Finite Group Databasebeta

← #13150368

Subgroups — #13150368

Order 1728.

1068 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D9C211yesmaximal, characteristic, complemented
2C2×#88662C211yesmaximal, characteristic, complemented
3#11721818C211yesmaximal, characteristic, complemented
4C9×#929C211yesmaximal, characteristic, complemented
5#11721285C211yesmaximal, characteristic, complemented
6#11721276C211yesmaximal, characteristic, complemented
7#11721245C211yesmaximal, characteristic, complemented
8#11721033C212yesmaximal, complemented
9#11721012C212yesmaximal, complemented
10#11721002C212yesmaximal, complemented
11#11720939C212yesmaximal, complemented
12#10594896—31nomaximal
13#10589758—31nomaximal
14C23×C3×D9C2211yescharacteristic
15C22×C3×C4×C9C2211yescharacteristic
16C2×#6233C2211yescharacteristic
17C2×C9×#67C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C2×C3×C4×D9—24no—
20C2×C3×C4×D9C2212yes—
21C2×C3×C4×D9C2212yes—
22C2×#6195C2211yescharacteristic, complemented
23C2×#6194C2212yes—
24#88662—24no—
25C2×D3×Dic9C2211yescharacteristic, complemented
26C2×#6192C2212yes—
27C2×(C3⋊D9)⋊C4C2211yescharacteristic, complemented
28C2×#6189C2211yescharacteristic, complemented
29#88551C2211yescharacteristic
30#88536C2212yescomplemented
31#88534C2212yescomplemented
32C9×#217C2211yescharacteristic
33C9×(C2×D3)⋊C4C2212yescomplemented
34C9×Dic3⋊C4C2212yescomplemented
35(C2×D9)⋊Dic3C2212yes—
36(C2×D9)⋊Dic3C2212yes—
37Dic9⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C22×C3×C4×D3—31no—
40C2×#3804—31no—
41#64138—31no—
42C3×#929D311yescharacteristic
43#63977—31no—
44#63976—31no—
45#63955—31no—
46#63906—32no—
47#63904—32no—
48#63879—32no—
49#63851—32no—
50C2×#3699—31no—
51C22×C4×D9D311yescharacteristic, complemented
52C9×#123—31no—
53#63499—31no—
54#63491—31no—
55#63491—61no—
56#63458—32no—
57#63455—32no—
58#63446—32no—
59#63444—31no—
60#611⋊C4—32no—
61#63439—31no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2311yescharacteristic
65C22×C3×D9—22no—
66C22×C3×D9—24no—
67C22×C3×D9C2312yes—
68C2×C3×C4×C9—22no—
69C2×C3×C4×C9C2312yes—
70#6233—22no—
71C2×C3⋊Dic9C2311yescharacteristic
72C2×C3⋊Dic9C2312yes—
73C9×#67—22no—
74C2×C9×Dic3C2311yescharacteristic
75C2×C9×Dic3C2312yes—
76C2×C3×Dic9D414yescomplemented
77C2×C3×Dic9C2311yescharacteristic
78C2×C3×Dic9C2311yescharacteristic
79C3×C4×D9—24no—
80C3×C4×D9—24no—
81#6195—22no—
82#6194—24no—
83D3×Dic9—22no—
84#6192—24no—
85(C3⋊D9)⋊C4—22no—
86#6189—22no—
87#5791—91no—
88#5656—91no—
89C23×C3×D3—31no—
