Finite Group Databasebeta

← #13147599

Subgroups — #13147599

Order 1728.

946 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88642C211yesmaximal, characteristic, complemented
2C2×D3×Dic18C211yesmaximal, characteristic, complemented
3#11721712C211yesmaximal, characteristic, complemented
4C4×C9×D12C211yesmaximal, characteristic, complemented
5C3×C4×Dic18C211yesmaximal, characteristic, complemented
6#11721038C211yesmaximal, characteristic, complemented
7#11721017C212yesmaximal, complemented
8#11721013C212yesmaximal, complemented
9#11720994C212yesmaximal, complemented
10#11720961C211yesmaximal, characteristic, complemented
11#11720956C212yesmaximal, complemented
12#10593899—31nomaximal
13#10589654—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C9×D12C2211yescharacteristic
17C2×C4×D3×C9C2211yescharacteristic
18C2×C4×D3×C9C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×#6195C2212yes—
21C2×#6195C2212yes—
22C2×D3×Dic9C2212yes—
23C2×(C3⋊D9)⋊C4C2212yes—
24C2×#6189C2212yes—
25#88642—24no—
26D3×Dic18—24no—
27C3×C42×C9C2211yescharacteristic
28#88536C2212yes—
29#88536C2212yes—
30C9×(C2×D3)⋊C4C2211yescharacteristic
31C9×(C2×D3)⋊C4C2211yescharacteristic
32C9×C4⋊Dic3C2211yescharacteristic
33C3×C4⋊Dic9C2211yescharacteristic
34C3×Dic9⋊C4C2212yes—
35C3×C4×Dic9C2212yes—
36(C2×D3)⋊Dic9C2212yes—
37#88445C2212yes—
38Dic9⋊Dic3C2212yes—
39C2×#3799—31no—
40C2×D3×Dic6—31no—
41#64084—31no—
42C3×C4×D12D311yescharacteristic
43C3×C4×Dic6—31no—
44#63909—31no—
45#63895—32no—
46#63886—32no—
47#63880—32no—
48#63857—31no—
49#63852—32no—
50C2×#3697—31no—
51C22×Dic18—31no—
52C4×D4×C9—31no—
53#63501—32no—
54#63492—31no—
55#63486—31no—
56#63458—32no—
57#63447—32no—
58#611⋊C4—32no—
59#63437—31no—
60#63437—61no—
61C4×Dic18D311yescharacteristic, complemented
62C22×C3⋊D9C2312yes—
63C22×D3×C9C2311yescharacteristic
64C22×D3×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3⋊Dic9C2312yes—
69C4⋊(C3⋊D9)—22no—
70#6226—22no—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2312yes—
75C9×D12—22no—
76C4×D3×C9—22no—
77C4×D3×C9—22no—
78C3×Dic18D414yescomplemented
79#6195—24no—
80#6195—24no—
81D3×Dic9—24no—
82(C3⋊D9)⋊C4—24no—
83#6189—24no—
84#5576—91no—
85#5552—91no—
86C2×C4⋊(C3⋊D3)—31no—
87C2×#634—31no—
88C2×C3×D12C2×D311yescharacteristic
89C2×C3×C4×D3C2×D311yescharacteristic
90C2×C3×C4×D3C2×D311yescharacteristic
91C2×C3×Dic6—31no—
92C2×#627—32no—
93C2×#626—32no—
94C2×#626—32no—
95C2×#624—32no—
96C2×D3×Dic3—32no—
97#3799—64no—
98D3×Dic6—64no—
99C32×C42C2×D311yescharacteristic
100#3752—32no—
101#3752—32no—
102C3×(C2×D3)⋊C4C2×D311yescharacteristic
103C3×(C2×D3)⋊C4C2×D311yescharacteristic
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3C2×D311yescharacteristic
106C3×Dic3⋊C4—32no—
107C3×C4×Dic3—32no—
108Dic3⋊Dic3—32no—
109#3722—32no—
110(C2×D3)⋊Dic3—32no—
111C2×D4×C9—31no—
112C22×C4×C9—31no—
113C22×C4×C9—31no—
114C2×#611—32no—
115C2×#611—32no—
116C22×Dic9—32no—
117#3697—64no—
118C2×D36—31no—
119C2×D36—61no—
120C2×C4×D9—32no—
121C2×Dic18—31no—
122C2×Dic18—32no—
123C2×Dic18—64no—
