Finite Group Databasebeta

← #13141123

Subgroups — #13141123

Order 1728.

846 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#611C211yesmaximal, characteristic
2C2×#88551C211yesmaximal, characteristic, complemented
3C2×C9×#217C211yesmaximal, characteristic, complemented
4C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
5C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
6C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591709—31nomaximal
9#10588641—31nomaximal
10C24×C3×C9C2211yescharacteristic
11C23×C3×D9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C2×C3×#611—22no—
15C2×C3×#611—22no—
16C2×C3×#611—22no—
17C22×C9×Dic3C2211yescharacteristic
18C22×C9×Dic3C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20#88551—22no—
21C9×#217—22no—
22(C2×D9)⋊Dic3—22no—
23(C2×D9)⋊Dic3—22no—
24Dic3⋊Dic9—22no—
25C22×C3×#67—31no—
26C2×#3757—31no—
27C2×C3×#217D311yescharacteristic
28C2×Dic3⋊Dic3—31no—
29C2×(C2×D3)⋊Dic3—31no—
30C2×(C2×D3)⋊Dic3—31no—
31C22×#611D311yescharacteristic
32#63579—31no—
33C2×C9×C22⋊C4—31no—
34C2×#3676—31no—
35C2×#3676—61no—
36C2×(C2×D9)⋊C4—31no—
37C2×(C2×D9)⋊C4—31no—
38C2×C4⋊Dic9—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—
55C2×C3⋊Dic9—21no—
56C2×C3⋊Dic9—21no—
57C2×C3⋊Dic9—21no—
58C2×C3⋊Dic9—21no—
59C2×C3⋊Dic9—21no—
60C2×C3⋊Dic9—21no—
61C3×#611—28no—
62C2×C9×Dic3—21no—
63C2×C9×Dic3—21no—
64C2×C9×Dic3—21no—
65C2×C9×Dic3—21no—
66C2×C9×Dic3—21no—
67C2×C9×Dic3D412yescomplemented
68C2×C3×Dic9—21no—
69C2×C3×Dic9—21no—
70C2×C3×Dic9D412yes—
71#5222—91no—
72#4961—91no—
73C24×C32C2×D311yescharacteristic
74C23×C3×D3—31no—
75C22×C3⋊Dic3—31no—
76C22×C3⋊Dic3—31no—
77C2×C3×#67—62no—
78C2×C3×#67—62no—
79C2×C3×#67—62no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3C2×D311yescharacteristic
82C22×C3×Dic3C2×D311yescharacteristic
83C24×C9—21no—
84C24×C9C2×D311yescharacteristic
85C23×D9C2×D311yescharacteristic
86#3757—62no—
87C3×#217—22no—
88Dic3⋊Dic3—62no—
89(C2×D3)⋊Dic3—62no—
90(C2×D3)⋊Dic3—62no—
91C22×C4×C9—31no—
92C22×C4×C9—31no—
93C2×#611—22no—
94C2×#611—22no—
95C2×#611—22no—
96C22×Dic9—31no—
97C22×Dic9—31no—
98C22×Dic9—61no—
99C22×Dic9—61no—
100C22×Dic9C2×D311yescharacteristic
101C9×C22⋊C4—62no—
102#3676—64no—
103#3676—62no—
104(C2×D9)⋊C4—62no—
105(C2×D9)⋊C4—62no—
106C4⋊Dic9—62no—
107C22×C3×C9—22no—
108C22×C3×C9—22no—
109C22×C3×C9—22no—
110C22×C3×C9—41no—
111C22×C3×C9—41no—
112C22×C3×C9—41no—
113C22×C3×C9—41no—
114C22×C3×C9C2×D411yescharacteristic
115C22×C3×C9C2×D411yescharacteristic
116C22×C3×C9C2×D411yescharacteristic
117C22×C3×C9C2×D411yescharacteristic
118C22×C3×C9#4111yescharacteristic
119C22×C3×C9#4111yescharacteristic
120C22×C3×C9#4111yescharacteristic
121C2×C3×D9—22no—
122C2×C3×D9—22no—
123C2×C3×D9—22no—
124C2×C3×D9—41no—
125C2×C3×D9—41no—
126C2×C3×D9—41no—
127C2×C3×D9—41no—
128C3⋊Dic9—41no—
129C3⋊Dic9—41no—
130C9×Dic3—22no—
131C9×Dic3—41no—
132C3×Dic9—22no—
133C22×C3×D4—91no—
134C22×#67—31no—
135C2×C3×C22⋊C4—31no—
136C2×#217—91no—
137C2×#217—91no—
138C2×#217—91no—
139C2×#217—91no—
140C2×#217—181no—
141C2×#217D911yescharacteristic, complemented
142C2×(C2×D3)⋊C4—91no—
143C2×(C2×D3)⋊C4—91no—
144C2×C4⋊Dic3—91no—
145C2×Dic3⋊C4—91no—
