Finite Group Databasebeta

← #13150365

Subgroups — #13150365

Order 1728.

1096 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C9×D12C211yesmaximal, characteristic, complemented
2C2×#88639C211yesmaximal, characteristic, complemented
3C4×#6233C211yesmaximal, characteristic, complemented
4C3×#63486C211yesmaximal, characteristic, complemented
5#11721287C212yesmaximal, complemented
6D3×#3676C212yesmaximal, complemented
7#11721045C211yesmaximal, characteristic, complemented
8#11721025C211yesmaximal, characteristic, complemented
9#11721017C212yesmaximal, complemented
10#11721015C212yesmaximal, complemented
11#11720967C211yesmaximal, characteristic, complemented
12#10594889—31nomaximal
13#10589908—31nomaximal
14C23×D3×C9C2212yes—
15C22×C3×C4×C9C2211yescharacteristic
16C2×#6233C2211yescharacteristic
17C2×C4×C3⋊D9C2211yescharacteristic, complemented
18C2×C9×D12—24no—
19C2×C9×D12C2211yescharacteristic
20C2×C9×D12C2212yes—
21C2×C9×D12C2211yescharacteristic
22C2×C3×Dic18C2211yescharacteristic, complemented
23C2×#6195C2212yescomplemented
24C2×#6195C2212yescomplemented
25C2×D3×Dic9C2212yescomplemented
26C2×D3×Dic9C2212yescomplemented
27#88639—24no—
28#88551C2211yescharacteristic
29#88536C2211yescharacteristic, complemented
30#88534C2211yescharacteristic, complemented
31C4×C3⋊Dic9C2211yescharacteristic, complemented
32C3×#3676C2212yes—
33C3×C4⋊Dic9C2211yescharacteristic, complemented
34C3×Dic9⋊C4C2212yescomplemented
35(C2×D3)⋊Dic9C2212yescomplemented
36(C2×D3)⋊Dic9C2212yescomplemented
37(C2×D3)⋊Dic9C2212yescomplemented
38(C2×D3)⋊Dic9C2212yescomplemented
39C22×C3×D12D311yescharacteristic
40C2×#3796—31no—
41C4×#638—31no—
42C3×#924—31no—
43#63977—32no—
44D3×#217—32no—
45#63916—31no—
46#63898—31no—
47#63886—32no—
48#63884—32no—
49#63863—31no—
50C22×D4×C9—31no—
51C2×#3699—31no—
52#63515—32no—
53C2×#3676—32no—
54#63501—32no—
55#63500—31no—
56#63498—31no—
57#63497—32no—
58D4×Dic9—31no—
59C4×#611—31no—
60C4×#611—61no—
61#63486D311yescharacteristic, complemented
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9—24no—
65C22×D3×C9—24no—
66C22×D3×C9—42no—
67C22×D3×C9C2312yes—
68C22×D3×C9C2312yes—
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72#6233—22no—
73C2×C3⋊Dic9C2311yescharacteristic
74C2×C3⋊Dic9C2311yescharacteristic
75C2×C3⋊Dic9C2311yescharacteristic
76C4×C3⋊D9—22no—
77C2×C3×Dic9C2312yes—
78C2×C3×Dic9C2312yes—
79C9×D12—24no—
80C9×D12—22no—
81C9×D12D414yescomplemented
82C3×Dic18—22no—
83#6195—24no—
84#6195—42no—
85D3×Dic9—24no—
86D3×Dic9—42no—
87#5806—91no—
88#5630—91no—
89C23×C3×D3C2×D312yes—
90C22×C32×C4C2×D311yescharacteristic
91C2×#638—31no—
92C2×C4×C3⋊D3—31no—
93C2×C3×D12—24no—
94C2×C3×D12C2×D311yescharacteristic
95C2×C3×D12C2×D312yes—
96C2×C3×D12C2×D311yescharacteristic
97C2×C3×Dic6—31no—
