Finite Group Databasebeta

← #13150481

Subgroups — #13150481

Order 1728.

1066 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C9×#235C211yesmaximal, characteristic, complemented
2C22×C3×C4×D9C211yesmaximal, characteristic, complemented
3#11721815C211yesmaximal, characteristic, complemented
4#11721285C212yesmaximal, complemented
5#11721265C212yesmaximal, complemented
6#11721036C212yesmaximal, complemented
7C4×#6194C214yesmaximal, complemented
8#11720944C212yesmaximal, complemented
9#10595022—31nomaximal
10#10589930—31nomaximal
11C23×C3×D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C2×C9×#67C2212yes—
14C22×C3×Dic9C2211yescharacteristic
15C9×#235—24no—
16C2×C9×D12C2211yescharacteristic, complemented
17C2×C4×D3×C9C2212yescomplemented
18C2×C3×C4×D9—24no—
19C2×C3×C4×D9C2212yes—
20C2×C3×C4×D9C2212yes—
21C2×C9×Dic6C2211yescharacteristic, complemented
22C2×#6194C2214yes—
23#88551C2212yes—
24C4⋊(C3⋊Dic9)C2212yescomplemented
25C4×C3⋊Dic9C2212yescomplemented
26(C2×D3)⋊Dic9C2214yescomplemented
27(C2×D9)⋊Dic3C2214yes—
28#88442C2214yescomplemented
29C2×C3×#235D311yescharacteristic
30C22×C3×C4×D3—31no—
31#64135—31no—
32#63976—32no—
33#63961—32no—
34#63908—32no—
35C4×#625—34no—
36#63856—32no—
37C2×C9×#41—31no—
38C22×C4×D9D311yescharacteristic, complemented
39#63508—32no—
40#63500—32no—
41#63499—32no—
42#63497—32no—
43C4×#611—34no—
44#63488—31no—
45#63488—61no—
46C23×C3×C9C2311yescharacteristic
47C22×D3×C9C2312yes—
48C22×C3×D9—22no—
49C22×C3×D9—24no—
50C22×C3×D9C2312yes—
51C2×C3×C4×C9D414yescomplemented
52C2×C3×C4×C9C2311yescharacteristic
53C2×C3×C4×C9C2311yescharacteristic
54C2×C3⋊Dic9C2314yes—
55C9×#67—24no—
56C2×C9×Dic3C2312yes—
57C2×C3×Dic9—22no—
58C2×C3×Dic9C2312yes—
59C9×D12—22no—
60C4×D3×C9—24no—
61C3×C4×D9—24no—
62C3×C4×D9—24no—
63C9×Dic6—22no—
64#6194—28no—
65#5828—91no—
66#5791—91no—
67C23×C3×D3—31no—
68C22×C32×C4C2×D311yescharacteristic
69C2×C3×#67C2×D312yes—
70C22×C3×Dic3—31no—
71C3×#235—24no—
72C2×C3×D12C2×D311yescharacteristic
73C2×C3×C4×D3—32no—
74C2×C3×C4×D3—32no—
75C2×C3×C4×D3—64no—
76C2×C3×C4×D3C2×D312yes—
77C2×C3×Dic6C2×D311yescharacteristic
78C2×#625—34no—
79C23×D9C2×D311yescharacteristic
80#3757—32no—
81C4⋊(C3⋊Dic3)—32no—
82C4×C3⋊Dic3—32no—
83#3724—34no—
84(C2×D3)⋊Dic3—34no—
85(C2×D3)⋊Dic3—34no—
86C9×#41—64no—
87C2×Q8×C9—31no—
88C2×D4×C9—32no—
89C2×D4×C9—31no—
90C22×C4×C9—21no—
91C22×C4×C9—32no—
92C22×C4×C9C2×D311yescharacteristic
93C2×#611—34no—
94C22×Dic9C2×D311yescharacteristic
95C2×C4×D9—24no—
96C2×C4×D9C2×D312yes—
97C2×C4×D9C2×D312yes—
