Finite Group Databasebeta

← C3×#10592507 #13162854

Subgroups — C3×#10592507 #13162854

Order 1728.

1080 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3832C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C2×C3×#3744C211yesmaximal, characteristic, complemented
4C3×#64106C211yesmaximal, characteristic, complemented
5C33×#103C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7C3×#63636C211yesmaximal, characteristic, complemented
8#10592507C311yesmaximal, characteristic, direct product factor
9C3×#4808—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C3×#3832—22no—
12C3×#3832—22no—
13C2×C3×C4⋊(C3⋊D3)—21no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3—21no—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C2×C33×C8C2211yescharacteristic
19C33×C4⋊C4C2211yescharacteristic
20C3×#3756—22no—
21C3×#3752—21no—
22C3×C4⋊(C3⋊Dic3)—21no—
23C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
24C3×#3744—22no—
25C2×#3832C2×C311yescharacteristic
26C2×C3×#239—34no—
27C2×#3756C2×C311yescharacteristic, complemented
28C2×#3744C2×C311yescharacteristic, complemented
29#64106C2×C311yescharacteristic, complemented
30C2×C3×#216—34no—
31C2×C3×#204—34no—
32C3×#897—34no—
33C32×#103—24no—
34C32×#103D314yescomplemented
35C32×#103C2×C311yescharacteristic, complemented
36#63647C2×C311yescharacteristic, complemented
37#63636C2×C311yescharacteristic, complemented
38C3×(C2×D3)⋊C8—34no—
39C3×#809—34no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C33×Q8—21no—
43C33×Q8D412yes—
44C2×C33×C4—21no—
45C2×C33×C4—21no—
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3D411yescharacteristic
49C3×C4⋊(C3⋊D3)—24no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)D412yes—
52C3×C4×C3⋊D3—24no—
53C3×C4×C3⋊D3—21no—
54C3×C4×C3⋊D3—21no—
55C3×#634—21no—
56C3×#634—21no—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×#452—91no—
60#4808—34no—
61C2×C32×Q8—24no—
62C2×C32×Q8C2×D314yes—
63C2×C32×Q8C22×C311yescharacteristic
64#3832—22no—
65#3832—22no—
66C2×C4⋊(C3⋊D3)—21no—
67C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C2×C4×C3⋊D3C22×C311yescharacteristic
70C2×#634C22×C311yescharacteristic
71C3×#239—68no—
72C3×#239—68no—
73C2×C3×D12—34no—
74C2×C3×D12—64no—
75C2×C3×C4×D3—34no—
76C2×C3×C4×D3—64no—
77C2×C3×Dic6—34no—
78C2×C32×C8—24no—
79C2×C32×C8C2×D314yes—
80C2×C32×C8C22×C311yescharacteristic
81C32×C4⋊C4—24no—
82C32×C4⋊C4C2×D314yes—
83C32×C4⋊C4C22×C311yescharacteristic
84#3756—22no—
85#3752—21no—
86C4⋊(C3⋊Dic3)—21no—
87C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
88#3744—22no—
89C3×#216—68no—
90C3×(C2×D3)⋊C4—64no—
91C3×C4⋊Dic3—64no—
92C2×C3×C3⋊C8—34no—
93C3×#204—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3×C3⋊D3—21no—
96C2×C3×C3⋊D3—22no—
97C2×C3×C3⋊D3—21no—
98C2×C3×C3⋊D3—21no—
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C33×C4—22no—
102C33×C4—41no—
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3×C3⋊Dic3—21no—
106C3×C3⋊Dic3—41no—
107C2×C3×#41—91no—
108C2×#239—34no—
109C2×C3×Q16—91no—
110C2×C3×SD16—91no—
111C3×C22⋊Q8—91no—
112C2×#216—34no—
113C2×#204—34no—
114#897—34no—
115C3×#103—24no—
116C3×#103C3×D314yescomplemented
117C3×#103C3⋊D311yescharacteristic, complemented
118C3×#102—91no—
119C3×C22⋊C8—91no—
120(C2×D3)⋊C8—34no—
121#809—34no—
122C22×C3⋊D3—21no—
123C22×C3⋊D3C3×D411yescharacteristic
