Finite Group Databasebeta

← #13148214

Subgroups — #13148214

Order 1728.

1066 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88663C211yesmaximal, characteristic, complemented
2C22×D3×Dic9C211yesmaximal, characteristic, complemented
3#11721723C211yesmaximal, characteristic, complemented
4C9×#882C211yesmaximal, characteristic, complemented
5C3×#63448C211yesmaximal, characteristic, complemented
6#11721286C211yesmaximal, characteristic, complemented
7#11721265C211yesmaximal, characteristic, complemented
8#11721028C212yesmaximal, complemented
9#11721026C212yesmaximal, complemented
10#11721025C212yesmaximal, complemented
11#11721016C212yesmaximal, complemented
12#10594082—31nomaximal
13#10589688—31nomaximal
14C23×D3×C9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C2×C9×#67C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C2×C9×D12C2212yes—
20C2×#6194C2212yes—
21C2×#6194C2211yescharacteristic, complemented
22#88663—24no—
23C2×D3×Dic9—24no—
24C2×D3×Dic9C2212yes—
25C2×D3×Dic9C2212yes—
26C2×D3×Dic9C2211yescharacteristic, complemented
27C2×Dic3×D9C2211yescharacteristic, complemented
28C2×#6189C2211yescharacteristic, complemented
29C3×C9×C22⋊C4C2211yescharacteristic
30#88551C2211yescharacteristic
31#88534C2212yescomplemented
32C4×C3⋊Dic9C2212yescomplemented
33C9×(C2×D3)⋊C4C2212yes—
34C3×(C2×D9)⋊C4C2212yescomplemented
35C3×C4⋊Dic9C2212yescomplemented
36(C2×D3)⋊Dic9C2212yes—
37(C2×D3)⋊Dic9C2212yescomplemented
38#88442C2212yescomplemented
39C2×#3805—31no—
40C22×D3×Dic3—31no—
41#64086—31no—
42C3×#886—31no—
43C3×#882D311yescharacteristic
44#63976—31no—
45#63961—31no—
46#63902—32no—
47#63899—32no—
48#63898—32no—
49#63885—32no—
50C23×Dic9—31no—
51C2×#3699—31no—
52C9×#120—31no—
53#63515—31no—
54#63501—32no—
55#63497—31no—
56D4×Dic9—32no—
57#63448D311yescharacteristic, complemented
58#611⋊C4—32no—
59#63440—32no—
60#63439—31no—
61#63439—61no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9—22no—
64C22×D3×C9—24no—
65C22×D3×C9C2311yescharacteristic
66C22×D3×C9C2312yes—
67C22×C3×D9C2311yescharacteristic
68C2×C3×C4×C9C2312yes—
69C2×C3⋊Dic9D414yescomplemented
70C2×C3⋊Dic9C2312yes—
71C2×C3⋊Dic9C2311yescharacteristic
72C2×C3⋊Dic9C2311yescharacteristic
73C9×#67—22no—
74C3×#611—22no—
75C2×C9×Dic3C2311yescharacteristic
76C2×C3×Dic9—22no—
77C2×C3×Dic9C2311yescharacteristic
78C2×C3×Dic9C2312yes—
79C9×D12—24no—
80#6194—24no—
81#6194—22no—
82D3×Dic9—24no—
83D3×Dic9—24no—
84D3×Dic9—22no—
85Dic3×D9—22no—
86#6189—22no—
87#5656—91no—
88#5586—91no—
