Finite Group Databasebeta

← #13150204

Subgroups — #13150204

Order 1728.

968 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88663C212yesmaximal, complemented
2C2×C4⋊(C3⋊Dic9)C211yesmaximal, characteristic, complemented
3C9×#930C211yesmaximal, characteristic, complemented
4C3×C4×#611C211yesmaximal, characteristic, complemented
5#11721266C212yesmaximal, complemented
6#11721036C211yesmaximal, characteristic, complemented
7#11721023C212yesmaximal, complemented
8D12×Dic9C211yesmaximal, characteristic, complemented
9#11720989C212yesmaximal, complemented
10D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
11#11720969C211yesmaximal, characteristic, complemented
12#10594907—31nomaximal
13#10589713—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2212yes—
16C2×C9×#67C2212yes—
17C2×C3×#611C2211yescharacteristic
18C2×C9×D12C2211yescharacteristic
19C2×C3×C4×D9C2211yescharacteristic
20C2×#6194C2212yes—
21#88663—28no—
22C2×D3×Dic9C2212yes—
23C2×Dic3×D9C2212yes—
24C2×#6189C2212yes—
25C4⋊(C3⋊Dic9)C2211yescharacteristic
26C4⋊(C3⋊Dic9)C2211yescharacteristic
27C4⋊(C3⋊Dic9)C2212yes—
28C3×#3676C2211yescharacteristic
29C9×(C2×D3)⋊C4C2212yes—
30C3×(C2×D9)⋊C4C2211yescharacteristic
31C9×C4⋊Dic3C2211yescharacteristic
32C3×Dic9⋊C4C2211yescharacteristic
33C3×C4×Dic9C2211yescharacteristic
34(C2×D3)⋊Dic9C2212yes—
35(C2×D3)⋊Dic9C2212yes—
36Dic9⋊Dic3C2212yes—
37Dic9⋊Dic3C2212yes—
38C2×#3805—32no—
39C2×C4⋊(C3⋊Dic3)—31no—
40C3×#930D311yescharacteristic
41C3×C4×#67—31no—
42#63962—32no—
43#63918—31no—
44#63908—31no—
45#63893—32no—
46Dic3×D12—31no—
47#63891—32no—
48D3×C4⋊Dic3—31no—
49C2×#3699—32no—
50C9×#121—31no—
51#63500—31no—
52#63497—32no—
53D4×Dic9—31no—
54C4×#611D311yescharacteristic, complemented
55C2×C4⋊Dic9—31no—
56C2×C4⋊Dic9—61no—
57C4⋊C4×D9—31no—
58#63450—31no—
59#63445—32no—
60#63440—32no—
61C23×C3×C9C2311yescharacteristic
62C22×D3×C9C2312yes—
63C22×C3×D9C2311yescharacteristic
64C2×C3×C4×C9—22no—
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3⋊Dic9—24no—
68C2×C3⋊Dic9C2312yes—
69C2×C3⋊Dic9C2312yes—
70C9×#67—24no—
71C3×#611D414yescomplemented
72C2×C9×Dic3C2312yes—
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2311yescharacteristic
75C2×C3×Dic9C2311yescharacteristic
76C9×D12—22no—
77C3×C4×D9—22no—
78#6194—24no—
79D3×Dic9—24no—
80Dic3×D9—24no—
81#6189—24no—
82#5611—91no—
83#5555—91no—
84C22×C32×C4C2×D311yescharacteristic
85C22×C3⋊Dic3—32no—
86C2×C3×#67—31no—
87C2×C3×#67C2×D312yes—
88C2×C3×D12C2×D311yescharacteristic
89C2×C3×C4×D3—31no—
90C2×#627—32no—
91C2×#625—32no—
92#3805—68no—
93C2×D3×Dic3—32no—
94C2×D3×Dic3—32no—
95C4⋊(C3⋊Dic3)—31no—
96C4⋊(C3⋊Dic3)—32no—
97C4⋊(C3⋊Dic3)—31no—
98C3×#217—31no—
99C3×(C2×D3)⋊C4—31no—
100C3×(C2×D3)⋊C4C2×D312yes—
101C3×C4⋊Dic3C2×D311yescharacteristic
102C3×Dic3⋊C4—31no—
103C3×C4×Dic3—31no—
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106(C2×D3)⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108C2×D4×C9—32no—
109C2×D4×C9—31no—
110C22×C4×C9—21no—
111C22×C4×C9C2×D311yescharacteristic
112C2×#611—32no—
113C2×#611C2×D311yescharacteristic
