Finite Group Databasebeta

← #13150178

Subgroups — #13150178

Order 1728.

1076 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D36C211yesmaximal, characteristic, complemented
2C2×#88641C211yesmaximal, characteristic, complemented
3#11721813C211yesmaximal, characteristic, complemented
4C4×C9×#67C211yesmaximal, characteristic, complemented
5#11721285C212yesmaximal, complemented
6D9×#217C212yesmaximal, complemented
7#11721037C211yesmaximal, characteristic, complemented
8Dic3×D36C211yesmaximal, characteristic, complemented
9#11720982C212yesmaximal, complemented
10#11720981C212yesmaximal, complemented
11#11720948C211yesmaximal, characteristic, complemented
12#10594886—31nomaximal
13#10589652—31nomaximal
14C23×C3×D9C2212yes—
15C22×C3×C4×C9C2211yescharacteristic
16C2×#6226C2211yescharacteristic, complemented
17C2×C9×#67C2211yescharacteristic
18C2×C4×D3×C9C2211yescharacteristic, complemented
19C2×C3×D36—24no—
20C2×C3×D36C2212yes—
21C2×C3×D36C2211yescharacteristic
22C2×C3×D36C2211yescharacteristic
23C2×#6194C2212yescomplemented
24C2×#6194C2212yescomplemented
25C2×Dic3×D9C2212yescomplemented
26C2×Dic3×D9C2212yescomplemented
27#88641—24no—
28#88551C2212yes—
29C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
30#88534C2212yescomplemented
31C9×#217C2211yescharacteristic
32C9×(C2×D3)⋊C4C2211yescharacteristic, complemented
33C9×Dic3⋊C4C2211yescharacteristic, complemented
34C4×C9×Dic3C2211yescharacteristic, complemented
35(C2×D9)⋊Dic3C2212yescomplemented
36(C2×D9)⋊Dic3C2212yescomplemented
37(C2×D9)⋊Dic3C2212yescomplemented
38(C2×D9)⋊Dic3C2212yescomplemented
39C22×C3×D12—31no—
40C2×#3795—31no—
41#64133—31no—
42C3×C4×#67D311yescharacteristic
43#63976—32no—
44D3×#217—32no—
45#63908—31no—
46Dic3×D12—31no—
47#63885—32no—
48#63884—32no—
49#63860—31no—
50C2×#3697—31no—
51C22×D36D311yescharacteristic, complemented
52C4×D4×C9—31no—
53#63499—32no—
54#63492—31no—
55#63486—31no—
56#63486—61no—
57#63456—32no—
58#63446—32no—
59C22⋊C4×D9—32no—
60#63437—31no—
61C4×D36—31no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9C2311yescharacteristic
64C22×C3×D9—24no—
65C22×C3×D9—24no—
66C22×C3×D9—42no—
67C22×C3×D9C2312yes—
68C22×C3×D9C2312yes—
69C2×C3×C4×C9—22no—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3×C4×C9C2311yescharacteristic
72C2×C3⋊Dic9C2312yes—
73C2×C3⋊Dic9C2312yes—
74#6226—22no—
75C9×#67—22no—
76C2×C9×Dic3C2311yescharacteristic
77C2×C9×Dic3C2311yescharacteristic
78C2×C9×Dic3C2311yescharacteristic
79C4×D3×C9—22no—
80C3×D36—24no—
81C3×D36—22no—
82C3×D36D414yescomplemented
83#6194—24no—
84#6194—42no—
85Dic3×D9—24no—
86Dic3×D9—42no—
87#5806—91no—
88#5550—91no—
89C23×C3×D3—32no—
90C22×C32×C4C2×D311yescharacteristic
91C2×#634—31no—
92C2×C3×#67C2×D311yescharacteristic
93C2×C3×D12—31no—
