Finite Group Databasebeta

← #13139903

Subgroups — #13139903

Order 1728.

882 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6194C211yesmaximal, characteristic
2C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
3C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic, complemented
6C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
7C2×#88442C211yesmaximal, characteristic, complemented
8#10591385—31nomaximal
9#10588637—31nomaximal
10C23×D3×C9C2211yescharacteristic
11C23×C3×D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×#6194—22no—
18C2×#6194—22no—
19C2×#6194C414yescomplemented
20C4×C3⋊Dic9—22no—
21C9×(C2×D3)⋊C4—22no—
22C3×(C2×D9)⋊C4—22no—
23(C2×D3)⋊Dic9—22no—
24(C2×D9)⋊Dic3—22no—
25#88442—22no—
26C22×#625—31no—
27C2×C4×C3⋊Dic3—31no—
28C2×C3×(C2×D3)⋊C4—31no—
29C2×C3×(C2×D3)⋊C4D311yescharacteristic
30C2×#3724—31no—
31C2×(C2×D3)⋊Dic3—31no—
32C2×(C2×D3)⋊Dic3—31no—
33C22×#611—31no—
34C2×C9×C22⋊C4—31no—
35C2×#3676—31no—
36C2×(C2×D9)⋊C4—31no—
37C2×(C2×D9)⋊C4D311yescharacteristic, complemented
38C2×Dic9⋊C4—31no—
39C2×C4×Dic9—31no—
40C2×C4×Dic9—61no—
41C23×C3×C9C2311yescharacteristic
42C22×D3×C9—21no—
43C22×D3×C9—21no—
44C22×D3×C9—21no—
45C22×D3×C9—21no—
46C22×D3×C9—21no—
47C22×D3×C9—21no—
48C22×D3×C9C2×C412yes—
49C22×C3×D9—21no—
50C22×C3×D9—21no—
51C22×C3×D9—21no—
52C22×C3×D9—21no—
53C22×C3×D9—21no—
54C22×C3×D9—21no—
55C22×C3×D9C2×C412yes—
56C2×C3×C4×C9—21no—
57C2×C3×C4×C9—21no—
58C2×C3×C4×C9D412yescomplemented
59C2×C3⋊Dic9—21no—
60C2×C3⋊Dic9—21no—
61C2×C3⋊Dic9C2×C412yes—
62C2×C3⋊Dic9D412yes—
63C2×C3⋊Dic9D412yes—
64C2×C3⋊Dic9D412yescomplemented
65C2×C9×Dic3—21no—
66C2×C9×Dic3—21no—
67C2×C9×Dic3—21no—
68C2×C3×Dic9—21no—
69C2×C3×Dic9—21no—
70C2×C3×Dic9—21no—
71#6194—28no—
72#4957—91no—
73#4957—91no—
74C23×C3×D3—31no—
75C23×C3×D3C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C22×C3⋊Dic3—31no—
78C22×C3⋊Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3C2×D311yescharacteristic
81C2×#625—34no—
82C2×#625—62no—
83C2×#625—62no—
84C24×C9—31no—
85C23×D9C2×D311yescharacteristic
86C4×C3⋊Dic3—62no—
87C3×(C2×D3)⋊C4—22no—
88C3×(C2×D3)⋊C4—62no—
89#3724—62no—
90(C2×D3)⋊Dic3—62no—
91(C2×D3)⋊Dic3—62no—
92C22×C4×C9—21no—
93C22×C4×C9—31no—
94C22×C4×C9C2×D311yescharacteristic
95C2×#611—34no—
96C2×#611—62no—
97C2×#611—62no—
98C22×Dic9—31no—
99C22×Dic9—31no—
100C22×Dic9—61no—
101C22×Dic9—61no—
102C22×Dic9C2×D311yescharacteristic
103C9×C22⋊C4—62no—
104#3676—62no—
105(C2×D9)⋊C4—22no—
106(C2×D9)⋊C4—62no—
107Dic9⋊C4—62no—
108C4×Dic9—62no—
109C4×Dic9—64no—
110C22×C3×C9C22×C411yescharacteristic
111C22×C3×C9C2×D411yescharacteristic
112C22×C3×C9C2×D411yescharacteristic
