Finite Group Databasebeta

← #13147553

Subgroups — #13147553

Order 1728.

724 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721709C211yesmaximal, characteristic, complemented
2C3×#63437C211yesmaximal, characteristic, complemented
3C9×#869C211yesmaximal, characteristic, complemented
4#11721007C211yesmaximal, characteristic
5#11721003C212yesmaximal, complemented
6#11720998C211yesmaximal, characteristic, complemented
7#11720988C212yesmaximal, complemented
8#11720984C212yesmaximal, complemented
9#11720946C212yesmaximal, complemented
10#11720936C211yesmaximal, characteristic, complemented
11#11720910C211yesmaximal, characteristic, complemented
12#10593955—31nomaximal
13#10589835—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C3×D36C2211yescharacteristic
16C2×C9×Dic6C2212yes—
17C2×C3×Dic18C2211yescharacteristic
18C2×#6192C2212yes—
19C2×Dic3×D9C2212yes—
20C3×C42×C9C2211yescharacteristic
21#88536C2211yescharacteristic
22#88536C2211yescharacteristic
23#88534C2211yescharacteristic
24#88534C2211yescharacteristic
25C4×C3⋊Dic9C2211yescharacteristic
26C3×(C2×D9)⋊C4C2212yes—
27C3×(C2×D9)⋊C4C2212yes—
28C9×C4⋊Dic3C2212yes—
29C9×C4⋊Dic3C2211yescharacteristic
30C9×C4⋊Dic3C2211yescharacteristic
31#88445C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64081—31no—
38C3×#875—31no—
39C3×#869D311yescharacteristic
40#63916—31no—
41#63907—32no—
42#63898—31no—
43#63890—32no—
44#63887—32no—
45#63871—31no—
46#63858—32no—
47#63843—31no—
48C9×C4⋊Q8—31no—
49#63509—32no—
50#63508—32no—
51#63460—32no—
52#63459—31no—
53#63457—31no—
54#63456—32no—
55#63455—31no—
56#63449—31no—
57#63437D311yescharacteristic, complemented
58#63434—31no—
59#63434—61no—
60C22×C3⋊D9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C4×C3⋊D9—22no—
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2312yes—
72C3×D36—22no—
73C9×Dic6—42no—
74C3×Dic18—22no—
75#6192—42no—
76Dic3×D9—42no—
77#5733—91no—
78#5682—91no—
79C2×C4×C3⋊D3—31no—
80C2×C3×D12—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6C2×D312yes—
83C2×#626—32no—
84C2×D3×Dic3—32no—
85C32×C42C2×D311yescharacteristic
86#3752—31no—
87#3752—31no—
88#3750—31no—
89#3750—31no—
90C4×C3⋊Dic3—31no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D312yes—
95C3×C4⋊Dic3C2×D311yescharacteristic
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98#3722—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101Dic32—32no—
102C2×Q8×C9—32no—
103C2×D36—32no—
104C2×D36C2×D311yescharacteristic
105C2×C4×D9—31no—
106C2×C4×D9—32no—
107C2×C4×D9—61no—
108C2×Dic18C2×D311yescharacteristic
109C9×C4⋊C4—31no—
110C9×C4⋊C4—31no—
111C9×C4⋊C4—32no—
112C42×C9—21no—
113C42×C9C2×D311yescharacteristic
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—31no—
117(C2×D9)⋊C4—31no—
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—61no—
120(C2×D9)⋊C4—61no—
121(C2×D9)⋊C4C2×D312yes—
122(C2×D9)⋊C4C2×D312yes—
123C4⋊Dic9—32no—
124Dic9⋊C4—31no—
125Dic9⋊C4—31no—
126Dic9⋊C4—32no—
127Dic9⋊C4—61no—
128Dic9⋊C4—61no—
129C4×Dic9—31no—
130C4×Dic9—32no—
131C4×Dic9—61no—
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—21no—
134C2×C3⋊D9—21no—
135C2×C3⋊D9—21no—
136C2×C3×D9—22no—
137C2×C3×D9—22no—
138C2×C3×D9—22no—
139C3×C4×C9—21no—
140C3×C4×C9—21no—
141C3×C4×C9#4112yes—
142C3⋊Dic9—21no—
143C3⋊Dic9—21no—
144C3⋊Dic9—21no—
145C9×Dic3—22no—
146C9×Dic3—22no—
147C3×Dic9—22no—
148C3×C4⋊Q8—31no—
149C3×#124—91no—
150#947—92no—
151#946—92no—
152#946—91no—
153Q8×Dic3—91no—
154#938—91no—
155D4×Dic3—91no—
156#898—92no—
157#897—91no—
158#895—91no—
159#894—92no—
