Finite Group Databasebeta

← #13141128

Subgroups — #13141128

Order 1728.

950 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×#6233C211yesmaximal, characteristic
2C2×C9×#217C211yesmaximal, characteristic, complemented
3C2×C3×#3676C211yesmaximal, characteristic, complemented
4C2×#88445C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic, complemented
6C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591712—31nomaximal
9#10588639—31nomaximal
10C24×C3×C9C2211yescharacteristic
11C23×C3⋊D9C2211yescharacteristic
12C2×#6233—22no—
13C2×#6233—22no—
14C2×#6233—22no—
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C9×#217—22no—
21C3×#3676—22no—
22#88445—22no—
23#88445—22no—
24Dic9⋊Dic3—22no—
25C22×#638—31no—
26C2×C3×#217—31no—
27C2×C3×#217D311yescharacteristic
28C2×Dic3⋊Dic3—31no—
29C2×#3722—31no—
30C2×#3722—31no—
31C22×#611—31no—
32C22×#611—61no—
33#63579—31no—
34C2×C9×C22⋊C4—31no—
35C2×#3676D311yescharacteristic, complemented
36C2×(C2×D9)⋊C4—31no—
37C2×(C2×D9)⋊C4—31no—
38C2×Dic9⋊C4—31no—
39C9⋊#95—31no—
40C23×C3×C9—21no—
41C23×C3×C9—21no—
42C23×C3×C9—21no—
43C23×C3×C9—21no—
44C23×C3×C9—21no—
45C23×C3×C9—21no—
46C23×C3×C9D412yes—
47C23×C3×C9C2311yescharacteristic
48C22×C3⋊D9—21no—
49C22×C3⋊D9—21no—
50C22×C3⋊D9—21no—
51C22×C3⋊D9—21no—
52C22×C3⋊D9—21no—
53C22×C3⋊D9—21no—
54C22×C3⋊D9D412yes—
55#6233—28no—
56C2×C3⋊Dic9—21no—
57C2×C3⋊Dic9—21no—
58C2×C3⋊Dic9D412yes—
59C2×C9×Dic3—21no—
60C2×C9×Dic3—21no—
61C2×C9×Dic3—21no—
62C2×C9×Dic3—21no—
63C2×C9×Dic3—21no—
64C2×C9×Dic3—21no—
65C2×C3×Dic9—21no—
66C2×C3×Dic9—21no—
67C2×C3×Dic9—21no—
68C2×C3×Dic9—21no—
69C2×C3×Dic9—21no—
70C2×C3×Dic9D412yescomplemented
71#4961—91no—
72#4959—91no—
73C24×C32C2×D311yescharacteristic
74C23×C3⋊D3—31no—
75C2×#638—62no—
76C2×#638—62no—
77C2×#638—62no—
78C22×C3⋊Dic3—31no—
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3C2×D311yescharacteristic
82C22×C3×Dic3C2×D311yescharacteristic
83C24×C9—21no—
84C24×C9C2×D311yescharacteristic
85C23×D9—31no—
86C23×D9—61no—
87C3×#217—22no—
88C3×#217—62no—
89Dic3⋊Dic3—62no—
90#3722—62no—
91#3722—62no—
92C22×C4×C9—31no—
93C22×C4×C9—31no—
94C2×#611—62no—
95C2×#611—62no—
96C2×#611—64no—
97C2×#611—64no—
98C2×#611—64no—
99C2×#611—62no—
100C22×Dic9—31no—
101C22×Dic9—61no—
102C22×Dic9C2×D311yescharacteristic
103C22×Dic9C2×D311yescharacteristic
104C9×C22⋊C4—62no—
105#3676—22no—
106(C2×D9)⋊C4—62no—
107(C2×D9)⋊C4—62no—
108Dic9⋊C4—62no—
109C22×C3×C9—22no—
110C22×C3×C9—22no—
111C22×C3×C9—22no—
112C22×C3×C9—41no—
113C22×C3×C9—41no—
114C22×C3×C9—41no—
115C22×C3×C9—41no—
116C22×C3×C9C2×D411yescharacteristic
117C22×C3×C9C2×D411yescharacteristic
118C22×C3×C9C2×D411yescharacteristic
119C22×C3×C9C2×D411yescharacteristic
120C22×C3×C9#4111yescharacteristic
121C22×C3×C9#4111yescharacteristic
122C22×C3×C9#4111yescharacteristic
123C2×C3⋊D9—22no—
124C2×C3⋊D9—22no—
125C2×C3⋊D9—22no—
126C2×C3⋊D9—41no—
127C2×C3⋊D9—41no—
128C2×C3⋊D9—41no—
129C2×C3⋊D9—41no—
130C3⋊Dic9—22no—
131C9×Dic3—41no—
132C9×Dic3—41no—
133C3×Dic9—22no—
134C3×Dic9—41no—
135C22×#67—91no—
136C22×#67—91no—
137C22×#67—181no—
138C2×C3×C22⋊C4—31no—
139C2×C3×C22⋊C4—91no—
140C2×#217—31no—
