Finite Group Databasebeta

← C3×#10592525 #13162864

Subgroups — C3×#10592525 #13162864

Order 1728.

1028 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
1C3×#64367C211yesmaximal, characteristic
2C3×#64160C211yesmaximal, characteristic, complemented
3C3×#64118C211yesmaximal, characteristic, complemented
4C3×#64078C211yesmaximal, characteristic, complemented
5C33×#104C211yesmaximal, characteristic, complemented
6C3×#63651C211yesmaximal, characteristic, complemented
7C3×#63632C211yesmaximal, characteristic, complemented
8#10592525C311yesmaximal, characteristic, direct product factor
9C3×#4826—34nomaximal
10C33×#41C2211yescharacteristic
11C3×Q8×C3⋊D3—21no—
12C3×#3830—21no—
13C3×#3830—22no—
14C3×#3828—21no—
15C3×#3828C2211yescharacteristic
16C2×C3×#634—21no—
17C2×C3×#634C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic
19C33×M16C2211yescharacteristic
20C33×C42C2211yescharacteristic
21C3×#3756—21no—
22C3×#3754—21no—
23C3×C4⋊(C3⋊Dic3)—22no—
24C3×#3748C2211yescharacteristic
25C3×#3746—21no—
26C3×#3744—21no—
27#64367C2×C311yescharacteristic
28C3×#1010—34no—
29#64160C2×C311yescharacteristic, complemented
30#64118C2×C311yescharacteristic, complemented
31#64078C2×C311yescharacteristic, complemented
32C3×#951—34no—
33C3×#909—34no—
34C3×#869—34no—
35C32×#104—24no—
36C32×#104D314yescomplemented
37C32×#104C2×C311yescharacteristic, complemented
38#63651C2×C311yescharacteristic, complemented
39#63632C2×C311yescharacteristic, complemented
40C3×#824—34no—
41C3×#805—34no—
42C33×Q8D411yescharacteristic
43C33×D4—21no—
44C33×D4D411yescharacteristic
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C3×#638—22no—
49C3×#638—21no—
50C2×C3×C3⋊Dic3—22no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3D412yes—
53C3×C4⋊(C3⋊D3)D411yescharacteristic
54C3×C4×C3⋊D3—22no—
55C3×C4×C3⋊D3—21no—
56C3×#634—22no—
57C3×#634—21no—
58C3×#634—21no—
59C3×#634—21no—
60C3×#634—21no—
61C3×#634D411yescharacteristic
62C33×C8—21no—
63C3×C3⋊(C3⋊C8)—21no—
64C3×#456—91no—
65#4826—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×#634—21no—
75C2×#634C22×C311yescharacteristic
76C2×#634C22×C311yescharacteristic
77C3×D3×Q8—64no—
78C3×#237—68no—
79C3×#237—64no—
80C3×#235—34no—
81C3×#235—64no—
82C2×C3×Dic6—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#3756—21no—
92#3754—21no—
93C4⋊(C3⋊Dic3)—22no—
94#3748C22×C311yescharacteristic
95#3746—21no—
96#3744—21no—
97C3×#216—64no—
98C3×#214—64no—
99C3×C4⋊Dic3—68no—
100C3×#208—34no—
101C3×Dic12—64no—
102C3×#204—64no—
103C22×C33—21no—
104C22×C33C2×D411yescharacteristic
105C2×C3×C3⋊D3—21no—
106C33×C4—22no—
107C33×C4—21no—
108C33×C4C2×D411yescharacteristic
109C33×C4C2×D411yescharacteristic
110C3×C3⋊Dic3—22no—
111C3×C3⋊Dic3—21no—
112C3×C3⋊Dic3—41no—
113C3×2−1+4—91no—
114#1010—34no—
115C3×#137—91no—
116C3×#137—91no—
117C3×C4⋊Q8—91no—
118#951—34no—
119#909—34no—
120#869—34no—
121C3×#104—24no—
122C3×#104—91no—
123C3×#104C3×D314yescomplemented
124C3×#104C3⋊D311yescharacteristic, complemented
125C3×#101—91no—
126#824—34no—
127#805—34no—
128C32×Q8—24no—
129C32×Q8D1214yes—
130C32×Q8C3×D411yescharacteristic
131C32×D4—21no—
132C32×D4—24no—
133C32×D4—24no—
134C32×D4—44no—
