Finite Group Databasebeta

← #13151139

Subgroups — #13151139

Order 1728.

1024 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×#88663C212yesmaximal, complemented
2D4×C3⋊Dic9C211yesmaximal, characteristic, complemented
3C9×#938C211yesmaximal, characteristic, complemented
4C3×#63502C211yesmaximal, characteristic, complemented
5#11721281C212yesmaximal, complemented
6#611⋊Dic3C212yesmaximal, complemented
7#11721257C212yesmaximal, complemented
8#11721037C211yesmaximal, characteristic, complemented
9C4×D3×Dic9C211yesmaximal, characteristic, complemented
10#11720998C211yesmaximal, characteristic, complemented
11#11720969C211yesmaximal, characteristic, complemented
12#10595133—31nomaximal
13#10589698—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2212yes—
16C2×C9×#67C2212yes—
17C2×C3×#611C2212yes—
18C2×C3×#611C2212yes—
19C2×C4×D3×C9C2211yescharacteristic
20C2×C3×D36C2211yescharacteristic
21C2×#6194C2212yes—
22#88663—28no—
23C2×D3×Dic9C2211yescharacteristic
24C2×D3×Dic9C2211yescharacteristic
25C2×Dic3×D9C2212yes—
26C2×#6189C2212yes—
27#88551C2212yes—
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29C4×C3⋊Dic9C2211yescharacteristic
30C9×#217C2212yes—
31C9×C4⋊Dic3C2211yescharacteristic
32C3×C4×Dic9C2211yescharacteristic
33(C2×D9)⋊Dic3C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2211yescharacteristic
37Dic3×Dic9C2211yescharacteristic
38C2×#3805—32no—
39D4×C3⋊Dic3—31no—
40C3×#940—31no—
41C3×#938D311yescharacteristic
42#67⋊Dic3—32no—
43#63970—32no—
44#63959—32no—
45#63918—31no—
46#63908—31no—
47#63898—31no—
48C4×D3×Dic3—31no—
49C2×#3699—32no—
50C9×#121—31no—
51#63502D311yescharacteristic, complemented
52#63501—32no—
53D4×Dic9—31no—
54D4×Dic9—61no—
55#63492—31no—
56C2×C4×Dic9—31no—
57#63455—31no—
58#63450—31no—
59#63447—32no—
60#611⋊C4—32no—
61C23×C3×C9C2312yes—
62C22×D3×C9C2311yescharacteristic
63C22×C3×D9C2312yes—
64C3×D4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9—24no—
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2311yescharacteristic
70C9×#67—24no—
71C3×#611—24no—
72C3×#611—24no—
73C2×C9×Dic3C2312yes—
74C2×C9×Dic3C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77C4×D3×C9—22no—
78C3×D36—22no—
79#6194—24no—
80D3×Dic9—22no—
81D3×Dic9D414yescomplemented
82Dic3×D9—24no—
83#6189—24no—
84#5716—91no—
85#5596—91no—
86C2×C32×D4C2×D311yescharacteristic
87C22×C3⋊Dic3—32no—
88C2×C3×#67—32no—
89C2×C3×#67—32no—
90C2×C3×#67C2×D312yes—
91C2×C3×D12—31no—
92C2×C3×C4×D3C2×D311yescharacteristic
93C2×#627—32no—
94C2×#625—32no—
95#3805—68no—
96C2×D3×Dic3—31no—
97C2×D3×Dic3—31no—
98C2×D3×Dic3—32no—
99#3757—32no—
100C4⋊(C3⋊Dic3)—31no—
101C4×C3⋊Dic3—31no—
102C3×#217C2×D312yes—
103C3×C4⋊Dic3C2×D311yescharacteristic
104C3×C4×Dic3—31no—
105Dic3⋊Dic3—32no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108Dic32—31no—
109Dic32—31no—
110C2×D4×C9—21no—
111C2×D4×C9—32no—
112C2×D4×C9C2×D311yescharacteristic
113C22×C4×C9—31no—
114C2×#611—32no—
115C2×#611C2×D312yes—
116C2×#611C2×D312yes—
117C22×Dic9—31no—
118C22×Dic9—32no—