98C2×#626—32no—
99C2×#626—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102#3796—64no—
103C24×C9—32no—
104#3757—31no—
105#3752—31no—
106#3750—31no—
107C4×C3⋊Dic3—31no—
108C3×#217—32no—
109C3×C4⋊Dic3—31no—
110C3×Dic3⋊C4—32no—
111(C2×D3)⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C2×D4×C9—31no—
116C2×D4×C9—32no—
117C2×D4×C9—31no—
118C2×D4×C9—64no—
119C22×C4×C9—21no—
120C22×C4×C9C2×D311yescharacteristic
121C2×#611—31no—
122C2×#611—32no—
123C2×#611—32no—
124C2×#611—61no—
125C22×Dic9—32no—
126C22×Dic9—32no—
127#3699—64no—
128C2×C4×D9—31no—
129C2×C4×D9—61no—
130C2×Dic18C2×D311yescharacteristic, complemented
131#3676—31no—
132#3676—32no—
133#3676—32no—
134#3676—32no—
135#3676—32no—
136#3676—61no—
137#3676C2×D312yes—
138(C2×D9)⋊C4—31no—
139(C2×D9)⋊C4—61no—
140C4⋊Dic9C2×D311yescharacteristic, complemented
141Dic9⋊C4—31no—
142Dic9⋊C4—61no—
143Dic9⋊C4C2×D312yescomplemented
144C4×Dic9—31no—
145C4×Dic9—61no—
146C22×C3×C9—21no—
147C22×C3×C9—21no—
148C22×C3×C9#4112yescomplemented
149C22×C3×C9C2411yescharacteristic
150C2×C3⋊D9—21no—
151C2×C3⋊D9—21no—
152C2×C3⋊D9—21no—
153C2×D3×C9—22no—
154C2×D3×C9—22no—
155C2×D3×C9—22no—
156C2×D3×C9—22no—
157C2×D3×C9—22no—
158C2×D3×C9—48no—
159C2×D3×C9C2×D414yes—
160C3×C4×C9—21no—
161C3×C4×C9C2×D412yes—
162C3⋊Dic9—21no—
163C3⋊Dic9—21no—
164C3⋊Dic9—21no—
165C3×Dic9—22no—
166C3×Dic9—22no—
167C22×C3×D4—31no—
168C2×#239—91no—
169C2×#237—91no—
170C22×D12D911yescharacteristic, complemented
171C3×C22⋊Q8—91no—
172#953—92no—
173C2×#217—92no—
174#947—91no—
175#939—92no—
176#938—91no—
177#936—91no—
178#935—92no—
179D4×Dic3—91no—
180C4×#67—91no—
181C4×#67—91no—
182C4×#67—181no—
183#924—31no—
184#895—92no—
185#895—91no—
186#894—92no—
187#893—91no—
188#882—92no—
189D3×C22⋊C4—92no—
190C23×C32C22×D311yescharacteristic
191C22×C3⋊D3—31no—
192C22×C3×D3—24no—
193C22×C3×D3—24no—
194C22×C3×D3—42no—
195C22×C3×D3C22×D312yes—
196C22×C3×D3C22×D312yes—
197C2×C32×C4—22no—
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200#638—62no—
201C2×C3⋊Dic3—31no—
202C2×C3⋊Dic3—31no—
203C2×C3⋊Dic3—31no—
204C4×C3⋊D3—62no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C3×D12—24no—
208C3×D12—22no—
209C3×D12#6714yes—
210C3×Dic6—62no—
211#626—64no—
212#626—122no—
213D3×Dic3—64no—
214D3×Dic3—122no—
215C23×C9—21no—
216C23×C9—32no—
217C23×C9—32no—
218C23×C9—64no—
219C23×C9—64no—
220C23×C9—122no—
221C23×C9C22×D311yescharacteristic
222C22×D9—31no—
223C22×D9—61no—
224D4×C9—34no—
225D4×C9—64no—
226D4×C9—62no—
227C2×C4×C9—22no—
228C2×C4×C9—21no—
229C2×C4×C9—21no—
230C2×C4×C9—42no—
231C2×C4×C9C22×D311yescharacteristic
232C2×C4×C9C22×D311yescharacteristic
233#611—64no—
234#611—64no—
235#611—62no—
236#611—122no—
237C2×Dic9—31no—
238C2×Dic9—31no—
239C2×Dic9—31no—
240C2×Dic9—61no—
241C2×Dic9—61no—
242C2×Dic9—61no—