141C2×#217D911yescharacteristic, complemented
142C2×(C2×D3)⋊C4—91no—
143C2×(C2×D3)⋊C4—91no—
144C2×(C2×D3)⋊C4—91no—
145C2×(C2×D3)⋊C4—91no—
146C2×C4⋊Dic3—91no—
147C2×Dic3⋊C4—91no—
148C3⋊#95—91no—
149C3⋊#95—91no—
150C23×C32—21no—
151C23×C32—21no—
152C23×C32—21no—
153C23×C32—21no—
154C23×C32—21no—
155C23×C32—21no—
156C23×C32#6712yes—
157C23×C32C22×D311yescharacteristic
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—
165#638—68no—
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C23×C9—22no—
182C23×C9—22no—
183C23×C9—21no—
184C23×C9—21no—
185C23×C9—21no—
186C23×C9—21no—
187C23×C9—21no—
188C23×C9—21no—
189C23×C9—22no—
190C23×C9—21no—
191C23×C9—41no—
192C23×C9—41no—
193C23×C9—41no—
194C23×C9—41no—
195C23×C9D1212yes—
196C23×C9C22×D311yescharacteristic
197C22×D9—32no—
198C22×D9—62no—
199C22×D9—62no—
200C22×D9—62no—
201C22×D9—61no—
202C22×D9—61no—
203C22×D9—61no—
204C22×D9—61no—
205C22×D9—61no—
206C22×D9—61no—
207C22×D9—121no—
208C22×D9—121no—
209C22×D9—121no—
210C22×D9—121no—
211C2×C4×C9—61no—
212C2×C4×C9—61no—
213C2×C4×C9—61no—
214C2×C4×C9—61no—
215C2×C4×C9—61no—
216C2×C4×C9—61no—
217#611—616no—
218#611—68no—
219C2×Dic9—21no—
220C2×Dic9—21no—
221C2×Dic9—21no—
222C2×Dic9—21no—
223C2×Dic9—21no—
224C2×Dic9—32no—
225C2×Dic9—62no—
226C2×Dic9—62no—
227C2×Dic9—62no—
228C2×Dic9—61no—
229C2×Dic9—61no—
230C2×Dic9#6712yescomplemented
231#394—271noSylow
232C2×C3×C9—42no—
233C2×C3×C9#12011yescharacteristic
234C2×C3×C9#12111yescharacteristic
235C2×C3×C9#12111yescharacteristic
236C2×C3×C9#12111yescharacteristic
237C2×C3×C9#12311yescharacteristic
238C2×C3×C9#12311yescharacteristic
239C2×C3×C9#12411yescharacteristic
240C3⋊D9—42no—
241C24×C3—21no—
242C24×C3C2×D911yescharacteristic
243C24×C3D3211yescharacteristic
244C23×D3—91no—
245C23×D3—91no—
246C23×D3—181no—
247C22×C3×C4—31no—
248C22×C3×C4—31no—
249C22×C3×C4—91no—
250C22×C3×C4—91no—
251C2×#67—182no—
252C2×#67—182no—
253C2×#67—182no—
254C2×#67—182no—
255C2×#67—184no—
256C2×#67—184no—
257C2×#67—184no—
258C2×#67—182no—
259C2×#67—182no—
260C22×Dic3—31no—
261C22×Dic3—31no—
262C22×Dic3—91no—
263C22×Dic3—91no—
264C22×Dic3—181no—
265C22×Dic3C2×D911yescharacteristic
266C22×Dic3C2×D911yescharacteristic
267C3×C22⋊C4—62no—
268C3×C22⋊C4—182no—
269#217—22no—
270#217—62no—
271(C2×D3)⋊C4—182no—
272(C2×D3)⋊C4—182no—
273(C2×D3)⋊C4—182no—
274(C2×D3)⋊C4—182no—
275C4⋊Dic3—182no—
276Dic3⋊C4—182no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—22no—
280C22×C32—41no—
281C22×C32—41no—
282C22×C32—41no—
283C22×C32—41no—
284C22×C32#23511yescharacteristic
285C22×C32#23511yescharacteristic
286C22×C32D3×D411yescharacteristic
287C22×C32D3×D411yescharacteristic
288C22×C32D3×D411yescharacteristic
289C22×C32#23711yescharacteristic
290C22×C32C2×#6711yescharacteristic
291C2×C3⋊D3—62no—
292C2×C3⋊D3—62no—
293C2×C3⋊D3—62no—
294C2×C3⋊D3—121no—
295C2×C3⋊D3—121no—
296C2×C3⋊D3—121no—
297C2×C3⋊D3—121no—
298C3⋊Dic3—62no—
299C3×Dic3—41no—
300C3×Dic3—41no—
301C3×Dic3—62no—
302C3×Dic3—121no—
303C22×C9—24no—
304C22×C9—22no—
305C22×C9—22no—
306C22×C9—22no—
307C22×C9—21no—
308C22×C9—21no—
309C22×C9—21no—
310C22×C9—21no—
311C22×C9—21no—
