Finite Group Databasebeta

← C3×#10592736 #13163079

Subgroups — C3×#10592736 #13163079

Order 1728.

1348 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic, complemented
2C33×#121C211yesmaximal, characteristic, complemented
3C2×C3×#3754C211yesmaximal, characteristic, complemented
4C2×C3×#3753C211yesmaximal, characteristic, complemented
5C3×#63657C211yesmaximal, characteristic, complemented
6C3×#63637C211yesmaximal, characteristic, complemented
7C3×#63636C211yesmaximal, characteristic, complemented
8#10592736C311yesmaximal, characteristic, direct product factor
9C3×#5037—34nomaximal
10C2×C33×D4—21no—
11C2×C33×D4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C2×C3×#638—21no—
14C3×#3828—22no—
15C3×#3828—22no—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3—21no—
18C2×C3×#634C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20C33×C22⋊C4—21no—
21C3×#3754—22no—
22C3×#3753—22no—
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
24C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
25C2×#3828C2×C311yescharacteristic, complemented
26C2×C3×#235—34no—
27C32×#121—24no—
28C32×#121D314yescomplemented
29C32×#121C2×C311yescharacteristic, complemented
30C2×#3754C2×C311yescharacteristic, complemented
31C2×#3753C2×C311yescharacteristic, complemented
32C2×C3×#214—34no—
33C2×C3×#213—34no—
34#63657C2×C311yescharacteristic, complemented
35#63637C2×C311yescharacteristic, complemented
36#63636C2×C311yescharacteristic, complemented
37C3×#830—34no—
38C3×#810—34no—
39C3×#809—34no—
40C23×C33—21no—
41C23×C33D411yescharacteristic
42C22×C3×C3⋊D3—21no—
43C33×D4—21no—
44C33×D4—21no—
45C33×D4—42no—
46C2×C33×C4—21no—
47C2×C33×C4—21no—
48C2×C33×C4—21no—
49C2×C33×C4D411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C3×#638—24no—
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)D412yescomplemented
55C3×C4×C3⋊D3—24no—
56C3×#634—21no—
57C3×#634D412yescomplemented
58C3×C3⋊(C3⋊C8)—21no—
59C3×C3⋊(C3⋊C8)—21no—
60C3×#449—91no—
61#5037—34no—
62C2×C32×D4—28no—
63C2×C32×D4—24no—
64C2×C32×D4—24no—
65C2×C32×D4—21no—
66C2×C32×D4C2×D314yes—
67C2×C32×D4C22×C311yescharacteristic
68C22×C32×C4—24no—
69C22×C32×C4C2×D314yes—
70C22×C32×C4C22×C311yescharacteristic
71C2×#638—21no—
72#3828—22no—
73#3828—22no—
74C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
75C2×C4×C3⋊D3—21no—
76C2×#634C22×C311yescharacteristic
77C2×C3×#67—64no—
78C3×#235—68no—
79C3×#235—68no—
80C2×C3×D12—34no—
81C2×C3×C4×D3—64no—
82C2×C3×Dic6—34no—
83C32×C4⋊C4—24no—
84C32×C4⋊C4C2×D314yes—
85C32×C4⋊C4C22×C311yescharacteristic
86C32×C22⋊C4—28no—
87C32×C22⋊C4—24no—
88C32×C22⋊C4—21no—
89#3754—22no—
90#3753—22no—
91C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
92C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