98#3676—32no—
99#3676—34no—
100#3676—62no—
101(C2×D9)⋊C4—34no—
102C4⋊Dic9—32no—
103C4⋊Dic9—62no—
104Dic9⋊C4—34no—
105C4×Dic9—32no—
106C4×Dic9—62no—
107C22×C3×C9—22no—
108C22×C3×C9C2×D412yes—
109C22×C3×C9C2411yescharacteristic
110C2×D3×C9—22no—
111C2×D3×C9—24no—
112C2×C3×D9—28no—
113C2×C3×D9—22no—
114C2×C3×D9—22no—
115C2×C3×D9#4114yescomplemented
116C3×C4×C9C2×D412yes—
117C3×C4×C9C2×D412yes—
118C3⋊Dic9—24no—
119C9×Dic3—22no—
120C3×Dic9—22no—
121C2×C3×#41—31no—
122C23×C3×C4—91no—
123C2×#235D911yescharacteristic, complemented
124C22×C4×D3—31no—
125#953—92no—
126#946—92no—
127#938—92no—
128#937—92no—
129#935—92no—
130#930—92no—
131#929—92no—
132C4×#67—94no—
133C4×#67—94no—
134#926—91no—
135#926—91no—
136#926—181no—
137#924—92no—
138C23×C32C22×D311yescharacteristic
139C22×C3×D3—32no—
140C22×C3×D3—64no—
141C22×C3×D3—62no—
142C22×C3×D3C22×D312yes—
143C2×C32×C4#6714yes—
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C3⋊Dic3—34no—
147C3×#67—24no—
148C2×C3×Dic3—32no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3C22×D312yes—
151C3×D12—22no—
152C3×C4×D3—24no—
153C3×C4×D3—64no—
154C3×C4×D3—64no—
155C3×Dic6—22no—
156#625—68no—
157C23×C9—21no—
158C23×C9—32no—
159C23×C9C22×D311yescharacteristic
160C22×D9—22no—
161C22×D9—24no—
162C22×D9C22×D312yes—
163Q8×C9—62no—
164D4×C9—64no—
165D4×C9—62no—
166C2×C4×C9—24no—
167C2×C4×C9—21no—
168C2×C4×C9—21no—
169C2×C4×C9—32no—
170C2×C4×C9—64no—
171C2×C4×C9#6714yescomplemented
172C2×C4×C9C22×D311yescharacteristic
173C2×C4×C9C22×D311yescharacteristic
174#611—68no—
175C2×Dic9—22no—
176C2×Dic9—34no—
177C2×Dic9—64no—
178C2×Dic9C22×D312yes—
179C4×D9—24no—
180C4×D9—24no—
181#525—271noSylow
182C2×C3×C9—22no—
183C2×C3×C9C22×D411yescharacteristic
184C2×C3×C9C2×#4112yes—
185D3×C9—42no—
186C3×D9—24no—
187C24×C3—91no—
188C23×D3—31no—
189C3×#41—64no—
190C2×C3×Q8—31no—
191C2×C3×D4—32no—
192C2×C3×D4—31no—
193C22×C3×C4—21no—
194C22×C3×C4—32no—
195C22×C3×C4—91no—
196C22×C3×C4—92no—
197C22×C3×C4—92no—
198C22×C3×C4—184no—
199C22×C3×C4C2×D911yescharacteristic
200C22×C3×C4D3211yescharacteristic
201C2×#67—94no—
202C2×#67—94no—
203C2×#67C2×D912yes—
204C22×Dic3—31no—
205#235—24no—
206C2×D12C2×D911yescharacteristic, complemented
207C2×C4×D3—32no—
208C2×C4×D3—32no—
209C2×C4×D3—64no—
210C2×C4×D3C2×D912yescomplemented
211C2×Dic6C2×D911yescharacteristic, complemented
212#217—94no—
213#217—92no—
214#217—92no—
215#217—94no—
216#217—182no—
217(C2×D3)⋊C4—94no—
218(C2×D3)⋊C4—94no—
219C4⋊Dic3—92no—
220C4⋊Dic3—92no—
221C4⋊Dic3—182no—
222Dic3⋊C4—94no—
223Dic3⋊C4—94no—
