Finite Group Databasebeta

← C3×#10592526 #13162865

Subgroups — C3×#10592526 #13162865

Order 1728.

1052 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64367C211yesmaximal, characteristic, complemented
2C3×#64158C211yesmaximal, characteristic, complemented
3C3×#64117C211yesmaximal, characteristic, complemented
4C3×#64084C211yesmaximal, characteristic, complemented
5C33×#104C211yesmaximal, characteristic, complemented
6C3×#63651C211yesmaximal, characteristic, complemented
7C3×#63632C211yesmaximal, characteristic, complemented
8#10592526C311yesmaximal, characteristic, direct product factor
9C3×#4827—34nomaximal
10C33×#41C2211yescharacteristic
11C3×Q8×C3⋊D3—21no—
12C3×#3830—21no—
13C3×#3830—22no—
14C3×#3828—21no—
15C3×#3828C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#634—21no—
18C2×C3×#634C2211yescharacteristic
19C33×M16C2211yescharacteristic
20C33×C42C2211yescharacteristic
21C3×#3755—21no—
22C3×#3753—21no—
23C3×#3752—22no—
24C3×#3748C2211yescharacteristic
25C3×#3745—21no—
26C3×#3744—21no—
27#64367C2×C311yescharacteristic, complemented
28C3×#1010—34no—
29#64158C2×C311yescharacteristic, complemented
30#64117C2×C311yescharacteristic, complemented
31#64084C2×C311yescharacteristic, complemented
32C3×#949—34no—
33C3×#908—34no—
34C3×#875—34no—
35C32×#104—24no—
36C32×#104D314yescomplemented
37C32×#104C2×C311yescharacteristic, complemented
38#63651C2×C311yescharacteristic, complemented
39#63632C2×C311yescharacteristic, complemented
40C3×#824—34no—
41C3×#805—34no—
42C22×C3×C3⋊D3—21no—
43C33×Q8D411yescharacteristic
44C33×D4—21no—
45C33×D4D411yescharacteristic
46C2×C33×C4—21no—
47C2×C33×C4—21no—
48C2×C33×C4C2311yescharacteristic
49C3×#638—22no—
50C3×#638—21no—
51C2×C3×C3⋊Dic3—22no—
52C2×C3×C3⋊Dic3D412yes—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)D411yescharacteristic
56C3×C4×C3⋊D3—22no—
57C3×C4×C3⋊D3—21no—
58C3×#634—22no—
59C3×#634—21no—
60C3×#634—21no—
61C3×#634D411yescharacteristic
62C33×C8—21no—
63C3×C3⋊(C3⋊C8)—21no—
64C3×#454—91no—
65#4827—34no—
66C32×#41—24no—
67C32×#41C2×D314yes—
68C32×#41C22×C311yescharacteristic
69Q8×C3⋊D3—21no—
70#3830—21no—
71#3830—22no—
72#3828—21no—
73#3828C22×C311yescharacteristic
74C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
75C2×#634—21no—
76C2×#634C22×C311yescharacteristic
77C3×D3×Q8—64no—
78C3×#237—68no—
79C3×#237—64no—
80C3×#235—34no—
81C3×#235—64no—
82C2×C3×D12—34no—
83C2×C3×Dic6—34no—
84C2×C3×Dic6—64no—
85C32×M16—24no—
86C32×M16C2×D314yes—
87C32×M16C22×C311yescharacteristic
88C32×C42—24no—
89C32×C42C2×D314yes—
90C32×C42C22×C311yescharacteristic
91#3755—21no—
92#3753—21no—
93#3752—22no—
94#3748C22×C311yescharacteristic
95#3745—21no—
96#3744—21no—
97C3×#215—64no—
98C3×#213—64no—
99C3×(C2×D3)⋊C4—68no—
100C3×#208—34no—
101C3×D24—64no—
102C3×#204—64no—
103C22×C33—21no—
104C22×C33C2×D411yescharacteristic
105C2×C3×C3⋊D3—21no—
106C2×C3×C3⋊D3—42no—
107C2×C3×C3⋊D3—41no—
108C33×C4—21no—
109C33×C4—41no—
110C33×C4C2×D411yescharacteristic
111C33×C4C2×D411yescharacteristic
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3—21no—
114C3×2−1+4—91no—
115#1010—34no—
116C3×AGL(1,ℤ/8)—91no—
117C3×AGL(1,ℤ/8)—91no—
118C3×#124—91no—
119#949—34no—
120#908—34no—
121#875—34no—
122C3×#104—24no—
123C3×#104—91no—
124C3×#104C3×D314yescomplemented
125C3×#104C3⋊D311yescharacteristic, complemented
126C3×#101—91no—
127#824—34no—
128#805—34no—
129C22×C3⋊D3—21no—