160#893—91no—
161#887—91no—
162#886—92no—
163#885—92no—
164#879—92no—
165#878—92no—
166#875—31no—
167#872—91no—
168#872—91no—
169#872—181no—
170#869D911yescharacteristic, complemented
171C22×C3⋊D3—31no—
172C22×C3×D3—32no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—31no—
179C4×C3⋊D3—62no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3C22×D312yes—
182C2×C3×Dic3C22×D312yes—
183C3×D12—62no—
184C3×Dic6—42no—
185C3×Dic6—62no—
186#626—122no—
187D3×Dic3—122no—
188C22×D9—31no—
189C22×D9—61no—
190C22×D9C22×D312yes—
191Q8×C9—122no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—21no—
195C2×C4×C9—32no—
196C2×C4×C9—32no—
197C2×C4×C9C22×D311yescharacteristic
198C2×C4×C9C22×D311yescharacteristic
199C2×C4×C9C22×D311yescharacteristic
200C2×Dic9—31no—
201C2×Dic9—31no—
202C2×Dic9—31no—
203C2×Dic9—61no—
204C2×Dic9—61no—
205C2×Dic9—61no—
206C2×Dic9C22×D312yes—
207D36—22no—
208D36—122no—
209C4×D9—64no—
210C4×D9—62no—
211C4×D9—122no—
212Dic18—22no—
213#542—271noSylow
214C2×C3×C9C2×#4111yescharacteristic
215C2×C3×C9H(5,𝔽2)11yescharacteristic
216C2×C3×C92−1+411yescharacteristic
217C3⋊D9—41no—
218C3×D9—42no—
219C2×C3×Q8—32no—
220C2×C3×Q8—91no—
221C2×C3×D4—91no—
222C2×#67—92no—
223C22×Dic3—92no—
224C2×D12—31no—
225C2×D12—92no—
226C2×C4×D3—91no—
227C2×C4×D3—91no—
228C2×C4×D3—92no—
229C2×C4×D3—181no—
230C2×Dic6—31no—
231C2×Dic6C2×D912yes—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—32no—
235C3×C22⋊C4—92no—
236C3×C22⋊C4—92no—
237C3×C42—21no—
238C3×C42C2×D911yescharacteristic
239C3×C42D3211yescharacteristic
240#217—92no—
241#217—92no—
242(C2×D3)⋊C4—32no—
243(C2×D3)⋊C4—32no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—91no—
248(C2×D3)⋊C4—91no—
249(C2×D3)⋊C4—91no—
250(C2×D3)⋊C4—91no—
251(C2×D3)⋊C4—92no—
252(C2×D3)⋊C4—181no—
253(C2×D3)⋊C4—181no—
254C4⋊Dic3—92no—
255C4⋊Dic3—92no—
256C4⋊Dic3C2×D912yes—
257C4⋊Dic3C2×D911yescharacteristic
258C4⋊Dic3C2×D911yescharacteristic
259Dic3⋊C4—92no—
260Dic3⋊C4—91no—
261Dic3⋊C4—91no—
262Dic3⋊C4—91no—
263Dic3⋊C4—91no—
264Dic3⋊C4—92no—
265Dic3⋊C4—181no—
266Dic3⋊C4—181no—
267C4×Dic3—92no—
268C4×Dic3—91no—
269C4×Dic3—91no—
270C4×Dic3—92no—
271C4×Dic3—181no—
272C22×C32C23×D311yescharacteristic
273C2×C3⋊D3—61no—
274C2×C3⋊D3—61no—
275C2×C3⋊D3—61no—
276C2×C3×D3—62no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4#23912yes—
282C3⋊Dic3—61no—
283C3⋊Dic3—61no—
284C3⋊Dic3—61no—
285C3×Dic3—22no—
286C3×Dic3—22no—
287C3×Dic3—62no—
288C22×C9—21no—
289C22×C9C23×D311yescharacteristic
290C2×D9—22no—
291C2×D9—22no—
292C2×D9—22no—
293C2×D9—62no—
294C2×D9—61no—
295C2×D9—61no—
296C2×D9—61no—
297C2×D9—121no—
298C2×D9—121no—
299C4×C9—21no—
300C4×C9—21no—
301C4×C9—22no—
302C4×C9—41no—
303C4×C9—41no—
304C4×C9—62no—
305C4×C9—62no—
306C4×C9#23712yes—
307Dic9—22no—
308Dic9—62no—
309Dic9—61no—
310Dic9—61no—
311Dic9—61no—
312Dic9—121no—
313Dic9—121no—
314C4⋊Q8—31no—
315#126—272no—
316#124—91no—
317#124—272no—
318#123—272no—
319C22⋊Q8—272no—
320C22⋊Q8—271no—
321#121—271no—
322C4×Q8—271no—
323C4×D4—271no—
324#117—271no—
325C3×C9#54211yescharacteristic, complemented, Sylow
326C23×C3—92no—
327C22×D3—32no—
328C22×D3—91no—
329C22×D3—91no—
330C22×D3—181no—
331C3×Q8—122no—
332C3×Q8—182no—
333C3×D4—182no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—32no—
338C2×C3×C4—32no—
339C2×C3×C4—92no—
340C2×C3×C4C22×D911yescharacteristic
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346#67—362no—
347C2×Dic3—32no—
348C2×Dic3—91no—
349C2×Dic3—91no—
350C2×Dic3—91no—
351C2×Dic3—91no—
352C2×Dic3—91no—
353C2×Dic3—91no—
354C2×Dic3—181no—
