Finite Group Databasebeta

← C3×#10592432 #13162771

Subgroups — C3×#10592432 #13162771

Order 1728.

1112 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C211yesmaximal, characteristic
2C3×#64133C211yesmaximal, characteristic, complemented
3C2×C3×#3746C211yesmaximal, characteristic, complemented
4C2×C3×#3744C211yesmaximal, characteristic, complemented
5C33×C22⋊C8C211yesmaximal, characteristic, complemented
6C3×#63648C211yesmaximal, characteristic, complemented
7C3×#63643C211yesmaximal, characteristic, complemented
8#10592432C311yesmaximal, characteristic, direct product factor
9C3×#4733—34nomaximal
10C22×C33×C4C2211yescharacteristic
11C2×C3×#638—21no—
12C3×#3828—22no—
13C3×#3828—22no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3—21no—
16C2×C3×#634C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C2×C33×C8C2211yescharacteristic
19C2×C33×C8C2211yescharacteristic
20C3×#3757—21no—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C3×#3750—21no—
23C3×#3746—22no—
24C3×#3744—22no—
25C2×#3828C2×C311yescharacteristic
26C2×C3×#235—34no—
27#64133C2×C311yescharacteristic, complemented
28C2×#3746C2×C311yescharacteristic, complemented
29C2×#3744C2×C311yescharacteristic, complemented
30C3×#924—34no—
31C2×C3×Dic12—34no—
32C2×C3×#204—34no—
33C32×C22⋊C8—24no—
34C32×C22⋊C8D314yescomplemented
35C32×C22⋊C8C2×C311yescharacteristic, complemented
36#63648C2×C311yescharacteristic, complemented
37#63643C2×C311yescharacteristic, complemented
38C3×#821—34no—
39C3×#816—34no—
40C23×C33D411yescharacteristic
41C22×C3×C3⋊D3—21no—
42C2×C33×C4—21no—
43C2×C33×C4—21no—
44C2×C33×C4D411yescharacteristic
45C2×C33×C4C2311yescharacteristic
46C3×#638—24no—
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)D412yes—
52C3×C4×C3⋊D3—24no—
53C3×#634—21no—
54C3×#634—21no—
55C3×#634—21no—
56C3×#634D412yes—
57C33×C8—21no—
58C33×C8—21no—
59C3×#452—91no—
60#4733—34no—
61C22×C32×C4—24no—
62C22×C32×C4C2×D314yes—
63C22×C32×C4C22×C311yescharacteristic
64C2×#638—21no—
65#3828—22no—
66#3828—22no—
67C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C2×#634C22×C311yescharacteristic
70C2×#634C22×C311yescharacteristic
71C2×C3×#67—64no—
72C3×#235—68no—
73C3×#235—68no—
74C2×C3×D12—34no—
75C2×C3×C4×D3—64no—
76C2×C3×Dic6—34no—
77C2×C3×Dic6—34no—
78C2×C32×C8—24no—
79C2×C32×C8—24no—
80C2×C32×C8C2×D314yes—
81C2×C32×C8C2×D314yes—
82C2×C32×C8C22×C311yescharacteristic
83C2×C32×C8C22×C311yescharacteristic
84#3757—21no—
85C4⋊(C3⋊Dic3)C22×C311yescharacteristic
86#3750—21no—
87#3746—22no—
88#3744—22no—
89C3×#217—64no—
90C3×C4⋊Dic3—34no—
91C3×Dic3⋊C4—64no—
92C3×Dic12—68no—
93C3×#204—68no—
94C22×C33—21no—
95C22×C33—21no—
96C22×C33—21no—
97C22×C33C2×D411yescharacteristic
98C2×C3×C3⋊D3—22no—
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C33×C4—21no—
102C33×C4C2×D411yescharacteristic
103C33×C4C2×D411yescharacteristic
104C3×C3⋊Dic3—22no—
105C3×C3⋊Dic3—41no—
106C3×C3⋊Dic3—41no—
107C2×C3×#41—91no—
108C2×#235—34no—
109C2×C3×Q16—91no—
110C2×C3×SD16—91no—
111C3×C22⋊Q8—91no—
112#924—34no—
113C2×Dic12—34no—
114C2×#204—34no—
115C3×#103—91no—
116C3×#102—91no—
117C3×C22⋊C8—24no—
118C3×C22⋊C8C3×D314yescomplemented
119C3×C22⋊C8C3⋊D311yescharacteristic, complemented
120#821—34no—
121#816—34no—
122C23×C32—24no—
123C23×C32D1214yes—
124C23×C32C3×D411yescharacteristic