135C32×D4D1214yes—
136C32×D4C3×D411yescharacteristic
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—24no—
142C2×C32×C4—44no—
143C2×C32×C4—44no—
144C2×C32×C4C22×D314yes—
145C2×C32×C4C23×C311yescharacteristic
146#638—22no—
147#638—21no—
148C2×C3⋊Dic3—22no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3C3×D412yes—
151C4⋊(C3⋊D3)C3×D411yescharacteristic
152C4×C3⋊D3—22no—
153C4×C3⋊D3—21no—
154#634—22no—
155#634—21no—
156#634—21no—
157#634—21no—
158#634—21no—
159#634C3×D411yescharacteristic
160C3×#67—68no—
161C3×#67—64no—
162C2×C3×Dic3—38no—
163C2×C3×Dic3—68no—
164C2×C3×Dic3—64no—
165C3×D12—34no—
166C3×C4×D3—68no—
167C3×C4×D3—64no—
168C3×Dic6—34no—
169C3×Dic6—68no—
170C3×Dic6—64no—
171C3×Dic6—64no—
172C3×Dic6—64no—
173C3×Dic6—64no—
174C32×C8—21no—
175C32×C8—24no—
176C32×C8—44no—
177C3⋊(C3⋊C8)—21no—
178C3×C3⋊C8—64no—
179#456—91noSylow
180C2×C33—21no—
181C2×C33—41no—
182C2×C33#12011yescharacteristic
183C3×C3⋊D3—41no—
184C3×#41—24no—
185C3×#41—91no—
186C3×#41—181no—
187C3×#41—182no—
188C3×#41—181no—
189C3×#41C2×C3×D314yes—
190C3×#41C2×C3⋊D311yescharacteristic
191C2×C3×Q8—91no—
192C2×C3×Q8—91no—
193C2×C3×Q8—181no—
194C2×C3×Q8—181no—
195D3×Q8—64no—
196#237—68no—
197#237—64no—
198#235—34no—
199#235—64no—
200C2×Dic6—34no—
201C2×Dic6—34no—
202C2×Dic6—64no—
203C3×Q16—181no—
204C3×Q16—181no—
205C3×SD16—181no—
206C3×SD16—181no—
207C3×M16—24no—
208C3×M16—91no—
209C3×M16C2×C3×D314yes—
210C3×M16C2×C3⋊D311yescharacteristic
211C3×C4⋊C4—182no—
212C3×C42—24no—
213C3×C42C2×C3×D314yes—
214C3×C42C2×C3⋊D311yescharacteristic
215#216—64no—
216#214—64no—
217C4⋊Dic3—68no—
218#208—34no—
219Dic12—64no—
220#204—64no—
221C22×C32—21no—
222C22×C32—24no—
223C22×C32—24no—
224C22×C32—44no—
225C22×C32D3×D414yes—
226C22×C32C2×C3×D411yescharacteristic
227C2×C3⋊D3—21no—
228C2×C3×D3—64no—
229C32×C4—22no—
230C32×C4—21no—
231C32×C4—24no—
232C32×C4—24no—
233C32×C4—28no—
234C32×C4—24no—
235C32×C4—44no—
236C32×C4—48no—
237C32×C4C2×D1214yes—
238C32×C4D3×D414yes—
239C32×C4C2×C3×D411yescharacteristic
240C32×C4C2×C3×D411yescharacteristic
241C3⋊Dic3—22no—
242C3⋊Dic3—21no—
243C3⋊Dic3—41no—
244C3×Dic3—64no—
245C3×Dic3—68no—
246C3×Dic3—124no—
2472−1+4—91no—
248#137—91no—
249#137—91no—
250C4⋊Q8—91no—
251#104—91no—
252#104C3×C3⋊D311yescharacteristic, complemented
253#101—91no—
254C33#45611yescharacteristic, complemented, Sylow
255C3×Q8—24no—
256C3×Q8—91no—
257C3×Q8—182no—
258C3×Q8—181no—
259C3×Q8—181no—
260C3×Q8—181no—
261C3×Q8—181no—
262C3×Q8C3×D1214yes—
263C3×Q8C4⋊(C3⋊D3)11yescharacteristic
264C3×D4—24no—
265C3×D4—21no—
266C3×D4—24no—
267C3×D4—44no—
268C3×D4—91no—
269C3×D4—182no—
270C3×D4—181no—
271C3×D4C3×D1214yes—
272C3×D4C4⋊(C3⋊D3)11yescharacteristic
273C2×C3×C4—24no—
274C2×C3×C4—21no—
275C2×C3×C4—24no—
276C2×C3×C4—21no—
277C2×C3×C4—24no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—92no—
281C2×C3×C4—182no—
282C2×C3×C4—182no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4C22×C3×D314yes—
286C2×C3×C4C22×C3⋊D311yescharacteristic
287#67—68no—
288#67—64no—
289C2×Dic3—38no—
290C2×Dic3—68no—
291C2×Dic3—64no—
292D12—34no—
293C4×D3—68no—
294C4×D3—64no—
295Dic6—34no—
296Dic6—68no—
297Dic6—64no—
298Dic6—64no—