94C2×C3×D12—31no—
95C2×C3×D12—32no—
96C2×C3×D12—64no—
97C2×C3×C4×D3C2×D311yescharacteristic
98C2×#625—32no—
99C2×#625—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—32no—
102#3795—64no—
103C23×D9C2×D312yes—
104#3757—32no—
105C4⋊(C3⋊Dic3)—31no—
106#3750—32no—
107C3×#217C2×D311yescharacteristic
108C3×(C2×D3)⋊C4C2×D311yescharacteristic
109C3×Dic3⋊C4C2×D311yescharacteristic
110C3×C4×Dic3C2×D311yescharacteristic
111(C2×D3)⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C2×D4×C9—31no—
116C22×C4×C9—21no—
117C22×C4×C9—31no—
118C22×C4×C9C2×D311yescharacteristic
119C2×#611—32no—
120C2×#611—32no—
121#3697—64no—
122C2×D36—24no—
123C2×D36C2×D311yescharacteristic
124C2×D36C2×D312yes—
125C2×D36C2×D311yescharacteristic
126C2×C4×D9—32no—
127C2×C4×D9—32no—
128C2×Dic18—31no—
129C2×Dic18—61no—
130C9×C4⋊C4—31no—
131C9×C22⋊C4—31no—
132C9×C22⋊C4—31no—
133C42×C9—31no—
134#3676—32no—
135#3676—62no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—32no—
140C4⋊Dic9—31no—
141C4⋊Dic9—61no—
142Dic9⋊C4—32no—
143Dic9⋊C4—62no—
144C22×C3×C9—21no—
145C22×C3×C9—21no—
146C22×C3×C9#4112yescomplemented
147C22×C3×C9C2411yescharacteristic
148C2×D3×C9—21no—
149C2×D3×C9—21no—
150C2×D3×C9—21no—
151C2×C3×D9—22no—
152C2×C3×D9—22no—
153C2×C3×D9—22no—
154C2×C3×D9—22no—
155C2×C3×D9—22no—
156C2×C3×D9—48no—
157C2×C3×D9C2×D414yes—
158C3×C4×C9—21no—
159C3×C4×C9C2×D412yes—
160C3⋊Dic9—22no—
161C3⋊Dic9—22no—
162C9×Dic3—21no—
163C9×Dic3—21no—
164C9×Dic3—21no—
165C22×C3×D4—91no—
166C2×#237—91no—
167C2×#235—91no—
168C22×D12—31no—
169C3×C4×D4—31no—
170#953—92no—
171C2×#217—92no—
172#939—92no—
173#938—91no—
174#937—92no—
175#936—91no—
176#935—92no—
177D4×Dic3—91no—
178#930—91no—
179C4×#67D911yescharacteristic, complemented
180#924—91no—
181#924—91no—
182#924—181no—
183#894—92no—
184#884—92no—
185D3×C22⋊C4—92no—
186#875—91no—
187C4×D12—91no—
188C23×C32C22×D311yescharacteristic
189C22×C3×D3—32no—
190C22×C3×D3—32no—
191C22×C3×D3—64no—
192C22×C3×D3—64no—
193C22×C3×D3—122no—
194C22×C3×D3C22×D311yescharacteristic
195C2×C32×C4—22no—
196C2×C32×C4C22×D311yescharacteristic
197C2×C32×C4C22×D311yescharacteristic
198C2×C3⋊Dic3—32no—
199C2×C3⋊Dic3—32no—
200#634—62no—
201C3×#67—22no—
202C2×C3×Dic3C22×D311yescharacteristic
203C2×C3×Dic3C22×D311yescharacteristic
204C2×C3×Dic3C22×D311yescharacteristic
205C3×D12—34no—
206C3×D12—64no—
207C3×D12—62no—
208C3×C4×D3—22no—
209#625—64no—
210#625—122no—
211D3×Dic3—64no—
212D3×Dic3—122no—
213C23×C9—21no—
214C23×C9—31no—
215C23×C9C22×D311yescharacteristic
216C22×D9—24no—
217C22×D9—24no—
218C22×D9—42no—
219C22×D9C22×D312yes—
220C22×D9C22×D312yes—
221D4×C9—62no—
222C2×C4×C9—22no—
223C2×C4×C9—21no—
224C2×C4×C9—21no—
225C2×C4×C9—31no—
226C2×C4×C9—31no—
227C2×C4×C9—31no—
228C2×C4×C9—42no—
