Finite Group Databasebeta

← C3×#10592715 #13163054

Subgroups — C3×#10592715 #13163054

Order 1728.

1120 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×D4C211yesmaximal, characteristic, complemented
2C2×C3×#3754C211yesmaximal, characteristic, complemented
3C2×C3×#3753C211yesmaximal, characteristic
4C3×#64084C211yesmaximal, characteristic, complemented
5C3×#63637C211yesmaximal, characteristic, complemented
6C3×#63636C211yesmaximal, characteristic, complemented
7C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
8#10592715C311yesmaximal, characteristic, direct product factor
9C3×#5016—34nomaximal
10C2×C33×D4C2211yescharacteristic
11C22×C33×C4—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C33×C4⋊C4C2211yescharacteristic
15C33×C22⋊C4—21no—
16C33×C42C2211yescharacteristic
17C3×#3754—22no—
18C3×#3753—22no—
19C3×#3752—21no—
20C3×#3752—21no—
21C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
22C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
23C32×C4×D4—24no—
24C32×C4×D4D314yescomplemented
25C32×C4×D4C2×C311yescharacteristic, complemented
26C2×#3754C2×C311yescharacteristic, complemented
27C2×#3753C2×C311yescharacteristic
28#64084C2×C311yescharacteristic, complemented
29C2×C3×#214—34no—
30C2×C3×#213—34no—
31C3×#875—34no—
32#63637C2×C311yescharacteristic, complemented
33#63636C2×C311yescharacteristic, complemented
34C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
35C3×#810—34no—
36C3×#809—34no—
37C3×C4⋊(C3⋊C8)—34no—
38C23×C33—21no—
39C22×C3×C3⋊D3—21no—
40C33×D4—21no—
41C33×D4D412yes—
42C2×C33×C4—21no—
43C2×C33×C4—42no—
44C2×C33×C4D411yescharacteristic
45C2×C33×C4D411yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C3×C4⋊(C3⋊D3)—21no—
49C3×C4⋊(C3⋊D3)—21no—
50C3×#634—21no—
51C3×#634—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C3×#463—91no—
55#5016—34no—
56C2×C32×D4—24no—
57C2×C32×D4C2×D314yes—
58C2×C32×D4C22×C311yescharacteristic
59C22×C32×C4—28no—
60C22×C32×C4—24no—
61C22×C32×C4—21no—
62C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
63C2×#634C22×C311yescharacteristic
64C2×C3×D12—34no—
65C2×C3×Dic6—34no—
66C32×C4⋊C4—24no—
67C32×C4⋊C4C2×D314yes—
68C32×C4⋊C4C22×C311yescharacteristic
69C32×C22⋊C4—28no—
70C32×C22⋊C4—24no—
71C32×C22⋊C4—21no—
72C32×C42—24no—
73C32×C42C2×D314yes—
74C32×C42C22×C311yescharacteristic
75#3754—22no—
76#3753—22no—
77#3752—21no—
78#3752—21no—
79C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C3×#214—68no—
82C3×#213—68no—
83C3×(C2×D3)⋊C4—64no—
84C3×(C2×D3)⋊C4—64no—
85C2×C3×C3⋊C8—34no—
86C2×C3×C3⋊C8—34no—
87C22×C33—22no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—41no—
92C2×C3×C3⋊D3—41no—
93C2×C3×C3⋊D3—41no—
94C33×C4—21no—
95C33×C4—21no—
96C33×C4—41no—
97C33×C4C2×D411yescharacteristic
98C33×C4#4111yescharacteristic
99C3×C3⋊Dic3—41no—
100C2×C3×SD16—91no—
101C2×C3×D8—91no—
102C3×#124—91no—
103C3×C4×D4—24no—
104C3×C4×D4C3×D314yescomplemented
105C3×C4×D4C3⋊D311yescharacteristic, complemented
106C2×#214—34no—
107C2×#213—34no—
108#875—34no—
109C3×C4⋊C8—91no—
110C3×#103—91no—
111C3×#102—91no—
112#810—34no—
113#809—34no—
114C4⋊(C3⋊C8)—34no—
115C23×C32—28no—
116C23×C32—24no—
117C23×C32—21no—
118C22×C3⋊D3—21no—
119C22×C3×D3—64no—
120C32×D4—28no—
121C32×D4—24no—
122C32×D4—21no—
123C32×D4—28no—
124C32×D4D1218yes—