90C22×C32×C4C2×D311yescharacteristic
91C2×#638—31no—
92C2×C3×#67C2×D311yescharacteristic
93C22×C3×Dic3—31no—
94C2×C3×C4×D3—32no—
95C2×C3×C4×D3—32no—
96C2×C3×C4×D3—64no—
97C2×#627—31no—
98C2×#626—31no—
99C2×#626—32no—
100C2×#625—32no—
101C2×#624—31no—
102#3804—64no—
103C2×D3×Dic3—31no—
104C23×D9C2×D311yescharacteristic
105#3757—31no—
106#3752—32no—
107#3750—32no—
108C3×#217C2×D311yescharacteristic
109C3×(C2×D3)⋊C4C2×D312yes—
110C3×Dic3⋊C4C2×D312yes—
111Dic3⋊Dic3—32no—
112(C2×D3)⋊Dic3—32no—
113(C2×D3)⋊Dic3—32no—
114Dic32—32no—
115C2×D4×C9—31no—
116C22×C4×C9—21no—
117C22×C4×C9C2×D311yescharacteristic
118C2×#611—31no—
119C2×#611—32no—
120C2×#611—31no—
121C2×#611—61no—
122C22×Dic9—31no—
123C22×Dic9C2×D311yescharacteristic
124#3699—64no—
125C2×D36—32no—
126C2×C4×D9—24no—
127C2×C4×D9—31no—
128C2×C4×D9C2×D312yes—
129C2×C4×D9C2×D312yes—
130C2×Dic18—31no—
131C9×C4⋊C4—32no—
132C9×C22⋊C4—31no—
133C9×C22⋊C4—32no—
134#3676—31no—
135#3676—61no—
136(C2×D9)⋊C4—32no—
137(C2×D9)⋊C4—32no—
138(C2×D9)⋊C4—32no—
139(C2×D9)⋊C4—62no—
140Dic9⋊C4—32no—
141Dic9⋊C4—32no—
142Dic9⋊C4—62no—
143C4×Dic9—32no—
144C22×C3×C9—22no—
145C22×C3×C9C2×D412yes—
146C22×C3×C9C2411yescharacteristic
147C2×C3⋊D9—21no—
148C2×C3⋊D9—22no—
149C2×D3×C9—21no—
150C2×D3×C9—22no—
151C2×C3×D9—28no—
152C2×C3×D9—22no—
153C2×C3×D9—22no—
154C2×C3×D9#4114yescomplemented
155C3×C4×C9—22no—
156C3⋊Dic9—21no—
157C3⋊Dic9—22no—
158C9×Dic3—21no—
159C9×Dic3—22no—
160C3×Dic9C2×D412yes—
161C3×Dic9C2×D412yes—
162C23×C3×C4—91no—
163C2×#237—91no—
164C2×#235—91no—
165C22×C4×D3—31no—
166C3×#123—31no—
167#953—91no—
168#953—91no—
169#937—91no—
170#930—92no—
171#929—91no—
172#929—91no—
173#929—181no—
174#929D911yescharacteristic, complemented
175C4×#67—92no—
176C4×#67—92no—
177#926—91no—
178#924—92no—
179#896—92no—
180#893—92no—
181#884—92no—
182#882—91no—
183#67⋊C4—92no—
184#877—91no—
185C23×C32C22×D311yescharacteristic
186C22×C3⋊D3—31no—
187C22×C3×D3—32no—
188C22×C3×D3—64no—
189C22×C3×D3—62no—
190C22×C3×D3C22×D311yescharacteristic
191C2×C32×C4—22no—
192C2×C32×C4C22×D312yes—
193#638—62no—
194C2×C3⋊Dic3—32no—
195C2×C3⋊Dic3—31no—
196C3×#67—22no—
197C2×C3×Dic3—34no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3C22×D312yes—
201C2×C3×Dic3C22×D311yescharacteristic
202C3×C4×D3—64no—
203C3×C4×D3—64no—
204#627—62no—
205#626—64no—
206#626—62no—
207#625—64no—
208#624—62no—
209D3×Dic3—62no—
210C23×C9—21no—
211C23×C9—31no—
212C23×C9C22×D311yescharacteristic
213C22×D9—22no—
214C22×D9—24no—
215C22×D9—31no—
216C22×D9—61no—
