Finite Group Databasebeta

← C3×#10590917 #13157883

Subgroups — C3×#10590917 #13157883

Order 1728.

582 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64113C211yesmaximal, characteristic
2C3×#63817C212yesmaximal, complemented
3C32×#819C211yesmaximal, characteristic, complemented
4C32×C8×Dic3C211yesmaximal, characteristic, complemented
5C3×#63570C212yesmaximal, complemented
6C3×#4895—31nomaximal
7C3×#4700—31nomaximal
8#10590917C311yesmaximal, characteristic, direct product factor
9C3×#3828C2212yes—
10C2×C33×C8C2211yescharacteristic
11C3×#3746C411yescharacteristic, complemented
12C3×#3745C411yescharacteristic, complemented
13C3×#3744C412yescomplemented
14C32×C4×Dic3C2211yescharacteristic
15C32×#208C2212yes—
16C2×C32×C3⋊C8C2211yescharacteristic
17C3×#3712—22no—
18C3×#3710—22no—
19#64113C2×C311yescharacteristic
20C3×#907—32no—
21C3×#904—31no—
22C3×#904—31no—
23C3×#904—61no—
24C3×#901—32no—
25#63817C2×C312yescomplemented
26C32×#108—31no—
27C32×C4×C8—31no—
28C3×#825—32no—
29C3×#819—21no—
30C3×#819D311yescharacteristic, complemented
31C3×#819C2×C311yescharacteristic, complemented
32C3×C8×Dic3—21no—
33C3×C8×Dic3D311yescharacteristic, complemented
34C3×C8×Dic3C2×C311yescharacteristic, complemented
35C3×#805—32no—
36#63570C2×C312yescomplemented
37C2×C33×C4C2311yescharacteristic
38C3×#638—22no—
39C3×C4⋊(C3⋊D3)C2×C412yes—
40C3×C4×C3⋊D3—22no—
41C3×#634C2×C412yes—
42C2×C32×Dic3—21no—
43C33×C8C2×C411yescharacteristic
44C33×C8C2×C411yescharacteristic
45C32×C3⋊C8—22no—
46C32×C3⋊C8D411yescharacteristic
47C32×C3⋊C8D411yescharacteristic
48C3×#443—91no—
49#4895—31no—
50#4700—31no—
51#3828C22×C312yes—
52C3×#235—32no—
53C3×#235—32no—
54C3×#235—62no—
55C32×M16—32no—
56C2×C32×C8—21no—
57C2×C32×C8—21no—
58C2×C32×C8—21no—
59C2×C32×C8—31no—
60C2×C32×C8—41no—
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C2×D311yescharacteristic
63C2×C32×C8C22×C311yescharacteristic
64C32×C42—31no—
65#3746C3×C411yescharacteristic, complemented
66#3745C3×C411yescharacteristic, complemented
67#3744C3×C412yescomplemented
68C3×C4×Dic3—21no—
69C3×C4×Dic3C2×D311yescharacteristic
70C3×C4×Dic3C22×C311yescharacteristic
71C3×#208—22no—
72C3×#208C2×D312yes—
73C3×#208C22×C312yes—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8C2×D311yescharacteristic
76C2×C3×C3⋊C8C22×C311yescharacteristic
77C3×Dic12—31no—
78C3×Dic12—31no—
79C3×Dic12—61no—
80C3×D24—31no—
81C3×D24—31no—
82C3×D24—61no—
83C3×#204—32no—
84C3×#204—32no—
85C3×#204—62no—
86C3×ΓL(1,𝔽25)—62no—
87C3×ΓL(1,𝔽25)—62no—
88C3×D3×C8—62no—
89C3×D3×C8—62no—
90#3712—22no—
91#3710—22no—
92C22×C33#4111yescharacteristic
93C2×C3×C3⋊D3—22no—
94C33×C4C22×C411yescharacteristic
95C33×C4C2×D411yescharacteristic
96C3×C3⋊Dic3—22no—
97C32×Dic3—22no—
98C3×#135—91no—
99C3×#131—92no—
100#907—32no—
101#904—31no—
102#904—31no—
103#904—61no—
104#901—32no—
105C3×#108—31no—
106C3×#108—31no—
107C3×#108—61no—
108C3×#104—92no—
109C3×C4×C8—31no—
110C3×C4×C8—31no—
111C3×C4×C8—61no—
112#825—32no—
113#819C3×D311yescharacteristic, complemented
114C8×Dic3C3×D311yescharacteristic, complemented
115#805—32no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—41no—
120C2×C32×C4—61no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D311yescharacteristic
123C2×C32×C4C23×C311yescharacteristic
124#638—22no—
125C4⋊(C3⋊D3)C2×C3×C412yes—
126C4×C3⋊D3—22no—
127#634C2×C3×C412yes—
128C3×#67—62no—
129C3×#67—62no—
