Finite Group Databasebeta

← #13149323

Subgroups — #13149323

Order 1728.

1066 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×D9C211yesmaximal, characteristic, complemented
2C2×#88639C211yesmaximal, characteristic, complemented
3#11721748C211yesmaximal, characteristic, complemented
4C9×#895C211yesmaximal, characteristic, complemented
5C3×#63459C211yesmaximal, characteristic, complemented
6#11721050C212yesmaximal, complemented
7#11721036C211yesmaximal, characteristic, complemented
8#11721028C212yesmaximal, complemented
9#11721025C212yesmaximal, complemented
10#11720960C211yesmaximal, characteristic, complemented
11#11720958C212yesmaximal, complemented
12#10594632—31nomaximal
13#10589708—31nomaximal
14C22×D3×D9C2211yescharacteristic
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C9×D12C2212yes—
17C2×C9×D12C2211yescharacteristic, complemented
18C2×C4×D3×C9C2211yescharacteristic
19C2×C3×C4×D9C2211yescharacteristic
20C2×C3×Dic18C2211yescharacteristic, complemented
21C2×#6195C2212yes—
22C2×#6194C2212yes—
23C2×D3×Dic9C2211yescharacteristic
24C2×D3×Dic9C2212yescomplemented
25C2×(C3⋊D9)⋊C4C2211yescharacteristic
26C2×Dic3×D9C2211yescharacteristic
27C4×D3×D9—24no—
28#88639—24no—
29C3×C9×C4⋊C4C2211yescharacteristic, complemented
30#88536C2212yes—
31C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
32C4×C3⋊Dic9C2212yescomplemented
33C9×(C2×D3)⋊C4C2212yescomplemented
34C3×(C2×D9)⋊C4C2212yes—
35C3×C4⋊Dic9C2212yescomplemented
36(C2×D3)⋊Dic9C2212yes—
37(C2×D3)⋊Dic9C2212yescomplemented
38#88442C2212yescomplemented
39C2×C4×D32—31no—
40C2×#3796—31no—
41#64101—31no—
42C3×#897—31no—
43C3×#895D311yescharacteristic
44#63922—32no—
45#63908—31no—
46#63902—32no—
47#63898—32no—
48#63856—31no—
49#63854—32no—
50C2×#3699—31no—
51C22×C4×D9—31no—
52C9×#121—31no—
53#63500—31no—
54#63499—32no—
55D4×Dic9—32no—
56#63486—31no—
57#63459D311yescharacteristic, complemented
58#63454—31no—
59#63454—61no—
60#63445—32no—
61#611⋊C4—32no—
62C22×C3⋊D9C2311yescharacteristic
63C22×D3×C9C2312yes—
64C22×D3×C9C2311yescharacteristic
65C22×C3×D9C2311yescharacteristic
66C2×D3×D9—22no—
67C2×D3×D9—22no—
68C2×D3×D9—22no—
69C2×C3×C4×C9C2312yes—
70C2×C3×C4×C9C2311yescharacteristic
71C2×C3⋊Dic9C2312yes—
72C2×C3⋊Dic9C2311yescharacteristic
73C4×C3⋊D9D414yescomplemented
74C2×C9×Dic3C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2312yes—
77C9×D12—24no—
78C9×D12—22no—
79C4×D3×C9—22no—
80C3×C4×D9—22no—
81C3×Dic18—22no—
82#6195—24no—
83#6194—24no—
84D3×Dic9—24no—
85D3×Dic9—22no—
86(C3⋊D9)⋊C4—22no—
87Dic3×D9—22no—
88#5627—91no—
89#5606—91no—
90C22×D32—31no—
91C2×C4×C3⋊D3—31no—
92C2×C3×D12C2×D312yes—
93C2×C3×D12C2×D311yescharacteristic
94C2×C3×C4×D3—31no—
95C2×C3×C4×D3C2×D311yescharacteristic