124C22×C3×D3—34no—
125C22×C3×D3—64no—
126C32×Q8—21no—
127C32×Q8—28no—
128C32×Q8—24no—
129C32×Q8—44no—
130C32×Q8D1218yes—
131C32×Q8C3×D412yes—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—24no—
135C2×C32×C4—24no—
136C2×C32×C4—24no—
137C2×C32×C4—44no—
138C2×C32×C4—44no—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C23×C311yescharacteristic
141C2×C3⋊Dic3—21no—
142C2×C3⋊Dic3C3×D411yescharacteristic
143C4⋊(C3⋊D3)—24no—
144C4⋊(C3⋊D3)—21no—
145C4⋊(C3⋊D3)C3×D412yes—
146C4×C3⋊D3—24no—
147C4×C3⋊D3—21no—
148C4×C3⋊D3—21no—
149#634—21no—
150#634—21no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—64no—
153C3×D12—38no—
154C3×D12—616no—
155C3×D12—64no—
156C3×C4×D3—616no—
157C3×C4×D3—64no—
158C3×C4×D3—64no—
159C3×Dic6—64no—
160C3×Dic6—64no—
161C32×C8—21no—
162C32×C8—24no—
163C32×C8—44no—
164C3⋊(C3⋊C8)—21no—
165C3×C3⋊C8—64no—
166#452—91noSylow
167C2×C33#12011yescharacteristic
168C2×C33#13511yescharacteristic
169C2×C33#13711yescharacteristic
170C3×C3⋊D3—41no—
171C3×C3⋊D3—42no—
172C3×#41—182no—
173C3×#41—182no—
174C2×C3×Q8—24no—
175C2×C3×Q8—91no—
176C2×C3×Q8C2×C3×D314yes—
177C2×C3×Q8C2×C3⋊D311yescharacteristic
178C2×C3×D4—91no—
179C2×C3×D4—181no—
180C22×C3×C4—91no—
181C22×C3×C4—181no—
182#239—68no—
183#239—68no—
184C2×D12—34no—
185C2×D12—64no—
186C2×C4×D3—34no—
187C2×C4×D3—64no—
188C2×Dic6—34no—
189C3×Q16—182no—
190C3×SD16—182no—
191C2×C3×C8—24no—
192C2×C3×C8—91no—
193C2×C3×C8C2×C3×D314yes—
194C2×C3×C8C2×C3⋊D311yescharacteristic
195C3×C4⋊C4—24no—
196C3×C4⋊C4—181no—
197C3×C4⋊C4C2×C3×D314yes—
198C3×C4⋊C4C2×C3⋊D311yescharacteristic
199C3×C22⋊C4—181no—
200#216—68no—
201(C2×D3)⋊C4—64no—
202C4⋊Dic3—64no—
203C2×C3⋊C8—34no—
204#204—68no—
205C22×C32—24no—
206C22×C32D3×D414yes—
207C22×C32C2×C3×D411yescharacteristic
208C2×C3⋊D3—21no—
209C2×C3⋊D3—22no—
210C2×C3⋊D3—21no—
211C2×C3⋊D3—21no—
212C2×C3⋊D3—41no—
213C2×C3⋊D3—41no—
214C2×C3×D3—68no—
215C2×C3×D3—64no—
216C2×C3×D3—64no—
217C2×C3×D3—64no—
218C2×C3×D3—124no—
219C2×C3×D3—124no—
220C32×C4—22no—
221C32×C4—24no—
222C32×C4—24no—
223C32×C4—28no—
224C32×C4—48no—
225C32×C4—41no—
226C32×C4—44no—
227C32×C4—84no—
228C32×C4C2×D1214yes—
229C32×C4D3×D414yes—
230C32×C4C2×C3×D411yescharacteristic
231C32×C4C2×C3×D411yescharacteristic
232C3⋊Dic3—21no—
233C3⋊Dic3—41no—
234C3×Dic3—64no—
235C3×Dic3—124no—
236C2×#41—91no—
237C2×Q16—91no—
238C2×SD16—91no—
239C22⋊Q8—91no—
240#103C3×C3⋊D311yescharacteristic, complemented
241#102—91no—
242C22⋊C8—91no—
243C33#45211yescharacteristic, complemented, Sylow
244C23×C3—91no—
245C23×C3—181no—
246C22×D3—34no—
247C22×D3—64no—
248C3×Q8—24no—
249C3×Q8—21no—
250C3×Q8—28no—
251C3×Q8—44no—
252C3×Q8—181no—
253C3×Q8—181no—
254C3×Q8C3×D1218yes—
255C3×Q8C4⋊(C3⋊D3)12yes—
256C3×D4—92no—
257C3×D4—184no—
258C3×D4—181no—
259C2×C3×C4—24no—
260C2×C3×C4—21no—
261C2×C3×C4—24no—
262C2×C3×C4—21no—
263C2×C3×C4—24no—
264C2×C3×C4—44no—
265C2×C3×C4—44no—
266C2×C3×C4—91no—
267C2×C3×C4—184no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4—181no—
271C2×C3×C4C22×C3×D314yes—
272C2×C3×C4C22×C3⋊D311yescharacteristic
273C2×Dic3—34no—
274C2×Dic3—64no—
275D12—38no—
276D12—616no—
277D12—64no—
278C4×D3—616no—
279C4×D3—64no—
280C4×D3—64no—
281Dic6—64no—
282Dic6—64no—
283C3×C8—24no—
284C3×C8—21no—
285C3×C8—44no—
286C3×C8—181no—
287C3⋊C8—64no—
288C2×C32—24no—