89C23×C3×D3C2×D311yescharacteristic
90C22×C3⋊Dic3—31no—
91C2×C3×#67—31no—
92C2×C3×#67C2×D311yescharacteristic
93C22×C3×Dic3—31no—
94C2×C3×D12C2×D312yes—
95C2×#627—31no—
96C2×#625—31no—
97C2×#625—32no—
98#3805—64no—
99C2×D3×Dic3—32no—
100C2×D3×Dic3—32no—
101C2×D3×Dic3—31no—
102C2×D3×Dic3—31no—
103C2×D3×Dic3—64no—
104C24×C9—31no—
105C32×C22⋊C4C2×D311yescharacteristic
106#3757—31no—
107#3750—32no—
108C4×C3⋊Dic3—32no—
109C3×(C2×D3)⋊C4—32no—
110C3×(C2×D3)⋊C4C2×D312yes—
111C3×C4⋊Dic3—32no—
112#3724—32no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115C2×D4×C9—31no—
116C2×D4×C9—32no—
117C2×#611—31no—
118C2×#611—32no—
119C2×#611C2×D311yescharacteristic
120C22×Dic9—31no—
121C22×Dic9—31no—
122C22×Dic9—32no—
123C22×Dic9—32no—
124C22×Dic9—64no—
125C22×Dic9—61no—
126C22×Dic9C2×D311yescharacteristic
127#3699—64no—
128C2×C4×D9—31no—
129C2×Dic18—31no—
130C9×C22⋊C4—21no—
131C9×C22⋊C4—32no—
132C9×C22⋊C4C2×D311yescharacteristic
133#3676—31no—
134#3676—32no—
135#3676—32no—
136#3676—61no—
137(C2×D9)⋊C4C2×D312yescomplemented
138C4⋊Dic9C2×D312yescomplemented
139Dic9⋊C4—32no—
140Dic9⋊C4—32no—
141Dic9⋊C4—62no—
142C4×Dic9—32no—
143C4×Dic9—62no—
144C22×C3×C9—22no—
145C22×C3×C9C2×D412yes—
146C22×C3×C9C2411yescharacteristic
147C2×D3×C9—28no—
148C2×D3×C9—22no—
149C2×D3×C9—22no—
150C2×D3×C9—21no—
151C2×D3×C9—22no—
152C2×D3×C9#4114yescomplemented
153C2×C3×D9—21no—
154C2×C3×D9—22no—
155C3×C4×C9—22no—
156C3⋊Dic9—22no—
157C3⋊Dic9C2×D412yes—
158C3⋊Dic9C2×D412yes—
159C9×Dic3—21no—
160C3×Dic9—21no—
161C3×Dic9—22no—
162C23×Dic3—91no—
163C2×#237—91no—
164C2×#237—91no—
165C22×C4×D3—91no—
166C3×#123—91no—
167C3×#120—31no—
168#953—91no—
169#939—92no—
170#937—91no—
171#935—91no—
172D4×Dic3—92no—
173#929—91no—
174#896—92no—
175#893—92no—
176#886—31no—
177#884—92no—
178#882D911yescharacteristic, complemented
179#67⋊C4—92no—
180#67⋊C4—92no—
181#878—92no—
182#877—91no—
183#877—91no—
184#877—181no—
185C23×C32C22×D311yescharacteristic
186C22×C3×D3—22no—
187C22×C3×D3—24no—
188C22×C3×D3—31no—
189C22×C3×D3C22×D311yescharacteristic
190C22×C3×D3C22×D312yes—
191C2×C32×C4C22×D312yes—
192C2×C3⋊Dic3—34no—
193C2×C3⋊Dic3—31no—
194C2×C3⋊Dic3—31no—
195C2×C3⋊Dic3—32no—
196C3×#67—22no—
197C3×#67—62no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3C22×D311yescharacteristic
202C3×D12—24no—
203#627—62no—
204#625—62no—
205#625—64no—
206D3×Dic3—62no—
207D3×Dic3—64no—
208D3×Dic3—64no—
209D3×Dic3—62no—
210C23×C9—21no—
211C23×C9—32no—
212C23×C9—31no—
213C23×C9—64no—
214C23×C9—62no—
