Finite Group Databasebeta

← #13147685

Subgroups — #13147685

Order 1728.

796 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721709C211yesmaximal, characteristic, complemented
2C4×C9×D12C211yesmaximal, characteristic, complemented
3C3×C4×D36C211yesmaximal, characteristic, complemented
4#11721045C211yesmaximal, characteristic, complemented
5#11721043C211yesmaximal, characteristic, complemented
6#11721034C212yesmaximal, complemented
7#11721033C212yesmaximal, complemented
8#11721032C212yesmaximal, complemented
9#11721031C212yesmaximal, complemented
10#11721007C211yesmaximal, characteristic, complemented
11#11720937C211yesmaximal, characteristic, complemented
12#10593984—31nomaximal
13#10589664—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×C3×D36C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C2×#6195C2212yes—
20C2×#6194C2212yes—
21C2×#6194C2212yes—
22C2×#6192C2212yes—
23C3×C42×C9C2211yescharacteristic
24#88536C2211yescharacteristic
25#88536C2211yescharacteristic
26#88534C2211yescharacteristic
27#88534C2211yescharacteristic
28C4×C3⋊Dic9C2211yescharacteristic
29C9×(C2×D3)⋊C4C2212yes—
30C3×(C2×D9)⋊C4C2212yes—
31C9×C4⋊Dic3C2211yescharacteristic
32C3×C4⋊Dic9C2211yescharacteristic
33(C2×D3)⋊Dic9C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic3⋊Dic9C2212yes—
36Dic9⋊Dic3C2212yes—
37#64081—31no—
38C3×C4×D12—31no—
39C3×C4×D12D311yescharacteristic
40#63916—31no—
41#63916—31no—
42#63914—31no—
43#63906—32no—
44#63906—32no—
45#63905—32no—
46#63905—32no—
47#63872—31no—
48C4×D4×C9—31no—
49#63502—31no—
50#63500—31no—
51#63492—32no—
52#63491—32no—
53#63457—31no—
54#63452—31no—
55#63446—32no—
56#63445—32no—
57C4×D36D311yescharacteristic, complemented
58#63434—31no—
59#63434—61no—
60C22×C3⋊D9C2311yescharacteristic
61C22×D3×C9C2312yes—
62C22×C3×D9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69C4×C3⋊D9—22no—
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2312yes—
72C9×D12—22no—
73C4×D3×C9—42no—
74C3×D36—22no—
75C3×C4×D9—42no—
76#6195—42no—
77#6194—42no—
78#6194—42no—
79#6192—42no—
80#5562—91no—
81#5562—91no—
82C2×C4×C3⋊D3—31no—
83C2×C3×D12—31no—
84C2×C3×D12C2×D311yescharacteristic
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3C2×D312yes—
87C2×#626—32no—
88C2×#626—32no—
89C2×#625—32no—
90C2×#625—32no—
91C32×C42C2×D311yescharacteristic
92#3752—31no—
93#3752—31no—
94#3750—31no—
95#3750—31no—
96C4×C3⋊Dic3—31no—
97C3×(C2×D3)⋊C4—32no—
98C3×(C2×D3)⋊C4C2×D312yes—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3C2×D311yescharacteristic
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—32no—
103(C2×D3)⋊Dic3—32no—
104(C2×D3)⋊Dic3—32no—
105C2×D4×C9—31no—
106C22×C4×C9—32no—
107C2×#611—32no—
108C2×#611—32no—
109C2×#611—32no—
110C2×D36—32no—
111C2×D36C2×D311yescharacteristic
112C2×C4×D9—31no—
113C2×C4×D9—61no—
114C2×C4×D9C2×D312yes—
115C9×C4⋊C4—31no—
116C9×C22⋊C4—32no—
