Finite Group Databasebeta

← #13150221

Subgroups — #13150221

Order 1728.

1226 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic, complemented
2C2×#88663C211yesmaximal, characteristic, complemented
3C9×#929C211yesmaximal, characteristic, complemented
4C3×#63491C211yesmaximal, characteristic, complemented
5#11721291C211yesmaximal, characteristic, complemented
6#11721278C211yesmaximal, characteristic, complemented
7#11721248C211yesmaximal, characteristic, complemented
8#11721041C212yesmaximal, complemented
9#11721014C212yesmaximal, complemented
10#11720978C212yesmaximal, complemented
11#11720923C212yesmaximal, complemented
12#10594920—31nomaximal
13#10589758—31nomaximal
14C23×C3⋊D9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C4×C3⋊D9—24no—
18C2×C4×C3⋊D9C2212yes—
19C2×C4×C3⋊D9C2212yes—
20C2×C9×#67C2211yescharacteristic
21C2×C3×#611C2211yescharacteristic
22C2×#6195C2212yes—
23C2×#6194C2211yescharacteristic, complemented
24#88663—24no—
25C2×D3×Dic9C2211yescharacteristic, complemented
26C2×#6192C2212yes—
27C2×Dic3×D9C2211yescharacteristic, complemented
28C2×#6189C2211yescharacteristic, complemented
29C9×#217C2211yescharacteristic
30C3×#3676C2211yescharacteristic
31C9×(C2×D3)⋊C4C2212yescomplemented
32C3×(C2×D9)⋊C4C2212yescomplemented
33C9×Dic3⋊C4C2212yescomplemented
34C3×Dic9⋊C4C2212yescomplemented
35#88445C2212yes—
36#88445C2212yes—
37#88442C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C22×C4×C3⋊D3—31no—
40C2×#3805—31no—
41C3×#929—31no—
42C3×#929D311yescharacteristic
43#63981—31no—
44#63981—31no—
45#63957—31no—
46#63912—32no—
47#63881—32no—
48#63881—32no—
49#63865—32no—
50C2×#3699—31no—
51C22×C4×D9—31no—
52C22×C4×D9—61no—
53C9×#123—31no—
54#63499—31no—
55#63491D311yescharacteristic, complemented
56#63458—32no—
57#63455—32no—
58#63446—32no—
59#63444—31no—
60#611⋊C4—32no—
61#63439—31no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9—22no—
64C22×C3⋊D9—24no—
65C22×C3⋊D9C2312yes—
66C22×D3×C9C2311yescharacteristic
67C22×C3×D9C2311yescharacteristic
68C2×C3×C4×C9—22no—
69C2×C3×C4×C9C2312yes—
70C2×C3⋊Dic9D414yescomplemented
71C2×C3⋊Dic9C2311yescharacteristic
72C2×C3⋊Dic9C2311yescharacteristic
73C4×C3⋊D9—24no—
74C4×C3⋊D9—24no—
75C9×#67—22no—
76C3×#611—22no—
77C2×C9×Dic3C2311yescharacteristic
78C2×C9×Dic3C2312yes—
79C2×C3×Dic9C2311yescharacteristic
80C2×C3×Dic9C2312yes—
81#6195—24no—
82#6194—22no—
83D3×Dic9—22no—
84#6192—24no—
85Dic3×D9—22no—
86#6189—22no—
87#5656—91no—
88#5656—91no—
89C23×C3⋊D3—31no—
90C22×C32×C4C2×D311yescharacteristic
91C22×C3⋊Dic3—31no—
92C2×C4×C3⋊D3—32no—