224C4×Dic3—92no—
225C4×Dic3—92no—
226C4×Dic3—182no—
227C22×C32—22no—
228C22×C32C2×#6712yes—
229C22×C32C23×D311yescharacteristic
230C2×C3×D3—24no—
231C2×C3×D3—22no—
232C2×C3×D3—34no—
233C2×C3×D3—68no—
234C2×C3×D3—62no—
235C2×C3×D3—62no—
236C32×C4C2×#6712yes—
237C32×C4C2×#6712yes—
238C3⋊Dic3—64no—
239C3×Dic3—22no—
240C3×Dic3—62no—
241C22×C9—22no—
242C22×C9—21no—
243C22×C9—22no—
244C22×C9—42no—
245C22×C9—64no—
246C22×C9—62no—
247C22×C9C2×#6712yes—
248C22×C9C23×D311yescharacteristic
249C2×D9—28no—
250C2×D9—22no—
251C2×D9—22no—
252C2×D9#23514yescomplemented
253C4×C9—22no—
254C4×C9—22no—
255C4×C9—62no—
256C4×C9C2×#6712yes—
257C4×C9C2×#6712yes—
258Dic9—22no—
259Dic9—64no—
260Dic9—124no—
261C2×#41—31no—
262C23×C4—91no—
263#123—272no—
264C22⋊Q8—272no—
265#121—272no—
266#120—272no—
267C4×D4—274no—
268#117—271no—
269C3×C9#52511yescharacteristic, complemented, Sylow
270C23×C3—21no—
271C23×C3—32no—
272C23×C3—92no—
273C23×C3—184no—
274C23×C3—182no—
275C23×C3C22×D911yescharacteristic
276C23×C3C2×D3211yescharacteristic
277C22×D3—32no—
278C22×D3—64no—
279C22×D3—62no—
280C22×D3C22×D912yes—
281C3×Q8—62no—
282C3×D4—64no—
283C3×D4—62no—
284C2×C3×C4—24no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—32no—
288C2×C3×C4—64no—
289C2×C3×C4—92no—
290C2×C3×C4—182no—
291C2×C3×C4—184no—
292C2×C3×C4—184no—
293C2×C3×C4#61114yescomplemented
294C2×C3×C4C22×D911yescharacteristic
295C2×C3×C4C22×D911yescharacteristic
296C2×C3×C4#62514yes—
297C2×C3×C4C2×D3211yescharacteristic
298C2×C3×C4C2×D3211yescharacteristic
299#67—24no—
300#67—188no—
301#67—188no—
302C2×Dic3—32no—
303C2×Dic3—62no—
304C2×Dic3—94no—
305C2×Dic3—94no—
306C2×Dic3—184no—
307C2×Dic3C22×D912yes—
308D12—22no—
309C4×D3—24no—
310C4×D3—64no—
311C4×D3—64no—
312Dic6—22no—
313C2×C32—22no—
314C2×C32D3×#4112yes—
315C2×C32C22×#6711yescharacteristic
316C3×D3—42no—
317C3×D3—64no—
318C2×C9—22no—
319C2×C9—22no—
320C2×C9—21no—
321C2×C9—42no—
322C2×C9—122no—
323C2×C9C2×#23512yes—
324C2×C9C22×#6711yescharacteristic
325D9—24no—
326C24—91no—
327#41—64no—
328C2×Q8—31no—
329C2×D4—32no—
330C2×D4—31no—
331C2×D4—274no—
332C22×C4—32no—
333C22×C4—91no—
334C22×C4—92no—
335C22×C4—92no—
336C22×C4—184no—
337C22×C4D3×D911yescharacteristic
338C4⋊C4—274no—
339C4⋊C4—272no—
340C22⋊C4—274no—
341C22⋊C4—274no—
342C22⋊C4—272no—
343C42—272no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—21no—
347C22×C3—22no—
348C22×C3—42no—
349C22×C3—64no—
350C22×C3—62no—
351C22×C3—94no—
352C22×C3—188no—
353C22×C3—182no—
354C22×C3—182no—
355C22×C3C2×#61112yes—