243C2×Dic9—64no—
244C2×Dic9—122no—
245C2×Dic9C22×D312yes—
246C2×Dic9C22×D312yes—
247C4×D9—62no—
248C4×D9—122no—
249Dic18—22no—
250#546—271noSylow
251C2×C3×C9—22no—
252C2×C3×C9C22×D411yescharacteristic
253C2×C3×C9C2×#4111yescharacteristic
254C2×C3×C9H(5,𝔽2)11yescharacteristic
255C3⋊D9—41no—
256D3×C9—24no—
257D3×C9—42no—
258C24×C3—32no—
259C23×D3C2×D912yes—
260C2×C3×Q8—91no—
261C2×C3×D4—31no—
262C2×C3×D4—32no—
263C2×C3×D4—31no—
264C2×C3×D4—64no—
265C22×C3×C4—21no—
266C22×C3×C4C2×D911yescharacteristic
267C22×C3×C4D3211yescharacteristic
268C2×#67—92no—
269C2×#67—92no—
270C2×#67—91no—
271C2×#67—91no—
272C2×#67—181no—
273C22×Dic3—92no—
274C22×Dic3—92no—
275#239—184no—
276#237—184no—
277C2×D12—24no—
278C2×D12—92no—
279C2×D12—92no—
280C2×D12C2×D912yes—
281C2×D12C2×D911yescharacteristic
282C2×D12C2×D911yescharacteristic
283C2×C4×D3—92no—
284C2×C4×D3—91no—
285C2×C4×D3—91no—
286C2×C4×D3—92no—
287C2×C4×D3—181no—
288C2×Dic6—31no—
289C3×C4⋊C4—91no—
290C3×C4⋊C4—92no—
291C3×C22⋊C4—92no—
292#217—32no—
293#217—92no—
294#217—92no—
295#217—92no—
296#217—92no—
297#217—91no—
298#217—91no—
299#217—181no—
300(C2×D3)⋊C4—92no—
301(C2×D3)⋊C4—92no—
302(C2×D3)⋊C4—92no—
303(C2×D3)⋊C4—92no—
304(C2×D3)⋊C4—91no—
305(C2×D3)⋊C4—91no—
306(C2×D3)⋊C4—181no—
307C4⋊Dic3—31no—
308Dic3⋊C4—32no—
309Dic3⋊C4—91no—
310Dic3⋊C4—91no—
311Dic3⋊C4—181no—
312C4×Dic3—91no—
313C4×Dic3—91no—
314C4×Dic3—181no—
315C22×C32—21no—
316C22×C32—21no—
317C22×C32#23712yes—
318C22×C32C23×D311yescharacteristic
319C2×C3⋊D3—61no—
320C2×C3⋊D3—61no—
321C2×C3⋊D3—61no—
322C2×C3×D3—22no—
323C2×C3×D3—22no—
324C2×C3×D3—22no—
325C2×C3×D3—22no—
326C2×C3×D3—22no—
327C2×C3×D3—48no—
328C2×C3×D3C2×#6714yes—
329C32×C4—21no—
330C32×C4C2×#6712yes—
331C3⋊Dic3—61no—
332C3⋊Dic3—61no—
333C3⋊Dic3—61no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C22×C9—21no—
337C22×C9—21no—
338C22×C9—21no—
339C22×C9—22no—
340C22×C9—34no—
341C22×C9—41no—
342C22×C9—41no—
343C22×C9—62no—
344C22×C9—62no—
345C22×C9—62no—
346C22×C9—62no—
347C22×C9—62no—
348C22×C9—128no—
349C22×C9#23912yescomplemented
350C22×C9C23×D311yescharacteristic
351C2×D9—62no—
352C2×D9—61no—
353C2×D9—61no—
354C2×D9—61no—
355C2×D9—121no—
356C2×D9—121no—
357C4×C9—21no—
358C4×C9—22no—
359C4×C9—41no—
360C4×C9D3×D412yes—
361Dic9—22no—
362Dic9—22no—
363Dic9—62no—
364Dic9—61no—
365Dic9—61no—
366Dic9—61no—
367Dic9—121no—
368Dic9—121no—
369C2×#41—271no—
370C22×D4—31no—
371#124—271no—
372#123—272no—
373C22⋊Q8—91no—
374#121—271no—
375#121—272no—
376#120—272no—
377C4×D4—271no—
378C4×D4—271no—
379C2×C22⋊C4—272no—
380C3×C9#54611yescharacteristic, complemented, Sylow
381C23×C3—21no—
382C23×C3—32no—
383C23×C3—32no—
384C23×C3—64no—
385C23×C3—64no—
386C23×C3—122no—
387C23×C3C22×D911yescharacteristic
388C23×C3C2×D3211yescharacteristic
389C22×D3—24no—
