Finite Group Databasebeta

← #13141139

Subgroups — #13141139

Order 1728.

1094 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×#6233C211yesmaximal, characteristic
2C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
3C2×C9×#217C211yesmaximal, characteristic, complemented
4C2×C3×#3676C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic, complemented
6#11720331C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591722—31nomaximal
9#10588636—31nomaximal
10C24×C3×C9C2211yescharacteristic
11C23×C3⋊D9C2211yescharacteristic
12C2×#6233—22no—
13C2×#6233—22no—
14C2×#6233C414yescomplemented
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×(C3⋊D9)⋊C4—22no—
21C2×(C3⋊D9)⋊C4—22no—
22C2×(C3⋊D9)⋊C4—22no—
23C9×#217—22no—
24C3×#3676—22no—
25#88445—22no—
26C22×#638—31no—
27C22×#624—31no—
28C2×C3×#217—31no—
29C2×C3×#217D311yescharacteristic
30C2×#3722—31no—
31C22×#611—31no—
32C22×#611—61no—
33C22×C4×D9—31no—
34#63579—31no—
35#63579—31no—
36C2×C9×C22⋊C4—31no—
37C2×#3676D311yescharacteristic, complemented
38C2×(C2×D9)⋊C4—31no—
39C9⋊#95—31no—
40C9⋊#95—31no—
41C23×C3×C9—21no—
42C23×C3×C9—21no—
43C23×C3×C9—21no—
44C23×C3×C9—21no—
45C23×C3×C9—21no—
46C23×C3×C9—21no—
47C23×C3×C9C2×C412yescomplemented
48C23×C3×C9C2311yescharacteristic
49C22×C3⋊D9—21no—
50C22×C3⋊D9—21no—
51C22×C3⋊D9—21no—
52C22×C3⋊D9—21no—
53C22×C3⋊D9C2×C412yes—
54C22×C3⋊D9D412yes—
55C22×C3⋊D9D412yes—
56#6233—28no—
57C2×C3⋊Dic9—21no—
58C2×C3⋊Dic9—21no—
59C2×C3⋊Dic9C2×C412yescomplemented
60C2×C9×Dic3—21no—
61C2×C9×Dic3—21no—
62C2×C9×Dic3—21no—
63C2×C9×Dic3—21no—
64C2×C9×Dic3—21no—
65C2×C9×Dic3D412yescomplemented
66C2×C3×Dic9—21no—
67C2×C3×Dic9—21no—
68C2×C3×Dic9—21no—
69C2×C3×Dic9—21no—
70C2×C3×Dic9—21no—
71C2×C3×Dic9D412yescomplemented
72(C3⋊D9)⋊C4—28no—
73#4956—91no—
74#4956—91no—
75C24×C32C2×D311yescharacteristic
76C23×C3⋊D3—31no—
77C2×#638—34no—
78C2×#638—62no—
79C2×#638—62no—
80C22×C3⋊Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C22×C3×Dic3C2×D311yescharacteristic
84C22×C3×Dic3C2×D311yescharacteristic
85C2×#624—62no—
86C2×#624—62no—
87C2×#624—62no—
88C24×C9—21no—
89C24×C9C2×D311yescharacteristic
90C23×D9—31no—
91C23×D9—61no—
92C3×#217—22no—
93C3×#217—62no—
94#3722—62no—
95C22×C4×C9—31no—
96C22×C4×C9—31no—
97C2×#611—34no—
98C2×#611—62no—
99C2×#611—62no—
100C2×#611—64no—
101C2×#611—64no—
102C2×#611—64no—
103C22×Dic9—31no—
104C22×Dic9—61no—
105C22×Dic9C2×D311yescharacteristic
106C22×Dic9C2×D311yescharacteristic
107C2×C4×D9—62no—
108C2×C4×D9—62no—
109C2×C4×D9—62no—
110C9×C22⋊C4—62no—
111#3676—22no—
112(C2×D9)⋊C4—62no—
113C22×C3×C9—22no—
114C22×C3×C9—22no—
115C22×C3×C9—22no—
116C22×C3×C9—41no—
117C22×C3×C9—41no—
118C22×C3×C9—41no—