229C2×C4×C9—62no—
230C2×C4×C9C22×D311yescharacteristic
231C2×C4×C9C22×D311yescharacteristic
232#611—64no—
233#611—122no—
234C2×Dic9—32no—
235C2×Dic9—32no—
236C2×Dic9—62no—
237C2×Dic9—62no—
238D36—24no—
239D36—22no—
240D36#6714yescomplemented
241C4×D9—64no—
242C4×D9—122no—
243Dic18—62no—
244Dic18—122no—
245#546—271noSylow
246C2×C3×C9—22no—
247C2×C3×C9C22×D411yescharacteristic
248C2×C3×C9C2×#4111yescharacteristic
249C2×C3×C9H(5,𝔽2)11yescharacteristic
250D3×C9—41no—
251C3×D9—24no—
252C3×D9—42no—
253C24×C3—92no—
254C23×D3—32no—
255C2×C3×D4—31no—
256C2×C3×D4—91no—
257C2×C3×D4—92no—
258C2×C3×D4—91no—
259C2×C3×D4—184no—
260C22×C3×C4—21no—
261C22×C3×C4—31no—
262C22×C3×C4C2×D911yescharacteristic
263C22×C3×C4D3211yescharacteristic
264C2×#67—92no—
265C2×#67—92no—
266C2×#67—92no—
267C2×#67—92no—
268C2×#67C2×D911yescharacteristic
269C22×Dic3—92no—
270C22×Dic3—92no—
271#237—184no—
272#235—184no—
273C2×D12—31no—
274C2×D12—31no—
275C2×D12—32no—
276C2×D12—64no—
277C2×C4×D3—92no—
278C2×C4×D3—92no—
279C2×C4×D3C2×D911yescharacteristic, complemented
280C2×Dic6—91no—
281C2×Dic6—91no—
282C2×Dic6—181no—
283C3×C4⋊C4—31no—
284C3×C22⋊C4—31no—
285C3×C22⋊C4—31no—
286C3×C42—31no—
287#217—92no—
288#217—92no—
289#217—92no—
290#217—92no—
291#217—92no—
292#217—92no—
293#217—182no—
294#217C2×D911yescharacteristic
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4C2×D911yescharacteristic, complemented
300C4⋊Dic3—91no—
301C4⋊Dic3—91no—
302C4⋊Dic3—181no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—182no—
306Dic3⋊C4C2×D911yescharacteristic, complemented
307C4×Dic3C2×D911yescharacteristic, complemented
308C22×C32—21no—
309C22×C32—21no—
310C22×C32#23512yes—
311C22×C32C23×D311yescharacteristic
312C2×C3×D3—21no—
313C2×C3×D3—21no—
314C2×C3×D3—21no—
315C2×C3×D3—34no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—128no—
322C32×C4—21no—
323C32×C4D3×D412yes—
324C3⋊Dic3—62no—
325C3⋊Dic3—62no—
326C3×Dic3—21no—
327C3×Dic3—21no—
328C3×Dic3—21no—
329C22×C9—21no—
330C22×C9—21no—
331C22×C9—21no—
332C22×C9—22no—
333C22×C9—41no—
334C22×C9—41no—
335C22×C9—61no—
336C22×C9—61no—
337C22×C9—61no—
338C22×C9#23712yescomplemented
339C22×C9C23×D311yescharacteristic
340C2×D9—22no—
341C2×D9—22no—
342C2×D9—22no—
343C2×D9—22no—
344C2×D9—22no—
345C2×D9—48no—
346C2×D9C2×#6714yes—
347C4×C9—21no—
348C4×C9—22no—
349C4×C9—41no—
350C4×C9—61no—
351C4×C9—61no—
352C4×C9—61no—
353C4×C9C2×#6712yes—
354Dic9—62no—
355Dic9—62no—
356Dic9—122no—
357Dic9—122no—
358C2×#41—271no—
359C22×D4—91no—
360#124—271no—
361#123—272no—
362C22⋊Q8—271no—
363#121—272no—
364#121—271no—
365#120—272no—
366C4×D4—31no—
367C4×D4—271no—
368C2×C22⋊C4—272no—
369C3×C9#54611yescharacteristic, complemented, Sylow
370C23×C3—21no—
371C23×C3—31no—
372C23×C3—92no—