113C22×C3×C9C2×D411yescharacteristic
114C22×C3×C9C2×D411yescharacteristic
115C22×C3×C9#4111yescharacteristic
116C22×C3×C9#4111yescharacteristic
117C2×D3×C9—22no—
118C2×D3×C9—22no—
119C2×D3×C9—22no—
120C2×D3×C9—41no—
121C2×D3×C9—41no—
122C2×D3×C9—41no—
123C2×D3×C9—41no—
124C2×C3×D9—22no—
125C2×C3×D9—22no—
126C2×C3×D9—22no—
127C2×C3×D9—41no—
128C2×C3×D9—41no—
129C2×C3×D9—41no—
130C2×C3×D9—41no—
131C3×C4×C9—22no—
132C3⋊Dic9—22no—
133C3⋊Dic9C22⋊C414yescomplemented
134C9×Dic3—41no—
135C3×Dic9—41no—
136C22×#67—91no—
137C22×#67—91no—
138C2×C3×C22⋊C4—31no—
139C2×C3×C22⋊C4—91no—
140C2×#217—91no—
141C2×#217—91no—
142C2×(C2×D3)⋊C4—31no—
143C2×(C2×D3)⋊C4—91no—
144C2×(C2×D3)⋊C4—91no—
145C2×(C2×D3)⋊C4D911yescharacteristic, complemented
146C2×Dic3⋊C4—91no—
147C2×Dic3⋊C4—91no—
148C2×C4×Dic3—91no—
149C2×C4×Dic3—91no—
150C2×C4×Dic3—181no—
151C23×C32C22×D311yescharacteristic
152C22×C3×D3—21no—
153C22×C3×D3—21no—
154C22×C3×D3—21no—
155C22×C3×D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—32no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3C4×D312yes—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4#6712yes—
169C2×C3⋊Dic3—32no—
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—32no—
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—61no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181#625—68no—
182C23×C9—21no—
183C23×C9—32no—
184C23×C9—61no—
185C23×C9—61no—
186C23×C9—61no—
187C23×C9—61no—
188C23×C9—61no—
189C23×C9—61no—
190C23×C9C22×D311yescharacteristic
191C22×D9—21no—
192C22×D9—21no—
193C22×D9—21no—
194C22×D9—21no—
195C22×D9—21no—
196C22×D9—21no—
197C22×D9C4×D312yes—
198C2×C4×C9—22no—
199C2×C4×C9—22no—
200C2×C4×C9—21no—
201C2×C4×C9—21no—
202C2×C4×C9—22no—
203C2×C4×C9—61no—
204C2×C4×C9—61no—
205C2×C4×C9—61no—
206C2×C4×C9#6712yescomplemented
207#611—68no—
208C2×Dic9—21no—
209C2×Dic9—21no—
210C2×Dic9—21no—
211C2×Dic9—32no—
212C2×Dic9—32no—
213C2×Dic9—32no—
214C2×Dic9—32no—
215C2×Dic9—62no—
216C2×Dic9—62no—
217C2×Dic9—62no—
218C2×Dic9—62no—
219C2×Dic9—62no—
220C2×Dic9—62no—
221C2×Dic9—61no—
222C2×Dic9—61no—
223#390—271noSylow
224C2×C3×C9C2×C22⋊C411yescharacteristic
225C2×C3×C9C4×D411yescharacteristic
226C2×C3×C9C4×D411yescharacteristic
227C2×C3×C9#12111yescharacteristic
228C2×C3×C9#12111yescharacteristic
229C2×C3×C9#12411yescharacteristic
230C2×C3×C9#12711yescharacteristic
231D3×C9—42no—
232C3×D9—42no—
233C24×C3—31no—
234C24×C3—91no—
235C23×D3—31no—
236C23×D3C2×D911yescharacteristic
237C22×C3×C4—21no—
238C22×C3×C4—31no—
239C22×C3×C4—91no—
240C22×C3×C4C2×D911yescharacteristic
241C22×C3×C4D3211yescharacteristic
242C2×#67—94no—
243C2×#67—94no—
244C2×#67—182no—
245C2×#67—182no—