119C22×Dic9—31no—
120C22×Dic9—62no—
121#3699—68no—
122C2×D36C2×D311yescharacteristic
123C2×C4×D9—32no—
124C2×Dic18—32no—
125C9×C4⋊C4—31no—
126C9×C22⋊C4—32no—
127#3676—32no—
128#3676—62no—
129(C2×D9)⋊C4—32no—
130(C2×D9)⋊C4—32no—
131C4⋊Dic9—31no—
132C4⋊Dic9—61no—
133Dic9⋊C4—32no—
134C4×Dic9—31no—
135C4×Dic9—31no—
136C4×Dic9—31no—
137C4×Dic9—61no—
138C4×Dic9C2×D311yescharacteristic
139C22×C3×C9—22no—
140C22×C3×C9—22no—
141C22×C3×C9—22no—
142C22×C3×C9C2411yescharacteristic
143C2×D3×C9—21no—
144C2×D3×C9—21no—
145C2×D3×C9C2×D412yes—
146C2×C3×D9—22no—
147C2×C3×D9—22no—
148C2×C3×D9—22no—
149C3×C4×C9#4112yescomplemented
150C3⋊Dic9—22no—
151C3⋊Dic9—21no—
152C3⋊Dic9C2×D412yes—
153C9×Dic3—22no—
154C9×Dic3—21no—
155C3×Dic9C2×D412yes—
156C3×Dic9#4112yescomplemented
157C2×#237—92no—
158C2×#237—92no—
159C3×#127—91no—
160C3×#121—31no—
161#940—31no—
162#939—92no—
163#938—92no—
164#938—91no—
165#938D911yescharacteristic, complemented
166#936—92no—
167D4×Dic3—91no—
168D4×Dic3—91no—
169D4×Dic3—92no—
170D4×Dic3—91no—
171D4×Dic3—181no—
172#930—91no—
173C2×C4×Dic3—91no—
174#893—91no—
175#888—91no—
176#885—92no—
177#67⋊C4—92no—
178C42×D3—91no—
179#869—91no—
180C23×C32C22×D312yes—
181C22×C3×D3—32no—
182C22×C3×D3C22×D311yescharacteristic
183C32×D4—22no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—64no—
189C3×#67—24no—
190C3×#67—64no—
191C3×#67—64no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3C22×D311yescharacteristic
195C2×C3×Dic3C22×D312yes—
196C3×D12—62no—
197C3×C4×D3—22no—
198#627—64no—
199#625—64no—
200D3×Dic3—34no—
201D3×Dic3—64no—
202D3×Dic3—62no—
203C23×C9—22no—
204C23×C9—31no—
205C23×C9C22×D312yes—
206C22×D9C22×D312yes—
207D4×C9—24no—
208D4×C9—22no—
209D4×C9—64no—
210C2×C4×C9—21no—
211C2×C4×C9—31no—
212C2×C4×C9—32no—
213C2×C4×C9—62no—
214C2×C4×C9C22×D311yescharacteristic
215#611—24no—
216#611—24no—
217#611—64no—
218C2×Dic9—31no—
219C2×Dic9—34no—
220C2×Dic9—31no—
221C2×Dic9—32no—
222C2×Dic9—61no—
223C2×Dic9—61no—
224C2×Dic9—62no—
225C2×Dic9—64no—
226C2×Dic9—62no—
227C2×Dic9—124no—
228C2×Dic9C22×D311yescharacteristic
229C2×Dic9C22×D311yescharacteristic
230D36—22no—
231C4×D9—64no—
232Dic18—64no—
233#532—271noSylow
234C2×C3×C9—42no—
235C2×C3×C9C22×D411yescharacteristic
236C2×C3×C9C2×#4111yescharacteristic
237C2×C3×C9C2×#4111yescharacteristic
238D3×C9—22no—
239C3×D9—42no—
240C2×C3×D4—21no—
241C2×C3×D4—32no—
242C2×C3×D4—92no—
243C2×C3×D4—92no—
244C2×C3×D4—91no—
245C2×C3×D4C2×D911yescharacteristic
246C2×C3×D4D3211yescharacteristic
247C22×C3×C4—31no—
248C2×#67—32no—
249C2×#67—32no—
250C2×#67—92no—
251C2×#67—92no—
252C2×#67C2×D912yes—
253C22×Dic3—91no—
254C22×Dic3—91no—
255C22×Dic3—92no—
256C22×Dic3—92no—
257C22×Dic3—92no—
258C22×Dic3—182no—
259#237—188no—
260#237—188no—
261C2×D12—31no—
262C2×C4×D3—91no—
263C2×C4×D3—92no—
264C2×C4×D3—91no—
265C2×C4×D3C2×D911yescharacteristic
266C2×Dic6—92no—
267C2×Dic6—92no—
268C3×C4⋊C4—31no—
269C3×C22⋊C4—32no—
270C3×C42—91no—
271#217—92no—
272#217—92no—
273#217—92no—
274#217—92no—
275#217—182no—