125C22×C3⋊D3—21no—
126C22×C3×D3—64no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4—44no—
134C2×C32×C4—44no—
135C2×C32×C4D1214yes—
136C2×C32×C4C3×D411yescharacteristic
137C2×C32×C4C22×D314yes—
138C2×C32×C4C23×C311yescharacteristic
139#638—24no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3—21no—
143C4⋊(C3⋊D3)—21no—
144C4⋊(C3⋊D3)C3×D412yes—
145C4×C3⋊D3—24no—
146#634—21no—
147#634—21no—
148#634—21no—
149#634C3×D412yes—
150C3×#67—616no—
151C2×C3×Dic3—64no—
152C2×C3×Dic3—64no—
153C2×C3×Dic3—64no—
154C3×D12—38no—
155C3×D12—64no—
156C3×C4×D3—616no—
157C3×Dic6—38no—
158C3×Dic6—64no—
159C3×Dic6—64no—
160C3×Dic6—64no—
161C32×C8—21no—
162C32×C8—21no—
163C32×C8—24no—
164C32×C8—24no—
165C32×C8—44no—
166C32×C8—44no—
167#452—91noSylow
168C2×C33—41no—
169C2×C33#12011yescharacteristic
170C2×C33#13511yescharacteristic
171C2×C33#13711yescharacteristic
172C3×C3⋊D3—42no—
173C3×#41—182no—
174C3×#41—182no—
175C2×C3×Q8—91no—
176C2×C3×Q8—91no—
177C2×C3×D4—91no—
178C2×C3×D4—181no—
179C22×C3×C4—24no—
180C22×C3×C4—181no—
181C22×C3×C4C2×C3×D314yes—
182C22×C3×C4C2×C3⋊D311yescharacteristic
183C2×#67—64no—
184#235—68no—
185#235—68no—
186C2×D12—34no—
187C2×C4×D3—64no—
188C2×Dic6—34no—
189C2×Dic6—34no—
190C3×Q16—182no—
191C3×SD16—182no—
192C2×C3×C8—24no—
193C2×C3×C8—24no—
194C2×C3×C8C2×C3×D314yes—
195C2×C3×C8C2×C3×D314yes—
196C2×C3×C8C2×C3⋊D311yescharacteristic
197C2×C3×C8C2×C3⋊D311yescharacteristic
198C3×C4⋊C4—91no—
199C3×C4⋊C4—181no—
200C3×C22⋊C4—181no—
201#217—64no—
202C4⋊Dic3—34no—
203Dic3⋊C4—64no—
204Dic12—68no—
205#204—68no—
206C22×C32—28no—
207C22×C32—24no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—24no—
212C22×C32—24no—
213C22×C32—24no—
214C22×C32—44no—
215C22×C32—44no—
216C22×C32C2×D1214yes—
217C22×C32C2×C3×D411yescharacteristic
218C2×C3⋊D3—22no—
219C2×C3⋊D3—41no—
220C2×C3⋊D3—41no—
221C2×C3×D3—68no—
222C2×C3×D3—124no—
223C2×C3×D3—124no—
224C32×C4—21no—
225C32×C4—24no—
226C32×C4—24no—
227C32×C4—24no—
228C32×C4—44no—
229C32×C4D3×D414yes—
230C32×C4D3×D414yes—
231C32×C4C2×C3×D411yescharacteristic
232C32×C4C2×C3×D411yescharacteristic
233C3⋊Dic3—22no—
234C3⋊Dic3—41no—
235C3⋊Dic3—41no—
236C3×Dic3—68no—
237C3×Dic3—124no—
238C3×Dic3—124no—
239C2×#41—91no—
240C2×Q16—91no—
241C2×SD16—91no—
242C22⋊Q8—91no—
243#103—91no—
244#102—91no—
245C22⋊C8C3×C3⋊D311yescharacteristic, complemented
246C33#45211yescharacteristic, complemented, Sylow
247C23×C3—24no—
248C23×C3—181no—
249C23×C3C3×D1214yes—
250C23×C3C4⋊(C3⋊D3)11yescharacteristic
251C22×D3—64no—
252C3×Q8—92no—
253C3×Q8—181no—
254C3×Q8—181no—
255C3×Q8—181no—
256C3×D4—92no—
257C3×D4—184no—
258C3×D4—181no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—44no—
266C2×C3×C4—44no—
267C2×C3×C4—184no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4C3×D1214yes—
272C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
273C2×C3×C4C22×C3×D314yes—
274C2×C3×C4C22×C3⋊D311yescharacteristic
275#67—616no—
276C2×Dic3—64no—
277C2×Dic3—64no—
278C2×Dic3—64no—
279D12—38no—
280D12—64no—
281C4×D3—616no—
282Dic6—38no—
283Dic6—64no—
284Dic6—64no—
285Dic6—64no—
286C3×C8—24no—
287C3×C8—24no—
288C3×C8—21no—
289C3×C8—21no—
290C3×C8—44no—
291C3×C8—44no—
292C2×C32—24no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—48no—
296C2×C32—44no—
297C2×C32—41no—
298C2×C32#88214yes—