130C22×C3×D3—64no—
131C32×Q8—24no—
132C32×Q8D1214yes—
133C32×Q8C3×D411yescharacteristic
134C32×D4—21no—
135C32×D4—24no—
136C32×D4—24no—
137C32×D4—44no—
138C32×D4D1214yes—
139C32×D4C3×D411yescharacteristic
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—24no—
143C2×C32×C4—24no—
144C2×C32×C4—24no—
145C2×C32×C4—44no—
146C2×C32×C4—44no—
147C2×C32×C4C22×D314yes—
148C2×C32×C4C23×C311yescharacteristic
149#638—22no—
150#638—21no—
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3C3×D412yes—
153C4⋊(C3⋊D3)—21no—
154C4⋊(C3⋊D3)—21no—
155C4⋊(C3⋊D3)C3×D411yescharacteristic
156C4×C3⋊D3—22no—
157C4×C3⋊D3—21no—
158#634—22no—
159#634—21no—
160#634—21no—
161#634C3×D411yescharacteristic
162C3×#67—68no—
163C3×#67—64no—
164C2×C3×Dic3—38no—
165C2×C3×Dic3—68no—
166C3×D12—34no—
167C3×D12—64no—
168C3×D12—64no—
169C3×C4×D3—68no—
170C3×C4×D3—64no—
171C3×Dic6—34no—
172C3×Dic6—68no—
173C3×Dic6—64no—
174C3×Dic6—64no—
175C32×C8—21no—
176C32×C8—24no—
177C32×C8—44no—
178C3⋊(C3⋊C8)—21no—
179C3×C3⋊C8—64no—
180#454—91noSylow
181C2×C33—21no—
182C2×C33—41no—
183C2×C33#12011yescharacteristic
184C3×C3⋊D3—41no—
185C3×C3⋊D3—81no—
186C3×#41—24no—
187C3×#41—91no—
188C3×#41—181no—
189C3×#41—182no—
190C3×#41—181no—
191C3×#41C2×C3×D314yes—
192C3×#41C2×C3⋊D311yescharacteristic
193C2×C3×Q8—91no—
194C2×C3×Q8—181no—
195C2×C3×Q8—181no—
196C2×C3×D4—91no—
197D3×Q8—64no—
198#237—68no—
199#237—64no—
200#235—34no—
201#235—64no—
202C2×D12—34no—
203C2×Dic6—34no—
204C2×Dic6—64no—
205C3×SD16—181no—
206C3×SD16—181no—
207C3×D8—181no—
208C3×D8—181no—
209C3×M16—24no—
210C3×M16—91no—
211C3×M16C2×C3×D314yes—
212C3×M16C2×C3⋊D311yescharacteristic
213C3×C22⋊C4—182no—
214C3×C42—24no—
215C3×C42C2×C3×D314yes—
216C3×C42C2×C3⋊D311yescharacteristic
217#215—64no—
218#213—64no—
219(C2×D3)⋊C4—68no—
220#208—34no—
221D24—64no—
222#204—64no—
223C22×C32—21no—
224C22×C32—24no—
225C22×C32—24no—
226C22×C32—44no—
227C22×C32D3×D414yes—
228C22×C32C2×C3×D411yescharacteristic
229C2×C3⋊D3—21no—
230C2×C3⋊D3—42no—
231C2×C3⋊D3—41no—
232C2×C3×D3—64no—
233C2×C3×D3—128no—
234C2×C3×D3—124no—
235C32×C4—21no—
236C32×C4—24no—
237C32×C4—24no—
238C32×C4—24no—
239C32×C4—44no—
240C32×C4—41no—
241C32×C4—44no—
242C32×C4—48no—
243C32×C4C2×D1214yes—
244C32×C4D3×D414yes—
245C32×C4C2×C3×D411yescharacteristic
246C32×C4C2×C3×D411yescharacteristic
247C3⋊Dic3—22no—
248C3⋊Dic3—21no—
249C3×Dic3—64no—
250C3×Dic3—68no—
2512−1+4—91no—
252AGL(1,ℤ/8)—91no—
253AGL(1,ℤ/8)—91no—
254#124—91no—
255#104—91no—
256#104C3×C3⋊D311yescharacteristic, complemented
257#101—91no—
258C33#45411yescharacteristic, complemented, Sylow
259C23×C3—181no—
260C22×D3—64no—
261C3×Q8—24no—
262C3×Q8—91no—
263C3×Q8—182no—
264C3×Q8—181no—
265C3×Q8—181no—
266C3×Q8C3×D1214yes—
267C3×Q8C4⋊(C3⋊D3)11yescharacteristic
268C3×D4—24no—
269C3×D4—21no—
270C3×D4—24no—
271C3×D4—44no—
272C3×D4—91no—
273C3×D4—182no—
274C3×D4—181no—
275C3×D4—181no—
276C3×D4—181no—
277C3×D4C3×D1214yes—
278C3×D4C4⋊(C3⋊D3)11yescharacteristic
279C2×C3×C4—24no—
280C2×C3×C4—21no—
281C2×C3×C4—24no—
282C2×C3×C4—21no—
283C2×C3×C4—24no—
284C2×C3×C4—44no—
285C2×C3×C4—44no—
286C2×C3×C4—92no—
287C2×C3×C4—182no—
288C2×C3×C4—182no—
289C2×C3×C4—181no—
290C2×C3×C4C22×C3×D314yes—
291C2×C3×C4C22×C3⋊D311yescharacteristic
292#67—68no—
293#67—64no—
294C2×Dic3—38no—
295C2×Dic3—68no—
296D12—34no—
297D12—64no—
298D12—64no—
299C4×D3—68no—
300C4×D3—64no—
301Dic6—34no—
302Dic6—68no—
