Finite Group Databasebeta

← #13148024

Subgroups — #13148024

Order 1728.

1078 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6194C211yesmaximal, characteristic, complemented
2C2×#88663C211yesmaximal, characteristic, complemented
3#11721723C211yesmaximal, characteristic, complemented
4C9×#882C211yesmaximal, characteristic, complemented
5C3×#63444C211yesmaximal, characteristic, complemented
6#11721286C211yesmaximal, characteristic, complemented
7#11721285C211yesmaximal, characteristic, complemented
8#11721044C212yesmaximal, complemented
9#11721043C212yesmaximal, complemented
10#11721016C212yesmaximal, complemented
11#11720982C212yesmaximal, complemented
12#10594192—31nomaximal
13#10589694—31nomaximal
14C23×D3×C9C2211yescharacteristic
15C23×C3×D9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C9×#67C2211yescharacteristic
18C2×C3×#611C2211yescharacteristic
19C2×C9×D12C2212yescomplemented
20C2×C3×D36C2212yescomplemented
21C2×#6194—24no—
22C2×#6194C2212yes—
23C2×#6194C2212yes—
24C2×#6194C2211yescharacteristic, complemented
25C2×#6194C2212yescomplemented
26C2×#6194C2212yescomplemented
27#88663—24no—
28C2×D3×Dic9C2211yescharacteristic, complemented
29C2×Dic3×D9C2211yescharacteristic, complemented
30C2×#6189C2211yescharacteristic, complemented
31C3×C9×C22⋊C4C2211yescharacteristic
32#88551C2211yescharacteristic
33#88534C2212yescomplemented
34C4×C3⋊Dic9C2212yescomplemented
35C9×(C2×D3)⋊C4C2212yescomplemented
36C3×(C2×D9)⋊C4C2212yescomplemented
37(C2×D3)⋊Dic9C2212yescomplemented
38(C2×D9)⋊Dic3C2212yescomplemented
39C22×#625—31no—
40C2×#3805—31no—
41#64086—31no—
42C3×#882—31no—
43C3×#882D311yescharacteristic
44#63976—31no—
45#63976—31no—
46#63915—32no—
47#63914—32no—
48#63885—32no—
49#63885—32no—
50C22×#611—31no—
51C2×#3699—31no—
52C9×#120—31no—
53#63515—31no—
54#63502—32no—
55#63501—32no—
56#63499—31no—
57#63447—32no—
58#63446—32no—
59#63444D311yescharacteristic, complemented
60#63439—31no—
61#63439—61no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9—22no—
64C22×D3×C9—42no—
65C22×D3×C9C2312yes—
66C22×D3×C9C2311yescharacteristic
67C22×C3×D9—22no—
68C22×C3×D9—42no—
69C22×C3×D9C2312yes—
70C22×C3×D9C2311yescharacteristic
71C2×C3×C4×C9C2312yes—
72C2×C3⋊Dic9D414yescomplemented
73C2×C3⋊Dic9C2312yes—
74C2×C3⋊Dic9C2311yescharacteristic
75C2×C3⋊Dic9C2311yescharacteristic
76C9×#67—22no—
77C3×#611—22no—
78C2×C9×Dic3C2311yescharacteristic
79C2×C3×Dic9C2311yescharacteristic
80C9×D12—42no—
81C3×D36—42no—
82#6194—22no—
83#6194—24no—
84#6194—24no—
85#6194—42no—
86#6194—42no—
87D3×Dic9—22no—
88Dic3×D9—22no—
89#6189—22no—
90#5592—91no—
91#5592—91no—
92C23×C3×D3—31no—
93C23×C3×D3C2×D311yescharacteristic