276#217C2×D912yes—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279C4⋊Dic3—91no—
280C4⋊Dic3—91no—
281C4⋊Dic3—92no—
282C4⋊Dic3—181no—
283C4⋊Dic3C2×D911yescharacteristic
284Dic3⋊C4—92no—
285C4×Dic3—31no—
286C4×Dic3—91no—
287C4×Dic3—91no—
288C4×Dic3—91no—
289C4×Dic3—91no—
290C4×Dic3—91no—
291C4×Dic3—91no—
292C4×Dic3—181no—
293C22×C32—22no—
294C22×C32—22no—
295C22×C32—22no—
296C22×C32C23×D311yescharacteristic
297C2×C3×D3—21no—
298C2×C3×D3—21no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3D3×D412yes—
303C32×C4#23712yes—
304C3⋊Dic3—32no—
305C3⋊Dic3—62no—
306C3⋊Dic3—61no—
307C3×Dic3—22no—
308C3×Dic3—21no—
309C3×Dic3—32no—
310C3×Dic3—32no—
311C22×C9—24no—
312C22×C9—22no—
313C22×C9—22no—
314C22×C9—22no—
315C22×C9—21no—
316C22×C9—32no—
317C22×C9—42no—
318C22×C9—42no—
319C22×C9—61no—
320C22×C9—61no—
321C22×C9C23×D311yescharacteristic
322C2×D9—22no—
323C2×D9—22no—
324C2×D9—22no—
325C4×C9—22no—
326C4×C9—62no—
327C4×C9—61no—
328C4×C9#23712yescomplemented
329Dic9—32no—
330Dic9—62no—
331Dic9—62no—
332Dic9—61no—
333Dic9—121no—
334Dic9—122no—
335Dic9D3×D412yes—
336Dic9#23712yescomplemented
337C2×#41—272no—
338C4⋊Q8—271no—
339#127—91no—
340#124—272no—
341#121—31no—
342#121—271no—
343#121—272no—
344C4×D4—272no—
345C4×D4—271no—
346C4×D4—271no—
347C2×C42—271no—
348C3×C9#53211yescharacteristic, complemented, Sylow
349C23×C3—22no—
350C23×C3—31no—
351C23×C3—92no—
352C23×C3C22×D912yes—
353C23×C3C2×D3212yes—
354C22×D3—32no—
355C22×D3C22×D911yescharacteristic
356C3×D4—24no—
357C3×D4—22no—
358C3×D4—22no—
359C3×D4—64no—
360C3×D4—184no—
361C3×D4—184no—
362C3×D4—182no—
363C2×C3×C4—21no—
364C2×C3×C4—31no—
365C2×C3×C4—32no—
366C2×C3×C4—62no—
367C2×C3×C4—91no—
368C2×C3×C4—91no—
369C2×C3×C4C22×D911yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371#67—24no—
372#67—64no—
373#67—64no—
374#67—184no—
375#67—184no—
376C2×Dic3—31no—
377C2×Dic3—31no—
378C2×Dic3—94no—
379C2×Dic3—91no—
380C2×Dic3—91no—
381C2×Dic3—91no—
382C2×Dic3—92no—
383C2×Dic3—91no—
384C2×Dic3—92no—
385C2×Dic3—181no—
386C2×Dic3—181no—
387C2×Dic3—182no—
388C2×Dic3—184no—
389C2×Dic3—182no—
390C2×Dic3—184no—
391C2×Dic3—184no—
392C2×Dic3—364no—
393C2×Dic3C22×D911yescharacteristic
394C2×Dic3C22×D912yes—
395D12—62no—
396C4×D3—22no—
397C4×D3—94no—
398C4×D3—184no—
399C4×D3—182no—
400Dic6—184no—
401Dic6—184no—
402C2×C32—42no—
403C2×C32C2×D3×D411yescharacteristic
404C2×C32C2×#23711yescharacteristic
405C2×C32D3×#4111yescharacteristic
406C3×D3—22no—
407C3×D3—122no—
408C2×C9—21no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9—44no—
412C2×C9—42no—
413C2×C9—62no—
414C2×C9C2×D3×D411yescharacteristic
415C2×C9C2×#23711yescharacteristic
416C2×C9C2×#23711yescharacteristic
417D9—42no—
418#41—548no—
419C2×Q8—272no—
420C2×D4—32no—
421C2×D4—92no—
422C2×D4—92no—
423C2×D4—91no—
424C2×D4—272no—
425C2×D4D3×D911yescharacteristic
426C22×C4—31no—
427C22×C4—272no—
428C22×C4—271no—
429C22×C4—271no—
430C22×C4—272no—
431C4⋊C4—31no—
432C4⋊C4—272no—
433C4⋊C4—271no—
434C22⋊C4—32no—
435C22⋊C4—272no—
436C22⋊C4—272no—
437C22⋊C4—272no—