312C22×C9—21no—
313C22×C9—21no—
314C22×C9—42no—
315C22×C9—42no—
316C22×C9—42no—
317C22×C9—42no—
318C22×C9—42no—
319C22×C9—42no—
320C22×C9—41no—
321C22×C9—41no—
322C22×C9—41no—
323C22×C9—41no—
324C22×C9C2×D1211yescharacteristic
325C22×C9#23511yescharacteristic
326C22×C9D3×D411yescharacteristic
327C22×C9D3×D411yescharacteristic
328C22×C9#23711yescharacteristic
329C22×C9#23711yescharacteristic
330C22×C9C2×#6711yescharacteristic
331C2×D9—64no—
332C2×D9—62no—
333C2×D9—62no—
334C2×D9—62no—
335C2×D9—122no—
336C2×D9—122no—
337C2×D9—122no—
338C2×D9—122no—
339C2×D9—122no—
340C2×D9—122no—
341C2×D9—121no—
342C2×D9—121no—
343C2×D9—121no—
344C2×D9—121no—
345C4×C9—121no—
346C4×C9—121no—
347Dic9—22no—
348Dic9—41no—
349Dic9—64no—
350Dic9—62no—
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—
358C3×C9#39411yescharacteristic, complemented, Sylow
359C23×C3—22no—
360C23×C3—22no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—21no—
373C23×C3—22no—
374C23×C3—21no—
375C23×C3—41no—
376C23×C3—41no—
377C23×C3—41no—
378C23×C3—41no—
379C23×C3#61112yes—
380C23×C3C22×D911yescharacteristic
381C23×C3#62612yes—
382C23×C3C2×D3211yescharacteristic
383C22×D3—92no—
384C22×D3—92no—
385C22×D3—182no—
386C22×D3—182no—
387C22×D3—182no—
388C22×D3—181no—
389C22×D3—181no—
390C22×D3—181no—
391C22×D3—181no—
392C22×D3—181no—
393C22×D3—181no—
394C22×D3—181no—
395C22×D3—181no—
396C22×D3—181no—
397C22×D3—181no—
398C22×D3—181no—
399C22×D3—181no—
400C22×D3—361no—
401C22×D3—361no—
402C22×D3—361no—
403C22×D3—361no—
404C2×C3×C4—61no—
405C2×C3×C4—61no—
406C2×C3×C4—61no—
407C2×C3×C4—61no—
408C2×C3×C4—61no—
409C2×C3×C4—61no—
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—188no—
417#67—1816no—
418#67—188no—
419C2×Dic3—21no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—32no—
426C2×Dic3—61no—
427C2×Dic3—61no—
428C2×Dic3—61no—
429C2×Dic3—61no—
430C2×Dic3—61no—
431C2×Dic3—92no—
432C2×Dic3—92no—
433C2×Dic3—182no—
434C2×Dic3—182no—
435C2×Dic3—182no—
436C2×Dic3—181no—
437C2×Dic3—181no—
438C2×Dic3—181no—
439C2×Dic3—181no—
440C2×C32—42no—
441C2×C32#88311yescharacteristic
442C2×C32#88411yescharacteristic
443C2×C32#88411yescharacteristic
444C2×C32#88511yescharacteristic
445C2×C32#92911yescharacteristic
446C2×C32#93711yescharacteristic
447C2×C32#93911yescharacteristic
448C3⋊D3—122no—
449C2×C9—21no—
450C2×C9—21no—
451C2×C9—21no—
452C2×C9—21no—
453C2×C9—21no—
454C2×C9—21no—
455C2×C9—21no—
456C2×C9—44no—
457C2×C9—42no—
458C2×C9#88211yescharacteristic
459C2×C9#88311yescharacteristic
460C2×C9#88511yescharacteristic
461C2×C9#88611yescharacteristic
462C2×C9#93011yescharacteristic
463C2×C9#93811yescharacteristic
464C2×C9#93911yescharacteristic
465D9—124no—
466D9—122no—
467C24—271no—
468C24D3×D911yescharacteristic
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—24no—
483C22×C3—22no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—21no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—41no—
509C22×C3—41no—
510C22×C3#369711yescharacteristic
511C22×C3#369711yescharacteristic
512C22×C3D4×D911yescharacteristic
513C22×C3D4×D911yescharacteristic
514C22×C3D4×D911yescharacteristic
