Finite Group Databasebeta

← #13141097

Subgroups — #13141097

Order 1728.

906 conjugacy classes of subgroups.

The subgroup lattice has 634 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×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
7C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
8#10591707—31nomaximal
9#10588637—31nomaximal
10C24×C3×C9C2211yescharacteristic
11C23×C3×D9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C2×C3×#611—22no—
15C2×C3×#611—22no—
16C2×C3×#611C414yescomplemented
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—
24Dic9⋊Dic3—22no—
25Dic3×Dic9—22no—
26C22×C3×#67—31no—
27C2×#3757—31no—
28C2×C3×#217D311yescharacteristic
29C2×Dic3⋊Dic3—31no—
30C2×(C2×D3)⋊Dic3—31no—
31C2×(C2×D3)⋊Dic3—31no—
32C2×Dic32—31no—
33C22×#611D311yescharacteristic
34C2×C9×C22⋊C4—31no—
35C2×#3676—31no—
36C2×#3676—61no—
37C2×(C2×D9)⋊C4—31no—
38C2×(C2×D9)⋊C4—31no—
39C2×Dic9⋊C4—31no—
40C2×C4×Dic9—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×C412yes—
48C23×C3×C9C2311yescharacteristic
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⋊Dic9—21no—
57C2×C3⋊Dic9—21no—
58C2×C3⋊Dic9—21no—
59C2×C3⋊Dic9—21no—
60C2×C3⋊Dic9—21no—
61C2×C3⋊Dic9D412yescomplemented
62C3×#611—28no—
63C2×C9×Dic3—21no—
64C2×C9×Dic3—21no—
65C2×C9×Dic3—21no—
66C2×C9×Dic3—21no—
67C2×C9×Dic3—21no—
68C2×C9×Dic3D412yescomplemented
69C2×C3×Dic9C2×C412yes—
70C2×C3×Dic9D412yes—
71C2×C3×Dic9D412yes—
72#5221—91no—
73#4957—91no—
74C24×C32C2×D311yescharacteristic
75C23×C3×D3—31no—
76C22×C3⋊Dic3—31no—
77C22×C3⋊Dic3—31no—
78C2×C3×#67—34no—
79C2×C3×#67—62no—
80C2×C3×#67—62no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3C2×D311yescharacteristic
83C22×C3×Dic3C2×D311yescharacteristic
84C24×C9—21no—
85C24×C9C2×D311yescharacteristic
86C23×D9C2×D311yescharacteristic
87#3757—62no—
88C3×#217—22no—
89Dic3⋊Dic3—62no—
90(C2×D3)⋊Dic3—62no—
91(C2×D3)⋊Dic3—62no—
92Dic32—62no—
93C22×C4×C9—31no—
94C22×C4×C9—31no—
95C2×#611—22no—
96C2×#611—22no—
97C2×#611Dic314yescomplemented
98C22×Dic9—31no—
99C22×Dic9—31no—
100C22×Dic9—61no—
101C22×Dic9—61no—
102C22×Dic9C2×D311yescharacteristic
103C9×C22⋊C4—62no—
104#3676—62no—
105#3676—64no—
106(C2×D9)⋊C4—62no—
107(C2×D9)⋊C4—62no—
108Dic9⋊C4—62no—
109C4×Dic9—62no—
110C22×C3×C9—22no—
111C22×C3×C9—22no—
112C22×C3×C9—22no—
113C22×C3×C9—41no—
114C22×C3×C9—41no—
115C22×C3×C9—41no—
116C22×C3×C9—41no—
117C22×C3×C9C22×C411yescharacteristic
118C22×C3×C9C2×D411yescharacteristic
119C22×C3×C9C2×D411yescharacteristic
120C22×C3×C9C2×D411yescharacteristic
121C22×C3×C9C2×D411yescharacteristic
122C22×C3×C9#4111yescharacteristic
123C22×C3×C9#4111yescharacteristic
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⋊Dic9—22no—
132C3⋊Dic9—41no—
133C9×Dic3—22no—
134C9×Dic3—41no—
135C3×Dic9C22⋊C414yescomplemented
136C22×C3×D4—91no—
137C22×#67—31no—
138C2×C3×C22⋊C4—31no—
139C2×#217—91no—
140C2×#217—91no—
141C2×#217—91no—
142C2×#217—91no—
143C2×#217—181no—
144C2×#217D911yescharacteristic, complemented
145C2×(C2×D3)⋊C4—91no—
146C2×(C2×D3)⋊C4—91no—
147C2×C4⋊Dic3—91no—
148C2×Dic3⋊C4—91no—
149C2×C4×Dic3—91no—
150C2×C4×Dic3—91no—
151C23×C32—21no—
152C23×C32—21no—
153C23×C32—21no—
154C23×C32—21no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32C4×D312yes—