130C3×#67—122no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—21no—
134C3×D12—32no—
135C3×D12—32no—
136C3×D12—62no—
137C3×C4×D3—62no—
138C3×C4×D3—62no—
139C3×C4×D3—122no—
140C3×Dic6—32no—
141C3×Dic6—32no—
142C3×Dic6—62no—
143C32×C8—21no—
144C32×C8—21no—
145C32×C8—21no—
146C32×C8—21no—
147C32×C8—21no—
148C32×C8—21no—
149C32×C8—31no—
150C32×C8—31no—
151C32×C8—41no—
152C32×C8—41no—
153C32×C8—62no—
154C32×C8C4×D311yescharacteristic
155C32×C8C4×D311yescharacteristic
156C32×C8C4×D311yescharacteristic
157C32×C8C4×D311yescharacteristic
158C32×C8C2×C3×C411yescharacteristic
159C32×C8C2×C3×C411yescharacteristic
160C3×C3⋊C8—22no—
161C3×C3⋊C8—22no—
162C3×C3⋊C8—21no—
163C3×C3⋊C8—21no—
164C3×C3⋊C8—42no—
165C3×C3⋊C8D1211yescharacteristic
166C3×C3⋊C8D1211yescharacteristic
167C3×C3⋊C8C3×D411yescharacteristic
168C3×C3⋊C8C3×D411yescharacteristic
169#443—91noSylow
170C2×C33—21no—
171C2×C33C4×D411yescharacteristic
172C3×C3⋊D3—42no—
173C3×#41—92no—
174#235—32no—
175#235—32no—
176#235—62no—
177C3×Q16—91no—
178C3×SD16—92no—
179C3×D8—91no—
180C3×M16—32no—
181C3×M16—32no—
182C3×M16—62no—
183C3×M16—182no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—31no—
188C2×C3×C8—31no—
189C2×C3×C8—41no—
190C2×C3×C8—61no—
191C2×C3×C8—182no—
192C2×C3×C8D3211yescharacteristic
193C2×C3×C8C2×C3×D311yescharacteristic
194C2×C3×C8C2×C3×D311yescharacteristic
195C3×C42—31no—
196C3×C42—31no—
197C3×C42—61no—
198C4×Dic3C2×C3×D311yescharacteristic
199#208C2×C3×D312yes—
200C2×C3⋊C8C2×C3×D311yescharacteristic
201Dic12—31no—
202Dic12—31no—
203Dic12—61no—
204D24—31no—
205D24—31no—
206D24—61no—
207#204—32no—
208#204—32no—
209#204—62no—
210ΓL(1,𝔽25)—62no—
211ΓL(1,𝔽25)—62no—
212D3×C8—62no—
213D3×C8—62no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—41no—
218C22×C32#23511yescharacteristic
219C22×C32#23911yescharacteristic
220C22×C32C3×#4111yescharacteristic
221C2×C3⋊D3—22no—
222C2×C3×D3—62no—
223C2×C3×D3—62no—
224C2×C3×D3—122no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—21no—
230C32×C4—21no—
231C32×C4—41no—
232C32×C4—41no—
233C32×C4—62no—
234C32×C4C2×C4×D311yescharacteristic
235C32×C4C2×C4×D311yescharacteristic
236C32×C4C2×D1211yescharacteristic
237C32×C4D3×D411yescharacteristic
238C32×C4C22×C3×C411yescharacteristic
239C32×C4C2×C3×D411yescharacteristic
240C3⋊Dic3—22no—
241C3×Dic3—24no—
242C3×Dic3—22no—
243C3×Dic3—22no—
244C3×Dic3—62no—
245C3×Dic3—62no—
246C3×Dic3—122no—
247#135—91no—
248#131—92no—
249#108—31no—
250#104—92no—
251C4×C8—31no—
252C33#44311yescharacteristic, complemented, Sylow
253C3×Q8—92no—
254C3×D4—92no—
255C3×D4—182no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—41no—
260C2×C3×C4—62no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—182no—
264C2×C3×C4C2×D3211yescharacteristic
265C2×C3×C4C22×C3×D311yescharacteristic
266C2×C3×C4C22×C3×D311yescharacteristic
267#67—62no—
268#67—62no—
269#67—122no—
270C2×Dic3—21no—
271D12—32no—
272D12—32no—
273D12—62no—
274C4×D3—62no—
275C4×D3—62no—
276C4×D3—122no—
277Dic6—32no—
278Dic6—32no—
279Dic6—62no—
280C3×C8—21no—
281C3×C8—21no—
282C3×C8—21no—
283C3×C8—21no—
284C3×C8—21no—
285C3×C8—21no—
286C3×C8—31no—
287C3×C8—31no—
288C3×C8—31no—
289C3×C8—31no—
290C3×C8—41no—
291C3×C8—41no—
292C3×C8—61no—
293C3×C8—61no—
294C3×C8—62no—
295C3×C8—62no—
296C3×C8—122no—
297C3×C8#62411yescharacteristic
298C3×C8#62411yescharacteristic
299C3×C8C3×C4×D311yescharacteristic
300C3×C8C3×C4×D311yescharacteristic