215C23×C9C22×D311yescharacteristic
216C22×D9C22×D311yescharacteristic
217D4×C9—64no—
218D4×C9—62no—
219C2×C4×C9—22no—
220C2×C4×C9—31no—
221C2×C4×C9C22×D312yes—
222#611—22no—
223#611—62no—
224#611—64no—
225C2×Dic9—22no—
226C2×Dic9—34no—
227C2×Dic9—31no—
228C2×Dic9—31no—
229C2×Dic9—32no—
230C2×Dic9—61no—
231C2×Dic9—61no—
232C2×Dic9—64no—
233C2×Dic9—62no—
234C2×Dic9—62no—
235C2×Dic9—64no—
236C2×Dic9—64no—
237C2×Dic9C22×D312yes—
238C2×Dic9C22×D311yescharacteristic
239C4×D9—62no—
240Dic18—62no—
241#525—271noSylow
242C2×C3×C9—22no—
243C2×C3×C9C22×D411yescharacteristic
244C2×C3×C9C2×#4112yes—
245D3×C9—24no—
246D3×C9—41no—
247C3×D9—41no—
248C24×C3—31no—
249C23×D3C2×D911yescharacteristic
250C2×C3×D4—31no—
251C2×C3×D4—32no—
252C2×C3×D4—91no—
253C22×C3×C4—91no—
254C2×#67—31no—
255C2×#67—92no—
256C2×#67—91no—
257C2×#67—92no—
258C2×#67—91no—
259C2×#67C2×D911yescharacteristic
260C22×Dic3—31no—
261C22×Dic3—92no—
262C22×Dic3—92no—
263C22×Dic3—91no—
264C22×Dic3—91no—
265C22×Dic3—91no—
266C22×Dic3—91no—
267C22×Dic3—181no—
268C22×Dic3—184no—
269#237—184no—
270#237—184no—
271C2×D12C2×D912yes—
272C2×C4×D3—91no—
273C2×C4×D3—92no—
274C2×C4×D3—92no—
275C2×C4×D3—91no—
276C2×C4×D3—184no—
277C2×Dic6—91no—
278C2×Dic6—91no—
279C3×C4⋊C4—92no—
280C3×C22⋊C4—21no—
281C3×C22⋊C4—32no—
282C3×C22⋊C4—92no—
283C3×C22⋊C4C2×D911yescharacteristic
284C3×C22⋊C4D3211yescharacteristic
285#217—92no—
286#217—92no—
287#217—91no—
288#217—91no—
289#217—181no—
290(C2×D3)⋊C4—32no—
291(C2×D3)⋊C4—92no—
292(C2×D3)⋊C4—92no—
293(C2×D3)⋊C4C2×D912yes—
294C4⋊Dic3—32no—
295Dic3⋊C4—92no—
296Dic3⋊C4—92no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—182no—
300C4×Dic3—92no—
301C4×Dic3—92no—
302C4×Dic3—182no—
303C22×C32—22no—
304C22×C32D3×D412yes—
305C22×C32C23×D311yescharacteristic
306C2×C3×D3—28no—
307C2×C3×D3—22no—
308C2×C3×D3—22no—
309C2×C3×D3—21no—
310C2×C3×D3—22no—
311C2×C3×D3—61no—
312C2×C3×D3—62no—
313C2×C3×D3#23714yes—
314C32×C4—22no—
315C3⋊Dic3—32no—
316C3⋊Dic3—32no—
317C3⋊Dic3—62no—
318C3×Dic3—21no—
319C3×Dic3—62no—
320C3×Dic3—61no—
321C22×C9—22no—
322C22×C9—21no—
323C22×C9—22no—
324C22×C9—34no—
325C22×C9—42no—
326C22×C9—68no—
327C22×C9—62no—
328C22×C9—62no—
329C22×C9—61no—
330C22×C9—62no—
331C22×C9D3×D412yes—
332C22×C9C23×D311yescharacteristic
333C2×D9—21no—
334C2×D9—22no—
335C4×C9—22no—
336C4×C9—42no—
337C4×C9—61no—
338Dic9—21no—
339Dic9—22no—
340Dic9—32no—
341Dic9—32no—
342Dic9—62no—
343Dic9—62no—
344Dic9—62no—
345Dic9—122no—
346C2×#41—271no—
347C23×C4—271no—
348#123—91no—
349#123—271no—