125C32×D4C3×D412yes—
126C2×C32×C4—28no—
127C2×C32×C4—24no—
128C2×C32×C4—21no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—24no—
132C2×C32×C4—416no—
133C2×C32×C4—48no—
134C2×C32×C4—42no—
135C2×C32×C4#6714yes—
136C2×C32×C4#6714yes—
137C2×C32×C4C3×D411yescharacteristic
138C2×C32×C4C3×D411yescharacteristic
139C2×C32×C4C22×D314yes—
140C2×C32×C4C23×C311yescharacteristic
141C2×C3⋊Dic3—21no—
142C4⋊(C3⋊D3)—21no—
143C4⋊(C3⋊D3)—21no—
144#634—21no—
145#634—21no—
146C2×C3×Dic3—64no—
147C3×D12—64no—
148C3×D12—64no—
149C3×Dic6—64no—
150C3×Dic6—64no—
151C3⋊(C3⋊C8)—21no—
152C3⋊(C3⋊C8)—21no—
153C3×C3⋊C8—64no—
154C3×C3⋊C8—64no—
155#463—91noSylow
156C2×C33—42no—
157C2×C33#12111yescharacteristic
158C2×C33#13511yescharacteristic
159C2×C33AGL(1,ℤ/8)11yescharacteristic
160C3×C3⋊D3—81no—
161C2×C3×Q8—91no—
162C2×C3×D4—24no—
163C2×C3×D4—91no—
164C2×C3×D4C2×C3×D314yes—
165C2×C3×D4C2×C3⋊D311yescharacteristic
166C22×C3×C4—28no—
167C22×C3×C4—24no—
168C22×C3×C4—21no—
169C2×D12—34no—
170C2×Dic6—34no—
171C3×SD16—182no—
172C3×D8—182no—
173C2×C3×C8—91no—
174C2×C3×C8—91no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4C2×C3×D314yes—
177C3×C4⋊C4C2×C3⋊D311yescharacteristic
178C3×C22⋊C4—28no—
179C3×C22⋊C4—24no—
180C3×C22⋊C4—21no—
181C3×C22⋊C4—181no—
182C3×C22⋊C4—181no—
183C3×C42—24no—
184C3×C42C2×C3×D314yes—
185C3×C42C2×C3⋊D311yescharacteristic
186#214—68no—
187#213—68no—
188(C2×D3)⋊C4—64no—
189(C2×D3)⋊C4—64no—
190C2×C3⋊C8—34no—
191C2×C3⋊C8—34no—
192C22×C32—216no—
193C22×C32—28no—
194C22×C32—22no—
195C22×C32—24no—
196C22×C32—48no—
197C22×C32—48no—
198C22×C32—44no—
199C22×C32—44no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32C2×#6714yes—
203C22×C32C2×C3×D411yescharacteristic
204C2×C3⋊D3—41no—
205C2×C3⋊D3—41no—
206C2×C3⋊D3—41no—
207C2×C3×D3—124no—
208C2×C3×D3—124no—
209C2×C3×D3—124no—
210C32×C4—28no—
211C32×C4—24no—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—24no—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—44no—
218C32×C4—48no—
219C32×C4—44no—
220C32×C4—41no—
221C32×C4C2×D1214yes—
222C32×C4#23514yes—
223C32×C4C2×C3×D411yescharacteristic
224C32×C4C3×#4111yescharacteristic
225C3⋊Dic3—41no—
226C3×Dic3—124no—
227C2×SD16—91no—
228C2×D8—91no—
229#124—91no—
230C4×D4C3×C3⋊D311yescharacteristic, complemented
231C4⋊C8—91no—
232#103—91no—
233#102—91no—
234C33#46311yescharacteristic, complemented, Sylow
235C23×C3—28no—
236C23×C3—24no—
237C23×C3—21no—
238C23×C3—181no—
239C22×D3—64no—
240C3×Q8—181no—
241C3×Q8—181no—
242C3×D4—28no—
243C3×D4—24no—
244C3×D4—21no—
245C3×D4—28no—
246C3×D4—181no—
247C3×D4—181no—
248C3×D4C3×D1218yes—
249C3×D4C4⋊(C3⋊D3)12yes—
250C2×C3×C4—28no—
251C2×C3×C4—24no—
252C2×C3×C4—21no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4—416no—
257C2×C3×C4—48no—
258C2×C3×C4—42no—
259C2×C3×C4—181no—
260C2×C3×C4C3×#6714yes—
261C2×C3×C4C3×#6714yes—
262C2×C3×C4#63811yescharacteristic
263C2×C3×C4#63811yescharacteristic
264C2×C3×C4C22×C3×D314yes—
265C2×C3×C4C22×C3⋊D311yescharacteristic
266C2×Dic3—64no—
267D12—64no—
268D12—64no—
269Dic6—64no—
270Dic6—64no—
271C3×C8—181no—
272C3×C8—181no—
273C3⋊C8—64no—
274C3⋊C8—64no—
275C2×C32—24no—
276C2×C32—24no—
277C2×C32—24no—
278C2×C32—416no—
279C2×C32—48no—
280C2×C32—42no—
281C2×C32#93014yes—
282C2×C32#93214yes—
283C2×C32#95014yes—
284C2×C32C3×#12111yescharacteristic