117C42×C9—21no—
118C42×C9C2×D311yescharacteristic
119#3676—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—31no—
122(C2×D9)⋊C4—31no—
123(C2×D9)⋊C4—61no—
124(C2×D9)⋊C4—61no—
125(C2×D9)⋊C4C2×D312yes—
126C4⋊Dic9—32no—
127C4⋊Dic9C2×D311yescharacteristic
128Dic9⋊C4—31no—
129Dic9⋊C4—31no—
130Dic9⋊C4—32no—
131Dic9⋊C4—61no—
132Dic9⋊C4—61no—
133C4×Dic9—31no—
134C4×Dic9—61no—
135C22×C3×C9C2411yescharacteristic
136C2×C3⋊D9—21no—
137C2×C3⋊D9—21no—
138C2×C3⋊D9—21no—
139C2×D3×C9—22no—
140C2×D3×C9—22no—
141C2×D3×C9—22no—
142C2×C3×D9—22no—
143C2×C3×D9—22no—
144C2×C3×D9—22no—
145C3×C4×C9—21no—
146C3×C4×C9—21no—
147C3×C4×C9#4112yes—
148C3⋊Dic9—21no—
149C3⋊Dic9—21no—
150C3⋊Dic9—21no—
151C9×Dic3—22no—
152C3×Dic9—22no—
153C3×C4×D4—31no—
154C3×C4×D4—91no—
155#940—91no—
156#940—91no—
157#938—91no—
158#938—91no—
159#930—92no—
160#930—92no—
161#929—92no—
162#929—92no—
163#895—91no—
164#895—91no—
165#890—91no—
166#890—91no—
167#884—92no—
168#884—92no—
169#883—92no—
170#883—92no—
171C4×D12—31no—
172C4×D12D911yescharacteristic, complemented
173#872—91no—
174#872—91no—
175#872—181no—
176C22×C3⋊D3—31no—
177C22×C3×D3—32no—
178C22×C3×D3C22×D312yes—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C3⋊Dic3—31no—
183C2×C3⋊Dic3—31no—
184C2×C3⋊Dic3—31no—
185C4×C3⋊D3—62no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3C22×D312yes—
188C3×D12—22no—
189C3×D12—62no—
190C3×C4×D3—42no—
191C3×C4×D3—122no—
192#626—122no—
193#626—122no—
194#625—122no—
195#625—122no—
196C23×C9—32no—
197C22×D9—31no—
198C22×D9—61no—
199C22×D9C22×D312yes—
200D4×C9—62no—
201C2×C4×C9—21no—
202C2×C4×C9—21no—
203C2×C4×C9—21no—
204C2×C4×C9—32no—
205C2×C4×C9—122no—
206C2×C4×C9C22×D311yescharacteristic
207C2×C4×C9C22×D311yescharacteristic
208C2×C4×C9C22×D311yescharacteristic
209#611—122no—
210#611—122no—
211#611—122no—
212C2×Dic9—31no—
213C2×Dic9—31no—
214C2×Dic9—31no—
215C2×Dic9—61no—
216C2×Dic9—61no—
217C2×Dic9—61no—
218C2×Dic9C22×D312yes—
219D36—22no—
220D36—122no—
221C4×D9—42no—
222C4×D9—64no—
223C4×D9—62no—
224#540—271noSylow
225C2×C3×C9C2×#4111yescharacteristic
226C2×C3×C9H(5,𝔽2)11yescharacteristic
227C2×C3×C9H(5,𝔽2)11yescharacteristic
228C3⋊D9—41no—
229D3×C9—42no—
230C3×D9—42no—
231C2×C3×D4—31no—
232C2×C3×D4—91no—
233C22×C3×C4—32no—
234C22×C3×C4—92no—
235C2×#67—92no—
236C2×#67—92no—
237C2×#67—92no—
238C2×#67—92no—
239C2×#67—92no—
240C2×#67—92no—
241C2×D12—31no—
242C2×D12—92no—
243C2×D12—92no—
244C2×D12C2×D911yescharacteristic
245C2×C4×D3—32no—
246C2×C4×D3—91no—
247C2×C4×D3—91no—
248C2×C4×D3—181no—
249C2×C4×D3C2×D912yes—
250C3×C4⋊C4—31no—
251C3×C4⋊C4—91no—
252C3×C22⋊C4—32no—
253C3×C22⋊C4—92no—
254C3×C42—21no—
255C3×C42C2×D911yescharacteristic
256C3×C42D3211yescharacteristic
257#217—92no—
258#217—92no—
259(C2×D3)⋊C4—32no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—91no—
263(C2×D3)⋊C4—91no—
264(C2×D3)⋊C4—91no—
265(C2×D3)⋊C4—91no—