350C22⋊Q8—272no—
351#121—272no—
352#120—31no—
353#120—271no—
354C4×D4—272no—
355C4×D4—272no—
356#117—271no—
357C3×C9#52511yescharacteristic, complemented, Sylow
358C23×C3—21no—
359C23×C3—32no—
360C23×C3—31no—
361C23×C3—64no—
362C23×C3—62no—
363C23×C3—91no—
364C23×C3C22×D911yescharacteristic
365C23×C3C2×D3211yescharacteristic
366C22×D3—22no—
367C22×D3—24no—
368C22×D3—31no—
369C22×D3C22×D911yescharacteristic
370C22×D3C22×D912yes—
371C3×D4—64no—
372C3×D4—62no—
373C3×D4—182no—
374C2×C3×C4—22no—
375C2×C3×C4—31no—
376C2×C3×C4—92no—
377C2×C3×C4—91no—
378C2×C3×C4—182no—
379C2×C3×C4C22×D912yes—
380C2×C3×C4C2×D3212yes—
381#67—22no—
382#67—62no—
383#67—182no—
384#67—182no—
385#67—184no—
386#67—184no—
387C2×Dic3—32no—
388C2×Dic3—31no—
389C2×Dic3—62no—
390C2×Dic3—91no—
391C2×Dic3—91no—
392C2×Dic3—94no—
393C2×Dic3—91no—
394C2×Dic3—91no—
395C2×Dic3—94no—
396C2×Dic3—92no—
397C2×Dic3—92no—
398C2×Dic3—181no—
399C2×Dic3—181no—
400C2×Dic3—184no—
401C2×Dic3—182no—
402C2×Dic3—182no—
403C2×Dic3—182no—
404C2×Dic3—184no—
405C2×Dic3—184no—
406C2×Dic3C22×D911yescharacteristic
407D12—24no—
408C4×D3—182no—
409C4×D3—184no—
410C4×D3—184no—
411C4×D3—182no—
412Dic6—182no—
413Dic6—182no—
414C2×C32—22no—
415C2×C32C2×D3×D411yescharacteristic
416C2×C32C2×#23712yes—
417C3×D3—24no—
418C3×D3—41no—
419C3×D3—121no—
420C2×C9—22no—
421C2×C9—21no—
422C2×C9—22no—
423C2×C9—42no—
424C2×C9—64no—
425C2×C9—121no—
426C2×C9C2×D3×D411yescharacteristic
427C2×C9D3×#4112yes—
428D9—41no—
429C24—31no—
430#41—544no—
431C2×Q8—271no—
432C2×D4—31no—
433C2×D4—32no—
434C2×D4—91no—
435C2×D4—272no—
436C2×D4—271no—
437C22×C4—91no—
438C22×C4—271no—
439C22×C4—272no—
440C22×C4—271no—
441C22×C4—272no—
442C22×C4—271no—
443C22×C4—544no—
444C4⋊C4—92no—
445C4⋊C4—272no—
446C4⋊C4—272no—
447C22⋊C4—32no—
448C22⋊C4—92no—
449C22⋊C4—272no—
450C22⋊C4—272no—
451C22⋊C4—271no—
452C22⋊C4D3×D911yescharacteristic
453C42—272no—
454C22×C3—22no—
455C22×C3—22no—
456C22×C3—21no—
457C22×C3—22no—
458C22×C3—34no—
459C22×C3—42no—
460C22×C3—68no—
461C22×C3—62no—
462C22×C3—62no—
463C22×C3—61no—
464C22×C3—62no—
465C22×C3—181no—
466C22×C3—182no—
467C22×C3D4×D912yes—
468C22×C3C23×D911yescharacteristic
469C22×C3#380212yes—
470C22×C3C22×D3211yescharacteristic
471C2×D3—28no—
472C2×D3—22no—
473C2×D3—22no—
474C2×D3—21no—
475C2×D3—22no—
476C2×D3—61no—
477C2×D3—62no—
478C2×D3#369914yescomplemented
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—42no—
482C3×C4—61no—
483C3×C4—181no—
484C3×C4—182no—
485Dic3—21no—
486Dic3—61no—
487Dic3—62no—