355C2×Dic3—181no—
356C2×Dic3—181no—
357C2×Dic3—362no—
358C2×Dic3C22×D912yes—
359C2×Dic3C22×D912yes—
360D12—62no—
361D12—362no—
362C4×D3—184no—
363C4×D3—182no—
364C4×D3—182no—
365C4×D3—362no—
366Dic6—42no—
367Dic6—62no—
368C2×C32#100411yescharacteristic
369C2×C32C2×#23911yescharacteristic
370C2×C32#100711yescharacteristic
371C3⋊D3—121no—
372C3×D3—122no—
373C2×C9—21no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9C2×#23711yescharacteristic
377C2×C9#100911yescharacteristic
378C2×C9#101011yescharacteristic
379D9—42no—
380D9—122no—
381D9—121no—
382C2×Q8—32no—
383C2×Q8—91no—
384C2×D4—91no—
385C2×D4—272no—
386C22×C4—272no—
387C22×C4—271no—
388C4⋊C4—31no—
389C4⋊C4—31no—
390C4⋊C4—32no—
391C4⋊C4—272no—
392C4⋊C4—272no—
393C4⋊C4—271no—
394C4⋊C4—271no—
395C22⋊C4—92no—
396C22⋊C4—92no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—271no—
401C22⋊C4—271no—
402C42—272no—
403C42—271no—
404C42D3×D911yescharacteristic
405C22×C3—21no—
406C22×C3—182no—
407C22×C3—182no—
408C22×C3—182no—
409C22×C3C23×D911yescharacteristic
410C22×C3C22×D3211yescharacteristic
411C2×D3—62no—
412C2×D3—62no—
413C2×D3—62no—
414C2×D3—182no—
415C2×D3—181no—
416C2×D3—181no—
417C2×D3—181no—
418C2×D3—181no—
419C2×D3—181no—
420C2×D3—181no—
421C2×D3—361no—
422C2×D3—361no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—22no—
428C3×C4—41no—
429C3×C4—41no—
430C3×C4—62no—
431C3×C4—62no—
432C3×C4—182no—
433C3×C4#370112yes—
434C3×C4#379612yes—
435Dic3—22no—
436Dic3—22no—
437Dic3—62no—
438Dic3—182no—
439Dic3—181no—
440Dic3—181no—
441Dic3—181no—
442Dic3—181no—
443Dic3—181no—
444Dic3—181no—
445Dic3—361no—
446Dic3—361no—
447C32#573311yescharacteristic
448C9—21no—
449C9#568211yescharacteristic, complemented
450C23—92no—
451C23—271no—
452Q8—122no—
453Q8—182no—
454D4—182no—
455D4—1082no—
456C2×C4—32no—
457C2×C4—32no—
458C2×C4—92no—
459C2×C4—271no—
460C2×C4—271no—
461C2×C4—271no—
462C2×C4—542no—
463C2×C4—1082no—
464C2×C4C2×D3×D911yescharacteristic
465C2×C4C2×D3×D911yescharacteristic
466C2×C4C2×D3×D911yescharacteristic
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—362no—
471C2×C3#6371011yescharacteristic
472C2×C3C2×#370111yescharacteristic
473C2×C3#6371311yescharacteristic
474C2×C3C2×#379611yescharacteristic
475C2×C3#6429711yescharacteristic
476C2×C3#6430711yescharacteristic
477D3—122no—
478D3—362no—
479D3—361no—
480D3—361no—
481C22—182no—
482C22—182no—
483C22—182no—
484C22—541no—
485C22—541no—
486C22—541no—
487C22C22×D3×D911yescharacteristic
488C4—21no—
489C4—21no—
490C4—62no—
491C4—62no—
492C4—182no—
493C4—541no—
494C4—541no—
495C4—541no—
496C4#8863412yes—
497C3—21no—
498C3#1058983511yescharacteristic, complemented
499C3#1059395511yescharacteristic
500C2—362no—
501C2—1081no—
502C2C2×#8863411yescharacteristic
503C2#1172245811yescharacteristic
504C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721007C2
hypercenterC42D3×D9
hypocenterC3×C9#542
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721709C2semidirect
2. C3×#63437C2semidirect
3. C9×#869C2semidirect
5. #11721003C2semidirect
6. #11720998C2semidirect
7. #11720988C2semidirect
8. #11720984C2semidirect
9. #11720946C2semidirect
10. #11720936C2semidirect
11. #11720910C2semidirect
57. #63437D3semidirect
170. #869D9semidirect
325. C3×C9#542semidirect
449. C9#5682semidirect
498. C3#10589835semidirect

Maximal subgroups

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

subgroupindex
#117217092
C3×#634372
C9×#8692
#117210072
#117210032
#117209982
#117209882
#117209842
#117209462
#117209362
#117209102
#105939553
#105898353