356C22×C3C23×D911yescharacteristic
357C22×C3C2×#62512yes—
358C22×C3C22×D3211yescharacteristic
359C2×D3—24no—
360C2×D3—22no—
361C2×D3—34no—
362C2×D3—68no—
363C2×D3—62no—
364C2×D3—62no—
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—62no—
368C3×C4—182no—
369C3×C4C2×#61112yes—
370C3×C4C2×#61112yes—
371C3×C4C2×#62512yes—
372C3×C4C2×#62512yes—
373Dic3—22no—
374Dic3—62no—
375Dic3—184no—
376Dic3—184no—
377Dic3—364no—
378C32#582811yescharacteristic
379C9—21no—
380C9#579111yescharacteristic, complemented
381C23—32no—
382C23—92no—
383C23—184no—
384C23—182no—
385C23C2×D3×D911yescharacteristic
386Q8—62no—
387D4—64no—
388D4—62no—
389D4—548no—
390C2×C4—32no—
391C2×C4—64no—
392C2×C4—92no—
393C2×C4—182no—
394C2×C4—184no—
395C2×C4—184no—
396C2×C4—274no—
397C2×C4#619414yescomplemented
398C2×C4C2×D3×D911yescharacteristic
399C2×C4C2×D3×D911yescharacteristic
400C2×C3—22no—
401C2×C3—22no—
402C2×C3—21no—
403C2×C3—22no—
404C2×C3—42no—
405C2×C3—122no—
406C2×C3—184no—
407C2×C3#41×D912yes—
408C2×C3C22×#61111yescharacteristic
409C2×C3D3×#23512yes—
410C2×C3C22×#62511yescharacteristic
411D3—42no—
412D3—64no—
413C22—22no—
414C22—62no—
415C22—64no—
416C22—94no—
417C22—188no—
418C22—182no—
419C22—182no—
420C22C2×#619412yes—
421C22C22×D3×D911yescharacteristic
422C4—62no—
423C4—182no—
424C4—544no—
425C4C2×#619412yes—
426C4C2×#619412yes—
427C3—21no—
428C3#1058993011yescharacteristic, complemented
429C3#1059502211yescharacteristic
430C2—22no—
431C2—122no—
432C2—184no—
433C2D9×#23512yes—
434C2C22×#619411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#525
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C9×#235C2semidirect
2. C22×C3×C4×D9C2semidirect
3. #11721815C2semidirect
4. #11721285C2semidirect
5. #11721265C2semidirect
6. #11721036C2semidirect
7. C4×#6194C2semidirect
8. #11720944C2semidirect
16. C2×C9×D12C22semidirect
17. C2×C4×D3×C9C22semidirect
21. C2×C9×Dic6C22semidirect
24. C4⋊(C3⋊Dic9)C22semidirect
25. C4×C3⋊Dic9C22semidirect
26. (C2×D3)⋊Dic9C22semidirect
28. #88442C22semidirect
38. C22×C4×D9D3semidirect
51. C2×C3×C4×C9D4semidirect
115. C2×C3×D9#41semidirect
123. C2×#235D9semidirect
171. C2×C4×C9#67semidirect
206. C2×D12C2×D9semidirect
210. C2×C4×D3C2×D9semidirect
211. C2×Dic6C2×D9semidirect
252. C2×D9#235semidirect
269. C3×C9#525semidirect
293. C2×C3×C4#611semidirect
380. C9#5791semidirect
397. C2×C4#6194semidirect
428. C3#10589930semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2352
C22×C3×C4×D92
#117218152
#117212852
#117212652
#117210362
C4×#61942
#117209442
#105950223
#105899303