124C2×Dic18—61no—
125C2×Dic18C2×D311yescharacteristic
126C9×C4⋊C4—31no—
127C9×C22⋊C4—31no—
128C9×C22⋊C4—31no—
129C42×C9—21no—
130C42×C9C2×D311yescharacteristic
131#3676—32no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4—32no—
134(C2×D9)⋊C4—32no—
135(C2×D9)⋊C4—62no—
136(C2×D9)⋊C4—62no—
137C4⋊Dic9C2×D311yescharacteristic
138Dic9⋊C4—32no—
139Dic9⋊C4C2×D312yes—
140C4×Dic9C2×D312yes—
141C22×C3×C9C2411yescharacteristic
142C2×C3⋊D9—22no—
143C2×C3⋊D9—22no—
144C2×C3⋊D9—22no—
145C2×D3×C9—21no—
146C2×D3×C9—21no—
147C2×D3×C9—21no—
148C2×D3×C9—21no—
149C2×D3×C9—21no—
150C2×D3×C9#4112yescomplemented
151C3×C4×C9—21no—
152C3×C4×C9—21no—
153C3×C4×C9C2×D412yes—
154C3⋊Dic9—22no—
155C9×Dic3—21no—
156C9×Dic3—21no—
157C3×Dic9—22no—
158C3×Dic9C2×D414yes—
159C2×#239—91no—
160C2×D3×Q8—91no—
161C2×#235—91no—
162C22×Dic6—91no—
163C3×C4×Q8—91no—
164C3×C4×D4—31no—
165#939—92no—
166#930—91no—
167#924—91no—
168#897—91no—
169#897—92no—
170#896—92no—
171#895—92no—
172#895—91no—
173#885—92no—
174#67⋊C4—92no—
175#875—91no—
176#875—91no—
177#875—92no—
178#875—181no—
179C4×D12—92no—
180C4×D12D911yescharacteristic, complemented
181C4×Dic6—31no—
182C22×C3⋊D3—32no—
183C22×C3×D3C22×D311yescharacteristic
184C22×C3×D3C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C3⋊Dic3—32no—
189C4⋊(C3⋊D3)—62no—
190#634—62no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3C22×D311yescharacteristic
194C2×C3×Dic3C22×D311yescharacteristic
195C3×D12—22no—
196C3×C4×D3—22no—
197C3×C4×D3—22no—
198C3×Dic6—34no—
199#627—64no—
200#626—64no—
201#626—64no—
202#624—64no—
203D3×Dic3—64no—
204C23×C9—31no—
205C23×C9—31no—
206C22×D9—32no—
207C22×D9—62no—
208D4×C9—62no—
209C2×C4×C9—21no—
210C2×C4×C9—21no—
211C2×C4×C9—21no—
212C2×C4×C9—31no—
213C2×C4×C9—31no—
214C2×C4×C9—62no—
215C2×C4×C9—62no—
216C2×C4×C9C22×D311yescharacteristic
217C2×C4×C9C22×D311yescharacteristic
218C2×C4×C9C22×D311yescharacteristic
219#611—64no—
220#611—64no—
221C2×Dic9—32no—
222C2×Dic9—62no—
223C2×Dic9—64no—
224C2×Dic9C22×D312yes—
225C2×Dic9C22×D312yes—
226D36—62no—
227D36—122no—
228C4×D9—64no—
229Dic18—62no—
230Dic18—64no—
231Dic18—122no—
232Dic18D1214yescomplemented
233#548—271noSylow
234C2×C3×C9C22×D411yescharacteristic
235C2×C3×C9C2×#4111yescharacteristic
236C2×C3×C92−1+411yescharacteristic
237C3⋊D9—42no—
238D3×C9—22no—
239D3×C9—41no—
240C2×C3×Q8—91no—
241C2×C3×D4—31no—
242C22×C3×C4—31no—
243C22×C3×C4—31no—
244C2×#67—92no—
245C2×#67—92no—
246C22×Dic3—92no—
247#239—184no—
248D3×Q8—184no—
249#235—184no—
250C2×D12—92no—
251C2×D12—92no—
252C2×D12—91no—
253C2×D12—91no—
254C2×D12—181no—
255C2×D12C2×D911yescharacteristic
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×C4×D3—92no—
259C2×C4×D3C2×D911yescharacteristic
260C2×C4×D3C2×D911yescharacteristic
261C2×Dic6—31no—
262C2×Dic6—92no—
263C2×Dic6—91no—
264C2×Dic6—91no—
265C2×Dic6—92no—
266C2×Dic6—181no—
267C2×Dic6—184no—
268C3×C4⋊C4—31no—
269C3×C4⋊C4—92no—
270C3×C4⋊C4—91no—
271C3×C22⋊C4—31no—
272C3×C22⋊C4—31no—
273C3×C42—21no—
274C3×C42—92no—
275C3×C42C2×D911yescharacteristic