94C22×C3⋊Dic3—31no—
95C2×C3×#67—31no—
96C2×C3×#67C2×D311yescharacteristic
97C2×C3×D12—32no—
98C2×C3×D12C2×D312yes—
99C2×#627—31no—
100C2×#625—31no—
101C2×#625—32no—
102C2×#625—32no—
103C2×#625—32no—
104C2×#625—32no—
105C2×#625—64no—
106#3805—64no—
107C2×D3×Dic3—31no—
108C2×D3×Dic3—31no—
109C24×C9—31no—
110C23×D9C2×D311yescharacteristic
111C32×C22⋊C4C2×D311yescharacteristic
112#3757—31no—
113#3750—32no—
114C4×C3⋊Dic3—32no—
115C3×(C2×D3)⋊C4—32no—
116C3×(C2×D3)⋊C4C2×D312yes—
117(C2×D3)⋊Dic3—32no—
118(C2×D3)⋊Dic3—32no—
119C2×D4×C9—31no—
120C2×D4×C9—32no—
121C2×#611—32no—
122C2×#611—31no—
123C2×#611—32no—
124C2×#611—32no—
125C2×#611—32no—
126C2×#611—64no—
127C2×#611C2×D311yescharacteristic
128C22×Dic9—31no—
129C22×Dic9—31no—
130C22×Dic9—61no—
131#3699—64no—
132C2×D36C2×D312yescomplemented
133C2×C4×D9—31no—
134C2×Dic18—31no—
135C9×C22⋊C4—21no—
136C9×C22⋊C4—32no—
137C9×C22⋊C4C2×D311yescharacteristic
138#3676—31no—
139#3676—32no—
140#3676—61no—
141(C2×D9)⋊C4—32no—
142(C2×D9)⋊C4C2×D312yescomplemented
143Dic9⋊C4—32no—
144Dic9⋊C4—62no—
145C4×Dic9—32no—
146C4×Dic9—62no—
147C22×C3×C9—22no—
148C22×C3×C9C2×D412yes—
149C22×C3×C9C2411yescharacteristic
150C2×D3×C9—21no—
151C2×D3×C9—22no—
152C2×D3×C9—22no—
153C2×D3×C9—22no—
154C2×D3×C9—22no—
155C2×D3×C9—44no—
156C2×C3×D9—21no—
157C2×C3×D9—22no—
158C2×C3×D9—22no—
159C2×C3×D9—22no—
160C2×C3×D9—22no—
161C2×C3×D9—44no—
162C3×C4×C9—22no—
163C3⋊Dic9—22no—
164C3⋊Dic9C2×D412yes—
165C3⋊Dic9C2×D412yes—
166C9×Dic3—21no—
167C3×Dic9—21no—
168C22×#67—91no—
169C22×#67—91no—
170C2×#237—91no—
171C2×#237—91no—
172C3×#120—31no—
173C3×#120—91no—
174#953—91no—
175#953—91no—
176#940—92no—
177#940—92no—
178#939—92no—
179#939—92no—
180#937—91no—
181#937—91no—
182#885—92no—
183#885—92no—
184#884—92no—
185#884—92no—
186#882—31no—
187#882D911yescharacteristic, complemented
188#877—91no—
189#877—91no—
190#877—181no—
191C23×C32C22×D311yescharacteristic
192C22×C3×D3—22no—
193C22×C3×D3—32no—
194C22×C3×D3—31no—
195C22×C3×D3—42no—
196C22×C3×D3—62no—
197C22×C3×D3—122no—
198C22×C3×D3C22×D312yes—
199C22×C3×D3C22×D311yescharacteristic
200C2×C32×C4C22×D312yes—
201C2×C3⋊Dic3—34no—
202C2×C3⋊Dic3—31no—
203C2×C3⋊Dic3—31no—
204C2×C3⋊Dic3—32no—
205C3×#67—22no—
206C3×#67—62no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3C22×D311yescharacteristic
209C3×D12—42no—
210C3×D12—122no—
211#627—62no—
212#625—62no—
213#625—64no—
214#625—64no—
215#625—122no—
216#625—122no—
217D3×Dic3—62no—
218D3×Dic3—62no—
219C23×C9—21no—
220C23×C9—32no—
221C23×C9—31no—
222C23×C9—62no—
223C23×C9—122no—
224C23×C9C22×D311yescharacteristic
225C22×D9—22no—
226C22×D9—42no—
227C22×D9C22×D312yes—
228C22×D9C22×D311yescharacteristic
