Finite Group Databasebeta

← #13150482

Subgroups — #13150482

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
1C22×C4×D3×C9C211yesmaximal, characteristic, complemented
2C2×C3×#3697C211yesmaximal, characteristic, complemented
3#11721815C211yesmaximal, characteristic, complemented
4#11721286C212yesmaximal, complemented
5#11721259C212yesmaximal, complemented
6#11721037C212yesmaximal, complemented
7C4×#6194C214yesmaximal, complemented
8#11720960C212yesmaximal, complemented
9#10595022—31nomaximal
10#10589893—31nomaximal
11C23×D3×C9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C2×C3×#611C2212yes—
14C22×C9×Dic3C2211yescharacteristic
15C2×C4×D3×C9—24no—
16C2×C4×D3×C9C2212yes—
17C2×C4×D3×C9C2212yes—
18C3×#3697—24no—
19C2×C3×D36C2211yescharacteristic, complemented
20C2×C3×C4×D9C2212yescomplemented
21C2×C3×Dic18C2211yescharacteristic, complemented
22C2×#6194C2214yes—
23#88551C2212yes—
24C4⋊(C3⋊Dic9)C2212yescomplemented
25C4×C3⋊Dic9C2212yescomplemented
26(C2×D3)⋊Dic9C2214yes—
27(C2×D9)⋊Dic3C2214yescomplemented
28#88442C2214yescomplemented
29C2×C3×#235—31no—
30C22×C3×C4×D3D311yescharacteristic
31#64135—31no—
32#63976—32no—
33#63961—32no—
34#63908—32no—
35C4×#625—34no—
36#63856—32no—
37C23×C4×C9—31no—
38C2×#3697D311yescharacteristic, complemented
39#63515—32no—
40#63492—32no—
41#63491—32no—
42C4×#611—34no—
43#63488—31no—
44#63488—61no—
45#63486—32no—
46C23×C3×C9C2311yescharacteristic
47C22×D3×C9—22no—
48C22×D3×C9—24no—
49C22×D3×C9C2312yes—
50C22×C3×D9C2312yes—
51C2×C3×C4×C9D414yescomplemented
52C2×C3×C4×C9C2311yescharacteristic
53C2×C3×C4×C9C2311yescharacteristic
54C2×C3⋊Dic9C2314yes—
55C3×#611—24no—
56C2×C9×Dic3—22no—
57C2×C9×Dic3C2312yes—
58C2×C3×Dic9C2312yes—
59C4×D3×C9—24no—
60C4×D3×C9—24no—
61C3×D36—22no—
62C3×C4×D9—24no—
63C3×Dic18—22no—
64#6194—28no—
65#5828—91no—
66#5791—91no—
67C23×C3×D3C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C2×C3×#67—32no—
70C22×C3×Dic3C2×D311yescharacteristic
71C3×#235—64no—
72C2×C3×D12—31no—
73C2×C3×C4×D3—24no—
74C2×C3×C4×D3—32no—
75C2×C3×C4×D3C2×D312yes—
76C2×C3×C4×D3C2×D312yes—
77C2×C3×Dic6—31no—
78C2×#625—34no—
79C24×C9—31no—
80#3757—32no—
81C4⋊(C3⋊Dic3)—32no—
82C4×C3⋊Dic3—32no—
83#3724—34no—
84(C2×D3)⋊Dic3—34no—
85(C2×D3)⋊Dic3—34no—
86C22×C4×C9—21no—
87C22×C4×C9—31no—
88C22×C4×C9—32no—
89C22×C4×C9—32no—
90C22×C4×C9—64no—
91C22×C4×C9C2×D311yescharacteristic
92C2×#611—34no—
93C2×#611C2×D312yes—
94#3697—24no—
95C2×D36C2×D311yescharacteristic, complemented
96C2×C4×D9C2×D312yescomplemented
97C2×Dic18C2×D311yescharacteristic, complemented
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—28no—
111C2×D3×C9—22no—
112C2×D3×C9—22no—
113C2×D3×C9#4114yescomplemented
114C2×C3×D9—22no—
115C2×C3×D9—24no—
116C3×C4×C9C2×D412yes—
117C3×C4×C9C2×D412yes—
118C3⋊Dic9—24no—
119C9×Dic3—22no—
120C3×Dic9—22no—