96C2×C3×Dic6—31no—
97C2×#626—32no—
98C2×#625—32no—
99C2×#624—31no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—31no—
102C2×D3×Dic3—31no—
103C4×D32—64no—
104#3796—64no—
105C23×D9—31no—
106C32×C4⋊C4C2×D311yescharacteristic
107#3752—32no—
108C4⋊(C3⋊Dic3)—31no—
109C4×C3⋊Dic3—32no—
110C3×(C2×D3)⋊C4—32no—
111C3×(C2×D3)⋊C4C2×D312yes—
112C3×C4⋊Dic3—32no—
113#3724—32no—
114(C2×D3)⋊Dic3—32no—
115(C2×D3)⋊Dic3—32no—
116C2×D4×C9—32no—
117C2×D4×C9—31no—
118C22×C4×C9—31no—
119C2×#611—32no—
120C2×#611—32no—
121C22×Dic9—31no—
122C22×Dic9—32no—
123#3699—64no—
124C2×C4×D9—31no—
125C2×C4×D9—31no—
126C2×C4×D9—31no—
127C2×C4×D9—64no—
128C2×C4×D9—61no—
129C2×C4×D9C2×D311yescharacteristic
130C2×Dic18C2×D311yescharacteristic, complemented
131C9×C4⋊C4—21no—
132C9×C4⋊C4C2×D311yescharacteristic, complemented
133C9×C22⋊C4—32no—
134#3676—32no—
135#3676—32no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—62no—
138(C2×D9)⋊C4C2×D312yes—
139C4⋊Dic9—31no—
140C4⋊Dic9—61no—
141C4⋊Dic9C2×D312yescomplemented
142Dic9⋊C4—32no—
143C4×Dic9—32no—
144C4×Dic9—62no—
145C22×C3×C9C2411yescharacteristic
146C2×C3⋊D9—21no—
147C2×C3⋊D9—21no—
148C2×C3⋊D9C2×D412yes—
149C2×D3×C9—21no—
150C2×D3×C9—21no—
151C2×D3×C9—22no—
152C2×D3×C9—22no—
153C2×D3×C9—22no—
154C2×D3×C9#4112yescomplemented
155C2×C3×D9—21no—
156C2×C3×D9—21no—
157C2×C3×D9#4112yescomplemented
158D3×D9—28no—
159C3×C4×C9—22no—
160C3×C4×C9C2×D412yes—
161C3⋊Dic9—22no—
162C3⋊Dic9C2×D412yes—
163C9×Dic3—21no—
164C3×Dic9—21no—
165C3×Dic9—22no—
166C2×#239—91no—
167C2×#237—91no—
168C22×C4×D3—91no—
169C22×C4×D3—91no—
170C3×C22⋊Q8—91no—
171C3×#121—31no—
172#946—91no—
173#938—91no—
174#937—92no—
175D4×Dic3—92no—
176#930—91no—
177#924—91no—
178#897—31no—
179#895D911yescharacteristic, complemented
180#894—92no—
181#893—92no—
182#892—91no—
183#892—91no—
184#892—181no—
185#883—92no—
186#882—92no—
187#67⋊C4—92no—
188#67⋊C4—92no—
189C22×C3⋊D3—31no—
190C22×C3×D3—31no—
191C22×C3×D3C22×D312yes—
192C22×C3×D3C22×D311yescharacteristic
193C2×D32—62no—
194C2×D32—62no—
195C2×D32—62no—
196C2×C32×C4C22×D312yes—
197C2×C32×C4C22×D311yescharacteristic
198C2×C3⋊Dic3—31no—
199C2×C3⋊Dic3—32no—
200C4×C3⋊D3—34no—
201C2×C3×Dic3—31no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3C22×D311yescharacteristic
204C3×D12—24no—
205C3×D12—22no—
206C3×C4×D3—22no—
207C3×C4×D3—62no—
208C3×Dic6—62no—
209#626—64no—
210#625—64no—
211#624—62no—
212D3×Dic3—62no—
213D3×Dic3—64no—
214D3×Dic3—62no—
215C23×C9—31no—
216C23×C9—32no—
217C22×D9—31no—
218C22×D9—61no—
219C22×D9—62no—
220C22×D9—62no—
221C22×D9—62no—
222C22×D9C22×D311yescharacteristic
223D4×C9—64no—
224D4×C9—62no—
225C2×C4×C9—21no—
226C2×C4×C9—22no—
227C2×C4×C9—31no—
228C2×C4×C9—62no—
229C2×C4×C9C22×D312yes—