390C22×D3—24no—
391C22×D3—42no—
392C22×D3—91no—
393C22×D3—91no—
394C22×D3—181no—
395C22×D3C22×D912yes—
396C22×D3C22×D912yes—
397C3×Q8—182no—
398C3×D4—34no—
399C3×D4—64no—
400C3×D4—62no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—42no—
406C2×C3×C4—92no—
407C2×C3×C4—92no—
408C2×C3×C4C22×D911yescharacteristic
409C2×C3×C4C22×D911yescharacteristic
410C2×C3×C4C2×D3211yescharacteristic
411C2×C3×C4C2×D3211yescharacteristic
412#67—184no—
413#67—184no—
414#67—182no—
415#67—182no—
416#67—362no—
417C2×Dic3—32no—
418C2×Dic3—32no—
419C2×Dic3—91no—
420C2×Dic3—91no—
421C2×Dic3—91no—
422C2×Dic3—91no—
423C2×Dic3—91no—
424C2×Dic3—91no—
425C2×Dic3—181no—
426C2×Dic3—181no—
427C2×Dic3—181no—
428C2×Dic3—184no—
429C2×Dic3—362no—
430D12—24no—
431D12—22no—
432D12—184no—
433D12—362no—
434D12#61114yescomplemented
435C4×D3—184no—
436C4×D3—182no—
437C4×D3—182no—
438C4×D3—362no—
439C4×D3—362no—
440Dic6—62no—
441C2×C32—22no—
442C2×C32C2×#23711yescharacteristic
443C2×C32#100411yescharacteristic
444C2×C32C22×#6711yescharacteristic
445C3⋊D3—121no—
446C3×D3—24no—
447C3×D3—42no—
448C2×C9—22no—
449C2×C9—21no—
450C2×C9—21no—
451C2×C9—21no—
452C2×C9—42no—
453C2×C9—64no—
454C2×C9—122no—
455C2×C9C2×D3×D411yescharacteristic
456C2×C9C2×#23911yescharacteristic
457C2×C9#100911yescharacteristic
458D9—122no—
459D9—121no—
460C24—32no—
461#41—544no—
462C2×Q8—91no—
463C2×D4—31no—
464C2×D4—32no—
465C2×D4—31no—
466C2×D4—64no—
467C2×D4—272no—
468C2×D4—272no—
469C2×D4—271no—
470C22×C4—272no—
471C22×C4—272no—
472C22×C4—271no—
473C22×C4D3×D911yescharacteristic
474C4⋊C4—91no—
475C4⋊C4—92no—
476C4⋊C4—271no—
477C22⋊C4—92no—
478C22⋊C4—272no—
479C22⋊C4—272no—
480C22⋊C4—272no—
481C22⋊C4—272no—
482C22⋊C4—271no—
483C22⋊C4—271no—
484C42—271no—
485C22×C3—21no—
486C22×C3—21no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—22no—
490C22×C3—21no—
491C22×C3—34no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—62no—
495C22×C3—62no—
496C22×C3—62no—
497C22×C3—62no—
498C22×C3—62no—
499C22×C3—128no—
500C22×C3#369912yescomplemented
501C22×C3C23×D911yescharacteristic
502C22×C3#379612yes—
503C22×C3C22×D3211yescharacteristic
504C2×D3—22no—
505C2×D3—22no—
506C2×D3—22no—
507C2×D3—22no—
508C2×D3—22no—
509C2×D3—48no—
510C2×D3—182no—
511C2×D3—181no—
512C2×D3—181no—
513C2×D3—181no—
514C2×D3—181no—
515C2×D3—181no—
516C2×D3—181no—
517C2×D3—361no—
518C2×D3—361no—
519C2×D3C2×#61114yes—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—22no—
523C3×C4—41no—
524C3×C4—182no—
525C3×C4—182no—
526C3×C4C2×#61112yes—
527C3×C4D3×#6712yes—
528Dic3—62no—
529Dic3—62no—
530Dic3—182no—
531Dic3—181no—
532Dic3—181no—
533Dic3—181no—
534Dic3—181no—
535Dic3—181no—
536Dic3—181no—
537Dic3—361no—
538Dic3—361no—
539C32#580611yescharacteristic
540C9—21no—
541C9#563011yescharacteristic, complemented