438C42—91no—
439C42—271no—
440C42—271no—
441C42—271no—
442C22×C3—24no—
443C22×C3—22no—
444C22×C3—22no—
445C22×C3—22no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—21no—
450C22×C3—32no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—61no—
454C22×C3—61no—
455C22×C3—182no—
456C22×C3—182no—
457C22×C3—182no—
458C22×C3C23×D911yescharacteristic
459C22×C3C22×D3211yescharacteristic
460C2×D3—21no—
461C2×D3—21no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3D4×D912yes—
466C3×C4—22no—
467C3×C4—62no—
468C3×C4—61no—
469C3×C4—92no—
470C3×C4—92no—
471C3×C4#369912yescomplemented
472C3×C4#380512yes—
473Dic3—22no—
474Dic3—21no—
475Dic3—32no—
476Dic3—32no—
477Dic3—92no—
478Dic3—92no—
479Dic3—182no—
480Dic3—182no—
481Dic3—181no—
482Dic3—182no—
483Dic3—181no—
484Dic3—362no—
485Dic3—361no—
486C32#559611yescharacteristic
487C9—21no—
488C9#571611yescharacteristic, complemented
489C23—31no—
490C23—92no—
491C23C2×D3×D912yes—
492Q8—544no—
493D4—22no—
494D4—64no—
495D4—184no—
496D4—184no—
497D4—182no—
498D4—544no—
499C2×C4—31no—
500C2×C4—32no—
501C2×C4—62no—
502C2×C4—91no—
503C2×C4—91no—
504C2×C4—274no—
505C2×C4—271no—
506C2×C4—271no—
507C2×C4—272no—
508C2×C4—544no—
509C2×C4—542no—
510C2×C4—544no—
511C2×C4C2×D3×D911yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—44no—
516C2×C3—42no—
517C2×C3—42no—
518C2×C3—62no—
519C2×C3—362no—
520C2×C3C2×D4×D911yescharacteristic
521C2×C3C2×#369911yescharacteristic
522C2×C3#41×D911yescharacteristic
523C2×C3D32×D411yescharacteristic
524C2×C3D3×#23711yescharacteristic
525C2×C3C2×#380511yescharacteristic
526D3—22no—
527D3—122no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—32no—
532C22—61no—
533C22—61no—
534C22—182no—
535C22—182no—
536C22—182no—
537C22C22×D3×D911yescharacteristic
538C4—61no—
539C4—62no—
540C4—92no—
541C4—92no—
542C4—272no—
543C4—541no—
544C4—542no—
545C4#8866312yescomplemented
546C3—21no—
547C3#1058969811yescharacteristic, complemented
548C3#1059513311yescharacteristic
549C2—42no—
550C2—62no—
551C2—362no—
552C2D3×D4×D911yescharacteristic
553C2D9×#23711yescharacteristic
554C2C2×#8866311yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#532
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88663C2semidirect
2. D4×C3⋊Dic9C2semidirect
3. C9×#938C2semidirect
4. C3×#63502C2semidirect
5. #11721281C2semidirect
6. #611⋊Dic3C2semidirect
7. #11721257C2semidirect
8. #11721037C2semidirect
9. C4×D3×Dic9C2semidirect
10. #11720998C2semidirect
11. #11720969C2semidirect
51. #63502D3semidirect
81. D3×Dic9D4semidirect
149. C3×C4×C9#41semidirect
156. C3×Dic9#41semidirect
165. #938D9semidirect
328. C4×C9#237semidirect
336. Dic9#237semidirect
348. C3×C9#532semidirect
471. C3×C4#3699semidirect
488. C9#5716semidirect
545. C4#88663semidirect
547. C3#10589698semidirect

Maximal subgroups

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

subgroupindex
C2×#886632
D4×C3⋊Dic92
C9×#9382
C3×#635022
#117212812
#611⋊Dic32
#117212572
#117210372
C4×D3×Dic92
#117209982
#117209692
#105951333
#105896983