217C22×D9C22×D312yes—
218D4×C9—62no—
219C2×C4×C9—22no—
220C2×C4×C9—22no—
221C2×C4×C9—32no—
222C2×C4×C9—31no—
223C2×C4×C9—42no—
224C2×C4×C9C22×D312yes—
225#611—62no—
226#611—64no—
227#611—62no—
228#611—122no—
229C2×Dic9—32no—
230C2×Dic9—31no—
231C2×Dic9—61no—
232C2×Dic9—62no—
233C2×Dic9—62no—
234C2×Dic9#6714yescomplemented
235C2×Dic9C22×D311yescharacteristic
236C2×Dic9C22×D311yescharacteristic
237D36—64no—
238C4×D9—24no—
239C4×D9—24no—
240C4×D9—62no—
241Dic18—62no—
242#525—271noSylow
243C2×C3×C9—22no—
244C2×C3×C9C22×D411yescharacteristic
245C2×C3×C9C2×#4112yes—
246C3⋊D9—41no—
247D3×C9—41no—
248C3×D9—24no—
249C24×C3—91no—
250C23×D3—31no—
251C2×C3×D4—31no—
252C22×C3×C4—21no—
253C22×C3×C4—91no—
254C22×C3×C4—92no—
255C22×C3×C4—92no—
256C22×C3×C4—184no—
257C22×C3×C4C2×D911yescharacteristic
258C22×C3×C4D3211yescharacteristic
259C2×#67—91no—
260C2×#67—92no—
261C2×#67—91no—
262C2×#67—91no—
263C2×#67—92no—
264C2×#67—92no—
265C2×#67—181no—
266C2×#67C2×D911yescharacteristic
267C22×Dic3—31no—
268C22×Dic3—91no—
269#237—184no—
270#235—184no—
271C2×D12—92no—
272C2×D12—91no—
273C2×C4×D3—32no—
274C2×C4×D3—32no—
275C2×C4×D3—64no—
276C2×C4×D3—91no—
277C2×C4×D3—91no—
278C2×C4×D3—91no—
279C2×Dic6—91no—
280C2×Dic6—91no—
281C3×C4⋊C4—32no—
282C3×C22⋊C4—31no—
283C3×C22⋊C4—32no—
284#217—91no—
285#217—91no—
286#217—92no—
287#217—92no—
288#217—181no—
289#217C2×D911yescharacteristic
290(C2×D3)⋊C4—92no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4—92no—
294(C2×D3)⋊C4—182no—
295(C2×D3)⋊C4C2×D912yescomplemented
296C4⋊Dic3—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—182no—
301Dic3⋊C4C2×D912yescomplemented
302C4×Dic3—92no—
303C4×Dic3—92no—
304C22×C32—22no—
305C22×C32D3×D412yes—
306C22×C32C23×D311yescharacteristic
307C2×C3⋊D3—61no—
308C2×C3⋊D3—62no—
309C2×C3×D3—21no—
310C2×C3×D3—22no—
311C2×C3×D3—34no—
312C2×C3×D3—68no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C32×C4—22no—
316C3⋊Dic3—61no—
317C3⋊Dic3—62no—
318C3×Dic3—21no—
319C3×Dic3—22no—
320C3×Dic3—32no—
321C3×Dic3—32no—
322C22×C9—22no—
323C22×C9—21no—
324C22×C9—22no—
325C22×C9—42no—
326C22×C9—62no—
327C22×C9—61no—
328C22×C9C2×#6712yes—
329C22×C9C23×D311yescharacteristic
330C2×D9—28no—
331C2×D9—22no—
332C2×D9—22no—
333C2×D9—62no—
334C2×D9—61no—
335C2×D9—121no—
336C2×D9—122no—
337C2×D9#23514yescomplemented
338C4×C9—22no—
339C4×C9—42no—
340C4×C9—62no—
341C4×C9—61no—
342Dic9—62no—
343Dic9—61no—
344Dic9—122no—
345Dic9—121no—
346Dic9C2×#6712yes—
347Dic9C2×#6712yes—
348C2×#41—271no—