93C2×C4×C3⋊D3—32no—
94C2×C4×C3⋊D3—64no—
95C2×C3×#67—31no—
96C2×C3×#67C2×D311yescharacteristic
97C2×#627—31no—
98C2×#626—32no—
99C2×#626—32no—
100C2×#625—31no—
101#3805—64no—
102C2×D3×Dic3—31no—
103C2×D3×Dic3—31no—
104C23×D9—31no—
105C23×D9—61no—
106C3×#217—31no—
107C3×#217C2×D311yescharacteristic
108C3×(C2×D3)⋊C4—32no—
109C3×(C2×D3)⋊C4C2×D312yes—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4C2×D312yes—
112#3724—32no—
113#3722—32no—
114#3722—32no—
115Dic32—32no—
116C2×D4×C9—31no—
117C22×C4×C9—21no—
118C22×C4×C9C2×D311yescharacteristic
119C2×#611—31no—
120C2×#611—32no—
121C2×#611C2×D311yescharacteristic
122C22×Dic9—31no—
123C22×Dic9—31no—
124C22×Dic9—61no—
125#3699—64no—
126C2×D36—32no—
127C2×C4×D9—31no—
128C2×C4×D9—32no—
129C2×C4×D9—32no—
130C2×C4×D9—64no—
131C2×C4×D9—62no—
132C2×C4×D9—62no—
133C2×C4×D9—68no—
134C2×Dic18—31no—
135C9×C4⋊C4—32no—
136C9×C22⋊C4—31no—
137C9×C22⋊C4—32no—
138#3676C2×D311yescharacteristic
139(C2×D9)⋊C4—32no—
140(C2×D9)⋊C4—32no—
141(C2×D9)⋊C4C2×D312yescomplemented
142Dic9⋊C4—32no—
143Dic9⋊C4C2×D312yescomplemented
144C4×Dic9—32no—
145C22×C3×C9—22no—
146C22×C3×C9C2×D412yes—
147C22×C3×C9C2411yescharacteristic
148C2×C3⋊D9—28no—
149C2×C3⋊D9—22no—
150C2×C3⋊D9—22no—
151C2×C3⋊D9#4114yescomplemented
152C2×D3×C9—21no—
153C2×D3×C9—22no—
154C2×C3×D9—21no—
155C2×C3×D9—22no—
156C3×C4×C9—22no—
157C3⋊Dic9C2×D412yes—
158C3⋊Dic9C2×D412yes—
159C9×Dic3—21no—
160C9×Dic3—22no—
161C3×Dic9—21no—
162C3×Dic9—22no—
163C2×#237—91no—
164C2×#237—91no—
165C22×C4×D3—91no—
166C22×C4×D3—91no—
167C22×C4×D3—181no—
168C3×#123—31no—
169C3×#123—91no—
170#937—91no—
171#937—91no—
172#929—31no—
173#929D911yescharacteristic, complemented
174#896—92no—
175#896—92no—
176#893—92no—
177#893—92no—
178#884—92no—
179#884—92no—
180#882—91no—
181#882—91no—
182#67⋊C4—92no—
183#67⋊C4—92no—
184#877—91no—
185#877—91no—
186C23×C32C22×D311yescharacteristic
187C22×C3⋊D3—32no—
188C22×C3⋊D3—64no—
189C22×C3⋊D3—62no—
190C22×C3×D3—31no—
191C22×C3×D3C22×D311yescharacteristic
192C2×C32×C4—22no—
193C2×C32×C4C22×D312yes—
194C2×C3⋊Dic3—34no—
195C2×C3⋊Dic3—31no—
196C2×C3⋊Dic3—31no—
197C4×C3⋊D3—64no—
198C4×C3⋊D3—64no—
199C3×#67—22no—
200C3×#67—62no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3C22×D312yes—
204C2×C3×Dic3C22×D311yescharacteristic
205#627—62no—
206#626—64no—
207#626—64no—
208#625—62no—
209D3×Dic3—62no—
210D3×Dic3—62no—
211C23×C9—21no—
212C23×C9—31no—
213C23×C9C22×D311yescharacteristic
214C22×D9—32no—
215C22×D9—62no—
216C22×D9—64no—
217C22×D9—64no—
218C22×D9—68no—
219C22×D9—62no—
220C22×D9C22×D311yescharacteristic
