Finite Group Databasebeta

← #13148834

Subgroups — #13148834

Order 1728.

940 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88641C211yesmaximal, characteristic, complemented
2C2×D9×Dic6C211yesmaximal, characteristic, complemented
3#11721743C211yesmaximal, characteristic, complemented
4C9×#897C211yesmaximal, characteristic, complemented
5C3×#63457C211yesmaximal, characteristic, complemented
6#11721044C212yesmaximal, complemented
7C4×#6194C211yesmaximal, characteristic, complemented
8#11720999C212yesmaximal, complemented
9#11720998C212yesmaximal, complemented
10#11720982C212yesmaximal, complemented
11#11720951C211yesmaximal, characteristic, complemented
12#10594525—31nomaximal
13#10589735—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C4×D3×C9C2211yescharacteristic
16C2×C3×D36C2212yes—
17C2×C3×D36C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×C9×Dic6C2211yescharacteristic
20C2×#6194C2211yescharacteristic
21C2×#6194C2212yes—
22C2×Dic3×D9C2212yes—
23C2×Dic3×D9C2212yes—
24C2×#6189C2212yes—
25#88641—24no—
26D9×Dic6—24no—
27C3×C9×C4⋊C4C2211yescharacteristic
28#88534C2212yes—
29C4×C3⋊Dic9C2212yes—
30C4×C3⋊Dic9C2211yescharacteristic
31C9×(C2×D3)⋊C4C2212yes—
32C3×(C2×D9)⋊C4C2212yes—
33C9×C4⋊Dic3C2212yes—
34(C2×D3)⋊Dic9C2211yescharacteristic
35(C2×D9)⋊Dic3C2212yes—
36(C2×D9)⋊Dic3C2211yescharacteristic
37(C2×D9)⋊Dic3C2212yes—
38#88442C2211yescharacteristic
39C2×#3795—31no—
40C2×D3×Dic6—31no—
41#64096—31no—
42C3×#897D311yescharacteristic
43C3×#895—31no—
44#63915—32no—
45C4×#625—31no—
46#63899—32no—
47#63898—32no—
48#63885—32no—
49#63863—31no—
50C2×Q8×D9—31no—
51C2×#3697—31no—
52C9×C22⋊Q8—31no—
53#63509—31no—
54C4×#611—31no—
55#63458—32no—
56#63457D311yescharacteristic, complemented
57#63455—32no—
58#63449—31no—
59#63449—61no—
60#63447—32no—
61#63446—32no—
62C22×D3×C9C2311yescharacteristic
63C22×C3×D9C2312yes—
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2312yes—
70#6226D414yescomplemented
71C2×C9×Dic3C2312yes—
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C4×D3×C9—22no—
75C3×D36—24no—
76C3×D36—22no—
77C3×C4×D9—22no—
78C9×Dic6—22no—
79#6194—22no—
80#6194—24no—
81Dic3×D9—24no—
82Dic3×D9—24no—
83#6189—24no—
84#5633—91no—
85#5598—91no—
86C2×#634—31no—
87C2×C3×D12—31no—
88C2×C3×D12—32no—
89C2×C3×C4×D3—31no—
90C2×C3×C4×D3C2×D311yescharacteristic
91C2×C3×Dic6C2×D311yescharacteristic
92C2×#627—32no—
93C2×#625—32no—
94C2×#625—31no—
95C2×D3×Dic3—32no—
96C2×D3×Dic3—32no—
97#3795—64no—
98D3×Dic6—64no—
99C32×C4⋊C4C2×D311yescharacteristic
100#3750—32no—
101C4×C3⋊Dic3—31no—
102C4×C3⋊Dic3—32no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4C2×D312yes—
105C3×C4⋊Dic3C2×D312yes—
106#3724—31no—
107(C2×D3)⋊Dic3—31no—
108(C2×D3)⋊Dic3—32no—
109(C2×D3)⋊Dic3—32no—
110(C2×D3)⋊Dic3—31no—
111C2×Q8×C9—31no—
112C22×C4×C9—31no—
113C2×#611—31no—
114C2×#611—32no—
115Q8×D9—64no—
116#3697—64no—
117C2×D36C2×D311yescharacteristic
118C2×D36C2×D312yes—
119C2×C4×D9—32no—
