Finite Group Databasebeta

← C3×#10592740 #13163076

Subgroups — C3×#10592740 #13163076

Order 1728.

1120 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×#121C211yesmaximal, characteristic
2C2×C3×#3754C211yesmaximal, characteristic, complemented
3C3×#64133C211yesmaximal, characteristic, complemented
4C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×#63637C211yesmaximal, characteristic, complemented
7C3×#63635C211yesmaximal, characteristic, complemented
8#10592740C311yesmaximal, characteristic, direct product factor
9C3×#5041—34nomaximal
10C2×C33×D4—21no—
11C2×C33×D4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C33×C22⋊C4—21no—
16C3×#3757—21no—
17C3×#3754—22no—
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C3×#3750—21no—
20C2×C3×C3⋊(C3⋊C8)—22no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×#121—24no—
24C32×#121D314yes—
25C32×#121C2×C311yescharacteristic
26C2×#3754C2×C311yescharacteristic, complemented
27#64133C2×C311yescharacteristic, complemented
28C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
29C2×C3×#214—34no—
30C3×#924—34no—
31C22×C3×C3⋊C8—34no—
32#63659C2×C311yescharacteristic, complemented
33#63637C2×C311yescharacteristic, complemented
34#63635C2×C311yescharacteristic, complemented
35C3×#832—34no—
36C3×#810—34no—
37C3×#808—34no—
38C23×C33—21no—
39C23×C33D411yescharacteristic
40C33×D4—21no—
41C33×D4—21no—
42C33×D4—42no—
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4—21no—
46C2×C33×C4D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C3×#634—21no—
51C3×#634—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54C3×#465—91no—
55#5041—34no—
56C2×C32×D4—28no—
57C2×C32×D4—24no—
58C2×C32×D4—24no—
59C2×C32×D4—21no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C22×C311yescharacteristic
62C22×C32×C4—24no—
63C22×C32×C4C2×D314yes—
64C22×C32×C4C22×C311yescharacteristic
65C2×#634C22×C311yescharacteristic
66C2×C3×Dic6—34no—
67C32×C4⋊C4—24no—
68C32×C4⋊C4C2×D314yes—
69C32×C4⋊C4C22×C311yescharacteristic
70C32×C22⋊C4—28no—
71C32×C22⋊C4—24no—
72C32×C22⋊C4—21no—
73#3757—21no—
74#3754—22no—
75C4⋊(C3⋊Dic3)C22×C311yescharacteristic
76#3750—21no—
77C2×C3⋊(C3⋊C8)—22no—
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C3×#217—64no—
81C3×#214—68no—
82C3×C4⋊Dic3—34no—
83C3×Dic3⋊C4—64no—
84C2×C3×C3⋊C8—34no—
85C2×C3×C3⋊C8—34no—
86C2×C3×C3⋊C8—68no—
87C22×C33—21no—
88C22×C33—21no—
89C22×C33—41no—
90C22×C33—41no—
91C22×C33—41no—
92C22×C33SD1612yescomplemented
93C22×C33C2×D411yescharacteristic
94C33×C4—21no—
95C33×C4—41no—
96C33×C4C2×D411yescharacteristic
97C33×C4#4111yescharacteristic
98C3×C3⋊Dic3—41no—
99C3×C3⋊Dic3—41no—
100C2×C3×SD16—91no—
101C22×C3×C8—91no—
102C3×C22⋊Q8—91no—
103C3×#121—24no—
104C3×#121C3×D314yes—
105C3×#121C3⋊D311yescharacteristic
106C2×#214—34no—
107#924—34no—
108C22×C3⋊C8—34no—
109C3×#106—91no—
110C3×#103—91no—
111C3×#102—91no—
112#832—34no—
113#810—34no—
114#808—34no—
115C23×C32—28no—
116C23×C32—24no—
117C23×C32—21no—
118C23×C32—24no—
119C23×C32#6714yes—
120C23×C32C3×D411yescharacteristic
121C32×D4—21no—
122C32×D4—21no—
123C32×D4—24no—
124C32×D4—24no—
125C32×D4—44no—
126C32×D4—44no—
127C32×D4—416no—
128C32×D4—48no—
129C32×D4—42no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—28no—
133C2×C32×C4—24no—