119C22×C3×C9—41no—
120C22×C3×C9C22×C411yescharacteristic
121C22×C3×C9C2×D411yescharacteristic
122C22×C3×C9C2×D411yescharacteristic
123C22×C3×C9C2×D411yescharacteristic
124C22×C3×C9C2×D411yescharacteristic
125C22×C3×C9#4111yescharacteristic
126C22×C3×C9#4111yescharacteristic
127C2×C3⋊D9—22no—
128C2×C3⋊D9—22no—
129C2×C3⋊D9—22no—
130C2×C3⋊D9—22no—
131C2×C3⋊D9—22no—
132C2×C3⋊D9—22no—
133C2×C3⋊D9C22⋊C414yescomplemented
134C3⋊Dic9—22no—
135C9×Dic3—22no—
136C9×Dic3—41no—
137C3×Dic9—22no—
138C3×Dic9—41no—
139C22×#67—91no—
140C22×#67—91no—
141C22×#67—181no—
142C22×C4×D3—91no—
143C22×C4×D3—91no—
144C2×C3×C22⋊C4—31no—
145C2×C3×C22⋊C4—91no—
146C2×#217—31no—
147C2×#217D911yescharacteristic, complemented
148C2×(C2×D3)⋊C4—91no—
149C2×(C2×D3)⋊C4—91no—
150C3⋊#95—91no—
151C3⋊#95—91no—
152C3⋊#95—91no—
153C3⋊#95—91no—
154C23×C32—21no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32—21no—
159C23×C32—21no—
160C23×C32C4×D312yes—
161C23×C32C22×D311yescharacteristic
162C22×C3⋊D3—32no—
163C22×C3⋊D3—32no—
164C22×C3⋊D3—32no—
165C22×C3⋊D3—61no—
166C22×C3⋊D3—61no—
167C22×C3⋊D3—61no—
168C22×C3⋊D3—61no—
169#638—68no—
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3#6712yes—
185#624—68no—
186C23×C9—22no—
187C23×C9—22no—
188C23×C9—21no—
189C23×C9—21no—
190C23×C9—21no—
191C23×C9—21no—
192C23×C9—21no—
193C23×C9—21no—
194C23×C9—22no—
195C23×C9—21no—
196C23×C9—41no—
197C23×C9—41no—
198C23×C9—41no—
199C23×C9—41no—
200C23×C9C4×D312yescomplemented
201C23×C9C22×D311yescharacteristic
202C22×D9—32no—
203C22×D9—32no—
204C22×D9—32no—
205C22×D9—62no—
206C22×D9—62no—
207C22×D9—62no—
208C22×D9—61no—
209C22×D9—61no—
210C22×D9—61no—
211C22×D9—61no—
212C22×D9—121no—
213C22×D9—121no—
214C22×D9—121no—
215C22×D9—121no—
216C2×C4×C9—32no—
217C2×C4×C9—61no—
218C2×C4×C9—61no—
219C2×C4×C9—61no—
220C2×C4×C9—61no—
221C2×C4×C9—61no—
222#611—68no—
223#611—616no—
224C2×Dic9—21no—
225C2×Dic9—21no—
226C2×Dic9—21no—
227C2×Dic9—21no—
228C2×Dic9—21no—
229C2×Dic9—32no—
230C2×Dic9—62no—
231C2×Dic9—62no—
232C2×Dic9—62no—
233C2×Dic9—61no—
234C2×Dic9—61no—
235C2×Dic9#6712yescomplemented
236C4×D9—68no—
237#386—271noSylow
238C2×C3×C9—42no—
239C2×C3×C9C2×C22⋊C411yescharacteristic
240C2×C3×C9C4×D411yescharacteristic
241C2×C3×C9C4×D411yescharacteristic
242C2×C3×C9#12011yescharacteristic
243C2×C3×C9#12111yescharacteristic
244C2×C3×C9#12111yescharacteristic
245C2×C3×C9#12311yescharacteristic
246C3⋊D9—24no—
247C24×C3—21no—
248C24×C3C2×D911yescharacteristic
249C24×C3D3211yescharacteristic
250C23×D3—91no—
251C23×D3—91no—
252C23×D3—181no—
253C22×C3×C4—31no—
254C22×C3×C4—31no—
255C22×C3×C4—91no—
256C22×C3×C4—91no—
257C2×#67—94no—
258C2×#67—94no—
259C2×#67—184no—
260C2×#67—182no—