121C2×C3×#41—91no—
122C23×C3×C4—31no—
123C2×#235—31no—
124C22×C4×D3D911yescharacteristic, complemented
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—22no—
140C22×C3×D3—24no—
141C22×C3×D3—32no—
142C22×C3×D3C22×D312yes—
143C2×C32×C4#6714yes—
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C3⋊Dic3—34no—
147C3×#67—64no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3C22×D312yes—
151C3×D12—62no—
152C3×C4×D3—24no—
153C3×C4×D3—24no—
154C3×C4×D3—64no—
155C3×Dic6—62no—
156#625—68no—
157C23×C9—21no—
158C23×C9—32no—
159C23×C9—64no—
160C23×C9—62no—
161C23×C9C22×D311yescharacteristic
162C22×D9C22×D312yes—
163C2×C4×C9—24no—
164C2×C4×C9—21no—
165C2×C4×C9—21no—
166C2×C4×C9—32no—
167C2×C4×C9—62no—
168C2×C4×C9—64no—
169C2×C4×C9—64no—
170C2×C4×C9#6714yescomplemented
171C2×C4×C9C22×D311yescharacteristic
172C2×C4×C9C22×D311yescharacteristic
173#611—24no—
174#611—68no—
175C2×Dic9—34no—
176C2×Dic9—64no—
177C2×Dic9C22×D312yes—
178D36—22no—
179C4×D9—24no—
180Dic18—22no—
181#525—271noSylow
182C2×C3×C9—22no—
183C2×C3×C9C22×D411yescharacteristic
184C2×C3×C9C2×#4112yes—
185D3×C9—24no—
186C3×D9—42no—
187C24×C3—31no—
188C23×D3C2×D911yescharacteristic
189C3×#41—184no—
190C2×C3×Q8—91no—
191C2×C3×D4—92no—
192C2×C3×D4—91no—
193C22×C3×C4—21no—
194C22×C3×C4—31no—
195C22×C3×C4—32no—
196C22×C3×C4—32no—
197C22×C3×C4—64no—
198C22×C3×C4—92no—
199C22×C3×C4C2×D911yescharacteristic
200C22×C3×C4D3211yescharacteristic
201C2×#67—32no—
202C2×#67—94no—
203C2×#67—94no—
204C22×Dic3C2×D911yescharacteristic
205#235—64no—
206C2×D12—31no—
207C2×C4×D3—24no—
208C2×C4×D3—32no—
209C2×C4×D3C2×D912yes—
210C2×C4×D3C2×D912yes—
211C2×Dic6—31no—
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—28no—
231C2×C3×D3—22no—
232C2×C3×D3—22no—
233C2×C3×D3—64no—
234C2×C3×D3—62no—
235C2×C3×D3#23514yes—
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—34no—
245C22×C9—42no—
246C22×C9—68no—
247C22×C9—62no—
248C22×C9—62no—
249C22×C9C2×#6712yes—
250C22×C9C23×D311yescharacteristic
251C2×D9—24no—
252C2×D9—22no—
253C4×C9—22no—
254C4×C9—22no—
255C4×C9—62no—
256C4×C9C2×#6712yes—
257C4×C9C2×#6712yes—
258Dic9—22no—
259Dic9—64no—
260Dic9—124no—
261C2×#41—91no—
262C23×C4—31no—
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—64no—
273C23×C3—62no—
274C23×C3—92no—
275C23×C3C22×D911yescharacteristic
276C23×C3C2×D3211yescharacteristic
277C22×D3—22no—
278C22×D3—24no—
279C22×D3—32no—
280C22×D3C22×D912yes—
281C3×Q8—182no—
282C3×D4—184no—
283C3×D4—182no—
284C2×C3×C4—24no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—32no—
288C2×C3×C4—62no—
289C2×C3×C4—64no—
290C2×C3×C4—64no—
291C2×C3×C4—92no—
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—64no—
300#67—188no—
301#67—188no—
302C2×Dic3—22no—
303C2×Dic3—32no—
304C2×Dic3—94no—
305C2×Dic3—94no—
306C2×Dic3—184no—
307C2×Dic3C22×D912yes—