93C3×#214—68no—
94C3×#213—68no—
95C2×C3×C3⋊C8—34no—
96C2×C3×C3⋊C8—34no—
97C22×C33—21no—
98C22×C33—21no—
99C22×C33—21no—
100C22×C33—41no—
101C22×C33—41no—
102C22×C33—41no—
103C22×C33C2×D411yescharacteristic
104C2×C3×C3⋊D3—22no—
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3—41no—
107C33×C4—21no—
108C33×C4—41no—
109C33×C4C2×D411yescharacteristic
110C33×C4C2×D411yescharacteristic
111C3×C3⋊Dic3—22no—
112C2×C3×#41—91no—
113C2×#235—34no—
114C2×C3×SD16—91no—
115C2×C3×D8—91no—
116C3×#121—24no—
117C3×#121C3×D314yescomplemented
118C3×#121C3⋊D311yescharacteristic, complemented
119C2×#214—34no—
120C2×#213—34no—
121C3×#103—91no—
122C3×#102—91no—
123C3×C22⋊C8—91no—
124#830—34no—
125#810—34no—
126#809—34no—
127C23×C32—28no—
128C23×C32—24no—
129C23×C32—21no—
130C23×C32—24no—
131C23×C32#6714yes—
132C23×C32C3×D411yescharacteristic
133C22×C3⋊D3—21no—
134C22×C3×D3—64no—
135C32×D4—21no—
136C32×D4—21no—
137C32×D4—24no—
138C32×D4—24no—
139C32×D4—44no—
140C32×D4—44no—
141C32×D4—416no—
142C32×D4—48no—
143C32×D4—42no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—28no—
147C2×C32×C4—24no—
148C2×C32×C4—21no—
149C2×C32×C4—24no—
150C2×C32×C4—24no—
151C2×C32×C4—24no—
152C2×C32×C4—24no—
153C2×C32×C4—44no—
154C2×C32×C4—44no—
155C2×C32×C4#6714yes—
156C2×C32×C4C3×D411yescharacteristic
157C2×C32×C4C22×D314yes—
158C2×C32×C4C23×C311yescharacteristic
159#638—24no—
160C2×C3⋊Dic3—21no—
161C4⋊(C3⋊D3)—21no—
162C4⋊(C3⋊D3)C3×D412yescomplemented
163C4×C3⋊D3—24no—
164#634—21no—
165#634C3×D412yescomplemented
166C3×#67—616no—
167C2×C3×Dic3—64no—
168C3×D12—38no—
169C3×D12—64no—
170C3×C4×D3—616no—
171C3×Dic6—38no—
172C3×Dic6—64no—
173C3⋊(C3⋊C8)—21no—
174C3⋊(C3⋊C8)—21no—
175C3×C3⋊C8—64no—
176C3×C3⋊C8—64no—
177#449—91noSylow
178C2×C33—41no—
179C2×C33—81no—
180C2×C33#12011yescharacteristic
181C2×C33#13511yescharacteristic
182C2×C33AGL(1,ℤ/8)11yescharacteristic
183C3×C3⋊D3—42no—
184C3×#41—182no—
185C3×#41—182no—
186C2×C3×Q8—91no—
187C2×C3×D4—28no—
188C2×C3×D4—24no—
189C2×C3×D4—21no—
190C2×C3×D4—24no—
191C2×C3×D4—91no—
192C2×C3×D4—181no—
193C2×C3×D4C2×C3×D314yes—
194C2×C3×D4C2×C3⋊D311yescharacteristic
195C22×C3×C4—24no—
196C22×C3×C4—181no—
197C22×C3×C4C2×C3×D314yes—
198C22×C3×C4C2×C3⋊D311yescharacteristic
199C2×#67—64no—
200#235—68no—
201#235—68no—
202C2×D12—34no—
203C2×C4×D3—64no—
204C2×Dic6—34no—
205C3×SD16—182no—
206C3×D8—182no—
207C2×C3×C8—91no—
208C2×C3×C8—91no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4C2×C3×D314yes—
211C3×C4⋊C4C2×C3⋊D311yescharacteristic
212C3×C22⋊C4—28no—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—21no—
215#214—68no—
216#213—68no—
217C2×C3⋊C8—34no—
218C2×C3⋊C8—34no—
219C22×C32—28no—
