3—24no—
506C22×C3—24no—
507C22×C3—48no—
508C22×C3—48no—
509C22×C3—48no—
510C22×C3—48no—
511C22×C3—48no—
512C22×C3—48no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—44no—
518C22×C3—44no—
519C22×C3—44no—
520C22×C3—44no—
521C22×C3#380414yes—
522C22×C3#380414yes—
523C22×C3C2×#62414yes—
524C22×C3D3×#6714yes—
525C22×C3D3×#6714yes—
526C22×C3#381114yes—
527C22×C3#381114yes—
528C22×C3C2×C4×C3⋊D311yescharacteristic
529C22×C3#382811yescharacteristic
530C22×C3D4×C3⋊D311yescharacteristic
531C22×C3D4×C3⋊D311yescharacteristic
532C22×C3D4×C3⋊D311yescharacteristic
533C22×C3#383011yescharacteristic
534C22×C3C2×#63811yescharacteristic
535C2×D3—94no—
536C2×D3—916no—
537C2×D3—1816no—
538C2×D3—182no—
539C2×D3—182no—
540C2×D3—182no—
541C2×D3—182no—
542C2×D3—182no—
543C2×D3—182no—
544C2×D3—188no—
545C2×D3—188no—
546C2×D3—188no—
547C2×D3—188no—
548C2×D3—188no—
549C2×D3—188no—
550C2×D3—368no—
551C2×D3—368no—
552C2×D3—368no—
553C2×D3—368no—
554C2×D3—368no—
555C2×D3—368no—
556C3×C4—68no—
557C3×C4—124no—
558C3×C4—182no—
559C3×C4—361no—
560Dic3—22no—
561Dic3—41no—
562Dic3—68no—
563Dic3—124no—
564Dic3—1816no—
565Dic3—182no—
566Dic3—188no—
567C32—24no—
568C32#495614yescomplemented
569C32#495611yescharacteristic, complemented
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—272no—
577C23—272no—
578C23—272no—
579C23—541no—
580C23—541no—
581C23—541no—
582C23—541no—
583C23#628912yescomplemented
584C23C2×D3×C3⋊D311yescharacteristic
585D4—548no—
586C2×C4—32no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—92no—
593C2×C4—181no—
594C2×C4—181no—
595C2×C4—181no—
596C2×C4—181no—
597C2×C4—181no—
598C2×C4—272no—
599C2×C4—548no—
600C2×C4—541no—
601C2×C4—541no—
602C2×C3—24no—
603C2×C3—24no—
604C2×C3—24no—
605C2×C3—24no—
606C2×C3—24no—
607C2×C3—24no—
608C2×C3—24no—
609C2×C3—416no—
610C2×C3—42no—
611C2×C3—48no—
612C2×C3#6395214yes—
613C2×C3#6395214yes—
614C2×C3#6395814yes—
615C2×C3#6395814yes—
616C2×C3#6397414yes—
617C2×C3#6397514yes—
618C2×C3#6398314yes—
619C2×C3C22⋊C4×C3⋊D311yescharacteristic
620C2×C3#6409011yescharacteristic
621C2×C3#6409211yescharacteristic
622C2×C3#6409311yescharacteristic
623C2×C3C4×#63811yescharacteristic
624C2×C3#6414611yescharacteristic
625C2×C3#6414711yescharacteristic
626D3—184no—
627D3—1816no—
628D3—3616no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—274no—
637C22—542no—
638C22—542no—
639C22—542no—
640C22—542no—
641C22—542no—
642C22—542no—
643C22#8902811yescharacteristic
644C22#8903111yescharacteristic
645C22C2×#628911yescharacteristic
646C22D3×#63811yescharacteristic
647C22C3⋊D3×#6711yescharacteristic
648C22#8903911yescharacteristic
649C22#8903911yescharacteristic
650C4—62no—
651C4—121no—
652C4—182no—
653C4—361no—
654C4—542no—
655C3—24no—
656C3#1059172214yescomplemented
657C3#1059265511yescharacteristic, complemented
658C2—42no—
659C2—544no—
660C2#1172359911yescharacteristic
661C2#1172361811yescharacteristic
662C2#1172362211yescharacteristic
663C2#1172362311yescharacteristic
664C2#1172362711yescharacteristic
665C2#1172362811yescharacteristic
666C2#1172364311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
FittingC24×C33C22
socle, cosocleC23×C33C23
prime radicalC22×#6312C2
hypercenterC24D3×C3⋊D3
hypocenterC33#386
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×#6289C2semidirect
3. C2×C3×#3757C2semidirect
4. C2×C32×#217C2semidirect
5. C2×#88804C2semidirect
6. #11722061C2semidirect
7. #11722061C2semidirect
14. C2×#6312C4semidirect
32. C2×#3757D3semidirect
35. C2×C3×#217D3semidirect
47. C23×C33C2×C4semidirect
59. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
65. C2×C3×C3⋊Dic3D4semidirect
71. C2×C32×Dic3D4semidirect
133. C2×C3⋊(C3⋊D3)C22⋊C4semidirect
147. C2×#217C3⋊D3semidirect
174. C23×C32C4×D3semidirect
175. C23×C32C4×D3semidirect
222. C2×C3⋊Dic3#67semidirect
235. C2×C3×Dic3#67semidirect
366. C33#386semidirect
387. C23×C3#624semidirect
388. C23×C3C4×C3⋊D3semidirect
447. C2×Dic3#638semidirect
568. C32#4956semidirect
569. C32#4956semidirect
583. C23#6289semidirect
656. C3#10591722semidirect
657. C3#10592655semidirect

Maximal subgroups

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

subgroupindex
C22×#63122
C22×#62892
C2×C3×#37572
C2×C32×#2172
C2×#888042
#117220612
#105926553
#105917223