349C23×C4—91no—
350#123—31no—
351#123—271no—
352C22⋊Q8—272no—
353#121—272no—
354#120—271no—
355#120—271no—
356C4×D4—272no—
357C4×D4—272no—
358#117—271no—
359C3×C9#52511yescharacteristic, complemented, Sylow
360C23×C3—21no—
361C23×C3—31no—
362C23×C3—92no—
363C23×C3—184no—
364C23×C3—182no—
365C23×C3C22×D911yescharacteristic
366C23×C3C2×D3211yescharacteristic
367C22×D3—32no—
368C22×D3—64no—
369C22×D3—62no—
370C22×D3—91no—
371C22×D3—91no—
372C22×D3—181no—
373C22×D3C22×D911yescharacteristic
374C3×D4—62no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—32no—
379C2×C3×C4—31no—
380C2×C3×C4—42no—
381C2×C3×C4—94no—
382C2×C3×C4—91no—
383C2×C3×C4—91no—
384C2×C3×C4—184no—
385C2×C3×C4—184no—
386C2×C3×C4C22×D912yes—
387C2×C3×C4C2×D3212yes—
388#67—22no—
389#67—184no—
390#67—182no—
391#67—184no—
392#67—184no—
393#67—182no—
394#67—182no—
395#67—362no—
396C2×Dic3—31no—
397C2×Dic3—31no—
398C2×Dic3—34no—
399C2×Dic3—92no—
400C2×Dic3—91no—
401C2×Dic3—92no—
402C2×Dic3—91no—
403C2×Dic3—181no—
404C2×Dic3—182no—
405C2×Dic3—182no—
406C2×Dic3C22×D912yes—
407C2×Dic3C22×D911yescharacteristic
408D12—184no—
409D12—182no—
410C4×D3—64no—
411C4×D3—64no—
412C4×D3—182no—
413C4×D3—182no—
414C4×D3—182no—
415Dic6—182no—
416Dic6—182no—
417C2×C32—22no—
418C2×C32C2×D3×D411yescharacteristic
419C2×C32D3×#4112yes—
420C3⋊D3—121no—
421C3×D3—41no—
422C3×D3—64no—
423C2×C9—22no—
424C2×C9—22no—
425C2×C9—21no—
426C2×C9—42no—
427C2×C9—121no—
428C2×C9C2×#23512yes—
429C2×C9C22×#6711yescharacteristic
430D9—24no—
431D9—121no—
432D9—241no—
433C24—91no—
434#41—544no—
435C2×Q8—271no—
436C2×D4—31no—
437C2×D4—272no—
438C2×D4—271no—
439C2×D4—272no—
440C2×D4—271no—
441C22×C4—91no—
442C22×C4—92no—
443C22×C4—92no—
444C22×C4—184no—
445C22×C4—271no—
446C22×C4—271no—
447C22×C4D3×D911yescharacteristic
448C4⋊C4—32no—
449C4⋊C4—272no—
450C4⋊C4—272no—
451C22⋊C4—31no—
452C22⋊C4—32no—
453C22⋊C4—272no—
454C22⋊C4—272no—
455C22⋊C4—271no—
456C22⋊C4—272no—
457C42—272no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—21no—
461C22×C3—22no—
462C22×C3—42no—
463C22×C3—62no—
464C22×C3—61no—
465C22×C3—94no—
466C22×C3—188no—
467C22×C3—182no—
468C22×C3—182no—
469C22×C3D4×D912yes—
470C22×C3C23×D911yescharacteristic
471C22×C3D3×#6712yes—
472C22×C3C22×D3211yescharacteristic
473C2×D3—21no—
474C2×D3—22no—
475C2×D3—34no—
476C2×D3—68no—
477C2×D3—62no—
478C2×D3—62no—
479C2×D3—182no—
480C2×D3—181no—
481C2×D3—181no—
482C2×D3—182no—
483C2×D3—361no—
484C2×D3—362no—
485C3×C4—22no—
486C3×C4—22no—
487C3×C4—42no—
488C3×C4—62no—
489C3×C4—61no—