289C2×C32—24no—
290C2×C32—24no—
291C2×C32#88214yes—
292C2×C32#91614yes—
293C2×C32#91814yes—
294C2×C32C3×#12011yescharacteristic
295C2×C32C3×#13511yescharacteristic
296C2×C32C3×#13711yescharacteristic
297C3⋊D3—41no—
298C3⋊D3—42no—
299C3×D3—124no—
300C3×D3—128no—
301#41—182no—
302#41—182no—
303C2×Q8—91no—
304C2×Q8C2×C3×C3⋊D311yescharacteristic
305C2×D4—91no—
306C2×D4—181no—
307C22×C4—91no—
308C22×C4—181no—
309Q16—182no—
310SD16—182no—
311C2×C8—91no—
312C2×C8C2×C3×C3⋊D311yescharacteristic
313C4⋊C4—181no—
314C4⋊C4C2×C3×C3⋊D311yescharacteristic
315C22⋊C4—181no—
316C22×C3—24no—
317C22×C3—182no—
318C22×C3—181no—
319C22×C3—181no—
320C22×C3—181no—
321C22×C3—361no—
322C22×C3—361no—
323C22×C3C3×D3×D414yes—
324C22×C3D4×C3⋊D311yescharacteristic
325C2×D3—68no—
326C2×D3—64no—
327C2×D3—64no—
328C2×D3—64no—
329C2×D3—124no—
330C2×D3—124no—
331C3×C4—28no—
332C3×C4—22no—
333C3×C4—24no—
334C3×C4—24no—
335C3×C4—48no—
336C3×C4—44no—
337C3×C4—41no—
338C3×C4—84no—
339C3×C4—181no—
340C3×C4—361no—
341C3×C4C2×C3×D1214yes—
342C3×C4C3×D3×D414yes—
343C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
344C3×C4D4×C3⋊D311yescharacteristic
345Dic3—64no—
346Dic3—124no—
347C32—24no—
348C32#480814yescomplemented
349C32C3×#45211yescharacteristic, complemented
350C23—91no—
351C23—181no—
352Q8—21no—
353Q8—181no—
354Q8—181no—
355Q8C3×C4⋊(C3⋊D3)12yes—
356D4—92no—
357D4—184no—
358D4—181no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—91no—
362C2×C4—184no—
363C2×C4—181no—
364C2×C4—181no—
365C2×C4—181no—
366C2×C4C22×C3×C3⋊D311yescharacteristic
367C8—21no—
368C8—181no—
369C2×C3—24no—
370C2×C3—24no—
371C2×C3—24no—
372C2×C3—361no—
373C2×C3—362no—
374C2×C3C3×#88214yes—
375C2×C3C3×#91614yes—
376C2×C3C3×#91814yes—
377C2×C3#6409111yescharacteristic
378C2×C3#6412511yescharacteristic
379C2×C3#6412711yescharacteristic
380D3—124no—
381D3—128no—
382C22—182no—
383C22—181no—
384C22—181no—
385C22—181no—
386C22—361no—
387C22—361no—
388C22C3×D4×C3⋊D311yescharacteristic
389C4—22no—
390C4—41no—
391C4—181no—
392C4—361no—
393C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
394C4C3×D4×C3⋊D311yescharacteristic
395C3—24no—
396C3C3×#480814yescomplemented
397C3#1059250711yescharacteristic, direct product factor
398C2—361no—
399C2—362no—
400C2C3×#6409111yescharacteristic
401C2C3×#6412511yescharacteristic
402C2C3×#6412711yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#103C2
socleC22×C33C2×D4
prime radicalC2×C3×#3832C2
hypercenterC3×#103C3⋊D3
hypocenterC32C3×#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×#103

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3756C2semidirect
3. C2×C3×#3744C2semidirect
4. C3×#64106C2semidirect
5. C33×#103C2semidirect
6. C3×#63647C2semidirect
7. C3×#63636C2semidirect
8. #10592507C3direct (N × H)
27. C2×#3756C2×C3semidirect
28. C2×#3744C2×C3semidirect
29. #64106C2×C3semidirect
34. C32×#103D3semidirect
35. C32×#103C2×C3semidirect
36. #63647C2×C3semidirect
37. #63636C2×C3semidirect
116. C3×#103C3×D3semidirect
117. C3×#103C3⋊D3semidirect
240. #103C3×C3⋊D3semidirect
243. C33#452semidirect
348. C32#4808semidirect
349. C32C3×#452semidirect
396. C3C3×#4808semidirect
397. C3#10592507direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C3×#37562
C2×C3×#37442
C3×#641062
C33×#1032
C3×#636472
C3×#636362
#105925073
C3×#48083