114C22×Dic9—32no—
115C22×Dic9—32no—
116C22×Dic9—62no—
117#3699—68no—
118C2×C4×D9—32no—
119C2×C4×D9C2×D311yescharacteristic
120C2×Dic18—32no—
121C9×C4⋊C4—31no—
122C9×C22⋊C4—32no—
123#3676—32no—
124#3676—32no—
125#3676C2×D311yescharacteristic
126(C2×D9)⋊C4C2×D311yescharacteristic
127C4⋊Dic9—31no—
128C4⋊Dic9—31no—
129C4⋊Dic9—32no—
130C4⋊Dic9—62no—
131C4⋊Dic9—61no—
132C4⋊Dic9—61no—
133Dic9⋊C4—32no—
134Dic9⋊C4—32no—
135Dic9⋊C4C2×D311yescharacteristic
136C4×Dic9C2×D311yescharacteristic
137C22×C3×C9—21no—
138C22×C3×C9—21no—
139C22×C3×C9C2×D412yes—
140C22×C3×C9C2411yescharacteristic
141C2×D3×C9—22no—
142C2×D3×C9—22no—
143C2×D3×C9—22no—
144C2×C3×D9—21no—
145C2×C3×D9—21no—
146C2×C3×D9C2×D412yes—
147C3×C4×C9—21no—
148C3×C4×C9#4112yescomplemented
149C3⋊Dic9—22no—
150C3⋊Dic9—22no—
151C9×Dic3—22no—
152C3×Dic9—21no—
153C3×Dic9—21no—
154C3×Dic9C2×D412yes—
155C2×#237—92no—
156C2×#237—92no—
157C3×#121—31no—
158C3×C4×D4—91no—
159#938—91no—
160#935—92no—
161D4×Dic3—91no—
162#930—91no—
163#930D911yescharacteristic, complemented
164C4×#67—31no—
165C2×C4⋊Dic3—91no—
166C2×C4⋊Dic3—91no—
167C2×C4⋊Dic3—91no—
168C2×C4⋊Dic3—181no—
169#897—92no—
170D3×C4⋊C4—91no—
171#888—91no—
172#886—92no—
173#886—92no—
174#883—92no—
175#878—92no—
176C4×D12—91no—
177#869—91no—
178C23×C32C22×D311yescharacteristic
179C22×C3×D3—31no—
180C22×C3×D3C22×D312yes—
181C2×C32×C4—22no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—32no—
185C2×C3⋊Dic3—32no—
186C2×C3⋊Dic3—64no—
187C3×#67—24no—
188C3×#67—34no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3C22×D312yes—
193C3×D12—22no—
194C3×C4×D3—62no—
195#627—64no—
196#625—64no—
197D3×Dic3—64no—
198D3×Dic3—64no—
199C23×C9—21no—
200C23×C9—32no—
201C23×C9C22×D311yescharacteristic
202C22×D9C22×D311yescharacteristic
203D4×C9—64no—
204D4×C9—62no—
205C2×C4×C9—24no—
206C2×C4×C9—22no—
207C2×C4×C9—21no—
208C2×C4×C9—21no—
209C2×C4×C9—32no—
210C2×C4×C9C22×D311yescharacteristic
211C2×C4×C9C22×D311yescharacteristic
212#611—64no—
213#611D1214yescomplemented
214C2×Dic9—32no—
215C2×Dic9—32no—
216C2×Dic9—62no—
217C2×Dic9—62no—
218C2×Dic9—64no—
219C2×Dic9—64no—
220C2×Dic9—124no—
221C2×Dic9C22×D311yescharacteristic
222C2×Dic9C22×D311yescharacteristic
223C2×Dic9C22×D311yescharacteristic
224C4×D9—22no—
225C4×D9—64no—
226Dic18—64no—
227#547—271noSylow
228C2×C3×C9—22no—
229C2×C3×C9C22×D411yescharacteristic
230C2×C3×C9C2×#4111yescharacteristic
231C2×C3×C92−1+411yescharacteristic
232D3×C9—42no—
233C3×D9—22no—
234C2×C3×D4—32no—
235C2×C3×D4—31no—
236C2×C3×D4—91no—
237C22×C3×C4—21no—
238C22×C3×C4—91no—
239C22×C3×C4C2×D911yescharacteristic
240C22×C3×C4D3211yescharacteristic
241C2×#67—31no—
242C2×#67—92no—
243C2×#67—92no—
244C2×#67C2×D912yes—
245C22×Dic3—92no—
246C22×Dic3—92no—
247C22×Dic3—92no—
248C22×Dic3—92no—
249C22×Dic3—182no—
250#237—188no—
251#237—188no—
252C2×D12C2×D911yescharacteristic
253C2×C4×D3—31no—
254C2×C4×D3—92no—
255C2×C4×D3—92no—
256C2×Dic6—92no—
257C2×Dic6—92no—
258C3×C4⋊C4—31no—
259C3×C4⋊C4—91no—
260C3×C22⋊C4—32no—
261C3×C22⋊C4—91no—
262C3×C22⋊C4—91no—