158C23×C32C22×D311yescharacteristic
159C22×C3×D3—32no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C22×C3×D3—61no—
164C22×C3×D3—61no—
165C22×C3×D3—61no—
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C3×#67—68no—
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—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3#6712yes—
182C23×C9—22no—
183C23×C9—22no—
184C23×C9—21no—
185C23×C9—21no—
186C23×C9—21no—
187C23×C9—21no—
188C23×C9—21no—
189C23×C9—21no—
190C23×C9—22no—
191C23×C9—21no—
192C23×C9—41no—
193C23×C9—41no—
194C23×C9—41no—
195C23×C9—41no—
196C23×C9C2×Dic312yes—
197C23×C9C22×D311yescharacteristic
198C22×D9—21no—
199C22×D9—21no—
200C22×D9—21no—
201C22×D9—21no—
202C22×D9—21no—
203C22×D9—21no—
204C22×D9C2×Dic312yes—
205C2×C4×C9—32no—
206C2×C4×C9—61no—
207C2×C4×C9—61no—
208C2×C4×C9—61no—
209C2×C4×C9—61no—
210C2×C4×C9—61no—
211#611—28no—
212C2×Dic9—32no—
213C2×Dic9—62no—
214C2×Dic9—61no—
215C2×Dic9—61no—
216C2×Dic9—62no—
217C2×Dic9—61no—
218C2×Dic9—61no—
219C2×Dic9—61no—
220C2×Dic9—121no—
221C2×Dic9—121no—
222C2×Dic9—121no—
223C2×Dic9—121no—
224C2×Dic9C2×Dic312yes—
225C2×Dic9#6712yes—
226C2×Dic9#6712yes—
227#390—271noSylow
228C2×C3×C9—42no—
229C2×C3×C9C2×C22⋊C411yescharacteristic
230C2×C3×C9C4×D411yescharacteristic
231C2×C3×C9C4×D411yescharacteristic
232C2×C3×C9#12111yescharacteristic
233C2×C3×C9#12111yescharacteristic
234C2×C3×C9#12411yescharacteristic
235C2×C3×C9#12711yescharacteristic
236C3×D9—42no—
237C24×C3—21no—
238C24×C3—91no—
239C24×C3C2×D911yescharacteristic
240C24×C3D3211yescharacteristic
241C23×D3—31no—
242C2×C3×D4—94no—
243C2×C3×D4—182no—
244C2×C3×D4—182no—
245C22×C3×C4—31no—
246C22×C3×C4—31no—
247C22×C3×C4—91no—
248C2×#67—34no—
249C2×#67—62no—
250C2×#67—62no—
251C22×Dic3—31no—
252C22×Dic3—91no—
253C22×Dic3—91no—
254C22×Dic3—91no—
255C22×Dic3—91no—
256C22×Dic3—181no—
257C22×Dic3—181no—
258C22×Dic3C2×D911yescharacteristic
259C22×Dic3C2×D911yescharacteristic
260C3×C22⋊C4—62no—
261#217—22no—
262#217—182no—
263#217—182no—
264#217—182no—
265#217—184no—
266#217—182no—
267(C2×D3)⋊C4—182no—
268(C2×D3)⋊C4—182no—
269C4⋊Dic3—182no—
270Dic3⋊C4—182no—
271C4×Dic3—182no—
272C4×Dic3—182no—
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32—41no—
277C22×C32—41no—
278C22×C32—41no—
279C22×C32—41no—
280C22×C32C2×C4×D311yescharacteristic
281C22×C32#23511yescharacteristic
282C22×C32D3×D411yescharacteristic
283C22×C32D3×D411yescharacteristic
284C22×C32D3×D411yescharacteristic
285C22×C32#23711yescharacteristic
286C22×C32C2×#6711yescharacteristic
287C2×C3×D3—62no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C2×C3×D3—121no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C2×C3×D3—121no—
294C3⋊Dic3—62no—
295C3⋊Dic3—121no—
296C3×Dic3—22no—
297C3×Dic3—34no—
298C3×Dic3—41no—
299C22×C9—24no—
300C22×C9—22no—
301C22×C9—22no—
302C22×C9—22no—
303C22×C9—21no—
304C22×C9—21no—
305C22×C9—21no—
306C22×C9—21no—
307C22×C9—21no—
308C22×C9—21no—
309C22×C9—21no—
310C22×C9—42no—
311C22×C9—42no—
312C22×C9—42no—
313C22×C9—42no—
314C22×C9—42no—
315C22×C9—42no—
316C22×C9—41no—
317C22×C9—41no—
318C22×C9—41no—
319C22×C9—41no—
320C22×C9D3×D411yescharacteristic
321C22×C9D3×D411yescharacteristic
322C22×C9#23711yescharacteristic
323C22×C9#23711yescharacteristic