120C2×C4×D9—32no—
121C2×C4×D9C2×D311yescharacteristic
122C2×Dic18—31no—
123C2×Dic18—32no—
124C2×Dic18—61no—
125C9×C4⋊C4—21no—
126C9×C4⋊C4—32no—
127C9×C4⋊C4C2×D311yescharacteristic
128C9×C22⋊C4—32no—
129#3676—31no—
130(C2×D9)⋊C4—31no—
131(C2×D9)⋊C4—32no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4C2×D312yes—
134Dic9⋊C4—32no—
135Dic9⋊C4—31no—
136Dic9⋊C4—62no—
137C4×Dic9—31no—
138C4×Dic9—32no—
139C4×Dic9—62no—
140C4×Dic9—61no—
141C22×C3×C9C2411yescharacteristic
142C2×D3×C9—21no—
143C2×D3×C9—21no—
144C2×D3×C9—21no—
145C2×C3×D9—21no—
146C2×C3×D9—21no—
147C2×C3×D9—22no—
148C2×C3×D9—22no—
149C2×C3×D9—22no—
150C2×C3×D9#4112yescomplemented
151C3×C4×C9—22no—
152C3×C4×C9C2×D412yes—
153C3⋊Dic9—21no—
154C3⋊Dic9—21no—
155C3⋊Dic9C2×D414yes—
156C9×Dic3—22no—
157C9×Dic3—21no—
158C3×Dic9—21no—
159C2×D3×Q8—91no—
160C2×#237—91no—
161C2×#235—91no—
162C22×Dic6—91no—
163C3×C22⋊Q8—31no—
164C3×#121—91no—
165#947—91no—
166#939—92no—
167#936—91no—
168D4×Dic3—92no—
169C4×#67—91no—
170C4×#67—91no—
171#897D911yescharacteristic, complemented
172#896—92no—
173#895—31no—
174#893—92no—
175#887—91no—
176#887—91no—
177#887—181no—
178#885—92no—
179#885—92no—
180#884—92no—
181#878—92no—
182C22×C3×D3—31no—
183C22×C3×D3—32no—
184C22×C3×D3C22×D311yescharacteristic
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—31no—
189C2×C3⋊Dic3—31no—
190#634—34no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3C22×D312yes—
193C2×C3×Dic3C22×D311yescharacteristic
194C3×D12—64no—
195C3×D12—62no—
196C3×C4×D3—22no—
197C3×C4×D3—62no—
198C3×Dic6—22no—
199#627—64no—
200#625—64no—
201#625—62no—
202D3×Dic3—64no—
203D3×Dic3—64no—
204C23×C9—31no—
205C22×D9C22×D312yes—
206C22×D9C22×D311yescharacteristic
207Q8×C9—62no—
208C2×C4×C9—21no—
209C2×C4×C9—22no—
210C2×C4×C9—31no—
211C2×C4×C9—32no—
212C2×C4×C9—62no—
213C2×C4×C9C22×D312yes—
214C2×C4×C9C22×D311yescharacteristic
215#611—64no—
216#611—62no—
217C2×Dic9—32no—
218C2×Dic9—31no—
219C2×Dic9—31no—
220C2×Dic9—61no—
221C2×Dic9—62no—
222C2×Dic9—61no—
223C2×Dic9C22×D311yescharacteristic
224D36—24no—
225D36—22no—
226C4×D9—22no—
227C4×D9—64no—
228C4×D9—64no—
229Dic18—34no—
230Dic18—64no—
231Dic18—64no—
232#548—271noSylow
233C2×C3×C9C22×D411yescharacteristic
234C2×C3×C9C2×#4111yescharacteristic
235C2×C3×C92−1+411yescharacteristic
236D3×C9—41no—
237C3×D9—22no—
238C3×D9—42no—
239C2×C3×Q8—31no—
240C2×C3×D4—92no—
241C2×C3×D4—91no—
242C22×C3×C4—31no—
243C22×C3×C4—91no—
244C2×#67—91no—
245C2×#67—92no—
246C2×#67—92no—
247C2×#67—91no—
248C22×Dic3—92no—
249C22×Dic3—92no—
250D3×Q8—184no—
251#237—184no—
252#235—184no—
253C2×D12—32no—
254C2×D12—31no—
255C2×C4×D3—31no—
256C2×C4×D3—92no—
257C2×C4×D3—92no—
258C2×C4×D3C2×D911yescharacteristic
259C2×Dic6—91no—
260C2×Dic6—91no—
261C2×Dic6—92no—
262C2×Dic6—92no—
263C2×Dic6—181no—
264C2×Dic6—184no—
265C2×Dic6C2×D911yescharacteristic
266C3×C4⋊C4—21no—
267C3×C4⋊C4—32no—
268C3×C4⋊C4C2×D911yescharacteristic
269C3×C4⋊C4D3211yescharacteristic