488Dic3—92no—
489Dic3—92no—
490Dic3—92no—
491Dic3—92no—
492Dic3—182no—
493Dic3—182no—
494Dic3—182no—
495Dic3—182no—
496Dic3—362no—
497C32#558611yescharacteristic
498C9—21no—
499C9#565611yescharacteristic, complemented
500C23—32no—
501C23—31no—
502C23—64no—
503C23—62no—
504C23—91no—
505C23C2×D3×D911yescharacteristic
506Q8—542no—
507D4—64no—
508D4—62no—
509D4—182no—
510D4—542no—
511D4—544no—
512C2×C4—31no—
513C2×C4—92no—
514C2×C4—91no—
515C2×C4—182no—
516C2×C4—271no—
517C2×C4—271no—
518C2×C4—274no—
519C2×C4—272no—
520C2×C4—542no—
521C2×C4—544no—
522C2×C4—544no—
523C2×C4—542no—
524C2×C4C2×D3×D912yes—
525C2×C3—22no—
526C2×C3—22no—
527C2×C3—22no—
528C2×C3—21no—
529C2×C3—42no—
530C2×C3—64no—
531C2×C3—121no—
532C2×C3—361no—
533C2×C3C2×D4×D911yescharacteristic
534C2×C3C2×#369912yes—
535C2×C3C2×#380211yescharacteristic
536C2×C3D3×#23712yes—
537D3—24no—
538D3—41no—
539D3—121no—
540C22—22no—
541C22—34no—
542C22—68no—
543C22—62no—
544C22—62no—
545C22—62no—
546C22—61no—
547C22—182no—
548C22—181no—
549C22#8864612yes—
550C22C22×D3×D911yescharacteristic
551C4—22no—
552C4—61no—
553C4—182no—
554C4—181no—
555C4—272no—
556C4—272no—
557C4—542no—
558C3—21no—
559C3#1058968811yescharacteristic, complemented
560C3#1059408211yescharacteristic
561C2—22no—
562C2—64no—
563C2—121no—
564C2—361no—
565C2C2×#8864611yescharacteristic
566C2D3×#369912yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
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. C2×#88663C2semidirect
2. C22×D3×Dic9C2semidirect
3. #11721723C2semidirect
4. C9×#882C2semidirect
5. C3×#63448C2semidirect
6. #11721286C2semidirect
7. #11721265C2semidirect
8. #11721028C2semidirect
9. #11721026C2semidirect
10. #11721025C2semidirect
11. #11721016C2semidirect
21. C2×#6194C22semidirect
26. C2×D3×Dic9C22semidirect
27. C2×Dic3×D9C22semidirect
28. C2×#6189C22semidirect
31. #88534C22semidirect
32. C4×C3⋊Dic9C22semidirect
34. C3×(C2×D9)⋊C4C22semidirect
35. C3×C4⋊Dic9C22semidirect
37. (C2×D3)⋊Dic9C22semidirect
38. #88442C22semidirect
57. #63448D3semidirect
69. C2×C3⋊Dic9D4semidirect
137. (C2×D9)⋊C4C2×D3semidirect
138. C4⋊Dic9C2×D3semidirect
152. C2×D3×C9#41semidirect
178. #882D9semidirect
357. C3×C9#525semidirect
478. C2×D3#3699semidirect
499. C9#5656semidirect
559. C3#10589688semidirect

Maximal subgroups

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

subgroupindex
C2×#886632
C22×D3×Dic92
#117217232
C9×#8822
C3×#634482
#117212862
#117212652
#117210282
#117210262
#117210252
#117210162
#105940823
#105896883