Finite Group Databasebeta

← C3×#10592710 #13163049

Subgroups — C3×#10592710 #13163049

Order 1728.

1120 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×D4C211yesmaximal, characteristic
2C2×C3×#3748C211yesmaximal, characteristic, complemented
3C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C3×#63657C211yesmaximal, characteristic, complemented
5C3×#63657C211yesmaximal, characteristic, complemented
6C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7C3×#63630C211yesmaximal, characteristic, complemented
8#10592710C311yesmaximal, characteristic, direct product factor
9C3×#5011—34nomaximal
10C2×C33×D4C411yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C33×C4⋊C4C411yescharacteristic
14C33×C22⋊C4C411yescharacteristic
15C33×C22⋊C4C411yescharacteristic
16C33×C42C2211yescharacteristic
17C3×#3748—22no—
18C2×C3×C3⋊(C3⋊C8)—22no—
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×C4×D4—24no—
24C32×C4×D4D314yes—
25C32×C4×D4C2×C311yescharacteristic
26C2×#3748C2×C311yescharacteristic, complemented
27C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
28C2×C3×#208—34no—
29C22×C3×C3⋊C8—34no—
30#63657C2×C311yescharacteristic, complemented
31#63657C2×C311yescharacteristic, complemented
32C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
33#63630C2×C311yescharacteristic, complemented
34C3×#830—34no—
35C3×#830—34no—
36C3×C4⋊(C3⋊C8)—34no—
37C3×#803—34no—
38C23×C33C2×C411yescharacteristic
39C23×C33C2×C411yescharacteristic
40C33×D4—22no—
41C2×C33×C4—21no—
42C2×C33×C4—21no—
43C2×C33×C4—21no—
44C2×C33×C4—21no—
45C2×C33×C4C2×C411yescharacteristic
46C2×C33×C4C2×C411yescharacteristic
47C2×C33×C4C2×C411yescharacteristic
48C2×C33×C4C2×C411yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C3×C3⋊(C3⋊C8)—21no—
51C3×C3⋊(C3⋊C8)—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C3×C3⋊(C3⋊C8)D412yescomplemented
54C3×#435—91no—
55#5011—34no—
56C2×C32×D4—24no—
57C2×C32×D4Dic314yes—
58C2×C32×D4C3×C411yescharacteristic
59C22×C32×C4—24no—
60C22×C32×C4—24no—
61C22×C32×C4C2×D314yes—
62C22×C32×C4C2×D314yes—
63C22×C32×C4C22×C311yescharacteristic
64C22×C32×C4C22×C311yescharacteristic
65C32×C4⋊C4—24no—
66C32×C4⋊C4Dic314yes—
67C32×C4⋊C4C3×C411yescharacteristic
68C32×C22⋊C4—24no—
69C32×C22⋊C4—24no—
70C32×C22⋊C4Dic314yes—
71C32×C22⋊C4Dic314yes—
72C32×C22⋊C4C3×C411yescharacteristic
73C32×C22⋊C4C3×C411yescharacteristic
74C32×C42—24no—
75C32×C42C2×D314yes—
76C32×C42C22×C311yescharacteristic
77#3748—22no—
78C2×C3⋊(C3⋊C8)—22no—
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
82C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
83C3×#208—68no—
84C2×C3×C3⋊C8—34no—
85C2×C3×C3⋊C8—34no—
86C2×C3×C3⋊C8—34no—
87C2×C3×C3⋊C8—34no—
88C2×C3×C3⋊C8—68no—
89C22×C33—21no—
90C22×C33—21no—
91C22×C33—21no—
92C22×C33—21no—
93C22×C33—21no—
94C22×C33M1612yescomplemented
95C22×C33C22×C411yescharacteristic
96C33×C4—21no—
97C33×C4—21no—
98C33×C4—21no—
99C33×C4—21no—
100C33×C4C2×D411yescharacteristic
101C33×C4#4111yescharacteristic
102C2×C3×M16—91no—
103C22×C3×C8—91no—
104C3×C4×D4—24no—
105C3×C4×D4C3×D314yes—
106C3×C4×D4C3⋊D311yescharacteristic
107C2×#208—34no—
108C22×C3⋊C8—34no—
109C3×C4⋊C8—91no—
110C3×C22⋊C8—91no—
111C3×C22⋊C8—91no—
112C3×#97—91no—
113#830—34no—
114#830—34no—
115C4⋊(C3⋊C8)—34no—
116#803—34no—
117C23×C32—24no—
118C23×C32—24no—
119C23×C32C2×Dic314yes—
120C23×C32C2×Dic314yes—
121C23×C32C2×C3×C411yescharacteristic
122C23×C32C2×C3×C411yescharacteristic
123C32×D4—216no—
124C32×D4—28no—
125C32×D4—22no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4—24no—
134C2×C32×C4—24no—