246C2×#67—182no—
247C2×#67—182no—
248C22×Dic3—31no—
249C22×Dic3—91no—
250C22×Dic3—91no—
251C22×Dic3—91no—
252C22×Dic3—91no—
253C22×Dic3—181no—
254C22×Dic3—181no—
255C22×Dic3C2×D911yescharacteristic
256C3×C22⋊C4—62no—
257C3×C22⋊C4—182no—
258#217—182no—
259#217—182no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—62no—
262(C2×D3)⋊C4—182no—
263(C2×D3)⋊C4—182no—
264Dic3⋊C4—182no—
265Dic3⋊C4—182no—
266C4×Dic3—182no—
267C4×Dic3—182no—
268C4×Dic3—184no—
269C22×C32C2×C4×D311yescharacteristic
270C22×C32#23511yescharacteristic
271C22×C32D3×D411yescharacteristic
272C22×C32D3×D411yescharacteristic
273C22×C32D3×D411yescharacteristic
274C22×C32#23711yescharacteristic
275C22×C32C2×#6711yescharacteristic
276C2×C3×D3—22no—
277C2×C3×D3—22no—
278C2×C3×D3—22no—
279C2×C3×D3—41no—
280C2×C3×D3—41no—
281C2×C3×D3—41no—
282C2×C3×D3—41no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C2×C3×D3—121no—
290C32×C4—22no—
291C3⋊Dic3—34no—
292C3⋊Dic3—62no—
293C3×Dic3—41no—
294C3×Dic3—121no—
295C22×C9—21no—
296C22×C9—21no—
297C22×C9—21no—
298C22×C9—21no—
299C22×C9—21no—
300C22×C9—21no—
301C22×C9—21no—
302C22×C9—62no—
303C22×C9—62no—
304C22×C9—62no—
305C22×C9—121no—
306C22×C9—121no—
307C22×C9—121no—
308C22×C9—121no—
309C22×C9C2×C4×D311yescharacteristic
310C22×C9#23511yescharacteristic
311C22×C9D3×D411yescharacteristic
312C22×C9D3×D411yescharacteristic
313C22×C9D3×D411yescharacteristic
314C22×C9#23711yescharacteristic
315C22×C9C2×#6711yescharacteristic
316C2×D9—22no—
317C2×D9—22no—
318C2×D9—22no—
319C2×D9—41no—
320C2×D9—41no—
321C2×D9—41no—
322C2×D9—41no—
323C4×C9—24no—
324C4×C9—22no—
325C4×C9—121no—
326Dic9—34no—
327Dic9—41no—
328Dic9—64no—
329Dic9—64no—
330Dic9—62no—
331C22×D4—271no—
332C2×C4⋊C4—271no—
333C2×C22⋊C4—31no—
334C2×C22⋊C4—91no—
335C2×C22⋊C4—271no—
336C2×C22⋊C4—271no—
337C2×C42—271no—
338C3×C9#39011yescharacteristic, complemented, Sylow
339C23×C3—21no—
340C23×C3—32no—
341C23×C3—61no—
342C23×C3—61no—
343C23×C3—61no—
344C23×C3—61no—
345C23×C3—61no—
346C23×C3—61no—
347C23×C3—92no—
348C23×C3—181no—
349C23×C3—181no—
350C23×C3—181no—
351C23×C3—181no—
352C23×C3—181no—
353C23×C3—181no—
354C23×C3C22×D911yescharacteristic
355C23×C3C2×D3211yescharacteristic
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—32no—
363C22×D3—61no—
364C22×D3—61no—
365C22×D3—61no—
366C22×D3—61no—
367C22×D3—61no—
368C22×D3—61no—
369C22×D3C4×D912yes—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—22no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—181no—
381C2×C3×C4—181no—
382C2×C3×C4—181no—
383C2×C3×C4#61112yescomplemented
384C2×C3×C4#62512yes—
385#67—188no—
386#67—188no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—61no—