266(C2×D3)⋊C4—181no—
267(C2×D3)⋊C4—181no—
268(C2×D3)⋊C4C2×D912yes—
269C4⋊Dic3—31no—
270C4⋊Dic3—92no—
271C4⋊Dic3—92no—
272C4⋊Dic3C2×D911yescharacteristic
273Dic3⋊C4—92no—
274Dic3⋊C4—91no—
275Dic3⋊C4—91no—
276Dic3⋊C4—91no—
277Dic3⋊C4—91no—
278Dic3⋊C4—92no—
279Dic3⋊C4—181no—
280Dic3⋊C4—181no—
281C4×Dic3—91no—
282C4×Dic3—91no—
283C4×Dic3—181no—
284C22×C32C23×D311yescharacteristic
285C2×C3⋊D3—61no—
286C2×C3⋊D3—61no—
287C2×C3⋊D3—61no—
288C2×C3×D3—22no—
289C2×C3×D3—22no—
290C2×C3×D3—22no—
291C2×C3×D3—62no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4#23512yes—
297C3⋊Dic3—61no—
298C3⋊Dic3—61no—
299C3⋊Dic3—61no—
300C3×Dic3—22no—
301C3×Dic3—62no—
302C22×C9—21no—
303C22×C9—62no—
304C22×C9—62no—
305C22×C9—62no—
306C22×C9C23×D311yescharacteristic
307C2×D9—22no—
308C2×D9—22no—
309C2×D9—22no—
310C2×D9—62no—
311C2×D9—61no—
312C2×D9—61no—
313C2×D9—61no—
314C2×D9—121no—
315C2×D9—121no—
316C4×C9—21no—
317C4×C9—21no—
318C4×C9—22no—
319C4×C9—41no—
320C4×C9—41no—
321C4×C9—62no—
322C4×C9#23512yes—
323Dic9—22no—
324Dic9—62no—
325Dic9—61no—
326Dic9—61no—
327Dic9—61no—
328Dic9—121no—
329Dic9—121no—
330#127—271no—
331#125—271no—
332#123—272no—
333#123—272no—
334#121—272no—
335#121—272no—
336#121—271no—
337#121—271no—
338C4×D4—31no—
339C4×D4—91no—
340#117—271no—
341C3×C9#54011yescharacteristic, complemented, Sylow
342C23×C3—32no—
343C23×C3—92no—
344C22×D3—32no—
345C22×D3—91no—
346C22×D3—91no—
347C22×D3—181no—
348C22×D3C22×D912yes—
349C3×D4—62no—
350C3×D4—182no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—32no—
355C2×C3×C4—92no—
356C2×C3×C4—122no—
357C2×C3×C4—362no—
358C2×C3×C4C22×D911yescharacteristic
359C2×C3×C4C22×D911yescharacteristic
360C2×C3×C4C22×D911yescharacteristic
361C2×C3×C4C2×D3211yescharacteristic
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C2×D3211yescharacteristic
364#67—362no—
365#67—362no—
366#67—362no—
367#67—362no—
368#67—362no—
369#67—362no—
370C2×Dic3—32no—
371C2×Dic3—91no—
372C2×Dic3—91no—
373C2×Dic3—91no—
374C2×Dic3—91no—
375C2×Dic3—91no—
376C2×Dic3—91no—
377C2×Dic3—181no—
378C2×Dic3—181no—
379C2×Dic3—181no—
380C2×Dic3C22×D912yes—
381D12—22no—
382D12—62no—
383D12—362no—
384D12—362no—
385C4×D3—42no—
386C4×D3—122no—
387C4×D3—184no—
388C4×D3—182no—
389C4×D3—182no—
390C2×C32C2×#23511yescharacteristic
391C2×C32#100411yescharacteristic
392C2×C32#100911yescharacteristic
393C3⋊D3—121no—
394C3×D3—42no—
395C3×D3—122no—
396C2×C9—21no—
397C2×C9—21no—
398C2×C9—21no—
399C2×C9—122no—
400C2×C9C2×#23511yescharacteristic
401C2×C9#100411yescharacteristic
402C2×C9#100911yescharacteristic
403D9—42no—
404D9—122no—
405D9—121no—
406C2×D4—31no—
407C2×D4—91no—
408C2×D4—272no—
409C2×D4—272no—
410C2×D4—272no—
411C2×D4—272no—
412C22×C4—32no—
413C22×C4—92no—
414C22×C4—271no—
415C4⋊C4—31no—
416C4⋊C4—91no—
417C4⋊C4—272no—
418C4⋊C4—272no—
419C4⋊C4—271no—
420C4⋊C4—271no—
421C22⋊C4—32no—
422C22⋊C4—92no—
423C22⋊C4—272no—