261C2×#67—182no—
262C2×#67—182no—
263C2×#67—182no—
264C2×#67—184no—
265C2×#67—184no—
266C22×Dic3—31no—
267C22×Dic3—31no—
268C22×Dic3—91no—
269C22×Dic3—91no—
270C22×Dic3—181no—
271C22×Dic3C2×D911yescharacteristic
272C22×Dic3C2×D911yescharacteristic
273C2×C4×D3—182no—
274C2×C4×D3—182no—
275C2×C4×D3—182no—
276C2×C4×D3—182no—
277C2×C4×D3—182no—
278C2×C4×D3—182no—
279C3×C22⋊C4—62no—
280C3×C22⋊C4—182no—
281#217—22no—
282#217—62no—
283(C2×D3)⋊C4—182no—
284(C2×D3)⋊C4—182no—
285C22×C32—22no—
286C22×C32—22no—
287C22×C32—22no—
288C22×C32—41no—
289C22×C32—41no—
290C22×C32—41no—
291C22×C32—41no—
292C22×C32C2×C4×D311yescharacteristic
293C22×C32#23511yescharacteristic
294C22×C32D3×D411yescharacteristic
295C22×C32D3×D411yescharacteristic
296C22×C32D3×D411yescharacteristic
297C22×C32#23711yescharacteristic
298C22×C32C2×#6711yescharacteristic
299C2×C3⋊D3—34no—
300C2×C3⋊D3—62no—
301C2×C3⋊D3—62no—
302C2×C3⋊D3—62no—
303C2×C3⋊D3—62no—
304C2×C3⋊D3—62no—
305C2×C3⋊D3—62no—
306C3⋊Dic3—62no—
307C3×Dic3—22no—
308C3×Dic3—41no—
309C3×Dic3—62no—
310C3×Dic3—121no—
311C22×C9—24no—
312C22×C9—22no—
313C22×C9—22no—
314C22×C9—22no—
315C22×C9—21no—
316C22×C9—21no—
317C22×C9—21no—
318C22×C9—21no—
319C22×C9—21no—
320C22×C9—21no—
321C22×C9—21no—
322C22×C9—42no—
323C22×C9—42no—
324C22×C9—42no—
325C22×C9—42no—
326C22×C9—42no—
327C22×C9—42no—
328C22×C9—41no—
329C22×C9—41no—
330C22×C9—41no—
331C22×C9—41no—
332C22×C9C2×C4×D311yescharacteristic
333C22×C9#23511yescharacteristic
334C22×C9D3×D411yescharacteristic
335C22×C9D3×D411yescharacteristic
336C22×C9D3×D411yescharacteristic
337C22×C9#23711yescharacteristic
338C22×C9C2×#6711yescharacteristic
339C2×D9—34no—
340C2×D9—64no—
341C2×D9—62no—
342C2×D9—62no—
343C2×D9—62no—
344C2×D9—62no—
345C2×D9—62no—
346C2×D9—62no—
347C2×D9—122no—
348C2×D9—122no—
349C2×D9—122no—
350C2×D9—122no—
351C2×D9—122no—
352C2×D9—122no—
353C4×C9—62no—
354C4×C9—121no—
355Dic9—22no—
356Dic9—41no—
357Dic9—64no—
358Dic9—62no—
359C22×D4—271no—
360C23×C4—271no—
361C2×C22⋊C4—31no—
362C2×C22⋊C4—91no—
363C2×C22⋊C4—271no—
364#95—271no—
365#95—271no—
366C3×C9#38611yescharacteristic, complemented, Sylow
367C23×C3—22no—
368C23×C3—22no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—22no—
382C23×C3—21no—
383C23×C3—41no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3—41no—
387C23×C3C4×D912yescomplemented
388C23×C3C22×D911yescharacteristic
389C23×C3#62412yes—
390C23×C3C2×D3211yescharacteristic
391C22×D3—92no—
392C22×D3—92no—
393C22×D3—92no—
394C22×D3—92no—
395C22×D3—92no—
396C22×D3—92no—
397C22×D3—182no—
398C22×D3—182no—
399C22×D3—182no—
400C22×D3—181no—
401C22×D3—181no—
402C22×D3—181no—
403C22×D3—181no—
404C22×D3—181no—