230C2×C4×C9C22×D311yescharacteristic
231#611—64no—
232#611—64no—
233C2×Dic9—31no—
234C2×Dic9—32no—
235C2×Dic9—61no—
236C2×Dic9—62no—
237C2×Dic9—62no—
238C2×Dic9—64no—
239C2×Dic9C22×D312yes—
240C2×Dic9C22×D311yescharacteristic
241C4×D9—22no—
242C4×D9—34no—
243C4×D9—64no—
244C4×D9—62no—
245C4×D9—62no—
246Dic18—22no—
247#525—271noSylow
248C2×C3×C9C22×D411yescharacteristic
249C2×C3×C9C2×#4111yescharacteristic
250C2×C3×C9C2×#4111yescharacteristic
251C3⋊D9—22no—
252D3×C9—22no—
253D3×C9—42no—
254C3×D9—22no—
255C23×D3—91no—
256C23×D3—91no—
257C2×C3×Q8—91no—
258C2×C3×D4—32no—
259C2×C3×D4—31no—
260C22×C3×C4—31no—
261C22×C3×C4—91no—
262C2×#67—92no—
263C2×#67—92no—
264C2×#67—92no—
265C22×Dic3—92no—
266C22×Dic3—91no—
267C22×Dic3—91no—
268#239—184no—
269#237—184no—
270C2×D12—92no—
271C2×D12C2×D912yes—
272C2×D12C2×D911yescharacteristic, complemented
273C2×C4×D3—31no—
274C2×C4×D3—91no—
275C2×C4×D3—91no—
276C2×C4×D3—91no—
277C2×C4×D3—91no—
278C2×C4×D3—92no—
279C2×C4×D3—91no—
280C2×C4×D3—91no—
281C2×C4×D3—181no—
282C2×C4×D3—184no—
283C2×C4×D3—184no—
284C2×C4×D3C2×D911yescharacteristic
285C2×Dic6—31no—
286C3×C4⋊C4—21no—
287C3×C4⋊C4—92no—
288C3×C4⋊C4C2×D911yescharacteristic, complemented
289C3×C4⋊C4D3211yescharacteristic
290C3×C22⋊C4—32no—
291C3×C22⋊C4—92no—
292#217—92no—
293#217—92no—
294(C2×D3)⋊C4—32no—
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298(C2×D3)⋊C4—92no—
299(C2×D3)⋊C4—182no—
300(C2×D3)⋊C4C2×D912yescomplemented
301C4⋊Dic3—32no—
302C4⋊Dic3—91no—
303C4⋊Dic3—91no—
304C4⋊Dic3—181no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307C4×Dic3—92no—
308C4×Dic3—92no—
309C4×Dic3—182no—
310C22×C32C23×D311yescharacteristic
311C2×C3⋊D3—32no—
312C2×C3⋊D3—61no—
313C2×C3⋊D3—61no—
314C2×C3×D3—21no—
315C2×C3×D3—21no—
316C2×C3×D3—22no—
317C2×C3×D3—22no—
318C2×C3×D3—22no—
319C2×C3×D3—32no—
320C2×C3×D3—61no—
321C2×C3×D3—61no—
322C2×C3×D3#23712yes—
323D32—68no—
324C32×C4—22no—
325C32×C4D3×D412yes—
326C3⋊Dic3—32no—
327C3⋊Dic3—62no—
328C3×Dic3—21no—
329C3×Dic3—61no—
330C3×Dic3—62no—
331C22×C9—21no—
332C22×C9—32no—
333C22×C9—61no—
334C22×C9—61no—
335C22×C9—62no—
336C22×C9—62no—
337C22×C9—62no—
338C22×C9C23×D311yescharacteristic
339C2×D9—21no—
340C2×D9—21no—
341C2×D9—32no—
342C2×D9—62no—
343C2×D9—68no—
344C2×D9—61no—
345C2×D9—61no—
346C2×D9—121no—
347C2×D9—121no—
348C2×D9#23912yescomplemented
349C4×C9—22no—
350C4×C9—22no—
351C4×C9—42no—
352C4×C9—61no—
353C4×C9D3×D412yes—
354Dic9—22no—
355Dic9—21no—
356Dic9—32no—
357Dic9—62no—
358Dic9—62no—
359Dic9—122no—
360C2×#41—271no—
361C23×C4—271no—
362#123—272no—
363C22⋊Q8—91no—
364C22⋊Q8—271no—
365#121—31no—
366#121—271no—
367#120—272no—
368C4×D4—272no—
369C4×D4—272no—
370#117—271no—
371C3×C9#52511yescharacteristic, complemented, Sylow