229D4×C9—62no—
230D4×C9—122no—
231C2×C4×C9—22no—
232C2×C4×C9—31no—
233C2×C4×C9C22×D312yes—
234#611—22no—
235#611—62no—
236#611—64no—
237#611—64no—
238#611—122no—
239#611—122no—
240C2×Dic9—34no—
241C2×Dic9—31no—
242C2×Dic9—31no—
243C2×Dic9—32no—
244C2×Dic9—61no—
245C2×Dic9—61no—
246C2×Dic9—64no—
247C2×Dic9—62no—
248C2×Dic9—62no—
249C2×Dic9C22×D311yescharacteristic
250D36—42no—
251C4×D9—62no—
252Dic18—62no—
253#535—271noSylow
254C2×C3×C9—22no—
255C2×C3×C9C22×D411yescharacteristic
256C2×C3×C9H(5,𝔽2)12yes—
257D3×C9—41no—
258D3×C9—42no—
259C3×D9—41no—
260C3×D9—42no—
261C24×C3—31no—
262C24×C3—91no—
263C23×D3—31no—
264C23×D3C2×D911yescharacteristic
265C2×C3×D4—31no—
266C2×C3×D4—32no—
267C2×C3×D4—91no—
268C2×C3×D4—92no—
269C2×#67—31no—
270C2×#67—92no—
271C2×#67—92no—
272C2×#67—91no—
273C2×#67—92no—
274C2×#67—92no—
275C2×#67—91no—
276C2×#67—92no—
277C2×#67—92no—
278C2×#67—92no—
279C2×#67—92no—
280C2×#67—184no—
281C2×#67—184no—
282C2×#67C2×D911yescharacteristic
283C22×Dic3—91no—
284C22×Dic3—91no—
285C22×Dic3—91no—
286C22×Dic3—91no—
287C22×Dic3—181no—
288#237—184no—
289#237—184no—
290C2×D12—32no—
291C2×D12C2×D912yescomplemented
292C2×C4×D3—91no—
293C2×C4×D3—91no—
294C2×Dic6—91no—
295C2×Dic6—91no—
296C3×C22⋊C4—21no—
297C3×C22⋊C4—32no—
298C3×C22⋊C4—92no—
299C3×C22⋊C4C2×D911yescharacteristic
300C3×C22⋊C4D3211yescharacteristic
301#217—92no—
302#217—91no—
303#217—91no—
304#217—92no—
305#217—181no—
306(C2×D3)⋊C4—32no—
307(C2×D3)⋊C4—92no—
308(C2×D3)⋊C4—92no—
309(C2×D3)⋊C4C2×D912yescomplemented
310Dic3⋊C4—92no—
311Dic3⋊C4—92no—
312Dic3⋊C4—182no—
313C4×Dic3—92no—
314C4×Dic3—92no—
315C4×Dic3—182no—
316C22×C32—22no—
317C22×C32D3×D412yes—
318C22×C32C23×D311yescharacteristic
319C2×C3×D3—21no—
320C2×C3×D3—22no—
321C2×C3×D3—22no—
322C2×C3×D3—22no—
323C2×C3×D3—22no—
324C2×C3×D3—44no—
325C2×C3×D3—61no—
326C2×C3×D3—62no—
327C2×C3×D3—62no—
328C2×C3×D3—62no—
329C2×C3×D3—62no—
330C2×C3×D3—124no—
331C32×C4—22no—
332C3⋊Dic3—32no—
333C3⋊Dic3—32no—
334C3⋊Dic3—62no—
335C3×Dic3—21no—
336C3×Dic3—61no—
337C22×C9—22no—
338C22×C9—21no—
339C22×C9—22no—
340C22×C9—42no—
341C22×C9—61no—
342C22×C9—62no—
343C22×C9—62no—
344C22×C9—62no—
345C22×C9—62no—
346C22×C9—124no—
347C22×C9D3×D412yes—
348C22×C9C23×D311yescharacteristic
349C2×D9—21no—
350C2×D9—22no—
351C2×D9—22no—
352C2×D9—22no—
353C2×D9—22no—
354C2×D9—44no—
355C4×C9—22no—
356C4×C9—42no—
357C4×C9—61no—
358Dic9—21no—
359Dic9—32no—
360Dic9—32no—
361Dic9—62no—
362Dic9—62no—
363Dic9—62no—
364Dic9—122no—
365C2×#41—271no—
366C22×D4—271no—
367#127—272no—
368#124—272no—
369#121—272no—
370#121—272no—
371#120—31no—
372#120—91no—
373#120—271no—
374#120—271no—
375#117—271no—