270C3×C22⋊C4—32no—
271C3×C22⋊C4—92no—
272#217—91no—
273#217—92no—
274#217—92no—
275#217—91no—
276(C2×D3)⋊C4—32no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—91no—
280(C2×D3)⋊C4—91no—
281(C2×D3)⋊C4C2×D912yes—
282C4⋊Dic3C2×D912yes—
283Dic3⋊C4—91no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—91no—
287Dic3⋊C4—182no—
288C4×Dic3—92no—
289C4×Dic3—91no—
290C4×Dic3—92no—
291C4×Dic3—91no—
292C4×Dic3—182no—
293C4×Dic3—181no—
294C22×C32C23×D311yescharacteristic
295C2×C3×D3—21no—
296C2×C3×D3—21no—
297C2×C3×D3—21no—
298C2×C3×D3—32no—
299C2×C3×D3—61no—
300C2×C3×D3—61no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C32×C4—22no—
305C32×C4D3×D412yes—
306C3⋊Dic3—34no—
307C3⋊Dic3—61no—
308C3⋊Dic3—61no—
309C3×Dic3—21no—
310C3×Dic3—22no—
311C3×Dic3—61no—
312C22×C9—21no—
313C22×C9—61no—
314C22×C9—61no—
315C22×C9—61no—
316C22×C9C23×D311yescharacteristic
317C2×D9—21no—
318C2×D9—21no—
319C2×D9—22no—
320C2×D9—22no—
321C2×D9—22no—
322C2×D9#23712yescomplemented
323C4×C9—22no—
324C4×C9—22no—
325C4×C9—42no—
326C4×C9—62no—
327C4×C9—61no—
328C4×C9D3×D412yes—
329Dic9—21no—
330Dic9—34no—
331Dic9—64no—
332Dic9—62no—
333Dic9—61no—
334Dic9—61no—
335Dic9—121no—
336C2×#41—271no—
337C22×Q8—271no—
338#124—272no—
339#124—271no—
340C22⋊Q8—31no—
341C22⋊Q8—272no—
342#121—91no—
343#121—272no—
344C4×Q8—271no—
345C4×D4—272no—
346C4×D4—271no—
347C3×C9#54811yescharacteristic, complemented, Sylow
348C23×C3—31no—
349C23×C3—91no—
350C23×C3—92no—
351C22×D3—31no—
352C22×D3—32no—
353C22×D3C22×D911yescharacteristic
354C3×Q8—62no—
355C3×D4—184no—
356C3×D4—182no—
357C2×C3×C4—21no—
358C2×C3×C4—22no—
359C2×C3×C4—31no—
360C2×C3×C4—32no—
361C2×C3×C4—62no—
362C2×C3×C4—91no—
363C2×C3×C4—182no—
364C2×C3×C4C22×D912yes—
365C2×C3×C4C22×D911yescharacteristic
366C2×C3×C4C2×D3212yes—
367C2×C3×C4C2×D3211yescharacteristic
368#67—184no—
369#67—184no—
370#67—182no—
371#67—182no—
372C2×Dic3—31no—
373C2×Dic3—92no—
374C2×Dic3—92no—
375C2×Dic3—91no—
376C2×Dic3—91no—
377C2×Dic3—91no—
378C2×Dic3—91no—
379C2×Dic3—181no—
380C2×Dic3—182no—
381C2×Dic3—181no—
382C2×Dic3—184no—
383C2×Dic3—184no—
384C2×Dic3C22×D911yescharacteristic
385C2×Dic3C22×D912yes—
386D12—64no—
387D12—62no—
388C4×D3—22no—
389C4×D3—62no—
390C4×D3—184no—
391C4×D3—184no—
392Dic6—22no—
393Dic6—94no—
394Dic6—94no—
395Dic6—184no—
396Dic6—184no—
397Dic6—184no—
398C2×C32C2×D3×D411yescharacteristic
399C2×C32#100711yescharacteristic
400C2×C32D3×#4111yescharacteristic
401C3×D3—41no—
402C3×D3—62no—
403C3×D3—122no—
404C2×C9—21no—
405C2×C9—21no—
406C2×C9—21no—
407C2×C9—121no—
408C2×C9C2×D3×D411yescharacteristic
409C2×C9C2×#23711yescharacteristic
410C2×C9#101011yescharacteristic
411D9—22no—
412D9—42no—
413#41—544no—
414C2×Q8—31no—
415C2×Q8—272no—
416C2×Q8—271no—
417C2×Q8—544no—
418C2×D4—92no—
419C2×D4—91no—
420C2×D4—271no—
421C2×D4—272no—
422C22×C4—31no—
423C22×C4—91no—
424C22×C4—272no—
425C22×C4—272no—
426C4⋊C4—32no—
427C4⋊C4—271no—
428C4⋊C4—272no—