372C23×C3—31no—
373C23×C3—32no—
374C23×C3—91no—
375C22×D3—31no—
376C22×D3—91no—
377C22×D3—91no—
378C22×D3—181no—
379C22×D3—182no—
380C22×D3—182no—
381C22×D3—182no—
382C22×D3—182no—
383C22×D3—182no—
384C22×D3—182no—
385C22×D3C22×D912yes—
386C22×D3C22×D911yescharacteristic
387C3×Q8—182no—
388C3×D4—64no—
389C3×D4—62no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—31no—
393C2×C3×C4—62no—
394C2×C3×C4—92no—
395C2×C3×C4—91no—
396C2×C3×C4—182no—
397C2×C3×C4C22×D912yes—
398C2×C3×C4C22×D911yescharacteristic
399C2×C3×C4C2×D3212yes—
400C2×C3×C4C2×D3211yescharacteristic
401#67—184no—
402#67—184no—
403#67—184no—
404C2×Dic3—32no—
405C2×Dic3—31no—
406C2×Dic3—91no—
407C2×Dic3—91no—
408C2×Dic3—92no—
409C2×Dic3—92no—
410C2×Dic3—181no—
411C2×Dic3—182no—
412C2×Dic3—182no—
413C2×Dic3—182no—
414C2×Dic3—184no—
415C2×Dic3C22×D911yescharacteristic
416D12—24no—
417D12—22no—
418D12—184no—
419C4×D3—22no—
420C4×D3—62no—
421C4×D3—94no—
422C4×D3—94no—
423C4×D3—184no—
424C4×D3—182no—
425C4×D3—182no—
426C4×D3—182no—
427C4×D3—182no—
428C4×D3—184no—
429Dic6—62no—
430C2×C32C2×D3×D411yescharacteristic
431C2×C32C2×#23711yescharacteristic
432C2×C32D3×#4111yescharacteristic
433C3⋊D3—62no—
434C3×D3—22no—
435C3×D3—42no—
436C3×D3—62no—
437C2×C9—21no—
438C2×C9—21no—
439C2×C9—21no—
440C2×C9—62no—
441C2×C9—122no—
442C2×C9C2×D3×D411yescharacteristic
443C2×C9C2×#23911yescharacteristic
444C2×C9D3×#4111yescharacteristic
445D9—22no—
446D9—62no—
447D9—122no—
448C24—271no—
449#41—544no—
450C2×Q8—91no—
451C2×D4—32no—
452C2×D4—31no—
453C2×D4—272no—
454C2×D4—272no—
455C22×C4—31no—
456C22×C4—91no—
457C22×C4—271no—
458C22×C4—271no—
459C22×C4—271no—
460C22×C4—272no—
461C22×C4—271no—
462C22×C4—544no—
463C4⋊C4—92no—
464C4⋊C4—272no—
465C4⋊C4—271no—
466C4⋊C4D3×D911yescharacteristic, complemented
467C22⋊C4—32no—
468C22⋊C4—92no—
469C22⋊C4—272no—
470C22⋊C4—272no—
471C22⋊C4—272no—
472C42—272no—
473C22×C3—21no—
474C22×C3—32no—
475C22×C3—61no—
476C22×C3—61no—
477C22×C3—62no—
478C22×C3—62no—
479C22×C3—62no—
480C22×C3—92no—
481C22×C3—181no—
482C22×C3—181no—
483C22×C3C23×D911yescharacteristic
484C22×C3C22×D3211yescharacteristic
485C2×D3—21no—
486C2×D3—21no—
487C2×D3—22no—
488C2×D3—22no—
489C2×D3—22no—
490C2×D3—32no—
491C2×D3—61no—
492C2×D3—61no—
493C2×D3—92no—
494C2×D3—92no—
495C2×D3—182no—
496C2×D3—188no—
497C2×D3—188no—
498C2×D3—181no—
499C2×D3—181no—
500C2×D3—181no—
501C2×D3—181no—
502C2×D3—361no—
503C2×D3—361no—
504C2×D3#369912yescomplemented
505C3×C4—22no—
506C3×C4—22no—
507C3×C4—22no—
508C3×C4—42no—
509C3×C4—61no—
510C3×C4—181no—
511C3×C4—182no—
512C3×C4D4×D912yes—
513C3×C4#380212yes—
514Dic3—21no—
515Dic3—62no—
516Dic3—61no—
517Dic3—92no—
518Dic3—92no—
519Dic3—182no—
520Dic3—182no—
521Dic3—182no—
522Dic3—362no—