220C22×C32—24no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—24no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—44no—
228C22×C32—44no—
229C22×C32—48no—
230C22×C32—48no—
231C22×C32—48no—
232C22×C32—44no—
233C22×C32—44no—
234C22×C32—44no—
235C22×C32—41no—
236C22×C32—41no—
237C22×C32—41no—
238C22×C32C2×#6714yes—
239C22×C32C2×C3×D411yescharacteristic
240C2×C3⋊D3—22no—
241C2×C3⋊D3—41no—
242C2×C3⋊D3—41no—
243C2×C3×D3—68no—
244C2×C3×D3—124no—
245C2×C3×D3—124no—
246C32×C4—21no—
247C32×C4—24no—
248C32×C4—24no—
249C32×C4—24no—
250C32×C4—44no—
251C32×C4—48no—
252C32×C4—44no—
253C32×C4—41no—
254C32×C4D3×D414yes—
255C32×C4D3×D414yes—
256C32×C4C2×C3×D411yescharacteristic
257C32×C4C2×C3×D411yescharacteristic
258C3⋊Dic3—22no—
259C3×Dic3—68no—
260C2×#41—91no—
261C2×SD16—91no—
262C2×D8—91no—
263#121C3×C3⋊D311yescharacteristic, complemented
264#103—91no—
265#102—91no—
266C22⋊C8—91no—
267C33#44911yescharacteristic, complemented, Sylow
268C23×C3—28no—
269C23×C3—24no—
270C23×C3—21no—
271C23×C3—24no—
272C23×C3—181no—
273C23×C3C3×#6714yes—
274C23×C3#63811yescharacteristic
275C22×D3—64no—
276C3×Q8—92no—
277C3×Q8—181no—
278C3×D4—24no—
279C3×D4—24no—
280C3×D4—21no—
281C3×D4—21no—
282C3×D4—44no—
283C3×D4—44no—
284C3×D4—416no—
285C3×D4—48no—
286C3×D4—42no—
287C3×D4—92no—
288C3×D4—184no—
289C3×D4—181no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—28no—
295C2×C3×C4—24no—
296C2×C3×C4—21no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—44no—
300C2×C3×C4—44no—
301C2×C3×C4—184no—
302C2×C3×C4—181no—
303C2×C3×C4C3×#6714yes—
304C2×C3×C4#63811yescharacteristic
305C2×C3×C4C22×C3×D314yes—
306C2×C3×C4C22×C3⋊D311yescharacteristic
307#67—616no—
308C2×Dic3—64no—
309D12—38no—
310D12—64no—
311C4×D3—616no—
312Dic6—38no—
313Dic6—64no—
314C3×C8—181no—
315C3×C8—181no—
316C3⋊C8—64no—
317C3⋊C8—64no—
318C2×C32—24no—
319C2×C32—24no—
320C2×C32—24no—
321C2×C32—48no—
322C2×C32—44no—
323C2×C32—41no—
324C2×C32—88no—
325C2×C32—84no—
326C2×C32—81no—
327C2×C32#93214yes—
328C2×C32#93714yes—
329C2×C32#95014yes—
330C2×C32C3×#12011yescharacteristic
331C2×C32C3×#13511yescharacteristic
332C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
333C3⋊D3—42no—
334C3×D3—128no—
335#41—182no—
336#41—182no—
337C2×Q8—91no—
338C2×D4—21no—
339C2×D4—91no—
340C2×D4—181no—
341C2×D4C2×C3×C3⋊D311yescharacteristic
342C22×C4—181no—
343C22×C4C2×C3×C3⋊D311yescharacteristic
344SD16—182no—
345D8—182no—
346C2×C8—91no—
347C2×C8—91no—
348C4⋊C4C2×C3×C3⋊D311yescharacteristic
349C22⋊C4—21no—
350C22×C3—28no—
351C22×C3—24no—
352C22×C3—24no—
353C22×C3—24no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—24no—
358C22×C3—44no—
359C22×C3—44no—
360C22×C3—48no—
361C22×C3—48no—
362C22×C3—48no—
363C22×C3—44no—
364C22×C3—44no—
365C22×C3—44no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—182no—