324C22×C9C22×Dic311yescharacteristic
325C22×C9C2×#6711yescharacteristic
326C22×C9C2×#6711yescharacteristic
327C2×D9—22no—
328C2×D9—22no—
329C2×D9—22no—
330C2×D9—41no—
331C2×D9—41no—
332C2×D9—41no—
333C2×D9—41no—
334C4×C9—62no—
335C4×C9—121no—
336Dic9—62no—
337Dic9—122no—
338Dic9—122no—
339Dic9—121no—
340Dic9#21714yescomplemented
341C22×D4—91no—
342C2×C4⋊C4—271no—
343C2×C22⋊C4—31no—
344C2×C22⋊C4—271no—
345C2×C22⋊C4—271no—
346C2×C22⋊C4—271no—
347C2×C42—271no—
348C3×C9#39011yescharacteristic, complemented, Sylow
349C23×C3—22no—
350C23×C3—22no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—21no—
357C23×C3—21no—
358C23×C3—21no—
359C23×C3—21no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—22no—
364C23×C3—21no—
365C23×C3—41no—
366C23×C3—41no—
367C23×C3—41no—
368C23×C3—41no—
369C23×C3—92no—
370C23×C3—181no—
371C23×C3—181no—
372C23×C3—181no—
373C23×C3—181no—
374C23×C3—181no—
375C23×C3—181no—
376C23×C3C4×D912yes—
377C23×C3C22×D911yescharacteristic
378C23×C3D3×Dic312yes—
379C23×C3C2×D3211yescharacteristic
380C22×D3—32no—
381C22×D3—61no—
382C22×D3—61no—
383C22×D3—61no—
384C22×D3—61no—
385C22×D3—61no—
386C22×D3—61no—
387C3×D4—188no—
388C2×C3×C4—32no—
389C2×C3×C4—61no—
390C2×C3×C4—61no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—92no—
395C2×C3×C4—92no—
396C2×C3×C4—92no—
397#67—68no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—32no—
404C2×Dic3—32no—
405C2×Dic3—32no—
406C2×Dic3—92no—
407C2×Dic3—92no—
408C2×Dic3—182no—
409C2×Dic3—181no—
410C2×Dic3—181no—
411C2×Dic3—181no—
412C2×Dic3—181no—
413C2×Dic3—182no—
414C2×Dic3—181no—
415C2×Dic3—181no—
416C2×Dic3—181no—
417C2×Dic3—181no—
418C2×Dic3—181no—
419C2×Dic3—181no—
420C2×Dic3—361no—
421C2×Dic3—361no—
422C2×Dic3—361no—
423C2×Dic3—361no—
424C2×Dic3#61112yescomplemented
425C2×C32—42no—
426C2×C32D3×C22⋊C411yescharacteristic
427C2×C32#67⋊C411yescharacteristic
428C2×C32#88411yescharacteristic
429C2×C32#88511yescharacteristic
430C2×C32C4×#6711yescharacteristic
431C2×C32#93911yescharacteristic
432C2×C32#94011yescharacteristic
433C3×D3—122no—
434C2×C9—21no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—21no—
438C2×C9—21no—
439C2×C9—21no—
440C2×C9—21no—
441C2×C9—44no—
442C2×C9—42no—
443C2×C9D4×Dic311yescharacteristic
444C2×C9D4×Dic311yescharacteristic
445C2×C9#93611yescharacteristic
446C2×C9#93811yescharacteristic
447C2×C9#93811yescharacteristic
448C2×C9#94011yescharacteristic
449C2×C9C2×#21711yescharacteristic
450D9—42no—
451C24—91no—
452C24D3×D911yescharacteristic
453C2×D4—94no—
454C2×D4—182no—
455C2×D4—182no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—91no—
459C22×C4—271no—
460C22×C4—271no—
461C4⋊C4—542no—
462C22⋊C4—62no—
463C22⋊C4—542no—
464C22⋊C4—542no—
465C22⋊C4—542no—
466C42—542no—
467C22×C3—24no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—21no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—21no—
479C22×C3—21no—
480C22×C3—21no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—41no—
495C22×C3—182no—
496C22×C3—182no—
497C22×C3—182no—
498C22×C3—361no—
499C22×C3—361no—
500C22×C3—361no—
501C22×C3—361no—
502C22×C3C2×C4×D911yescharacteristic
503C22×C3#369711yescharacteristic
504C22×C3D4×D911yescharacteristic
505C22×C3D4×D911yescharacteristic
506C22×C3D4×D911yescharacteristic