405C22×D3—181no—
406C22×D3—181no—
407C22×D3—181no—
408C22×D3—361no—
409C22×D3—361no—
410C22×D3—361no—
411C22×D3—361no—
412C2×C3×C4—32no—
413C2×C3×C4—61no—
414C2×C3×C4—61no—
415C2×C3×C4—61no—
416C2×C3×C4—61no—
417C2×C3×C4—61no—
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—188no—
426#67—1816no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—32no—
433C2×Dic3—61no—
434C2×Dic3—61no—
435C2×Dic3—61no—
436C2×Dic3—61no—
437C2×Dic3—61no—
438C2×Dic3—92no—
439C2×Dic3—92no—
440C2×Dic3—182no—
441C2×Dic3—182no—
442C2×Dic3—182no—
443C2×Dic3—181no—
444C2×Dic3—181no—
445C2×Dic3—181no—
446C2×Dic3—181no—
447C2×Dic3#61112yescomplemented
448C4×D3—188no—
449C4×D3—188no—
450C2×C32—42no—
451C2×C32D3×C22⋊C411yescharacteristic
452C2×C32#67⋊C411yescharacteristic
453C2×C32#88311yescharacteristic
454C2×C32#88411yescharacteristic
455C2×C32C4×#6711yescharacteristic
456C2×C32#93711yescharacteristic
457C2×C32#93811yescharacteristic
458C3⋊D3—64no—
459C2×C9—21no—
460C2×C9—21no—
461C2×C9—21no—
462C2×C9—21no—
463C2×C9—21no—
464C2×C9—21no—
465C2×C9—21no—
466C2×C9—44no—
467C2×C9—42no—
468C2×C9D3×C22⋊C411yescharacteristic
469C2×C9#67⋊C411yescharacteristic
470C2×C9#88311yescharacteristic
471C2×C9#88411yescharacteristic
472C2×C9C4×#6711yescharacteristic
473C2×C9#93711yescharacteristic
474C2×C9#93811yescharacteristic
475D9—64no—
476D9—124no—
477C24—271no—
478C24D3×D911yescharacteristic
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—24no—
494C22×C3—22no—
495C22×C3—22no—
496C22×C3—22no—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—21no—
501C22×C3—21no—
502C22×C3—21no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—21no—
507C22×C3—42no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—41no—
514C22×C3—41no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—41no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3C2×C4×D911yescharacteristic
522C22×C3#369711yescharacteristic
523C22×C3D4×D911yescharacteristic
524C22×C3D4×D911yescharacteristic
525C22×C3D4×D911yescharacteristic
526C22×C3#369911yescharacteristic
527C22×C3C2×#61111yescharacteristic
528C22×C3#380411yescharacteristic
529C22×C3#380411yescharacteristic
530C22×C3C2×#62411yescharacteristic
531C22×C3D3×#6711yescharacteristic
532C22×C3D3×#6711yescharacteristic
533C22×C3#381111yescharacteristic
534C22×C3#381111yescharacteristic
535C2×D3—94no—
536C2×D3—94no—
537C2×D3—184no—
538C2×D3—182no—
539C2×D3—182no—
540C2×D3—182no—
541C2×D3—182no—
542C2×D3—182no—
543C2×D3—182no—
544C2×D3—182no—
545C2×D3—182no—
546C2×D3—182no—
547C2×D3—182no—
548C2×D3—182no—
549C2×D3—182no—
550C2×D3—362no—
551C2×D3—362no—
552C2×D3—362no—
553C2×D3—362no—
554C2×D3—362no—
555C2×D3—362no—