370C22×C3—361no—
371C22×C3—361no—
372C22×C3C2×C3×#6714yes—
373C22×C3C2×#63811yescharacteristic
374C2×D3—68no—
375C2×D3—124no—
376C2×D3—124no—
377C3×C4—24no—
378C3×C4—21no—
379C3×C4—24no—
380C3×C4—24no—
381C3×C4—44no—
382C3×C4—48no—
383C3×C4—44no—
384C3×C4—41no—
385C3×C4—182no—
386C3×C4C3×D3×D414yes—
387C3×C4C3×D3×D414yes—
388C3×C4D4×C3⋊D311yescharacteristic
389C3×C4D4×C3⋊D311yescharacteristic
390Dic3—68no—
391C32—24no—
392C32#503714yescomplemented
393C32C3×#44911yescharacteristic, complemented
394C23—21no—
395C23—181no—
396C23C3×#63811yescharacteristic
397Q8—92no—
398Q8—181no—
399D4—21no—
400D4—21no—
401D4—42no—
402D4—92no—
403D4—184no—
404D4—181no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—184no—
409C2×C4—181no—
410C2×C4C3×#63811yescharacteristic
411C2×C4C22×C3×C3⋊D311yescharacteristic
412C8—181no—
413C8—181no—
414C2×C3—24no—
415C2×C3—24no—
416C2×C3—24no—
417C2×C3—48no—
418C2×C3—44no—
419C2×C3—41no—
420C2×C3—88no—
421C2×C3—84no—
422C2×C3—81no—
423C2×C3—362no—
424C2×C3C3×#93214yes—
425C2×C3C3×#93714yes—
426C2×C3C3×#95014yes—
427C2×C3#6414111yescharacteristic
428C2×C3#6414611yescharacteristic
429C2×C3#6415911yescharacteristic
430D3—128no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—182no—
438C22—361no—
439C22—361no—
440C22C2×C3×#63811yescharacteristic
441C4—21no—
442C4—41no—
443C4—182no—
444C4C3×D4×C3⋊D311yescharacteristic
445C4C3×D4×C3⋊D311yescharacteristic
446C3—24no—
447C3C3×#503714yescomplemented
448C3#1059273611yescharacteristic, direct product factor
449C2—41no—
450C2—81no—
451C2—362no—
452C2C3×#6414111yescharacteristic
453C2C3×#6414611yescharacteristic
454C2C3×#6415911yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#121C2
socleC22×C33C2×D4
hypercenterC3×#121C3⋊D3
hypocenterC32C3×#449
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3828C2semidirect
2. C33×#121C2semidirect
3. C2×C3×#3754C2semidirect
4. C2×C3×#3753C2semidirect
5. C3×#63657C2semidirect
6. C3×#63637C2semidirect
7. C3×#63636C2semidirect
8. #10592736C3direct (N × H)
25. C2×#3828C2×C3semidirect
28. C32×#121D3semidirect
29. C32×#121C2×C3semidirect
30. C2×#3754C2×C3semidirect
31. C2×#3753C2×C3semidirect
34. #63657C2×C3semidirect
35. #63637C2×C3semidirect
36. #63636C2×C3semidirect
54. C3×C4⋊(C3⋊D3)D4semidirect
57. C3×#634D4semidirect
117. C3×#121C3×D3semidirect
118. C3×#121C3⋊D3semidirect
162. C4⋊(C3⋊D3)C3×D4semidirect
165. #634C3×D4semidirect
263. #121C3×C3⋊D3semidirect
267. C33#449semidirect
392. C32#5037semidirect
393. C32C3×#449semidirect
447. C3C3×#5037semidirect
448. C3#10592736direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C33×#1212
C2×C3×#37542
C2×C3×#37532
C3×#636572
C3×#636372
C3×#636362
#105927363
C3×#50373