301C3×C8C3×C4×D311yescharacteristic
302C3×C8C3×C4×D311yescharacteristic
303C3⋊C8—22no—
304C3⋊C8C3×D1211yescharacteristic
305C3⋊C8C3×D1211yescharacteristic
306C2×C32—21no—
307C2×C32—22no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—41no—
314C2×C32—41no—
315C2×C32—41no—
316C2×C32—42no—
317C2×C32C4×D1211yescharacteristic
318C2×C32#89311yescharacteristic
319C2×C32C3×C4×D411yescharacteristic
320C3⋊D3—42no—
321C3×D3—122no—
322C3×D3—122no—
323C3×D3—242no—
324#41—92no—
325Q16—91no—
326SD16—92no—
327D8—91no—
328M16—32no—
329M16—182no—
330C2×C8—31no—
331C2×C8—182no—
332C2×C8C3×D3211yescharacteristic
333C42—31no—
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—41no—
338C22×C3—182no—
339C22×C3#379911yescharacteristic
340C22×C3C3×#23511yescharacteristic
341C22×C3C3×#23911yescharacteristic
342C2×D3—62no—
343C2×D3—62no—
344C2×D3—122no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—41no—
352C3×C4—41no—
353C3×C4—64no—
354C3×C4—62no—
355C3×C4—62no—
356C3×C4—182no—
357C3×C4D3×D1211yescharacteristic
358C3×C4C2×#62411yescharacteristic
359C3×C4C2×C3×C4×D311yescharacteristic
360C3×C4C2×C3×C4×D311yescharacteristic
361C3×C4C2×C3×D1211yescharacteristic
362C3×C4C3×D3×D411yescharacteristic
363Dic3—22no—
364Dic3—62no—
365Dic3—62no—
366Dic3—122no—
367C32—21no—
368C32—21no—
369C32—21no—
370C32—41no—
371C32#470011yescharacteristic, complemented
372C32#489511yescharacteristic, complemented
373C32C3×#44311yescharacteristic, complemented
374Q8—92no—
375D4—92no—
376D4—182no—
377C2×C4—61no—
378C2×C4—182no—
379C2×C4C2×C3×D3211yescharacteristic
380C8—31no—
381C8—31no—
382C8—62no—
383C8C3×#62411yescharacteristic
384C8C3×#62411yescharacteristic
385C2×C3—22no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—42no—
395C2×C3—41no—
396C2×C3—362no—
397C2×C3#6389411yescharacteristic
398C2×C3C3×C4×D1211yescharacteristic
399C2×C3C3×#89311yescharacteristic
400D3—122no—
401D3—122no—
402D3—242no—
403C22—182no—
404C22C3×#379911yescharacteristic
405C4—62no—
406C4—182no—
407C4C3×D3×D1211yescharacteristic
408C4C2×C3×#62411yescharacteristic
409C3—21no—
410C3—21no—
411C3—21no—
412C3—41no—
413C3#1059091711yescharacteristic, direct product factor
414C3C3×#470011yescharacteristic, complemented
415C3C3×#489511yescharacteristic, complemented
416C2—21no—
417C2—362no—
418C2C3×#6389411yescharacteristic
subgroupG/H
centerC3×C4D3×D12
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC2×C33C4×D4
prime radicalC3×#64113C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×#443
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×#63817C2semidirect
3. C32×#819C2semidirect
4. C32×C8×Dic3C2semidirect
5. C3×#63570C2semidirect
8. #10590917C3direct (N × H)
11. C3×#3746C4semidirect
12. C3×#3745C4semidirect
13. C3×#3744C4semidirect
25. #63817C2×C3semidirect
30. C3×#819D3semidirect
31. C3×#819C2×C3semidirect
33. C3×C8×Dic3D3semidirect
34. C3×C8×Dic3C2×C3semidirect
36. #63570C2×C3semidirect
65. #3746C3×C4semidirect
66. #3745C3×C4semidirect
67. #3744C3×C4semidirect
113. #819C3×D3semidirect
114. C8×Dic3C3×D3semidirect
252. C33#443semidirect
371. C32#4700semidirect
372. C32#4895semidirect
373. C32C3×#443semidirect
413. C3#10590917direct (N × H)
414. C3C3×#4700semidirect
415. C3C3×#4895semidirect

Maximal subgroups

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

subgroupindex
C3×#641132
C3×#638172
C32×#8192
C32×C8×Dic32
C3×#635702
C3×#48953
C3×#47003
#105909173