556C3×C4—62no—
557C3×C4—121no—
558C3×C4—182no—
559C3×C4—361no—
560Dic3—22no—
561Dic3—41no—
562Dic3—62no—
563Dic3—121no—
564Dic3—184no—
565Dic3—182no—
566Dic3—182no—
567C32#495611yescharacteristic
568C9—21no—
569C9#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(C3⋊D9)⋊C412yescomplemented
584C23C2×D3×D911yescharacteristic
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—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—21no—
608C2×C3—21no—
609C2×C3—44no—
610C2×C3—42no—
611C2×C3—42no—
612C2×C3C22⋊C4×D911yescharacteristic
613C2×C3#611⋊C411yescharacteristic
614C2×C3#6344511yescharacteristic
615C2×C3#6344611yescharacteristic
616C2×C3C4×#61111yescharacteristic
617C2×C3#6349911yescharacteristic
618C2×C3#6350011yescharacteristic
619C2×C3#6395211yescharacteristic
620C2×C3#6395211yescharacteristic
621C2×C3#6395811yescharacteristic
622C2×C3#6395811yescharacteristic
623C2×C3#6397411yescharacteristic
624C2×C3#6397511yescharacteristic
625C2×C3#6398311yescharacteristic
626D3—184no—
627D3—184no—
628D3—364no—
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#8865811yescharacteristic
644C22C2×(C3⋊D9)⋊C411yescharacteristic
645C22#8866211yescharacteristic
646C22D3×#61111yescharacteristic
647C22D9×#6711yescharacteristic
648C22#8866811yescharacteristic
649C22#8866811yescharacteristic
650C4—62no—
651C4—121no—
652C4—182no—
653C4—361no—
654C4—542no—
655C3—21no—
656C3#1058863611yescharacteristic, complemented
657C3#1059172211yescharacteristic
658C2—42no—
659C2—544no—
660C2#1172124011yescharacteristic
661C2#1172124111yescharacteristic
662C2#1172125611yescharacteristic
663C2#1172126211yescharacteristic
664C2#1172127411yescharacteristic
665C2#1172127511yescharacteristic
666C2#1172129211yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC22×#6233C2
cosocleC23×C3×C9C23
hypercenterC24D3×D9
hypocenterC3×C9#386
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×(C3⋊D9)⋊C4C2semidirect
3. C2×C9×#217C2semidirect
4. C2×C3×#3676C2semidirect
5. C2×#88445C2semidirect
6. #11720331C2semidirect
7. #11720331C2semidirect
14. C2×#6233C4semidirect
37. C2×#3676D3semidirect
47. C23×C3×C9C2×C4semidirect
59. C2×C3⋊Dic9C2×C4semidirect
65. C2×C9×Dic3D4semidirect
71. C2×C3×Dic9D4semidirect
133. C2×C3⋊D9C22⋊C4semidirect
147. C2×#217D9semidirect
200. C23×C9C4×D3semidirect
235. C2×Dic9#67semidirect
366. C3×C9#386semidirect
387. C23×C3C4×D9semidirect
447. C2×Dic3#611semidirect
569. C9#4956semidirect
583. C23(C3⋊D9)⋊C4semidirect
656. C3#10588636semidirect

Maximal subgroups

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

subgroupindex
C22×#62332
C22×(C3⋊D9)⋊C42
C2×C9×#2172
C2×C3×#36762
C2×#884452
#117203312
#105917223
#105886363