308D12—62no—
309C4×D3—24no—
310C4×D3—24no—
311C4×D3—64no—
312Dic6—62no—
313C2×C32—22no—
314C2×C32C2×#23512yes—
315C2×C32C22×#6711yescharacteristic
316C3×D3—24no—
317C3×D3—122no—
318C2×C9—22no—
319C2×C9—22no—
320C2×C9—21no—
321C2×C9—42no—
322C2×C9—64no—
323C2×C9D3×#4112yes—
324C2×C9C22×#6711yescharacteristic
325D9—42no—
326C24—31no—
327#41—184no—
328C2×Q8—91no—
329C2×D4—92no—
330C2×D4—91no—
331C2×D4—274no—
332C22×C4—31no—
333C22×C4—32no—
334C22×C4—32no—
335C22×C4—64no—
336C22×C4—92no—
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—34no—
349C22×C3—42no—
350C22×C3—68no—
351C22×C3—62no—
352C22×C3—62no—
353C22×C3—184no—
354C22×C3—182no—
355C22×C3C2×#61112yes—
356C22×C3C23×D911yescharacteristic
357C22×C3C2×#62512yes—
358C22×C3C22×D3211yescharacteristic
359C2×D3—28no—
360C2×D3—22no—
361C2×D3—22no—
362C2×D3—64no—
363C2×D3—62no—
364C2×D3#369714yescomplemented
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#579111yescharacteristic
379C9—21no—
380C9#582811yescharacteristic, complemented
381C23—32no—
382C23—64no—
383C23—62no—
384C23—92no—
385C23C2×D3×D911yescharacteristic
386Q8—182no—
387D4—184no—
388D4—182no—
389D4—548no—
390C2×C4—32no—
391C2×C4—62no—
392C2×C4—64no—
393C2×C4—64no—
394C2×C4—92no—
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—64no—
406C2×C3—362no—
407C2×C3C2×#369712yes—
408C2×C3C22×#61111yescharacteristic
409C2×C3D3×#23512yes—
410C2×C3C22×#62511yescharacteristic
411D3—24no—
412D3—122no—
413C22—22no—
414C22—34no—
415C22—68no—
416C22—62no—
417C22—62no—
418C22—182no—
419C22—184no—
420C22C2×#619412yes—
421C22C22×D3×D911yescharacteristic
422C4—62no—
423C4—182no—
424C4—544no—
425C4C2×#619412yes—
426C4C2×#619412yes—
427C3—21no—
428C3#1058989311yescharacteristic, complemented
429C3#1059502211yescharacteristic
430C2—22no—
431C2—64no—
432C2—362no—
433C2D3×#369712yes—
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. C22×C4×D3×C9C2semidirect
2. C2×C3×#3697C2semidirect
3. #11721815C2semidirect
4. #11721286C2semidirect
5. #11721259C2semidirect
6. #11721037C2semidirect
7. C4×#6194C2semidirect
8. #11720960C2semidirect
19. C2×C3×D36C22semidirect
20. C2×C3×C4×D9C22semidirect
21. C2×C3×Dic18C22semidirect
24. C4⋊(C3⋊Dic9)C22semidirect
25. C4×C3⋊Dic9C22semidirect
27. (C2×D9)⋊Dic3C22semidirect
28. #88442C22semidirect
38. C2×#3697D3semidirect
51. C2×C3×C4×C9D4semidirect
95. C2×D36C2×D3semidirect
96. C2×C4×D9C2×D3semidirect
97. C2×Dic18C2×D3semidirect
113. C2×D3×C9#41semidirect
124. C22×C4×D3D9semidirect
170. C2×C4×C9#67semidirect
269. C3×C9#525semidirect
293. C2×C3×C4#611semidirect
364. C2×D3#3697semidirect
380. C9#5828semidirect
397. C2×C4#6194semidirect
428. C3#10589893semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D3×C92
C2×C3×#36972
#117218152
#117212862
#117212592
#117210372
C4×#61942
#117209602
#105950223
#105898933