134C2×C32×C4—21no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—44no—
140C2×C32×C4—44no—
141C2×C32×C4#6714yes—
142C2×C32×C4C3×D411yescharacteristic
143C2×C32×C4C22×D314yes—
144C2×C32×C4C23×C311yescharacteristic
145C2×C3⋊Dic3—21no—
146C2×C3⋊Dic3—21no—
147#634—21no—
148#634—21no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3—64no—
151C3×Dic6—64no—
152C3×Dic6—64no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)C3×D412yescomplemented
155C3×C3⋊C8—38no—
156C3×C3⋊C8—64no—
157#465—91noSylow
158C2×C33—22no—
159C2×C33—81no—
160C2×C33#12111yescharacteristic
161C2×C33C2×SD1611yescharacteristic
162C2×C33#13511yescharacteristic
163C2×C3×Q8—91no—
164C2×C3×D4—28no—
165C2×C3×D4—24no—
166C2×C3×D4—21no—
167C2×C3×D4—24no—
168C2×C3×D4C2×C3×D314yes—
169C2×C3×D4C2×C3⋊D311yescharacteristic
170C22×C3×C4—24no—
171C22×C3×C4C2×C3×D314yes—
172C22×C3×C4C2×C3⋊D311yescharacteristic
173C2×Dic6—34no—
174C3×SD16—182no—
175C2×C3×C8—91no—
176C2×C3×C8—91no—
177C2×C3×C8—182no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—91no—
180C3×C4⋊C4—181no—
181C3×C4⋊C4C2×C3×D314yes—
182C3×C4⋊C4C2×C3⋊D311yescharacteristic
183C3×C22⋊C4—28no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—21no—
186C3×C22⋊C4—181no—
187#217—64no—
188#214—68no—
189C4⋊Dic3—34no—
190Dic3⋊C4—64no—
191C2×C3⋊C8—34no—
192C2×C3⋊C8—34no—
193C2×C3⋊C8—68no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—28no—
197C22×C32—24no—
198C22×C32—24no—
199C22×C32—24no—
200C22×C32—44no—
201C22×C32—44no—
202C22×C32—48no—
203C22×C32—48no—
204C22×C32—48no—
205C22×C32—44no—
206C22×C32—44no—
207C22×C32—44no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32#21418yescomplemented
212C22×C32C3×SD1612yescomplemented
213C22×C32C2×#6714yes—
214C22×C32C2×C3×D411yescharacteristic
215C32×C4—21no—
216C32×C4—24no—
217C32×C4—24no—
218C32×C4—24no—
219C32×C4—44no—
220C32×C4—48no—
221C32×C4—44no—
222C32×C4—41no—
223C32×C4D3×D414yes—
224C32×C4#23714yes—
225C32×C4C2×C3×D411yescharacteristic
226C32×C4C3×#4111yescharacteristic
227C3⋊Dic3—41no—
228C3⋊Dic3—41no—
229C3×Dic3—124no—
230C3×Dic3—124no—
231C2×SD16—91no—
232C22×C8—91no—
233C22⋊Q8—91no—
234#121C3×C3⋊D311yescharacteristic
235#106—91no—
236#103—91no—
237#102—91no—
238C33#46511yescharacteristic, complemented, Sylow
239C23×C3—28no—
240C23×C3—24no—
241C23×C3—21no—
242C23×C3—24no—
243C23×C3C3×#6714yes—
244C23×C3#63811yescharacteristic
245C3×Q8—181no—
246C3×Q8—181no—
247C3×D4—24no—
248C3×D4—24no—
249C3×D4—21no—
250C3×D4—21no—
251C3×D4—44no—
252C3×D4—44no—
253C3×D4—416no—
254C3×D4—48no—
255C3×D4—42no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—28no—
261C2×C3×C4—24no—
262C2×C3×C4—21no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—44no—
266C2×C3×C4—44no—
267C2×C3×C4—181no—
268C2×C3×C4—181no—
269C2×C3×C4C3×#6714yes—
270C2×C3×C4#63811yescharacteristic
271C2×C3×C4C22×C3×D314yes—
272C2×C3×C4C22×C3⋊D311yescharacteristic
273C2×Dic3—64no—
274C2×Dic3—64no—
275Dic6—64no—
276Dic6—64no—
277C3×C8—92no—
278C3×C8—181no—
279C3⋊C8—38no—
280C3⋊C8—64no—
281C2×C32—22no—
282C2×C32—24no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32—28no—
286C2×C32—48no—
287C2×C32—88no—
288C2×C32—84no—
289C2×C32—81no—
290C2×C32C2×#21414yes—
291C2×C32#93914yes—
292C2×C32#95014yes—
293C2×C32C3×#12111yescharacteristic
294C2×C32C2×C3×SD1611yescharacteristic