276C3×C42D3211yescharacteristic
277#217—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—92no—
284(C2×D3)⋊C4—92no—
285(C2×D3)⋊C4—182no—
286(C2×D3)⋊C4—182no—
287(C2×D3)⋊C4C2×D911yescharacteristic
288(C2×D3)⋊C4C2×D911yescharacteristic
289C4⋊Dic3—31no—
290C4⋊Dic3—92no—
291C4⋊Dic3C2×D911yescharacteristic
292Dic3⋊C4—32no—
293Dic3⋊C4—92no—
294C4×Dic3—32no—
295C22×C32C23×D311yescharacteristic
296C2×C3⋊D3—62no—
297C2×C3⋊D3—62no—
298C2×C3⋊D3—62no—
299C2×C3×D3—21no—
300C2×C3×D3—21no—
301C2×C3×D3—21no—
302C2×C3×D3—21no—
303C2×C3×D3—21no—
304C2×C3×D3#23512yes—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4D3×D412yes—
308C3⋊Dic3—62no—
309C3×Dic3—21no—
310C3×Dic3—21no—
311C3×Dic3—34no—
312C3×Dic3—62no—
313C22×C9—21no—
314C22×C9—32no—
315C22×C9—61no—
316C22×C9—61no—
317C22×C9—61no—
318C22×C9—61no—
319C22×C9—61no—
320C22×C9C23×D311yescharacteristic
321C2×D9—62no—
322C2×D9—62no—
323C2×D9—62no—
324C2×D9—122no—
325C2×D9—122no—
326C2×D9—122no—
327C4×C9—21no—
328C4×C9—21no—
329C4×C9—22no—
330C4×C9—41no—
331C4×C9—41no—
332C4×C9—61no—
333C4×C9—61no—
334C4×C9C2×D1212yes—
335Dic9—22no—
336Dic9—62no—
337Dic9—122no—
338Dic9C2×D1214yes—
339C2×#41—271no—
340C22×Q8—271no—
341#124—272no—
342#124—271no—
343C22⋊Q8—272no—
344C22⋊Q8—271no—
345#121—271no—
346#121—272no—
347C4×Q8—91no—
348C4×D4—31no—
349C4×D4—272no—
350C3×C9#54811yescharacteristic, complemented, Sylow
351C23×C3—31no—
352C23×C3—31no—
353C22×D3—92no—
354C22×D3—92no—
355C22×D3—182no—
356C22×D3C22×D911yescharacteristic
357C22×D3C22×D911yescharacteristic
358C3×Q8—94no—
359C3×D4—62no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—62no—
366C2×C3×C4—62no—
367C2×C3×C4—92no—
368C2×C3×C4—92no—
369C2×C3×C4C22×D911yescharacteristic
370C2×C3×C4C22×D911yescharacteristic
371C2×C3×C4C22×D911yescharacteristic
372C2×C3×C4C2×D3211yescharacteristic
373C2×C3×C4C2×D3211yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375#67—184no—
376#67—184no—
377C2×Dic3—32no—
378C2×Dic3—32no—
379C2×Dic3—92no—
380C2×Dic3—92no—
381C2×Dic3—182no—
382C2×Dic3—184no—
383C2×Dic3C22×D911yescharacteristic
384C2×Dic3C22×D911yescharacteristic
385D12—22no—
386D12—184no—
387D12—184no—
388D12—182no—
389D12—182no—
390D12—362no—
391C4×D3—22no—
392C4×D3—22no—
393C4×D3—184no—
394C4×D3—184no—
395C4×D3—184no—
396Dic6—34no—
397Dic6—184no—
398Dic6—184no—
399Dic6—182no—
400Dic6—182no—
401Dic6—362no—
402C2×C32C2×#23511yescharacteristic
403C2×C32C2×D3×D411yescharacteristic
404C2×C32#101011yescharacteristic
405C3⋊D3—122no—
406C3×D3—22no—
407C3×D3—41no—
408C2×C9—21no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9—62no—
412C2×C9—121no—
413C2×C9C22×D1211yescharacteristic
414C2×C9#100711yescharacteristic
415C2×C9D3×#4111yescharacteristic
416D9—122no—
417D9—242no—
418#41—544no—
419C2×Q8—91no—
420C2×Q8—272no—
421C2×Q8—271no—
422C2×Q8—544no—
423C2×D4—31no—
424C2×D4—272no—
425C2×D4—272no—
426C2×D4—271no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—272no—
430C22×C4—272no—
431C4⋊C4—31no—
432C4⋊C4—92no—
433C4⋊C4—91no—
434C4⋊C4—272no—
435C22⋊C4—31no—