429C4⋊C4D3×D911yescharacteristic
430C22⋊C4—32no—
431C22⋊C4—92no—
432C22⋊C4—271no—
433C22⋊C4—272no—
434C22⋊C4—272no—
435C22⋊C4—271no—
436C42—272no—
437C42—271no—
438C22×C3—21no—
439C22×C3—61no—
440C22×C3—61no—
441C22×C3—61no—
442C22×C3—92no—
443C22×C3—181no—
444C22×C3—181no—
445C22×C3—182no—
446C22×C3—182no—
447C22×C3—182no—
448C22×C3C23×D911yescharacteristic
449C22×C3C22×D3211yescharacteristic
450C2×D3—21no—
451C2×D3—21no—
452C2×D3—21no—
453C2×D3—32no—
454C2×D3—61no—
455C2×D3—61no—
456C2×D3—62no—
457C2×D3—62no—
458C2×D3—62no—
459C3×C4—22no—
460C3×C4—22no—
461C3×C4—22no—
462C3×C4—42no—
463C3×C4—62no—
464C3×C4—61no—
465C3×C4—181no—
466C3×C4D4×D912yes—
467C3×C4#380212yes—
468Dic3—22no—
469Dic3—21no—
470Dic3—61no—
471Dic3—94no—
472Dic3—94no—
473Dic3—184no—
474Dic3—182no—
475Dic3—181no—
476Dic3—181no—
477Dic3—181no—
478Dic3—181no—
479Dic3—361no—
480C32#563311yescharacteristic
481C9—21no—
482C9#559811yescharacteristic, complemented
483C23—31no—
484C23—91no—
485C23—92no—
486Q8—62no—
487Q8—274no—
488Q8—544no—
489D4—184no—
490D4—182no—
491D4—544no—
492D4—542no—
493C2×C4—32no—
494C2×C4—31no—
495C2×C4—62no—
496C2×C4—91no—
497C2×C4—182no—
498C2×C4—272no—
499C2×C4—271no—
500C2×C4—271no—
501C2×C4—544no—
502C2×C4—544no—
503C2×C4C2×D3×D912yes—
504C2×C4C2×D3×D911yescharacteristic
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—121no—
509C2×C3—182no—
510C2×C3—362no—
511C2×C3C2×D4×D911yescharacteristic
512C2×C3#6371311yescharacteristic
513C2×C3#41×D911yescharacteristic
514C2×C3C2×#380211yescharacteristic
515C2×C3D3×#23711yescharacteristic
516C2×C3#6431511yescharacteristic
517D3—41no—
518D3—62no—
519D3—122no—
520C22—61no—
521C22—61no—
522C22—61no—
523C22—92no—
524C22—181no—
525C22—181no—
526C22—182no—
527C22—182no—
528C22—182no—
529C22C22×D3×D911yescharacteristic
530C4—22no—
531C4—61no—
532C4—62no—
533C4—181no—
534C4—274no—
535C4—541no—
536C4—541no—
537C4#8864612yes—
538C3—21no—
539C3#1058973511yescharacteristic, complemented
540C3#1059452511yescharacteristic
541C2—121no—
542C2—182no—
543C2—362no—
544C2C2×#8864611yescharacteristic
545C2D9×#23711yescharacteristic
546C2#1172251011yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#548
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×#88641C2semidirect
2. C2×D9×Dic6C2semidirect
3. #11721743C2semidirect
4. C9×#897C2semidirect
5. C3×#63457C2semidirect
6. #11721044C2semidirect
7. C4×#6194C2semidirect
8. #11720999C2semidirect
9. #11720998C2semidirect
10. #11720982C2semidirect
11. #11720951C2semidirect
56. #63457D3semidirect
70. #6226D4semidirect
150. C2×C3×D9#41semidirect
171. #897D9semidirect
322. C2×D9#237semidirect
347. C3×C9#548semidirect
482. C9#5598semidirect
539. C3#10589735semidirect

Maximal subgroups

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

subgroupindex
C2×#886412
C2×D9×Dic62
#117217432
C9×#8972
C3×#634572
#117210442
C4×#61942
#117209992
#117209982
#117209822
#117209512
#105945253
#105897353