373C23×C3—92no—
374C23×C3—184no—
375C23×C3—184no—
376C23×C3—362no—
377C23×C3C22×D911yescharacteristic
378C23×C3C2×D3211yescharacteristic
379C22×D3—32no—
380C22×D3—32no—
381C22×D3—64no—
382C22×D3—64no—
383C22×D3—122no—
384C22×D3C22×D911yescharacteristic
385C3×D4—62no—
386C3×D4—94no—
387C3×D4—184no—
388C3×D4—182no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—31no—
394C2×C3×C4—31no—
395C2×C3×C4—31no—
396C2×C3×C4—42no—
397C2×C3×C4—62no—
398C2×C3×C4C22×D911yescharacteristic
399C2×C3×C4C22×D911yescharacteristic
400C2×C3×C4C2×D3211yescharacteristic
401C2×C3×C4C2×D3211yescharacteristic
402#67—22no—
403#67—184no—
404#67—184no—
405#67—362no—
406#67—362no—
407C2×Dic3—92no—
408C2×Dic3—92no—
409C2×Dic3—92no—
410C2×Dic3—92no—
411C2×Dic3—182no—
412C2×Dic3—182no—
413C2×Dic3—184no—
414C2×Dic3—362no—
415C2×Dic3C22×D911yescharacteristic
416C2×Dic3C22×D911yescharacteristic
417C2×Dic3C22×D911yescharacteristic
418D12—34no—
419D12—64no—
420D12—62no—
421C4×D3—22no—
422C4×D3—184no—
423C4×D3—362no—
424Dic6—182no—
425Dic6—182no—
426Dic6—362no—
427C2×C32—22no—
428C2×C32C2×#23511yescharacteristic
429C2×C32C2×D3×D411yescharacteristic
430C2×C32#100911yescharacteristic
431C3×D3—41no—
432C3×D3—64no—
433C3×D3—122no—
434C2×C9—22no—
435C2×C9—21no—
436C2×C9—21no—
437C2×C9—21no—
438C2×C9—42no—
439C2×C9—121no—
440C2×C9C2×#23711yescharacteristic
441C2×C9#100411yescharacteristic
442C2×C9C22×#6711yescharacteristic
443D9—24no—
444D9—42no—
445C24—92no—
446#41—544no—
447C2×Q8—271no—
448C2×D4—31no—
449C2×D4—91no—
450C2×D4—92no—
451C2×D4—91no—
452C2×D4—184no—
453C2×D4—272no—
454C2×D4—272no—
455C22×C4—31no—
456C22×C4—272no—
457C22×C4—272no—
458C22×C4D3×D911yescharacteristic
459C4⋊C4—31no—
460C4⋊C4—271no—
461C4⋊C4—272no—
462C22⋊C4—31no—
463C22⋊C4—31no—
464C22⋊C4—272no—
465C22⋊C4—272no—
466C22⋊C4—272no—
467C22⋊C4—272no—
468C22⋊C4—272no—
469C42—31no—
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—22no—
475C22×C3—21no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—61no—
479C22×C3—61no—
480C22×C3—61no—
481C22×C3—94no—
482C22×C3—182no—
483C22×C3—182no—
484C22×C3—182no—
485C22×C3—182no—
486C22×C3—182no—
487C22×C3—368no—
488C22×C3#369712yescomplemented
489C22×C3C23×D911yescharacteristic
490C22×C3#379512yes—
491C22×C3C22×D3211yescharacteristic
492C2×D3—21no—
493C2×D3—21no—
494C2×D3—21no—
495C2×D3—34no—
496C2×D3—62no—
497C2×D3—62no—
498C2×D3—62no—
499C2×D3—62no—
500C2×D3—62no—
501C2×D3—128no—
502C3×C4—21no—
503C3×C4—21no—
504C3×C4—22no—
505C3×C4—41no—
506C3×C4—61no—
507C3×C4—61no—
508C3×C4—61no—
509C3×C4D4×D912yes—
510C3×C4D3×#6712yes—
511Dic3—21no—
512Dic3—21no—
513Dic3—21no—
514Dic3—182no—
515Dic3—182no—
516Dic3—182no—
517Dic3—182no—
518Dic3—362no—
519Dic3—362no—
520C32#555011yescharacteristic
521C9—21no—