436C22⋊C4—31no—
437C22⋊C4—272no—
438C22⋊C4—272no—
439C22⋊C4—272no—
440C22⋊C4—272no—
441C42—92no—
442C42D3×D911yescharacteristic
443C22×C3—21no—
444C22×C3—32no—
445C22×C3—61no—
446C22×C3—61no—
447C22×C3—61no—
448C22×C3—61no—
449C22×C3—61no—
450C22×C3C23×D911yescharacteristic
451C22×C3C22×D3211yescharacteristic
452C2×D3—21no—
453C2×D3—21no—
454C2×D3—21no—
455C2×D3—21no—
456C2×D3—21no—
457C2×D3—182no—
458C2×D3—182no—
459C2×D3—182no—
460C2×D3—182no—
461C2×D3—182no—
462C2×D3—182no—
463C2×D3—362no—
464C2×D3—362no—
465C2×D3—362no—
466C2×D3#369712yescomplemented
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—22no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—61no—
475C3×C4—61no—
476C3×C4—94no—
477C3×C4—182no—
478C3×C4D4×D912yes—
479C3×C4D3×D1212yes—
480Dic3—21no—
481Dic3—21no—
482Dic3—34no—
483Dic3—62no—
484Dic3—182no—
485Dic3—182no—
486Dic3—362no—
487C32#555211yescharacteristic
488C9—21no—
489C9#557611yescharacteristic, complemented
490C23—31no—
491C23—31no—
492C23—272no—
493Q8—94no—
494Q8—544no—
495Q8—542no—
496D4—62no—
497D4—544no—
498D4—544no—
499D4—542no—
500C2×C4—31no—
501C2×C4—31no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—92no—
505C2×C4—92no—
506C2×C4—272no—
507C2×C4—544no—
508C2×C4—544no—
509C2×C4C2×D3×D911yescharacteristic
510C2×C4C2×D3×D911yescharacteristic
511C2×C4C2×D3×D911yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—62no—
516C2×C3—121no—
517C2×C3C2×#369711yescharacteristic
518C2×C3C2×D4×D911yescharacteristic
519C2×C3#6371611yescharacteristic
520C2×C3#6429711yescharacteristic
521C2×C3C2×D3×D1211yescharacteristic
522C2×C3D3×#23511yescharacteristic
523D3—22no—
524D3—41no—
525D3—362no—
526D3—362no—
527D3—722no—
528C22—32no—
529C22—61no—
530C22—61no—
531C22—61no—
532C22—61no—
533C22—61no—
534C22—542no—
535C22—542no—
536C22—542no—
537C22C22×D3×D911yescharacteristic
538C4—21no—
539C4—21no—
540C4—61no—
541C4—61no—
542C4—94no—
543C4—182no—
544C4—542no—
545C4D9×D1212yes—
546C3—21no—
547C3#1058965411yescharacteristic, complemented
548C3#1059389911yescharacteristic
549C2—62no—
550C2—121no—
551C2—1082no—
552C2#1172246411yescharacteristic
553C2C2×D9×D1211yescharacteristic
554C2D3×#369711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#548
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88642C2semidirect
2. C2×D3×Dic18C2semidirect
3. #11721712C2semidirect
4. C4×C9×D12C2semidirect
5. C3×C4×Dic18C2semidirect
6. #11721038C2semidirect
7. #11721017C2semidirect
8. #11721013C2semidirect
9. #11720994C2semidirect
10. #11720961C2semidirect
11. #11720956C2semidirect
61. C4×Dic18D3semidirect
78. C3×Dic18D4semidirect
150. C2×D3×C9#41semidirect
180. C4×D12D9semidirect
232. Dic18D12semidirect
350. C3×C9#548semidirect
466. C2×D3#3697semidirect
489. C9#5576semidirect
547. C3#10589654semidirect

Maximal subgroups

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

subgroupindex
C2×#886422
C2×D3×Dic182
#117217122
C4×C9×D122
C3×C4×Dic182
#117210382
#117210172
#117210132
#117209942
#117209612
#117209562
#105938993
#105896543