490C3×C4—92no—
491C3×C4—92no—
492Dic3—21no—
493Dic3—22no—
494Dic3—32no—
495Dic3—32no—
496Dic3—182no—
497Dic3—181no—
498Dic3—181no—
499Dic3—182no—
500Dic3—361no—
501Dic3—362no—
502C32#565611yescharacteristic
503C9—21no—
504C9#579111yescharacteristic, complemented
505C23—31no—
506C23—92no—
507C23—184no—
508C23—182no—
509C23—271no—
510C23C2×D3×D911yescharacteristic
511Q8—542no—
512D4—62no—
513D4—544no—
514D4—542no—
515D4—544no—
516D4—542no—
517C2×C4—22no—
518C2×C4—32no—
519C2×C4—31no—
520C2×C4—91no—
521C2×C4—91no—
522C2×C4—94no—
523C2×C4—184no—
524C2×C4—184no—
525C2×C4—272no—
526C2×C4—271no—
527C2×C4—542no—
528C2×C4—542no—
529C2×C4C2×D3×D912yes—
530C2×C3—22no—
531C2×C3—22no—
532C2×C3—21no—
533C2×C3—22no—
534C2×C3—42no—
535C2×C3—121no—
536C2×C3—184no—
537C2×C3C2×D4×D911yescharacteristic
538C2×C3#41×D912yes—
539C2×C3D3×#23512yes—
540C2×C3C2×D3×#6711yescharacteristic
541D3—41no—
542D3—64no—
543D3—361no—
544D3—361no—
545D3—721no—
546C22—22no—
547C22—61no—
548C22—62no—
549C22—94no—
550C22—188no—
551C22—182no—
552C22—182no—
553C22—541no—
554C22—542no—
555C22D9×#6712yes—
556C22C22×D3×D911yescharacteristic
557C4—22no—
558C4—61no—
559C4—62no—
560C4—92no—
561C4—92no—
562C4—541no—
563C4—542no—
564C3—21no—
565C3#1058975811yescharacteristic, complemented
566C3#1059489611yescharacteristic
567C2—22no—
568C2—121no—
569C2—184no—
570C2—1081no—
571C2D9×#23512yes—
572C2C2×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#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×C3×C4×D9C2semidirect
2. C2×#88662C2semidirect
3. #11721818C2semidirect
4. C9×#929C2semidirect
5. #11721285C2semidirect
6. #11721276C2semidirect
7. #11721245C2semidirect
8. #11721033C2semidirect
9. #11721012C2semidirect
10. #11721002C2semidirect
11. #11720939C2semidirect
22. C2×#6195C22semidirect
25. C2×D3×Dic9C22semidirect
27. C2×(C3⋊D9)⋊C4C22semidirect
28. C2×#6189C22semidirect
30. #88536C22semidirect
31. #88534C22semidirect
33. C9×(C2×D3)⋊C4C22semidirect
34. C9×Dic3⋊C4C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
51. C22×C4×D9D3semidirect
76. C2×C3×Dic9D4semidirect
154. C2×C3×D9#41semidirect
174. #929D9semidirect
234. C2×Dic9#67semidirect
295. (C2×D3)⋊C4C2×D9semidirect
301. Dic3⋊C4C2×D9semidirect
337. C2×D9#235semidirect
359. C3×C9#525semidirect
504. C9#5791semidirect
565. C3#10589758semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D92
C2×#886622
#117218182
C9×#9292
#117212852
#117212762
#117212452
#117210332
#117210122
#117210022
#117209392
#105948963
#105897583