299C2×C32#90414yes—
300C2×C32#90914yes—
301C2×C32C3×#12011yescharacteristic
302C2×C32C3×#13511yescharacteristic
303C2×C32C3×#13711yescharacteristic
304C3⋊D3—42no—
305C3×D3—128no—
306#41—182no—
307#41—182no—
308C2×Q8—91no—
309C2×Q8—91no—
310C2×D4—91no—
311C2×D4—181no—
312C22×C4—181no—
313C22×C4C2×C3×C3⋊D311yescharacteristic
314Q16—182no—
315SD16—182no—
316C2×C8C2×C3×C3⋊D311yescharacteristic
317C2×C8C2×C3×C3⋊D311yescharacteristic
318C4⋊C4—91no—
319C4⋊C4—181no—
320C22⋊C4—181no—
321C22×C3—28no—
322C22×C3—24no—
323C22×C3—24no—
324C22×C3—24no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—24no—
329C22×C3—44no—
330C22×C3—44no—
331C22×C3—182no—
332C22×C3—361no—
333C22×C3—361no—
334C22×C3C2×C3×D1214yes—
335C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
336C2×D3—68no—
337C2×D3—124no—
338C2×D3—124no—
339C3×C4—24no—
340C3×C4—21no—
341C3×C4—24no—
342C3×C4—24no—
343C3×C4—44no—
344C3×C4—182no—
345C3×C4—361no—
346C3×C4—361no—
347C3×C4C3×D3×D414yes—
348C3×C4C3×D3×D414yes—
349C3×C4D4×C3⋊D311yescharacteristic
350C3×C4D4×C3⋊D311yescharacteristic
351Dic3—68no—
352Dic3—124no—
353Dic3—124no—
354C32—24no—
355C32#473314yescomplemented
356C32C3×#45211yescharacteristic, complemented
357C23—181no—
358C23C3×C4⋊(C3⋊D3)11yescharacteristic
359Q8—92no—
360Q8—181no—
361Q8—181no—
362Q8—181no—
363D4—92no—
364D4—184no—
365D4—181no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—184no—
369C2×C4—181no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
373C2×C4C22×C3×C3⋊D311yescharacteristic
374C8—21no—
375C8—21no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3—24no—
379C2×C3—48no—
380C2×C3—44no—
381C2×C3—41no—
382C2×C3—362no—
383C2×C3C3×#88214yes—
384C2×C3C3×#90414yes—
385C2×C3C3×#90914yes—
386C2×C3#6409111yescharacteristic
387C2×C3#6411311yescharacteristic
388C2×C3#6411811yescharacteristic
389D3—128no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—182no—
394C22—361no—
395C22—361no—
396C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
397C4—21no—
398C4—182no—
399C4—361no—
400C4—361no—
401C4C3×D4×C3⋊D311yescharacteristic
402C4C3×D4×C3⋊D311yescharacteristic
403C3—24no—
404C3C3×#473314yescomplemented
405C3#1059243211yescharacteristic, direct product factor
406C2—41no—
407C2—362no—
408C2C3×#6409111yescharacteristic
409C2C3×#6411311yescharacteristic
410C2C3×#6411811yescharacteristic
subgroupG/H
centerC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊C8C2
socleC22×C33C2×D4
prime radicalC2×C3×#3828C2
hypercenterC3×C22⋊C8C3⋊D3
hypocenterC32C3×#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C3×#64133C2semidirect
3. C2×C3×#3746C2semidirect
4. C2×C3×#3744C2semidirect
5. C33×C22⋊C8C2semidirect
6. C3×#63648C2semidirect
7. C3×#63643C2semidirect
8. #10592432C3direct (N × H)
27. #64133C2×C3semidirect
28. C2×#3746C2×C3semidirect
29. C2×#3744C2×C3semidirect
34. C32×C22⋊C8D3semidirect
35. C32×C22⋊C8C2×C3semidirect
36. #63648C2×C3semidirect
37. #63643C2×C3semidirect
118. C3×C22⋊C8C3×D3semidirect
119. C3×C22⋊C8C3⋊D3semidirect
245. C22⋊C8C3×C3⋊D3semidirect
246. C33#452semidirect
355. C32#4733semidirect
356. C32C3×#452semidirect
404. C3C3×#4733semidirect
405. C3#10592432direct (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
C3×#641332
C2×C3×#37462
C2×C3×#37442
C33×C22⋊C82
C3×#636482
C3×#636432
#105924323
C3×#47333