542C23—32no—
543C23—32no—
544C23—64no—
545C23—64no—
546C23—122no—
547C23—271no—
548C23C2×D3×D911yescharacteristic
549Q8—182no—
550D4—34no—
551D4—64no—
552D4—62no—
553D4—544no—
554D4—542no—
555D4—1082no—
556C2×C4—22no—
557C2×C4—92no—
558C2×C4—92no—
559C2×C4—271no—
560C2×C4—271no—
561C2×C4—271no—
562C2×C4—544no—
563C2×C4—542no—
564C2×C4—1082no—
565C2×C4C2×D3×D911yescharacteristic
566C2×C4C2×D3×D911yescharacteristic
567C2×C3—22no—
568C2×C3—22no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—42no—
573C2×C3—64no—
574C2×C3—122no—
575C2×C3C2×#369911yescharacteristic
576C2×C3#6371011yescharacteristic
577C2×C3C22×#61111yescharacteristic
578C2×C3C2×#379611yescharacteristic
579C2×C3#6430711yescharacteristic
580C2×C3C2×D3×#6711yescharacteristic
581D3—24no—
582D3—42no—
583D3—362no—
584D3—361no—
585D3—361no—
586C22—21no—
587C22—21no—
588C22—34no—
589C22—62no—
590C22—62no—
591C22—62no—
592C22—62no—
593C22—62no—
594C22—128no—
595C22—541no—
596C22—541no—
597C22—541no—
598C22#8863912yescomplemented
599C22C22×D3×D911yescharacteristic
600C4—21no—
601C4—182no—
602C4—182no—
603C4—541no—
604C4—541no—
605C4—541no—
606C4D3×#61112yes—
607C3—21no—
608C3#1058990811yescharacteristic, complemented
609C3#1059488911yescharacteristic
610C2—22no—
611C2—64no—
612C2—122no—
613C2—1081no—
614C2C2×#8863911yescharacteristic
615C2#1172247611yescharacteristic
616C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#546
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×C9×D12C2semidirect
2. C2×#88639C2semidirect
3. C4×#6233C2semidirect
4. C3×#63486C2semidirect
5. #11721287C2semidirect
6. D3×#3676C2semidirect
7. #11721045C2semidirect
8. #11721025C2semidirect
9. #11721017C2semidirect
10. #11721015C2semidirect
11. #11720967C2semidirect
17. C2×C4×C3⋊D9C22semidirect
22. C2×C3×Dic18C22semidirect
23. C2×#6195C22semidirect
24. C2×#6195C22semidirect
25. C2×D3×Dic9C22semidirect
26. C2×D3×Dic9C22semidirect
29. #88536C22semidirect
30. #88534C22semidirect
31. C4×C3⋊Dic9C22semidirect
33. C3×C4⋊Dic9C22semidirect
34. C3×Dic9⋊C4C22semidirect
35. (C2×D3)⋊Dic9C22semidirect
36. (C2×D3)⋊Dic9C22semidirect
37. (C2×D3)⋊Dic9C22semidirect
38. (C2×D3)⋊Dic9C22semidirect
61. #63486D3semidirect
81. C9×D12D4semidirect
130. C2×Dic18C2×D3semidirect
140. C4⋊Dic9C2×D3semidirect
143. Dic9⋊C4C2×D3semidirect
148. C22×C3×C9#41semidirect
170. C22×D12D9semidirect
349. C22×C9#239semidirect
380. C3×C9#546semidirect
434. D12#611semidirect
500. C22×C3#3699semidirect
541. C9#5630semidirect
598. C22#88639semidirect
608. C3#10589908semidirect

Maximal subgroups

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

subgroupindex
C22×C9×D122
C2×#886392
C4×#62332
C3×#634862
#117212872
D3×#36762
#117210452
#117210252
#117210172
#117210152
#117209672
#105948893
#105899083