221D4×C9—62no—
222C2×C4×C9—24no—
223C2×C4×C9—22no—
224C2×C4×C9—22no—
225C2×C4×C9—32no—
226C2×C4×C9—31no—
227C2×C4×C9C22×D312yes—
228#611—22no—
229#611—64no—
230#611—62no—
231C2×Dic9—34no—
232C2×Dic9—31no—
233C2×Dic9—31no—
234C2×Dic9—64no—
235C2×Dic9—61no—
236C2×Dic9—61no—
237C2×Dic9—62no—
238C2×Dic9C22×D312yes—
239C2×Dic9C22×D311yescharacteristic
240D36—64no—
241C4×D9—64no—
242C4×D9—64no—
243C4×D9—62no—
244C4×D9—68no—
245C4×D9—68no—
246Dic18—62no—
247#525—271noSylow
248C2×C3×C9—22no—
249C2×C3×C9C22×D411yescharacteristic
250C2×C3×C9C2×#4112yes—
251C3⋊D9—24no—
252D3×C9—41no—
253C3×D9—41no—
254C23×D3—91no—
255C23×D3—91no—
256C23×D3—181no—
257C2×C3×D4—31no—
258C2×C3×D4—91no—
259C22×C3×C4—21no—
260C22×C3×C4C2×D911yescharacteristic
261C22×C3×C4D3211yescharacteristic
262C2×#67—31no—
263C2×#67—92no—
264C2×#67—91no—
265C2×#67—91no—
266C2×#67—92no—
267C2×#67C2×D911yescharacteristic
268C22×Dic3—91no—
269C22×Dic3—91no—
270C22×Dic3—91no—
271C22×Dic3—91no—
272C22×Dic3—181no—
273#237—184no—
274#237—184no—
275C2×D12—92no—
276C2×D12—92no—
277C2×C4×D3—91no—
278C2×C4×D3—91no—
279C2×C4×D3—92no—
280C2×C4×D3—92no—
281C2×C4×D3—92no—
282C2×C4×D3—92no—
283C2×C4×D3—182no—
284C2×C4×D3—182no—
285C2×C4×D3—184no—
286C2×C4×D3—184no—
287C2×C4×D3—188no—
288C2×Dic6—91no—
289C2×Dic6—91no—
290C3×C4⋊C4—32no—
291C3×C4⋊C4—92no—
292C3×C22⋊C4—31no—
293C3×C22⋊C4—32no—
294C3×C22⋊C4—91no—
295C3×C22⋊C4—92no—
296#217—31no—
297#217C2×D911yescharacteristic
298(C2×D3)⋊C4—32no—
299(C2×D3)⋊C4—92no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4C2×D912yescomplemented
304Dic3⋊C4—32no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307Dic3⋊C4C2×D912yescomplemented
308C4×Dic3—92no—
309C4×Dic3—92no—
310C22×C32—22no—
311C22×C32D3×D412yes—
312C22×C32C23×D311yescharacteristic
313C2×C3⋊D3—34no—
314C2×C3⋊D3—68no—
315C2×C3⋊D3—62no—
316C2×C3⋊D3—62no—
317C2×C3×D3—21no—
318C2×C3×D3—22no—
319C2×C3×D3—61no—
320C2×C3×D3—62no—
321C32×C4—22no—
322C3⋊Dic3—32no—
323C3⋊Dic3—32no—
324C3×Dic3—21no—
325C3×Dic3—22no—
326C3×Dic3—61no—
327C3×Dic3—62no—
328C22×C9—24no—
329C22×C9—22no—
330C22×C9—21no—
331C22×C9—22no—
332C22×C9—61no—
333C22×C9—62no—
334C22×C9D3×D412yes—
335C22×C9C23×D311yescharacteristic
336C2×D9—21no—
337C2×D9—22no—
338C2×D9—34no—
339C2×D9—64no—
340C2×D9—64no—
341C2×D9—616no—
342C2×D9—68no—
343C2×D9—64no—
344C2×D9—62no—
345C2×D9—62no—
346C4×C9—24no—
347C4×C9—22no—
348C4×C9—61no—
349C4×C9—62no—
350Dic9—21no—
351Dic9—22no—
352Dic9—32no—
353Dic9—32no—
354Dic9—62no—
355Dic9—62no—
356C2×#41—271no—
357C23×C4—271no—
358#123—31no—