146C3⋊#95—91no—
147C3⋊#95—91no—
148C23×C32—21no—
149C23×C32—21no—
150C23×C32—21no—
151C23×C32—21no—
152C23×C32—21no—
153C23×C32—21no—
154C23×C32D1212yes—
155C23×C32C22×D311yescharacteristic
156C22×C3×D3—32no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C3×#67—68no—
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#6712yes—
179C23×C9—22no—
180C23×C9—22no—
181C23×C9—21no—
182C23×C9—21no—
183C23×C9—21no—
184C23×C9—21no—
185C23×C9—21no—
186C23×C9—21no—
187C23×C9—22no—
188C23×C9—21no—
189C23×C9—41no—
190C23×C9—41no—
191C23×C9—41no—
192C23×C9—41no—
193C23×C9#6712yes—
194C23×C9C22×D311yescharacteristic
195C22×D9—21no—
196C22×D9—21no—
197C22×D9—21no—
198C22×D9—21no—
199C22×D9—21no—
200C22×D9—21no—
201C22×D9#6712yes—
202C2×C4×C9—32no—
203C2×C4×C9—61no—
204C2×C4×C9—61no—
205C2×C4×C9—61no—
206C2×C4×C9—61no—
207C2×C4×C9—61no—
208#611—28no—
209C2×Dic9—21no—
210C2×Dic9—21no—
211C2×Dic9—62no—
212C2×Dic9—62no—
213C2×Dic9—61no—
214C2×Dic9—61no—
215C2×Dic9—61no—
216C2×Dic9—61no—
217C2×Dic9—61no—
218C2×Dic9—61no—
219C2×Dic9—121no—
220C2×Dic9—121no—
221C2×Dic9—121no—
222C2×Dic9—121no—
223C2×Dic9#6712yes—
224#394—271noSylow
225C2×C3×C9—42no—
226C2×C3×C9#12011yescharacteristic
227C2×C3×C9#12111yescharacteristic
228C2×C3×C9#12111yescharacteristic
229C2×C3×C9#12111yescharacteristic
230C2×C3×C9#12311yescharacteristic
231C2×C3×C9#12311yescharacteristic
232C2×C3×C9#12411yescharacteristic
233C3×D9—42no—
234C24×C3—21no—
235C24×C3—91no—
236C24×C3C2×D911yescharacteristic
237C24×C3D3211yescharacteristic
238C23×D3—31no—
239C2×C3×D4—182no—
240C2×C3×D4—182no—
241C2×C3×D4—182no—
242C22×C3×C4—31no—
243C22×C3×C4—31no—
244C22×C3×C4—91no—
245C2×#67—62no—
246C2×#67—62no—
247C2×#67—62no—
248C22×Dic3—31no—
249C22×Dic3—91no—
250C22×Dic3—91no—
251C22×Dic3—91no—
252C22×Dic3—91no—
253C22×Dic3—181no—
254C22×Dic3—181no—
255C22×Dic3C2×D911yescharacteristic
256C22×Dic3C2×D911yescharacteristic
257C3×C22⋊C4—62no—
258#217—22no—
259#217—182no—
260#217—184no—
261#217—182no—
262#217—182no—
263#217—182no—
264(C2×D3)⋊C4—182no—
265(C2×D3)⋊C4—182no—
266C4⋊Dic3—182no—
267Dic3⋊C4—182no—
268C22×C32—22no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32—41no—
272C22×C32—41no—
273C22×C32—41no—
274C22×C32—41no—
275C22×C32C2×D1211yescharacteristic
276C22×C32#23511yescharacteristic
277C22×C32D3×D411yescharacteristic
278C22×C32D3×D411yescharacteristic
279C22×C32#23711yescharacteristic
280C22×C32#23711yescharacteristic
281C22×C32C2×#6711yescharacteristic
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C3⋊Dic3—121no—
290C3⋊Dic3—121no—
291C3×Dic3—22no—
292C3×Dic3—41no—
293C3×Dic3—62no—
294C22×C9—24no—
295C22×C9—22no—
296C22×C9—22no—
297C22×C9—22no—
298C22×C9—21no—
299C22×C9—21no—
300C22×C9—21no—
301C22×C9—21no—
302C22×C9—21no—
303C22×C9—21no—
304C22×C9—21no—
305C22×C9—42no—
306C22×C9—42no—
307C22×C9—42no—
308C22×C9—42no—
309C22×C9—42no—
310C22×C9—42no—
311C22×C9—41no—
312C22×C9—41no—
313C22×C9—41no—
314C22×C9—41no—
315C22×C9D3×D411yescharacteristic
316C22×C9#23711yescharacteristic
317C22×C9#23711yescharacteristic
318C22×C9#23711yescharacteristic
319C22×C9C2×#6711yescharacteristic
320C22×C9C2×#6711yescharacteristic