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—44no—
142C2×C32×C4—44no—
143C2×C32×C4C2×Dic314yes—
144C2×C32×C4C2×Dic314yes—
145C2×C32×C4C2×Dic314yes—
146C2×C32×C4C2×Dic314yes—
147C2×C32×C4C2×C3×C411yescharacteristic
148C2×C32×C4C2×C3×C411yescharacteristic
149C2×C32×C4C2×C3×C411yescharacteristic
150C2×C32×C4C2×C3×C411yescharacteristic
151C2×C32×C4C22×D314yes—
152C2×C32×C4C23×C311yescharacteristic
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)—21no—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)C3×D412yescomplemented
157C3×C3⋊C8—38no—
158C3×C3⋊C8—64no—
159C3×C3⋊C8—64no—
160C3×C3⋊C8—64no—
161#435—91noSylow
162C2×C33—22no—
163C2×C33—41no—
164C2×C33C4×D411yescharacteristic
165C2×C33C2×M1611yescharacteristic
166C2×C33#13111yescharacteristic
167C2×C3×D4—24no—
168C2×C3×D4C3×Dic314yes—
169C2×C3×D4C3⋊Dic311yescharacteristic
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4C2×C3×D314yes—
173C22×C3×C4C2×C3×D314yes—
174C22×C3×C4C2×C3⋊D311yescharacteristic
175C22×C3×C4C2×C3⋊D311yescharacteristic
176C3×M16—182no—
177C2×C3×C8—91no—
178C2×C3×C8—91no—
179C2×C3×C8—91no—
180C2×C3×C8—91no—
181C2×C3×C8—182no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4C3×Dic314yes—
184C3×C4⋊C4C3⋊Dic311yescharacteristic
185C3×C22⋊C4—24no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4C3×Dic314yes—
188C3×C22⋊C4C3×Dic314yes—
189C3×C22⋊C4C3⋊Dic311yescharacteristic
190C3×C22⋊C4C3⋊Dic311yescharacteristic
191C3×C42—24no—
192C3×C42C2×C3×D314yes—
193C3×C42C2×C3⋊D311yescharacteristic
194#208—68no—
195C2×C3⋊C8—34no—
196C2×C3⋊C8—34no—
197C2×C3⋊C8—34no—
198C2×C3⋊C8—34no—
199C2×C3⋊C8—68no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—28no—
203C22×C32—24no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—28no—
208C22×C32—24no—
209C22×C32—24no—
210C22×C32—24no—
211C22×C32—24no—
212C22×C32—24no—
213C22×C32—44no—
214C22×C32—44no—
215C22×C32—44no—
216C22×C32—44no—
217C22×C32#20818yescomplemented
218C22×C32C3×M1612yescomplemented
219C22×C32C22×Dic314yes—
220C22×C32C22×C3×C411yescharacteristic
221C32×C4—28no—
222C32×C4—24no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—24no—
228C32×C4—24no—
229C32×C4—24no—
230C32×C4—24no—
231C32×C4—24no—
232C32×C4—44no—
233C32×C4—44no—
234C32×C4—44no—
235C32×C4D3×D414yes—
236C32×C4#23714yes—
237C32×C4C2×C3×D411yescharacteristic
238C32×C4C3×#4111yescharacteristic
239C2×M16—91no—
240C22×C8—91no—
241C4×D4C3×C3⋊D311yescharacteristic
242C4⋊C8—91no—
243C22⋊C8—91no—
244C22⋊C8—91no—
245#97—91no—
246C33#43511yescharacteristic, complemented, Sylow
247C23×C3—24no—
248C23×C3—24no—
249C23×C3C2×C3×Dic314yes—
250C23×C3C2×C3×Dic314yes—
251C23×C3C2×C3⋊Dic311yescharacteristic
252C23×C3C2×C3⋊Dic311yescharacteristic
253C3×D4—216no—
254C3×D4—28no—
255C3×D4—22no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—44no—
270C2×C3×C4—44no—
271C2×C3×C4—44no—
272C2×C3×C4—44no—
273C2×C3×C4C2×C3×Dic314yes—
274C2×C3×C4C2×C3×Dic314yes—
275C2×C3×C4C2×C3×Dic314yes—
276C2×C3×C4C2×C3×Dic314yes—
277C2×C3×C4C2×C3⋊Dic311yescharacteristic
278C2×C3×C4C2×C3⋊Dic311yescharacteristic
279C2×C3×C4C2×C3⋊Dic311yescharacteristic
280C2×C3×C4C2×C3⋊Dic311yescharacteristic
281C2×C3×C4C22×C3×D314yes—
282C2×C3×C4C22×C3⋊D311yescharacteristic
283C3×C8—92no—
284C3×C8—181no—
285C3×C8—181no—
286C3×C8—181no—
287C3⋊C8—38no—
288C3⋊C8—64no—
289C3⋊C8—64no—
290C3⋊C8—64no—
291C2×C32—22no—
292C2×C32—24no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—28no—
296C2×C32—48no—
297C2×C32—48no—
298C2×C32—44no—
299C2×C32—41no—
300C2×C32C2×#20814yes—
301C2×C32D4×Dic314yes—
302C2×C32#94814yes—