515C22×C3#369911yescharacteristic
516C22×C3C2×#61111yescharacteristic
517C22×C3#380411yescharacteristic
518C22×C3#380411yescharacteristic
519C22×C3#380411yescharacteristic
520C22×C3C2×#62611yescharacteristic
521C22×C3D3×#6711yescharacteristic
522C22×C3#381111yescharacteristic
523C22×C3#381111yescharacteristic
524C2×D3—184no—
525C2×D3—182no—
526C2×D3—182no—
527C2×D3—182no—
528C2×D3—182no—
529C2×D3—182no—
530C2×D3—182no—
531C2×D3—362no—
532C2×D3—362no—
533C2×D3—362no—
534C2×D3—362no—
535C2×D3—362no—
536C2×D3—362no—
537C2×D3—361no—
538C2×D3—361no—
539C2×D3—361no—
540C2×D3—361no—
541C2×D3—361no—
542C2×D3—361no—
543C2×D3—361no—
544C2×D3—361no—
545C3×C4—121no—
546C3×C4—121no—
547C3×C4—182no—
548C3×C4—361no—
549Dic3—41no—
550Dic3—41no—
551Dic3—62no—
552Dic3—121no—
553Dic3—184no—
554Dic3—182no—
555Dic3—182no—
556C32#495911yescharacteristic
557C9—21no—
558C9#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#619512yes—
573C23C2×D3×D911yescharacteristic
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—21no—
591C2×C3—21no—
592C2×C3—21no—
593C2×C3—21no—
594C2×C3—21no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—44no—
598C2×C3—42no—
599C2×C3—42no—
600C2×C3#6344511yescharacteristic
601C2×C3#6344611yescharacteristic
602C2×C3#6344611yescharacteristic
603C2×C3#6344711yescharacteristic
604C2×C3#6349111yescharacteristic
605C2×C3#6349911yescharacteristic
606C2×C3#6350111yescharacteristic
607C2×C3#6395311yescharacteristic
608C2×C3#6395811yescharacteristic
609C2×C3#6396611yescharacteristic
610C2×C3#6397111yescharacteristic
611C2×C3#6397511yescharacteristic
612C2×C3#6397811yescharacteristic
613C2×C3#6398111yescharacteristic
614D3—364no—
615D3—362no—
616D3—362no—
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#8865811yescharacteristic
632C22#8865811yescharacteristic
633C22#8866211yescharacteristic
634C22C2×#619511yescharacteristic
635C22D9×#6711yescharacteristic
636C22#8866811yescharacteristic
637C22#8866811yescharacteristic
638C4—121no—
639C4—121no—
640C4—182no—
641C4—361no—
642C4—542no—
643C3—21no—
644C3#1058863911yescharacteristic, complemented
645C3#1059171211yescharacteristic
646C2—42no—
647C2—1082no—
648C2#1172124211yescharacteristic
649C2#1172125111yescharacteristic
650C2#1172125611yescharacteristic
651C2#1172127111yescharacteristic
652C2#1172127511yescharacteristic
653C2#1172128811yescharacteristic
654C2#1172129111yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C2×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC22×#6233C2
hypercenterC24D3×D9
hypocenterC3×C9#394
LCSG ▷ C23×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×C9×#217C2semidirect
3. C2×C3×#3676C2semidirect
4. C2×#88445C2semidirect
5. C2×#88445C2semidirect
6. C2×Dic9⋊Dic3C2semidirect
7. #11720331C2semidirect
35. C2×#3676D3semidirect
70. C2×C3×Dic9D4semidirect
141. C2×#217D9semidirect
230. C2×Dic9#67semidirect
358. C3×C9#394semidirect
558. C9#4961semidirect
644. C3#10588639semidirect

Maximal subgroups

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

subgroupindex
C22×#62332
C2×C9×#2172
C2×C3×#36762
C2×#884452
C2×Dic9⋊Dic32
#117203312
#105917123
#105886393