321C22×C9C2×#6711yescharacteristic
322C2×D9—22no—
323C2×D9—22no—
324C2×D9—22no—
325C2×D9—41no—
326C2×D9—41no—
327C2×D9—41no—
328C2×D9—41no—
329C4×C9—62no—
330C4×C9—121no—
331Dic9—22no—
332Dic9—122no—
333Dic9—122no—
334Dic9—121no—
335Dic9—121no—
336C22×D4—91no—
337C2×C4⋊C4—271no—
338C2×C22⋊C4—31no—
339C2×C22⋊C4—271no—
340C2×C22⋊C4—271no—
341C2×C22⋊C4—271no—
342#95—271no—
343C3×C9#39411yescharacteristic, complemented, Sylow
344C23×C3—22no—
345C23×C3—22no—
346C23×C3—21no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—21no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—22no—
359C23×C3—21no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3—41no—
364C23×C3—92no—
365C23×C3—181no—
366C23×C3—181no—
367C23×C3—181no—
368C23×C3—181no—
369C23×C3—181no—
370C23×C3—181no—
371C23×C3D3612yes—
372C23×C3C22×D911yescharacteristic
373C23×C3#62612yes—
374C23×C3C2×D3211yescharacteristic
375C22×D3—32no—
376C22×D3—61no—
377C22×D3—61no—
378C22×D3—61no—
379C22×D3—61no—
380C22×D3—61no—
381C22×D3—61no—
382C3×D4—188no—
383C2×C3×C4—32no—
384C2×C3×C4—61no—
385C2×C3×C4—61no—
386C2×C3×C4—61no—
387C2×C3×C4—61no—
388C2×C3×C4—61no—
389C2×C3×C4—92no—
390C2×C3×C4—181no—
391C2×C3×C4—181no—
392#67—68no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—21no—
398C2×Dic3—32no—
399C2×Dic3—61no—
400C2×Dic3—61no—
401C2×Dic3—182no—
402C2×Dic3—182no—
403C2×Dic3—181no—
404C2×Dic3—181no—
405C2×Dic3—181no—
406C2×Dic3—181no—
407C2×Dic3—181no—
408C2×Dic3—181no—
409C2×Dic3—181no—
410C2×Dic3—181no—
411C2×Dic3—181no—
412C2×Dic3—181no—
413C2×Dic3—181no—
414C2×Dic3—181no—
415C2×Dic3—361no—
416C2×Dic3—361no—
417C2×Dic3—361no—
418C2×Dic3—361no—
419C2×Dic3#61112yescomplemented
420C2×C32—42no—
421C2×C32#88211yescharacteristic
422C2×C32#88311yescharacteristic
423C2×C32#88511yescharacteristic
424C2×C32#88611yescharacteristic
425C2×C32#93011yescharacteristic
426C2×C32#93811yescharacteristic
427C2×C32#93911yescharacteristic
428C3×D3—122no—
429C2×C9—21no—
430C2×C9—21no—
431C2×C9—21no—
432C2×C9—21no—
433C2×C9—21no—
434C2×C9—21no—
435C2×C9—21no—
436C2×C9—44no—
437C2×C9—42no—
438C2×C9#93511yescharacteristic
439C2×C9#93511yescharacteristic
440C2×C9#93611yescharacteristic
441C2×C9#93811yescharacteristic
442C2×C9#93911yescharacteristic
443C2×C9#93911yescharacteristic
444C2×C9#95311yescharacteristic
445D9—42no—
446C24—91no—
447C24D3×D911yescharacteristic
448C2×D4—182no—
449C2×D4—182no—
450C2×D4—182no—
451C22×C4—31no—
452C22×C4—31no—
453C22×C4—91no—
454C22×C4—271no—
455C22×C4—271no—
456C4⋊C4—542no—
457C22⋊C4—62no—
458C22⋊C4—542no—
459C22⋊C4—542no—
460C22⋊C4—542no—
461C22×C3—24no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—22no—
467C22×C3—22no—
468C22×C3—21no—
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—21no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—41no—
482C22×C3—41no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3—41no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—182no—
490C22×C3—182no—
491C22×C3—182no—
492C22×C3—361no—
493C22×C3—361no—
494C22×C3—361no—
495C22×C3—361no—
496C22×C3C2×D3611yescharacteristic
497C22×C3#369711yescharacteristic
498C22×C3D4×D911yescharacteristic
499C22×C3D4×D911yescharacteristic