376C3×C9#53511yescharacteristic, complemented, Sylow
377C23×C3—21no—
378C23×C3—32no—
379C23×C3—31no—
380C23×C3—62no—
381C23×C3—92no—
382C23×C3—91no—
383C23×C3—122no—
384C23×C3—182no—
385C23×C3—362no—
386C23×C3C22×D911yescharacteristic
387C23×C3C2×D3211yescharacteristic
388C22×D3—22no—
389C22×D3—32no—
390C22×D3—31no—
391C22×D3—42no—
392C22×D3—62no—
393C22×D3—122no—
394C22×D3C22×D912yes—
395C22×D3C22×D911yescharacteristic
396C3×D4—62no—
397C3×D4—122no—
398C3×D4—182no—
399C3×D4—362no—
400C2×C3×C4—22no—
401C2×C3×C4—31no—
402C2×C3×C4—91no—
403C2×C3×C4C22×D912yes—
404C2×C3×C4C2×D3212yes—
405#67—22no—
406#67—62no—
407#67—182no—
408#67—184no—
409#67—184no—
410#67—182no—
411#67—184no—
412#67—184no—
413#67—362no—
414#67—362no—
415#67—362no—
416#67—362no—
417C2×Dic3—31no—
418C2×Dic3—91no—
419C2×Dic3—91no—
420C2×Dic3—94no—
421C2×Dic3—91no—
422C2×Dic3—91no—
423C2×Dic3—94no—
424C2×Dic3—92no—
425C2×Dic3—92no—
426C2×Dic3—181no—
427C2×Dic3—181no—
428C2×Dic3—184no—
429C2×Dic3—182no—
430C2×Dic3—182no—
431C2×Dic3—182no—
432C2×Dic3C22×D911yescharacteristic
433D12—42no—
434D12—122no—
435C4×D3—182no—
436C4×D3—182no—
437Dic6—182no—
438Dic6—182no—
439C2×C32—22no—
440C2×C32C2×D3×D411yescharacteristic
441C2×C32#100412yes—
442C3×D3—41no—
443C3×D3—42no—
444C3×D3—121no—
445C3×D3—122no—
446C2×C9—22no—
447C2×C9—21no—
448C2×C9—22no—
449C2×C9—42no—
450C2×C9—121no—
451C2×C9—122no—
452C2×C9C2×D3×D411yescharacteristic
453C2×C9#100412yes—
454D9—41no—
455D9—42no—
456C24—31no—
457C24—91no—
458#41—544no—
459C2×Q8—271no—
460C2×D4—31no—
461C2×D4—32no—
462C2×D4—91no—
463C2×D4—92no—
464C2×D4—272no—
465C2×D4—272no—
466C2×D4—271no—
467C2×D4—272no—
468C2×D4—272no—
469C2×D4—544no—
470C22×C4—271no—
471C22×C4—271no—
472C22×C4—271no—
473C4⋊C4—272no—
474C22⋊C4—32no—
475C22⋊C4—92no—
476C22⋊C4—272no—
477C22⋊C4—272no—
478C22⋊C4—271no—
479C22⋊C4D3×D911yescharacteristic
480C42—272no—
481C22×C3—22no—
482C22×C3—22no—
483C22×C3—21no—
484C22×C3—22no—
485C22×C3—42no—
486C22×C3—61no—
487C22×C3—62no—
488C22×C3—62no—
489C22×C3—62no—
490C22×C3—62no—
491C22×C3—124no—
492C22×C3—181no—
493C22×C3—182no—
494C22×C3—182no—
495C22×C3—182no—
496C22×C3—182no—
497C22×C3—364no—
498C22×C3D4×D912yes—
499C22×C3C23×D911yescharacteristic
500C22×C3#380212yes—
501C22×C3C22×D3211yescharacteristic
502C2×D3—21no—
503C2×D3—22no—
504C2×D3—22no—
505C2×D3—22no—
506C2×D3—22no—
507C2×D3—44no—
508C2×D3—61no—
509C2×D3—62no—
510C2×D3—62no—
511C2×D3—62no—
512C2×D3—62no—
513C2×D3—124no—
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—42no—
517C3×C4—61no—
518C3×C4—181no—
519Dic3—21no—
520Dic3—61no—
521Dic3—92no—
522Dic3—92no—
523Dic3—92no—
524Dic3—92no—
525Dic3—182no—
526Dic3—182no—
527Dic3—182no—
528Dic3—182no—