285C2×C32C3×#13511yescharacteristic
286C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
287C3⋊D3—81no—
288C3×D3—244no—
289C2×Q8—91no—
290C2×D4—91no—
291C2×D4C2×C3×C3⋊D311yescharacteristic
292C22×C4—21no—
293SD16—182no—
294D8—182no—
295C2×C8—91no—
296C2×C8—91no—
297C4⋊C4C2×C3×C3⋊D311yescharacteristic
298C22⋊C4—21no—
299C22⋊C4—181no—
300C22⋊C4—181no—
301C42C2×C3×C3⋊D311yescharacteristic
302C22×C3—216no—
303C22×C3—28no—
304C22×C3—22no—
305C22×C3—24no—
306C22×C3—48no—
307C22×C3—48no—
308C22×C3—44no—
309C22×C3—44no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—361no—
313C22×C3—361no—
314C22×C3—361no—
315C22×C3C2×C3×#6714yes—
316C22×C3C2×#63811yescharacteristic
317C2×D3—124no—
318C2×D3—124no—
319C2×D3—124no—
320C3×C4—28no—
321C3×C4—24no—
322C3×C4—24no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—24no—
326C3×C4—24no—
327C3×C4—44no—
328C3×C4—48no—
329C3×C4—44no—
330C3×C4—41no—
331C3×C4—361no—
332C3×C4C2×C3×D1214yes—
333C3×C4C3×#23514yes—
334C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
335C3×C4#382811yescharacteristic
336Dic3—124no—
337C32—24no—
338C32#501614yescomplemented
339C32C3×#46311yescharacteristic, complemented
340C23—21no—
341C23—181no—
342Q8—181no—
343Q8—181no—
344D4—21no—
345D4—181no—
346D4—181no—
347D4C3×C4⋊(C3⋊D3)12yes—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—181no—
351C2×C4C3×#63811yescharacteristic
352C2×C4C3×#63811yescharacteristic
353C2×C4C22×C3×C3⋊D311yescharacteristic
354C8—181no—
355C8—181no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—24no—
359C2×C3—416no—
360C2×C3—48no—
361C2×C3—42no—
362C2×C3—721no—
363C2×C3C3×#93014yes—
364C2×C3C3×#93214yes—
365C2×C3C3×#95014yes—
366C2×C3#6413911yescharacteristic
367C2×C3#6414111yescharacteristic
368C2×C3#6415911yescharacteristic
369D3—244no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—361no—
374C22—361no—
375C22—361no—
376C22C2×C3×#63811yescharacteristic
377C4—21no—
378C4—21no—
379C4—41no—
380C4—361no—
381C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
382C4C3×#382811yescharacteristic
383C3—24no—
384C3C3×#501614yescomplemented
385C3#1059271511yescharacteristic, direct product factor
386C2—42no—
387C2—721no—
388C2C3×#6413911yescharacteristic
389C2C3×#6414111yescharacteristic
390C2C3×#6415911yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C2×D4
prime radicalC2×C3×#3753C2
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#463
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C33×C4×D4C2semidirect
2. C2×C3×#3754C2semidirect
4. C3×#64084C2semidirect
5. C3×#63637C2semidirect
6. C3×#63636C2semidirect
7. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
8. #10592715C3direct (N × H)
24. C32×C4×D4D3semidirect
25. C32×C4×D4C2×C3semidirect
26. C2×#3754C2×C3semidirect
28. #64084C2×C3semidirect
32. #63637C2×C3semidirect
33. #63636C2×C3semidirect
34. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
104. C3×C4×D4C3×D3semidirect
105. C3×C4×D4C3⋊D3semidirect
230. C4×D4C3×C3⋊D3semidirect
234. C33#463semidirect
338. C32#5016semidirect
339. C32C3×#463semidirect
384. C3C3×#5016semidirect
385. C3#10592715direct (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×#37542
C2×C3×#37532
C3×#640842
C3×#636372
C3×#636362
C3×C4⋊(C3⋊(C3⋊C8))2
#105927153
C3×#50163