523C32#560611yescharacteristic
524C9—21no—
525C9#562711yescharacteristic, complemented
526C23—31no—
527C23—32no—
528C23—91no—
529C23—271no—
530C23—542no—
531C23—542no—
532C23—542no—
533Q8—182no—
534D4—64no—
535D4—62no—
536D4—544no—
537D4—544no—
538C2×C4—31no—
539C2×C4—62no—
540C2×C4—92no—
541C2×C4—91no—
542C2×C4—182no—
543C2×C4—271no—
544C2×C4—274no—
545C2×C4—272no—
546C2×C4—542no—
547C2×C4—542no—
548C2×C4—542no—
549C2×C4—544no—
550C2×C4C2×D3×D912yes—
551C2×C4C2×D3×D911yescharacteristic
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—62no—
556C2×C3—122no—
557C2×C3—182no—
558C2×C3C2×D4×D911yescharacteristic
559C2×C3C2×#369911yescharacteristic
560C2×C3#41×D911yescharacteristic
561C2×C3C2×#380211yescharacteristic
562C2×C3D3×#23711yescharacteristic
563C2×C3D3×#23911yescharacteristic
564D3—22no—
565D3—42no—
566D3—62no—
567D3—182no—
568D3—182no—
569D3—362no—
570C22—32no—
571C22—61no—
572C22—61no—
573C22—62no—
574C22—62no—
575C22—62no—
576C22—92no—
577C22—181no—
578C22—181no—
579C22—272no—
580C22—548no—
581C22—541no—
582C22—541no—
583C22C22×D3×D911yescharacteristic
584C4—22no—
585C4—61no—
586C4—181no—
587C4—182no—
588C4—272no—
589C4—542no—
590C4#8864612yes—
591C3—21no—
592C3#1058970811yescharacteristic, complemented
593C3#1059463211yescharacteristic
594C2—62no—
595C2—122no—
596C2—182no—
597C2—542no—
598C2C2×#8864611yescharacteristic
599C2D3×#369911yescharacteristic
600C2D9×#23911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#525
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D3×D9C2semidirect
2. C2×#88639C2semidirect
3. #11721748C2semidirect
4. C9×#895C2semidirect
5. C3×#63459C2semidirect
6. #11721050C2semidirect
7. #11721036C2semidirect
8. #11721028C2semidirect
9. #11721025C2semidirect
10. #11720960C2semidirect
11. #11720958C2semidirect
17. C2×C9×D12C22semidirect
20. C2×C3×Dic18C22semidirect
24. C2×D3×Dic9C22semidirect
29. C3×C9×C4⋊C4C22semidirect
31. C4⋊(C3⋊Dic9)C22semidirect
32. C4×C3⋊Dic9C22semidirect
33. C9×(C2×D3)⋊C4C22semidirect
35. C3×C4⋊Dic9C22semidirect
37. (C2×D3)⋊Dic9C22semidirect
38. #88442C22semidirect
57. #63459D3semidirect
73. C4×C3⋊D9D4semidirect
130. C2×Dic18C2×D3semidirect
132. C9×C4⋊C4C2×D3semidirect
141. C4⋊Dic9C2×D3semidirect
154. C2×D3×C9#41semidirect
157. C2×C3×D9#41semidirect
179. #895D9semidirect
272. C2×D12C2×D9semidirect
288. C3×C4⋊C4C2×D9semidirect
300. (C2×D3)⋊C4C2×D9semidirect
348. C2×D9#239semidirect
371. C3×C9#525semidirect
466. C4⋊C4D3×D9semidirect
504. C2×D3#3699semidirect
525. C9#5627semidirect
592. C3#10589708semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×D92
C2×#886392
#117217482
C9×#8952
C3×#634592
#117210502
#117210362
#117210282
#117210252
#117209602
#117209582
#105946323
#105897083