299Dic6—64no—
300Dic6—64no—
301C3×C8—24no—
302C3×C8—21no—
303C3×C8—44no—
304C3×C8—181no—
305C3⋊C8—64no—
306C2×C32—21no—
307C2×C32—24no—
308C2×C32—24no—
309C2×C32—44no—
310C2×C32—41no—
311C2×C32—44no—
312C2×C32—84no—
313C2×C32#88214yes—
314C2×C32C3×#12011yescharacteristic
315C3⋊D3—41no—
316C3×D3—124no—
317#41—91no—
318#41—181no—
319#41—182no—
320#41—181no—
321#41C2×C3×C3⋊D311yescharacteristic
322C2×Q8—91no—
323C2×Q8—91no—
324C2×Q8—181no—
325C2×Q8—181no—
326Q16—181no—
327Q16—181no—
328SD16—181no—
329SD16—181no—
330M16—91no—
331M16C2×C3×C3⋊D311yescharacteristic
332C4⋊C4—182no—
333C42C2×C3×C3⋊D311yescharacteristic
334C22×C3—24no—
335C22×C3—21no—
336C22×C3—24no—
337C22×C3—44no—
338C22×C3—181no—
339C22×C3C3×D3×D414yes—
340C22×C3D4×C3⋊D311yescharacteristic
341C2×D3—64no—
342C3×C4—22no—
343C3×C4—28no—
344C3×C4—24no—
345C3×C4—21no—
346C3×C4—24no—
347C3×C4—24no—
348C3×C4—44no—
349C3×C4—48no—
350C3×C4—182no—
351C3×C4—181no—
352C3×C4—361no—
353C3×C4C2×C3×D1214yes—
354C3×C4C3×D3×D414yes—
355C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
356C3×C4D4×C3⋊D311yescharacteristic
357Dic3—64no—
358Dic3—68no—
359Dic3—124no—
360C32—24no—
361C32#482614yescomplemented
362C32C3×#45611yescharacteristic, complemented
363Q8—91no—
364Q8—182no—
365Q8—181no—
366Q8—181no—
367Q8—181no—
368Q8—181no—
369Q8C3×C4⋊(C3⋊D3)11yescharacteristic
370D4—21no—
371D4—91no—
372D4—182no—
373D4—181no—
374D4C3×C4⋊(C3⋊D3)11yescharacteristic
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—92no—
378C2×C4—182no—
379C2×C4—182no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4C22×C3×C3⋊D311yescharacteristic
383C8—21no—
384C8—181no—
385C2×C3—21no—
386C2×C3—24no—
387C2×C3—24no—
388C2×C3—44no—
389C2×C3—41no—
390C2×C3—44no—
391C2×C3—84no—
392C2×C3—361no—
393C2×C3C3×#88214yes—
394C2×C3#6409111yescharacteristic
395D3—124no—
396C22—21no—
397C22—181no—
398C22C3×D4×C3⋊D311yescharacteristic
399C4—22no—
400C4—21no—
401C4—182no—
402C4—181no—
403C4—361no—
404C4C2×C3×C4⋊(C3⋊D3)11yescharacteristic
405C4C3×D4×C3⋊D311yescharacteristic
406C3—24no—
407C3C3×#482614yescomplemented
408C3#1059252511yescharacteristic, direct product factor
409C2—21no—
410C2—41no—
411C2—361no—
412C2C3×#6409111yescharacteristic
subgroupG/H
centerC2×C3#64091
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#104C2
socleC2×C33#120
prime radicalC3×#64367C2
hypercenterC3×#104C3⋊D3
hypocenterC32C3×#456
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
2. C3×#64160C2semidirect
3. C3×#64118C2semidirect
4. C3×#64078C2semidirect
5. C33×#104C2semidirect
6. C3×#63651C2semidirect
7. C3×#63632C2semidirect
8. #10592525C3direct (N × H)
29. #64160C2×C3semidirect
30. #64118C2×C3semidirect
31. #64078C2×C3semidirect
36. C32×#104D3semidirect
37. C32×#104C2×C3semidirect
38. #63651C2×C3semidirect
39. #63632C2×C3semidirect
123. C3×#104C3×D3semidirect
124. C3×#104C3⋊D3semidirect
252. #104C3×C3⋊D3semidirect
254. C33#456semidirect
361. C32#4826semidirect
362. C32C3×#456semidirect
407. C3C3×#4826semidirect
408. C3#10592525direct (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×#641602
C3×#641182
C3×#640782
C33×#1042
C3×#636512
C3×#636322
#105925253
C3×#48263