522C9#580611yescharacteristic, complemented
523C23—31no—
524C23—92no—
525C23—92no—
526C23—184no—
527C23—184no—
528C23—362no—
529C23C2×D3×D911yescharacteristic
530Q8—542no—
531D4—62no—
532D4—94no—
533D4—184no—
534D4—182no—
535D4—544no—
536D4—1082no—
537C2×C4—22no—
538C2×C4—31no—
539C2×C4—31no—
540C2×C4—31no—
541C2×C4—62no—
542C2×C4—272no—
543C2×C4—272no—
544C2×C4—544no—
545C2×C4—1082no—
546C2×C4C2×D3×D911yescharacteristic
547C2×C4C2×D3×D911yescharacteristic
548C2×C3—22no—
549C2×C3—22no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—42no—
554C2×C3—121no—
555C2×C3—184no—
556C2×C3—362no—
557C2×C3C2×#369711yescharacteristic
558C2×C3C2×D4×D911yescharacteristic
559C2×C3#6371511yescharacteristic
560C2×C3C2×#379511yescharacteristic
561C2×C3#6430711yescharacteristic
562C2×C3C2×D3×#6711yescharacteristic
563D3—41no—
564D3—64no—
565D3—122no—
566C22—21no—
567C22—21no—
568C22—61no—
569C22—61no—
570C22—61no—
571C22—94no—
572C22—182no—
573C22—182no—
574C22—182no—
575C22—182no—
576C22—182no—
577C22—368no—
578C22#8864112yescomplemented
579C22C22×D3×D911yescharacteristic
580C4—21no—
581C4—61no—
582C4—61no—
583C4—61no—
584C4—542no—
585C4—542no—
586C4D9×#6712yes—
587C3—21no—
588C3#1058965211yescharacteristic, complemented
589C3#1059488611yescharacteristic
590C2—22no—
591C2—121no—
592C2—184no—
593C2—362no—
594C2C2×#8864111yescharacteristic
595C2#1172247711yescharacteristic
596C2C2×D9×#6711yescharacteristic
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×C3×D36C2semidirect
2. C2×#88641C2semidirect
3. #11721813C2semidirect
4. C4×C9×#67C2semidirect
5. #11721285C2semidirect
6. D9×#217C2semidirect
7. #11721037C2semidirect
8. Dic3×D36C2semidirect
9. #11720982C2semidirect
10. #11720981C2semidirect
11. #11720948C2semidirect
16. C2×#6226C22semidirect
18. C2×C4×D3×C9C22semidirect
23. C2×#6194C22semidirect
24. C2×#6194C22semidirect
25. C2×Dic3×D9C22semidirect
26. C2×Dic3×D9C22semidirect
29. C4⋊(C3⋊Dic9)C22semidirect
30. #88534C22semidirect
32. C9×(C2×D3)⋊C4C22semidirect
33. C9×Dic3⋊C4C22semidirect
34. C4×C9×Dic3C22semidirect
35. (C2×D9)⋊Dic3C22semidirect
36. (C2×D9)⋊Dic3C22semidirect
37. (C2×D9)⋊Dic3C22semidirect
38. (C2×D9)⋊Dic3C22semidirect
51. C22×D36D3semidirect
82. C3×D36D4semidirect
146. C22×C3×C9#41semidirect
179. C4×#67D9semidirect
240. D36#67semidirect
279. C2×C4×D3C2×D9semidirect
299. (C2×D3)⋊C4C2×D9semidirect
306. Dic3⋊C4C2×D9semidirect
307. C4×Dic3C2×D9semidirect
338. C22×C9#237semidirect
369. C3×C9#546semidirect
488. C22×C3#3697semidirect
522. C9#5806semidirect
578. C22#88641semidirect
588. C3#10589652semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D362
C2×#886412
#117218132
C4×C9×#672
#117212852
D9×#2172
#117210372
Dic3×D362
#117209822
#117209812
#117209482
#105948863
#105896523