529Dic3—362no—
530C32#559211yescharacteristic
531C9—21no—
532C9#559211yescharacteristic, complemented
533C23—32no—
534C23—31no—
535C23—62no—
536C23—92no—
537C23—91no—
538C23—122no—
539C23—182no—
540C23—362no—
541C23C2×D3×D911yescharacteristic
542Q8—542no—
543D4—62no—
544D4—122no—
545D4—182no—
546D4—362no—
547D4—544no—
548D4—544no—
549D4—542no—
550D4—1082no—
551D4—1082no—
552C2×C4—31no—
553C2×C4—91no—
554C2×C4—271no—
555C2×C4—271no—
556C2×C4—274no—
557C2×C4—272no—
558C2×C4—542no—
559C2×C4—542no—
560C2×C4C2×D3×D912yes—
561C2×C3—22no—
562C2×C3—22no—
563C2×C3—22no—
564C2×C3—21no—
565C2×C3—42no—
566C2×C3—122no—
567C2×C3—121no—
568C2×C3—362no—
569C2×C3—361no—
570C2×C3C2×D4×D911yescharacteristic
571C2×C3#6371012yes—
572C2×C3C2×#380211yescharacteristic
573C2×C3#6431312yes—
574D3—42no—
575D3—41no—
576D3—122no—
577D3—121no—
578C22—22no—
579C22—62no—
580C22—62no—
581C22—62no—
582C22—61no—
583C22—62no—
584C22—124no—
585C22—182no—
586C22—182no—
587C22—182no—
588C22—181no—
589C22—182no—
590C22—364no—
591C22#8864612yes—
592C22C22×D3×D911yescharacteristic
593C4—22no—
594C4—61no—
595C4—181no—
596C4—272no—
597C4—272no—
598C4—542no—
599C3—21no—
600C3#1058969411yescharacteristic, complemented
601C3#1059419211yescharacteristic
602C2—22no—
603C2—122no—
604C2—121no—
605C2—362no—
606C2—361no—
607C2C2×#8864611yescharacteristic
608C2#1172249912yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
hypercenterC22⋊C4D3×D9
hypocenterC3×C9#535
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#6194C2semidirect
2. C2×#88663C2semidirect
3. #11721723C2semidirect
4. C9×#882C2semidirect
5. C3×#63444C2semidirect
6. #11721286C2semidirect
7. #11721285C2semidirect
8. #11721044C2semidirect
9. #11721043C2semidirect
10. #11721016C2semidirect
11. #11720982C2semidirect
19. C2×C9×D12C22semidirect
20. C2×C3×D36C22semidirect
24. C2×#6194C22semidirect
25. C2×#6194C22semidirect
26. C2×#6194C22semidirect
28. C2×D3×Dic9C22semidirect
29. C2×Dic3×D9C22semidirect
30. C2×#6189C22semidirect
33. #88534C22semidirect
34. C4×C3⋊Dic9C22semidirect
35. C9×(C2×D3)⋊C4C22semidirect
36. C3×(C2×D9)⋊C4C22semidirect
37. (C2×D3)⋊Dic9C22semidirect
38. (C2×D9)⋊Dic3C22semidirect
59. #63444D3semidirect
72. C2×C3⋊Dic9D4semidirect
132. C2×D36C2×D3semidirect
142. (C2×D9)⋊C4C2×D3semidirect
187. #882D9semidirect
291. C2×D12C2×D9semidirect
309. (C2×D3)⋊C4C2×D9semidirect
376. C3×C9#535semidirect
532. C9#5592semidirect
600. C3#10589694semidirect

Maximal subgroups

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

subgroupindex
C22×#61942
C2×#886632
#117217232
C9×#8822
C3×#634442
#117212862
#117212852
#117210442
#117210432
#117210162
#117209822
#105941923
#105896943