303Dic6—64no—
304Dic6—64no—
305C3×C8—24no—
306C3×C8—21no—
307C3×C8—44no—
308C3×C8—181no—
309C3⋊C8—64no—
310C2×C32—21no—
311C2×C32—24no—
312C2×C32—24no—
313C2×C32—44no—
314C2×C32—41no—
315C2×C32—44no—
316C2×C32—84no—
317C2×C32#88214yes—
318C2×C32C3×#12011yescharacteristic
319C3⋊D3—41no—
320C3⋊D3—81no—
321C3×D3—124no—
322C3×D3—244no—
323#41—91no—
324#41—181no—
325#41—182no—
326#41—181no—
327#41C2×C3×C3⋊D311yescharacteristic
328C2×Q8—91no—
329C2×Q8—181no—
330C2×Q8—181no—
331C2×D4—91no—
332SD16—181no—
333SD16—181no—
334D8—181no—
335D8—181no—
336M16—91no—
337M16C2×C3×C3⋊D311yescharacteristic
338C22⋊C4—182no—
339C42C2×C3×C3⋊D311yescharacteristic
340C22×C3—24no—
341C22×C3—21no—
342C22×C3—24no—
343C22×C3—44no—
344C22×C3—181no—
345C22×C3—362no—
346C22×C3—361no—
347C22×C3C3×D3×D414yes—
348C22×C3D4×C3⋊D311yescharacteristic
349C2×D3—64no—
350C2×D3—128no—
351C2×D3—124no—
352C3×C4—24no—
353C3×C4—21no—
354C3×C4—24no—
355C3×C4—24no—
356C3×C4—44no—
357C3×C4—41no—
358C3×C4—44no—
359C3×C4—48no—
360C3×C4—182no—
361C3×C4—181no—
362C3×C4C2×C3×D1214yes—
363C3×C4C3×D3×D414yes—
364C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
365C3×C4D4×C3⋊D311yescharacteristic
366Dic3—64no—
367Dic3—68no—
368C32—24no—
369C32#482714yescomplemented
370C32C3×#45411yescharacteristic, complemented
371C23—181no—
372Q8—91no—
373Q8—182no—
374Q8—181no—
375Q8—181no—
376Q8C3×C4⋊(C3⋊D3)11yescharacteristic
377D4—21no—
378D4—91no—
379D4—182no—
380D4—181no—
381D4—181no—
382D4—181no—
383D4C3×C4⋊(C3⋊D3)11yescharacteristic
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—92no—
387C2×C4—182no—
388C2×C4—182no—
389C2×C4—181no—
390C2×C4C22×C3×C3⋊D311yescharacteristic
391C8—21no—
392C8—181no—
393C2×C3—21no—
394C2×C3—24no—
395C2×C3—24no—
396C2×C3—44no—
397C2×C3—41no—
398C2×C3—44no—
399C2×C3—84no—
400C2×C3—361no—
401C2×C3—721no—
402C2×C3C3×#88214yes—
403C2×C3#6409111yescharacteristic
404D3—124no—
405D3—244no—
406C22—21no—
407C22—181no—
408C22—362no—
409C22—361no—
410C22C3×D4×C3⋊D311yescharacteristic
411C4—21no—
412C4—41no—
413C4—182no—
414C4—181no—
415C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
416C4C3×D4×C3⋊D311yescharacteristic
417C3—24no—
418C3C3×#482714yescomplemented
419C3#1059252611yescharacteristic, direct product factor
420C2—21no—
421C2—41no—
422C2—361no—
423C2—721no—
424C2C3×#6409111yescharacteristic
subgroupG/H
centerC2×C3#64091
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#104C2
socleC2×C33#120
hypercenterC3×#104C3⋊D3
hypocenterC32C3×#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#104

Complements

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

NH ≅ G/Ntype
1. C3×#64367C2semidirect
2. C3×#64158C2semidirect
3. C3×#64117C2semidirect
4. C3×#64084C2semidirect
5. C33×#104C2semidirect
6. C3×#63651C2semidirect
7. C3×#63632C2semidirect
8. #10592526C3direct (N × H)
27. #64367C2×C3semidirect
29. #64158C2×C3semidirect
30. #64117C2×C3semidirect
31. #64084C2×C3semidirect
36. C32×#104D3semidirect
37. C32×#104C2×C3semidirect
38. #63651C2×C3semidirect
39. #63632C2×C3semidirect
124. C3×#104C3×D3semidirect
125. C3×#104C3⋊D3semidirect
256. #104C3×C3⋊D3semidirect
258. C33#454semidirect
369. C32#4827semidirect
370. C32C3×#454semidirect
418. C3C3×#4827semidirect
419. C3#10592526direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643672
C3×#641582
C3×#641172
C3×#640842
C33×#1042
C3×#636512
C3×#636322
#105925263
C3×#48273