263C3×C42—91no—
264#217—31no—
265#217—92no—
266#217—92no—
267(C2×D3)⋊C4—31no—
268(C2×D3)⋊C4—92no—
269(C2×D3)⋊C4—92no—
270(C2×D3)⋊C4C2×D912yes—
271C4⋊Dic3—91no—
272C4⋊Dic3—91no—
273C4⋊Dic3—92no—
274C4⋊Dic3—92no—
275C4⋊Dic3—91no—
276C4⋊Dic3—91no—
277C4⋊Dic3—92no—
278C4⋊Dic3—92no—
279C4⋊Dic3—181no—
280C4⋊Dic3—181no—
281C4⋊Dic3—182no—
282C4⋊Dic3C2×D911yescharacteristic
283Dic3⋊C4—31no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286C4×Dic3—31no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32D3×D412yes—
290C22×C32C23×D311yescharacteristic
291C2×C3×D3—22no—
292C2×C3×D3—22no—
293C2×C3×D3—22no—
294C2×C3×D3—32no—
295C2×C3×D3—61no—
296C2×C3×D3—61no—
297C32×C4—21no—
298C32×C4#23712yes—
299C3⋊Dic3—62no—
300C3⋊Dic3—62no—
301C3×Dic3—22no—
302C3×Dic3—32no—
303C3×Dic3—61no—
304C3×Dic3—61no—
305C22×C9—22no—
306C22×C9—22no—
307C22×C9—21no—
308C22×C9—21no—
309C22×C9—21no—
310C22×C9—22no—
311C22×C9—62no—
312C22×C9—62no—
313C22×C9—62no—
314C22×C9C2×D1212yes—
315C22×C9C23×D311yescharacteristic
316C2×D9—21no—
317C2×D9—21no—
318C2×D9C2×D1212yes—
319C4×C9—22no—
320C4×C9—21no—
321C4×C9—22no—
322C4×C9—62no—
323C4×C9#23712yescomplemented
324Dic9—21no—
325Dic9—21no—
326Dic9—62no—
327Dic9—62no—
328Dic9—122no—
329Dic9—122no—
330Dic9C2×D1212yes—
331C2×#41—272no—
332C4⋊Q8—271no—
333#123—272no—
334#123—272no—
335C22⋊Q8—272no—
336#121—31no—
337#121—271no—
338C4×D4—91no—
339C4×D4—271no—
340C2×C4⋊C4—271no—
341C2×C4⋊C4—271no—
342C3×C9#54711yescharacteristic, complemented, Sylow
343C23×C3—21no—
344C23×C3—32no—
345C23×C3—91no—
346C23×C3C22×D911yescharacteristic
347C23×C3C2×D3211yescharacteristic
348C22×D3—31no—
349C22×D3C22×D912yes—
350C3×D4—64no—
351C3×D4—62no—
352C3×D4—94no—
353C2×C3×C4—24no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—32no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—91no—
362C2×C3×C4—182no—
363C2×C3×C4C22×D911yescharacteristic
364C2×C3×C4C22×D911yescharacteristic
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C2×D3211yescharacteristic
367#67—24no—
368#67—34no—
369#67—184no—
370#67—184no—
371C2×Dic3—31no—
372C2×Dic3—31no—
373C2×Dic3—31no—
374C2×Dic3—92no—
375C2×Dic3—92no—
376C2×Dic3—92no—
377C2×Dic3—92no—
378C2×Dic3—182no—
379C2×Dic3—182no—
380C2×Dic3—184no—
381C2×Dic3—184no—
382C2×Dic3—184no—
383C2×Dic3—184no—
384C2×Dic3—364no—
385C2×Dic3C22×D912yes—
386D12—22no—
387C4×D3—62no—
388C4×D3—184no—
389C4×D3—184no—
390Dic6—184no—
391Dic6—184no—
392C2×C32—22no—
393C2×C32C2×D3×D411yescharacteristic
394C2×C32C2×#23711yescharacteristic
395C2×C32#101011yescharacteristic
396C3×D3—42no—
397C3×D3—62no—
398C2×C9—24no—
399C2×C9—22no—
400C2×C9—21no—
401C2×C9—21no—
402C2×C9—21no—
403C2×C9—122no—
404C2×C9C22×D1211yescharacteristic
405C2×C9C2×#23711yescharacteristic
406C2×C9#101011yescharacteristic
407D9—22no—
408#41—548no—
409C2×Q8—272no—
410C2×D4—32no—
411C2×D4—31no—
412C2×D4—91no—
413C2×D4—272no—
414C22×C4—91no—
415C22×C4—272no—
416C22×C4—272no—
417C22×C4—272no—
418C22×C4D3×D911yescharacteristic
419C4⋊C4—31no—
420C4⋊C4—91no—
421C4⋊C4—272no—
422C4⋊C4—272no—