424C22⋊C4—272no—
425C22⋊C4—271no—
426C22⋊C4—271no—
427C42—271no—
428C42D3×D911yescharacteristic
429C22×C3—21no—
430C22×C3—62no—
431C22×C3—62no—
432C22×C3—62no—
433C22×C3—182no—
434C22×C3—182no—
435C22×C3—182no—
436C22×C3C23×D911yescharacteristic
437C22×C3C22×D3211yescharacteristic
438C2×D3—22no—
439C2×D3—22no—
440C2×D3—22no—
441C2×D3—62no—
442C2×D3—62no—
443C2×D3—62no—
444C2×D3—182no—
445C2×D3—181no—
446C2×D3—181no—
447C2×D3—181no—
448C2×D3—181no—
449C2×D3—181no—
450C2×D3—181no—
451C2×D3—361no—
452C2×D3—361no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—22no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—62no—
461C3×C4—182no—
462C3×C4#369712yes—
463C3×C4#379812yes—
464Dic3—22no—
465Dic3—62no—
466Dic3—182no—
467Dic3—181no—
468Dic3—181no—
469Dic3—181no—
470Dic3—181no—
471Dic3—181no—
472Dic3—181no—
473Dic3—361no—
474Dic3—361no—
475C32#556211yescharacteristic
476C9—21no—
477C9#556211yescharacteristic, complemented
478C23—32no—
479C23—92no—
480C23—271no—
481D4—62no—
482D4—182no—
483D4—1082no—
484D4—1082no—
485D4—1082no—
486D4—1082no—
487C2×C4—32no—
488C2×C4—92no—
489C2×C4—122no—
490C2×C4—271no—
491C2×C4—271no—
492C2×C4—271no—
493C2×C4—362no—
494C2×C4—542no—
495C2×C4C2×D3×D911yescharacteristic
496C2×C4C2×D3×D911yescharacteristic
497C2×C4C2×D3×D911yescharacteristic
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—122no—
502C2×C3—362no—
503C2×C3C2×#369711yescharacteristic
504C2×C3#6371011yescharacteristic
505C2×C3#6371511yescharacteristic
506C2×C3C2×#379811yescharacteristic
507C2×C3#6430711yescharacteristic
508C2×C3#6430711yescharacteristic
509D3—42no—
510D3—122no—
511D3—362no—
512D3—361no—
513D3—361no—
514C22—62no—
515C22—62no—
516C22—62no—
517C22—182no—
518C22—182no—
519C22—182no—
520C22—541no—
521C22—541no—
522C22—541no—
523C22C22×D3×D911yescharacteristic
524C4—21no—
525C4—21no—
526C4—62no—
527C4—182no—
528C4—541no—
529C4—541no—
530C4—541no—
531C4#8864012yes—
532C3—21no—
533C3#1058966411yescharacteristic, complemented
534C3#1059398411yescharacteristic
535C2—122no—
536C2—362no—
537C2—1081no—
538C2C2×#8864011yescharacteristic
539C2#1172247611yescharacteristic
540C2#1172247711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#540
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. C4×C9×D12C2semidirect
3. C3×C4×D36C2semidirect
4. #11721045C2semidirect
5. #11721043C2semidirect
6. #11721034C2semidirect
7. #11721033C2semidirect
8. #11721032C2semidirect
9. #11721031C2semidirect
10. #11721007C2semidirect
11. #11720937C2semidirect
57. C4×D36D3semidirect
172. C4×D12D9semidirect
341. C3×C9#540semidirect
477. C9#5562semidirect
533. C3#10589664semidirect

Maximal subgroups

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

subgroupindex
#117217092
C4×C9×D122
C3×C4×D362
#117210452
#117210432
#117210342
#117210332
#117210322
#117210312
#117210072
#117209372
#105939843
#105896643