393C2×Dic3—92no—
394C2×Dic3—92no—
395C2×Dic3—92no—
396C2×Dic3—92no—
397C2×Dic3—92no—
398C2×Dic3—92no—
399C2×Dic3—92no—
400C2×Dic3—92no—
401C2×Dic3—182no—
402C2×Dic3—182no—
403C2×Dic3—182no—
404C2×Dic3—182no—
405C2×Dic3—182no—
406C2×Dic3—182no—
407C2×Dic3—181no—
408C2×Dic3—181no—
409C2×Dic3—181no—
410C2×Dic3—181no—
411C2×C32D3×C22⋊C411yescharacteristic
412C2×C32#67⋊C411yescharacteristic
413C2×C32#88411yescharacteristic
414C2×C32#88511yescharacteristic
415C2×C32C4×#6711yescharacteristic
416C2×C32#93911yescharacteristic
417C2×C32#94011yescharacteristic
418C3×D3—42no—
419C3×D3—122no—
420C2×C9—21no—
421C2×C9—21no—
422C2×C9—21no—
423C2×C9—21no—
424C2×C9—21no—
425C2×C9—21no—
426C2×C9—21no—
427C2×C9—122no—
428C2×C9D3×C22⋊C411yescharacteristic
429C2×C9#67⋊C411yescharacteristic
430C2×C9#88411yescharacteristic
431C2×C9#88511yescharacteristic
432C2×C9C4×#6711yescharacteristic
433C2×C9#93911yescharacteristic
434C2×C9#94011yescharacteristic
435D9—42no—
436C24—31no—
437C24—91no—
438C2×D4—274no—
439C2×D4—542no—
440C2×D4—542no—
441C22×C4—31no—
442C22×C4—91no—
443C22×C4—271no—
444C22×C4—271no—
445C22×C4D3×D911yescharacteristic
446C4⋊C4—542no—
447C22⋊C4—62no—
448C22⋊C4—182no—
449C22⋊C4—542no—
450C22⋊C4—542no—
451C42—542no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—62no—
460C22×C3—62no—
461C22×C3—62no—
462C22×C3—121no—
463C22×C3—121no—
464C22×C3—121no—
465C22×C3—121no—
466C22×C3—182no—
467C22×C3—182no—
468C22×C3—182no—
469C22×C3—361no—
470C22×C3—361no—
471C22×C3—361no—
472C22×C3—361no—
473C22×C3C2×C4×D911yescharacteristic
474C22×C3#369711yescharacteristic
475C22×C3D4×D911yescharacteristic
476C22×C3D4×D911yescharacteristic
477C22×C3D4×D911yescharacteristic
478C22×C3#369911yescharacteristic
479C22×C3C2×#61111yescharacteristic
480C22×C3#379811yescharacteristic
481C22×C3C4×D3211yescharacteristic
482C22×C3#380211yescharacteristic
483C22×C3#380211yescharacteristic
484C22×C3#380511yescharacteristic
485C22×C3C2×#62511yescharacteristic
486C22×C3#381111yescharacteristic
487C2×D3—22no—
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—41no—
494C2×D3—62no—
495C2×D3—62no—
496C2×D3—62no—
497C2×D3—121no—
498C2×D3—121no—
499C2×D3—121no—
500C2×D3—121no—
501C3×C4—24no—
502C3×C4—22no—
503C3×C4—22no—
504C3×C4—121no—
505C3×C4—361no—
506Dic3—41no—
507Dic3—94no—
508Dic3—94no—
509Dic3—121no—
510Dic3—184no—
511Dic3—184no—
512Dic3—182no—
513Dic3—182no—
514C32#495711yescharacteristic
515C9—21no—
516C9#495711yescharacteristic, complemented
517C23—32no—
518C23—61no—
519C23—61no—
520C23—61no—
521C23—61no—
522C23—61no—
523C23—61no—
524C23—92no—
525C23—181no—
526C23—181no—