295C2×C32C3×#13511yescharacteristic
296C2×Q8—91no—
297C2×D4—21no—
298C2×D4C2×C3×C3⋊D311yescharacteristic
299C22×C4C2×C3×C3⋊D311yescharacteristic
300SD16—182no—
301C2×C8—91no—
302C2×C8—91no—
303C2×C8—182no—
304C4⋊C4—91no—
305C4⋊C4—181no—
306C4⋊C4C2×C3×C3⋊D311yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—181no—
309C22×C3—24no—
310C22×C3—24no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—28no—
314C22×C3—24no—
315C22×C3—44no—
316C22×C3—44no—
317C22×C3—48no—
318C22×C3—48no—
319C22×C3—48no—
320C22×C3—44no—
321C22×C3—44no—
322C22×C3—44no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3C3×#21418yescomplemented
327C22×C3#375412yescomplemented
328C22×C3C2×C3×#6714yes—
329C22×C3C2×#63811yescharacteristic
330C3×C4—24no—
331C3×C4—21no—
332C3×C4—24no—
333C3×C4—24no—
334C3×C4—44no—
335C3×C4—48no—
336C3×C4—44no—
337C3×C4—41no—
338C3×C4—361no—
339C3×C4—361no—
340C3×C4C3×D3×D414yes—
341C3×C4C3×#23714yes—
342C3×C4D4×C3⋊D311yescharacteristic
343C3×C4#383011yescharacteristic
344Dic3—124no—
345Dic3—124no—
346C32—24no—
347C32#504114yescomplemented
348C32C3×#46511yescharacteristic, complemented
349C23—21no—
350C23C3×#63811yescharacteristic
351Q8—181no—
352Q8—181no—
353D4—21no—
354D4—21no—
355D4—42no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—181no—
360C2×C4—181no—
361C2×C4C3×#63811yescharacteristic
362C2×C4C22×C3×C3⋊D311yescharacteristic
363C8—92no—
364C8—181no—
365C2×C3—28no—
366C2×C3—22no—
367C2×C3—24no—
368C2×C3—24no—
369C2×C3—24no—
370C2×C3—48no—
371C2×C3—88no—
372C2×C3—84no—
373C2×C3—81no—
374C2×C3C2×C3×#21414yes—
375C2×C3C3×#93914yes—
376C2×C3C3×#95014yes—
377C2×C3C2×#375411yescharacteristic
378C2×C3#6414811yescharacteristic
379C2×C3#6415911yescharacteristic
380C22—21no—
381C22—21no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22C3×#375412yescomplemented
386C22C2×C3×#63811yescharacteristic
387C4—21no—
388C4—41no—
389C4—361no—
390C4—361no—
391C4C3×D4×C3⋊D311yescharacteristic
392C4C3×#383011yescharacteristic
393C3—24no—
394C3C3×#504114yescomplemented
395C3#1059274011yescharacteristic, direct product factor
396C2—22no—
397C2—81no—
398C2C2×C3×#375411yescharacteristic
399C2C3×#6414811yescharacteristic
400C2C3×#6415911yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
Fitting, prime radicalC33×#121C2
socleC22×C33C2×D4
hypercenterC3×#121C3⋊D3
hypocenterC32C3×#465
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
2. C2×C3×#3754C2semidirect
3. C3×#64133C2semidirect
4. C22×C3×C3⋊(C3⋊C8)C2semidirect
5. C3×#63659C2semidirect
6. C3×#63637C2semidirect
7. C3×#63635C2semidirect
8. #10592740C3direct (N × H)
26. C2×#3754C2×C3semidirect
27. #64133C2×C3semidirect
28. C22×C3⋊(C3⋊C8)C2×C3semidirect
32. #63659C2×C3semidirect
33. #63637C2×C3semidirect
34. #63635C2×C3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
92. C22×C33SD16semidirect
154. C3⋊(C3⋊C8)C3×D4semidirect
211. C22×C32#214semidirect
212. C22×C32C3×SD16semidirect
238. C33#465semidirect
326. C22×C3C3×#214semidirect
327. C22×C3#3754semidirect
347. C32#5041semidirect
348. C32C3×#465semidirect
385. C22C3×#3754semidirect
394. C3C3×#5041semidirect
395. C3#10592740direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1212
C2×C3×#37542
C3×#641332
C22×C3×C3⋊(C3⋊C8)2
C3×#636592
C3×#636372
C3×#636352
#105927403
C3×#50413