359#123—91no—
360C22⋊Q8—272no—
361#121—272no—
362#120—271no—
363#120—271no—
364C4×D4—272no—
365C4×D4—272no—
366#117—271no—
367C3×C9#52511yescharacteristic, complemented, Sylow
368C23×C3—21no—
369C23×C3—31no—
370C23×C3—91no—
371C23×C3C22×D911yescharacteristic
372C23×C3C2×D3211yescharacteristic
373C22×D3—31no—
374C22×D3—92no—
375C22×D3—92no—
376C22×D3—182no—
377C22×D3—184no—
378C22×D3—184no—
379C22×D3—184no—
380C22×D3—188no—
381C22×D3—182no—
382C22×D3—182no—
383C22×D3C22×D911yescharacteristic
384C3×D4—62no—
385C3×D4—182no—
386C2×C3×C4—24no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—32no—
391C2×C3×C4—31no—
392C2×C3×C4—92no—
393C2×C3×C4—91no—
394C2×C3×C4C22×D912yes—
395C2×C3×C4C2×D3212yes—
396#67—22no—
397#67—62no—
398#67—184no—
399#67—184no—
400#67—182no—
401#67—182no—
402C2×Dic3—32no—
403C2×Dic3—31no—
404C2×Dic3—91no—
405C2×Dic3—91no—
406C2×Dic3—94no—
407C2×Dic3—91no—
408C2×Dic3—91no—
409C2×Dic3—94no—
410C2×Dic3—181no—
411C2×Dic3—181no—
412C2×Dic3—184no—
413C2×Dic3—182no—
414C2×Dic3—182no—
415C2×Dic3C22×D912yes—
416C2×Dic3C22×D911yescharacteristic
417D12—184no—
418D12—184no—
419C4×D3—184no—
420C4×D3—184no—
421C4×D3—184no—
422C4×D3—184no—
423C4×D3—182no—
424C4×D3—182no—
425C4×D3—188no—
426C4×D3—188no—
427Dic6—182no—
428Dic6—182no—
429C2×C32—22no—
430C2×C32C2×D3×D411yescharacteristic
431C2×C32D3×#4112yes—
432C3⋊D3—64no—
433C3×D3—41no—
434C3×D3—121no—
435C2×C9—24no—
436C2×C9—22no—
437C2×C9—22no—
438C2×C9—21no—
439C2×C9—121no—
440C2×C9C2×D3×D411yescharacteristic
441C2×C9D3×#4112yes—
442D9—41no—
443D9—68no—
444D9—64no—
445C24—271no—
446#41—544no—
447C2×Q8—271no—
448C2×D4—31no—
449C2×D4—91no—
450C2×D4—272no—
451C2×D4—272no—
452C2×D4—271no—
453C22×C4—271no—
454C22×C4—271no—
455C22×C4—271no—
456C22×C4—272no—
457C22×C4—272no—
458C22×C4—544no—
459C22×C4D3×D911yescharacteristic
460C4⋊C4—32no—
461C4⋊C4—92no—
462C4⋊C4—272no—
463C22⋊C4—31no—
464C22⋊C4—32no—
465C22⋊C4—91no—
466C22⋊C4—92no—
467C22⋊C4—272no—
468C22⋊C4—272no—
469C42—272no—
470C22×C3—24no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—21no—
474C22×C3—22no—
475C22×C3—62no—
476C22×C3—61no—
477C22×C3—181no—
478C22×C3—182no—
479C22×C3D4×D912yes—
480C22×C3C23×D911yescharacteristic
481C22×C3#381112yes—
482C22×C3C22×D3211yescharacteristic
483C2×D3—21no—
484C2×D3—22no—
485C2×D3—62no—
486C2×D3—61no—
487C2×D3—94no—
488C2×D3—94no—
489C2×D3—184no—
490C2×D3—184no—
491C2×D3—188no—
492C2×D3—1816no—
493C2×D3—188no—
494C2×D3—184no—
495C2×D3—182no—
496C2×D3—182no—
497C2×D3—182no—
498C2×D3—182no—
499C3×C4—24no—
500C3×C4—22no—
501C3×C4—22no—