303C2×C32C3×C4×D411yescharacteristic
304C2×C32C2×C3×M1611yescharacteristic
305C2×C32C3×#13111yescharacteristic
306C2×D4C3×C3⋊Dic311yescharacteristic
307C22×C4C2×C3×C3⋊D311yescharacteristic
308C22×C4C2×C3×C3⋊D311yescharacteristic
309M16—182no—
310C2×C8—91no—
311C2×C8—91no—
312C2×C8—91no—
313C2×C8—91no—
314C2×C8—182no—
315C4⋊C4C3×C3⋊Dic311yescharacteristic
316C22⋊C4C3×C3⋊Dic311yescharacteristic
317C22⋊C4C3×C3⋊Dic311yescharacteristic
318C42C2×C3×C3⋊D311yescharacteristic
319C22×C3—24no—
320C22×C3—24no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—28no—
324C22×C3—24no—
325C22×C3—24no—
326C22×C3—24no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—28no—
331C22×C3—24no—
332C22×C3—44no—
333C22×C3—44no—
334C22×C3—44no—
335C22×C3—44no—
336C22×C3C3×#20818yescomplemented
337C22×C3#374812yescomplemented
338C22×C3C22×C3×Dic314yes—
339C22×C3C22×C3⋊Dic311yescharacteristic
340C3×C4—28no—
341C3×C4—24no—
342C3×C4—24no—
343C3×C4—24no—
344C3×C4—24no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—24no—
350C3×C4—24no—
351C3×C4—44no—
352C3×C4—44no—
353C3×C4—44no—
354C3×C4C3×D3×D414yes—
355C3×C4C3×#23714yes—
356C3×C4D4×C3⋊D311yescharacteristic
357C3×C4#383011yescharacteristic
358C32—24no—
359C32#501114yescomplemented
360C32C3×#43511yescharacteristic, complemented
361C23C2×C3×C3⋊Dic311yescharacteristic
362C23C2×C3×C3⋊Dic311yescharacteristic
363D4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4C2×C3×C3⋊Dic311yescharacteristic
369C2×C4C2×C3×C3⋊Dic311yescharacteristic
370C2×C4C2×C3×C3⋊Dic311yescharacteristic
371C2×C4C2×C3×C3⋊Dic311yescharacteristic
372C2×C4C22×C3×C3⋊D311yescharacteristic
373C8—92no—
374C8—181no—
375C8—181no—
376C8—181no—
377C2×C3—28no—
378C2×C3—22no—
379C2×C3—24no—
380C2×C3—24no—
381C2×C3—24no—
382C2×C3—48no—
383C2×C3—48no—
384C2×C3—44no—
385C2×C3—41no—
386C2×C3C2×C3×#20814yes—
387C2×C3C3×D4×Dic314yes—
388C2×C3C3×#94814yes—
389C2×C3C2×#374811yescharacteristic
390C2×C3D4×C3⋊Dic311yescharacteristic
391C2×C3#6415711yescharacteristic
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22C3×#374812yescomplemented
398C22C22×C3×C3⋊Dic311yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4C3×D4×C3⋊D311yescharacteristic
404C4C3×#383011yescharacteristic
405C3—24no—
406C3C3×#501114yescomplemented
407C3#1059271011yescharacteristic, direct product factor
408C2—22no—
409C2—41no—
410C2C2×C3×#374811yescharacteristic
411C2C3×D4×C3⋊Dic311yescharacteristic
412C2C3×#6415711yescharacteristic
subgroupG/H
centerC2×C3×C4C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C22×C4
prime radicalC2×C33×D4C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#435
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. C22×C3×C3⋊(C3⋊C8)C2semidirect
4. C3×#63657C2semidirect
5. C3×#63657C2semidirect
6. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
7. C3×#63630C2semidirect
8. #10592710C3direct (N × H)
26. C2×#3748C2×C3semidirect
27. C22×C3⋊(C3⋊C8)C2×C3semidirect
30. #63657C2×C3semidirect
31. #63657C2×C3semidirect
32. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
33. #63630C2×C3semidirect
53. C3×C3⋊(C3⋊C8)D4semidirect
94. C22×C33M16semidirect
156. C3⋊(C3⋊C8)C3×D4semidirect
217. C22×C32#208semidirect
218. C22×C32C3×M16semidirect
246. C33#435semidirect
336. C22×C3C3×#208semidirect
337. C22×C3#3748semidirect
359. C32#5011semidirect
360. C32C3×#435semidirect
397. C22C3×#3748semidirect
406. C3C3×#5011semidirect
407. C3#10592710direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4×D42
C2×C3×#37482
C22×C3×C3⋊(C3⋊C8)2
C3×#636572
C3×C4⋊(C3⋊(C3⋊C8))2
C3×#636302
#105927103
C3×#50113