500C22×C3#369911yescharacteristic
501C22×C3#369911yescharacteristic
502C22×C3C2×#61111yescharacteristic
503C22×C3#380411yescharacteristic
504C22×C3#380511yescharacteristic
505C22×C3#380511yescharacteristic
506C22×C3C2×#62611yescharacteristic
507C22×C3D3×#6711yescharacteristic
508C22×C3D3×#6711yescharacteristic
509C22×C3D3×#6711yescharacteristic
510C2×D3—62no—
511C2×D3—62no—
512C2×D3—62no—
513C2×D3—121no—
514C2×D3—121no—
515C2×D3—121no—
516C2×D3—121no—
517C3×C4—62no—
518C3×C4—121no—
519C3×C4—182no—
520Dic3—22no—
521Dic3—41no—
522Dic3—62no—
523Dic3—362no—
524Dic3—362no—
525Dic3—361no—
526Dic3—361no—
527Dic3—361no—
528Dic3—361no—
529C32#496111yescharacteristic
530C9—21no—
531C9#522211yescharacteristic, complemented
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—92no—
539C23—181no—
540C23—181no—
541C23—181no—
542C23—181no—
543C23—181no—
544C23—181no—
545C23#619212yes—
546C23C2×D3×D911yescharacteristic
547D4—188no—
548C2×C4—32no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—92no—
555C2×C4—181no—
556C2×C4—181no—
557C2×C4—541no—
558C2×C4—541no—
559C2×C4—541no—
560C2×C4—541no—
561C2×C4—541no—
562C2×C4—541no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—44no—
571C2×C3—42no—
572C2×C3—42no—
573C2×C3—362no—
574C2×C3#6344411yescharacteristic
575C2×C3#6344511yescharacteristic
576C2×C3#6344711yescharacteristic
577C2×C3#6344811yescharacteristic
578C2×C3#6349211yescharacteristic
579C2×C3#6350011yescharacteristic
580C2×C3#6350111yescharacteristic
581C2×C3#6395911yescharacteristic
582C2×C3#6396211yescharacteristic
583C2×C3#6396911yescharacteristic
584C2×C3#6397011yescharacteristic
585C2×C3#6397011yescharacteristic
586C2×C3#6397711yescharacteristic
587C2×C3#6397811yescharacteristic
588D3—122no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—182no—
597C22—182no—
598C22—182no—
599C22—361no—
600C22—361no—
601C22—361no—
602C22—361no—
603C22#8865811yescharacteristic
604C22C2×#619211yescharacteristic
605C22#8866311yescharacteristic
606C22#8866311yescharacteristic
607C22D3×#61111yescharacteristic
608C22D9×#6711yescharacteristic
609C22D9×#6711yescharacteristic
610C4—62no—
611C4—121no—
612C4—182no—
613C4—1081no—
614C4—1081no—
615C3—21no—
616C3#1058864111yescharacteristic, complemented
617C3#1059170911yescharacteristic
618C2—42no—
619C2—362no—
620C2#1172125711yescharacteristic
621C2#1172126011yescharacteristic
622C2#1172126911yescharacteristic
623C2#1172127011yescharacteristic
624C2#1172127611yescharacteristic
625C2#1172127711yescharacteristic
626C2#1172128111yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C2×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC22×C3×#611C2
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×#88551C2semidirect
3. C2×C9×#217C2semidirect
4. C2×(C2×D9)⋊Dic3C2semidirect
5. C2×(C2×D9)⋊Dic3C2semidirect
6. C2×Dic3⋊Dic9C2semidirect
7. #11720331C2semidirect
67. C2×C9×Dic3D4semidirect
141. C2×#217D9semidirect
343. C3×C9#394semidirect
419. C2×Dic3#611semidirect
531. C9#5222semidirect
616. C3#10588641semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6112
C2×#885512
C2×C9×#2172
C2×(C2×D9)⋊Dic32
C2×Dic3⋊Dic92
#117203312
#105917093
#105886413