423C4⋊C4—271no—
424C4⋊C4—271no—
425C4⋊C4—272no—
426C22⋊C4—32no—
427C22⋊C4—91no—
428C22⋊C4—91no—
429C22⋊C4—272no—
430C22⋊C4—272no—
431C42—91no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—22no—
440C22×C3—62no—
441C22×C3—62no—
442C22×C3—62no—
443C22×C3—92no—
444C22×C3—181no—
445C22×C3—181no—
446C22×C3D4×D912yes—
447C22×C3C23×D911yescharacteristic
448C22×C3D3×D1212yes—
449C22×C3C22×D3211yescharacteristic
450C2×D3—22no—
451C2×D3—22no—
452C2×D3—22no—
453C2×D3—32no—
454C2×D3—61no—
455C2×D3—61no—
456C3×C4—22no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—22no—
460C3×C4—62no—
461C3×C4—92no—
462C3×C4—181no—
463C3×C4—181no—
464C3×C4#369912yescomplemented
465C3×C4#380512yes—
466Dic3—22no—
467Dic3—32no—
468Dic3—61no—
469Dic3—61no—
470Dic3—182no—
471Dic3—182no—
472Dic3—182no—
473Dic3—182no—
474Dic3—362no—
475Dic3—362no—
476C32#561111yescharacteristic
477C9—21no—
478C9#555511yescharacteristic, complemented
479C23—32no—
480C23—91no—
481C23C2×D3×D911yescharacteristic
482Q8—544no—
483D4—64no—
484D4—62no—
485D4—94no—
486D4—544no—
487C2×C4—22no—
488C2×C4—32no—
489C2×C4—91no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—182no—
493C2×C4—272no—
494C2×C4—272no—
495C2×C4—544no—
496C2×C4—544no—
497C2×C4—544no—
498C2×C4C2×D3×D911yescharacteristic
499C2×C4C2×D3×D911yescharacteristic
500C2×C3—24no—
501C2×C3—22no—
502C2×C3—22no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—122no—
507C2×C3—182no—
508C2×C3C2×D4×D911yescharacteristic
509C2×C3C2×#369911yescharacteristic
510C2×C3#6371611yescharacteristic
511C2×C3#6429811yescharacteristic
512C2×C3C2×D3×D1211yescharacteristic
513C2×C3C2×#380511yescharacteristic
514D3—42no—
515D3—62no—
516C22—21no—
517C22—21no—
518C22—62no—
519C22—62no—
520C22—62no—
521C22—92no—
522C22—181no—
523C22—181no—
524C22D9×D1212yes—
525C22C22×D3×D911yescharacteristic
526C4—21no—
527C4—62no—
528C4—92no—
529C4—181no—
530C4—181no—
531C4—542no—
532C4—542no—
533C4#8866312yescomplemented
534C3—21no—
535C3#1058971311yescharacteristic, complemented
536C3#1059490711yescharacteristic
537C2—22no—
538C2—122no—
539C2—182no—
540C2#1172246511yescharacteristic
541C2C2×D9×D1211yescharacteristic
542C2C2×#8866311yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#547
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. C2×C4⋊(C3⋊Dic9)C2semidirect
3. C9×#930C2semidirect
4. C3×C4×#611C2semidirect
5. #11721266C2semidirect
6. #11721036C2semidirect
7. #11721023C2semidirect
8. D12×Dic9C2semidirect
9. #11720989C2semidirect
10. D9×C4⋊Dic3C2semidirect
11. #11720969C2semidirect
54. C4×#611D3semidirect
71. C3×#611D4semidirect
148. C3×C4×C9#41semidirect
163. #930D9semidirect
213. #611D12semidirect
323. C4×C9#237semidirect
342. C3×C9#547semidirect
464. C3×C4#3699semidirect
478. C9#5555semidirect
533. C4#88663semidirect
535. C3#10589713semidirect

Maximal subgroups

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

subgroupindex
C2×#886632
C2×C4⋊(C3⋊Dic9)2
C9×#9302
C3×C4×#6112
#117212662
#117210362
#117210232
D12×Dic92
#117209892
D9×C4⋊Dic32
#117209692
#105949073
#105897133