502C3×C4—62no—
503C3×C4—61no—
504C3×C4—181no—
505C3×C4—182no—
506Dic3—21no—
507Dic3—22no—
508Dic3—62no—
509Dic3—61no—
510Dic3—92no—
511Dic3—92no—
512Dic3—92no—
513Dic3—92no—
514Dic3—182no—
515Dic3—182no—
516C32#565611yescharacteristic
517C9—21no—
518C9#565611yescharacteristic, complemented
519C23—31no—
520C23—91no—
521C23—272no—
522C23—544no—
523C23—542no—
524C23C2×D3×D911yescharacteristic
525Q8—542no—
526D4—62no—
527D4—182no—
528D4—544no—
529D4—544no—
530D4—542no—
531C2×C4—22no—
532C2×C4—32no—
533C2×C4—31no—
534C2×C4—92no—
535C2×C4—91no—
536C2×C4—271no—
537C2×C4—271no—
538C2×C4—274no—
539C2×C4—544no—
540C2×C4—544no—
541C2×C4—542no—
542C2×C4—542no—
543C2×C4C2×D3×D912yes—
544C2×C3—24no—
545C2×C3—22no—
546C2×C3—22no—
547C2×C3—21no—
548C2×C3—22no—
549C2×C3—121no—
550C2×C3—361no—
551C2×C3C2×D4×D911yescharacteristic
552C2×C3#41×D912yes—
553C2×C3#6430612yes—
554C2×C3C2×#381111yescharacteristic
555D3—41no—
556D3—121no—
557D3—188no—
558D3—184no—
559D3—184no—
560C22—22no—
561C22—61no—
562C22—62no—
563C22—181no—
564C22—182no—
565C22—274no—
566C22—548no—
567C22—542no—
568C22—542no—
569C22#8866812yes—
570C22C22×D3×D911yescharacteristic
571C4—22no—
572C4—61no—
573C4—62no—
574C4—181no—
575C4—182no—
576C4—272no—
577C4—272no—
578C3—21no—
579C3#1058975811yescharacteristic, complemented
580C3#1059492011yescharacteristic
581C2—22no—
582C2—121no—
583C2—361no—
584C2—544no—
585C2#1172247512yes—
586C2C2×#8866811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#525
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×C4×C3⋊D9C2semidirect
2. C2×#88663C2semidirect
3. C9×#929C2semidirect
4. C3×#63491C2semidirect
5. #11721291C2semidirect
6. #11721278C2semidirect
7. #11721248C2semidirect
8. #11721041C2semidirect
9. #11721014C2semidirect
10. #11720978C2semidirect
11. #11720923C2semidirect
23. C2×#6194C22semidirect
25. C2×D3×Dic9C22semidirect
27. C2×Dic3×D9C22semidirect
28. C2×#6189C22semidirect
31. C9×(C2×D3)⋊C4C22semidirect
32. C3×(C2×D9)⋊C4C22semidirect
33. C9×Dic3⋊C4C22semidirect
34. C3×Dic9⋊C4C22semidirect
37. #88442C22semidirect
38. Dic3×Dic9C22semidirect
55. #63491D3semidirect
70. C2×C3⋊Dic9D4semidirect
141. (C2×D9)⋊C4C2×D3semidirect
143. Dic9⋊C4C2×D3semidirect
151. C2×C3⋊D9#41semidirect
173. #929D9semidirect
303. (C2×D3)⋊C4C2×D9semidirect
307. Dic3⋊C4C2×D9semidirect
367. C3×C9#525semidirect
518. C9#5656semidirect
579. C3#10589758semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C2×#886632
C9×#9292
C3×#634912
#117212912
#117212782
#117212482
#117210412
#117210142
#117209782
#117209232
#105949203
#105897583