527C23—181no—
528C23—181no—
529C23—181no—
530C23—181no—
531C23C2×D3×D911yescharacteristic
532D4—548no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—181no—
539C2×C4—181no—
540C2×C4—181no—
541C2×C4—272no—
542C2×C4—272no—
543C2×C4—272no—
544C2×C4—272no—
545C2×C4—541no—
546C2×C4—541no—
547C2×C4#619412yescomplemented
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—122no—
556C2×C3—362no—
557C2×C3C22⋊C4×D911yescharacteristic
558C2×C3#611⋊C411yescharacteristic
559C2×C3#6344611yescharacteristic
560C2×C3#6344711yescharacteristic
561C2×C3C4×#61111yescharacteristic
562C2×C3#6350111yescharacteristic
563C2×C3#6350211yescharacteristic
564C2×C3C4×#62511yescharacteristic
565C2×C3#6390211yescharacteristic
566C2×C3#6391211yescharacteristic
567C2×C3#6391411yescharacteristic
568C2×C3#6391511yescharacteristic
569C2×C3#6392111yescharacteristic
570C2×C3#6398011yescharacteristic
571D3—42no—
572D3—122no—
573C22—62no—
574C22—62no—
575C22—62no—
576C22—121no—
577C22—121no—
578C22—121no—
579C22—121no—
580C22—182no—
581C22—182no—
582C22—182no—
583C22—361no—
584C22—361no—
585C22—361no—
586C22—361no—
587C22#8864011yescharacteristic
588C22C4×D3×D911yescharacteristic
589C22#8864611yescharacteristic
590C22#8864611yescharacteristic
591C22#8866311yescharacteristic
592C22C2×#619411yescharacteristic
593C22#8866811yescharacteristic
594C4—22no—
595C4—121no—
596C4—274no—
597C4—361no—
598C4—542no—
599C3—21no—
600C3#1058863711yescharacteristic, complemented
601C3#1059138511yescharacteristic
602C2—122no—
603C2—362no—
604C2C4×#619411yescharacteristic
605C2#1172102811yescharacteristic
606C2#1172104111yescharacteristic
607C2#1172104311yescharacteristic
608C2#1172104411yescharacteristic
609C2#1172104811yescharacteristic
610C2#1172129011yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×#6194C2
cosocleC23×C3×C9C23
hypercenterC22×C4D3×D9
hypocenterC3×C9#390
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C3⋊Dic9C2semidirect
3. C2×C9×(C2×D3)⋊C4C2semidirect
4. C2×C3×(C2×D9)⋊C4C2semidirect
5. C2×(C2×D3)⋊Dic9C2semidirect
6. C2×(C2×D9)⋊Dic3C2semidirect
7. C2×#88442C2semidirect
19. C2×#6194C4semidirect
37. C2×(C2×D9)⋊C4D3semidirect
58. C2×C3×C4×C9D4semidirect
64. C2×C3⋊Dic9D4semidirect
133. C3⋊Dic9C22⋊C4semidirect
145. C2×(C2×D3)⋊C4D9semidirect
206. C2×C4×C9#67semidirect
338. C3×C9#390semidirect
383. C2×C3×C4#611semidirect
516. C9#4957semidirect
547. C2×C4#6194semidirect
600. C3#10588637semidirect

Maximal subgroups

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

subgroupindex
C22×#61942
C2×C4×C3⋊Dic92
C2×C9×(C2×D3)⋊C42
C2×C3×(C2×D9)⋊C42
C2×(C2×D3)⋊Dic92
C2×(C2×D9)⋊Dic32
C2×#884422
#105913853
#105886373