507C22×C3#369911yescharacteristic
508C22×C3C2×#61111yescharacteristic
509C22×C3C2×D3×Dic311yescharacteristic
510C22×C3#380411yescharacteristic
511C22×C3#380511yescharacteristic
512C22×C3D3×#6711yescharacteristic
513C22×C3D3×#6711yescharacteristic
514C22×C3D3×#6711yescharacteristic
515C22×C3#381111yescharacteristic
516C2×D3—62no—
517C2×D3—62no—
518C2×D3—62no—
519C2×D3—121no—
520C2×D3—121no—
521C2×D3—121no—
522C2×D3—121no—
523C3×C4—62no—
524C3×C4—94no—
525C3×C4—121no—
526Dic3—22no—
527Dic3—34no—
528Dic3—41no—
529Dic3—182no—
530Dic3—182no—
531Dic3—362no—
532Dic3—362no—
533Dic3—361no—
534Dic3—361no—
535C32#495711yescharacteristic
536C9—21no—
537C9#522111yescharacteristic, complemented
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—92no—
545C23—181no—
546C23—181no—
547C23—181no—
548C23—181no—
549C23—181no—
550C23—181no—
551C23Dic3×D912yes—
552C23C2×D3×D911yescharacteristic
553D4—188no—
554C2×C4—32no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—92no—
561C2×C4—92no—
562C2×C4—92no—
563C2×C4—272no—
564C2×C4—541no—
565C2×C4—541no—
566C2×C4—541no—
567C2×C4—541no—
568C2×C4—541no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—44no—
577C2×C3—42no—
578C2×C3—42no—
579C2×C3—362no—
580C2×C3C22⋊C4×D911yescharacteristic
581C2×C3#611⋊C411yescharacteristic
582C2×C3#6344611yescharacteristic
583C2×C3#6344711yescharacteristic
584C2×C3C4×#61111yescharacteristic
585C2×C3#6350111yescharacteristic
586C2×C3#6350211yescharacteristic
587C2×C3#6395811yescharacteristic
588C2×C3#6395911yescharacteristic
589C2×C3D3×#21711yescharacteristic
590C2×C3#6397011yescharacteristic
591C2×C3Dic3×#6711yescharacteristic
592C2×C3#67⋊Dic311yescharacteristic
593C2×C3#6397911yescharacteristic
594D3—122no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—182no—
603C22—182no—
604C22—182no—
605C22—361no—
606C22—361no—
607C22—361no—
608C22—361no—
609C22C2×Dic3×D911yescharacteristic
610C22#8865811yescharacteristic
611C22#8866311yescharacteristic
612C22D3×#61111yescharacteristic
613C22D9×#6711yescharacteristic
614C22D9×#6711yescharacteristic
615C22#8866811yescharacteristic
616C4—62no—
617C4—94no—
618C4—121no—
619C4—542no—
620C4—1081no—
621C3—21no—
622C3#1058863711yescharacteristic, complemented
623C3#1059170711yescharacteristic
624C2—42no—
625C2—362no—
626C2#1172125611yescharacteristic
627C2#1172125711yescharacteristic
628C2D9×#21711yescharacteristic
629C2Dic3×#61111yescharacteristic
630C2#611⋊Dic311yescharacteristic
631C2#1172128111yescharacteristic
632C2#1172128911yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC22×C3×C9C22×C4
FrattiniC23×C3C2×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC22×C3×#611C2
cosocleC23×C3×C9C23
hypercenterC24D3×D9
hypocenterC3×C9#390
LCSG ▷ C22×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×Dic9⋊Dic3C2semidirect
7. C2×Dic3×Dic9C2semidirect
16. C2×C3×#611C4semidirect
61. C2×C3⋊Dic9D4semidirect
68. C2×C9×Dic3D4semidirect
97. C2×#611Dic3semidirect
135. C3×Dic9C22⋊C4semidirect
144. C2×#217D9semidirect
340. Dic9#217semidirect
348. C3×C9#390semidirect
424. C2×Dic3#611semidirect
537. C9#5221semidirect
622. C3#10588637semidirect

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×Dic9⋊Dic32
C2×Dic3×Dic92
#105917073
#105886373