Finite Group Databasebeta

← C3×#10590918 #13157884

Subgroups — C3×#10590918 #13157884

Order 1728.

570 conjugacy classes of subgroups.

The subgroup lattice has 410 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×#815C211yesmaximal, characteristic, complemented
5C3×#63570C212yesmaximal, complemented
6C3×#4894—31nomaximal
7C3×#4717—31nomaximal
8#10590918C311yesmaximal, 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×#97—31no—
28C3×#825—32no—
29C3×#819—21no—
30C3×#819D311yescharacteristic, complemented
31C3×#819C2×C311yescharacteristic, complemented
32C3×#815—21no—
33C3×#815D311yescharacteristic, complemented
34C3×#815C2×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⋊C8D412yes—
47C3×#444—91no—
48#4894—31no—
49#4717—31no—
50#3828C22×C312yes—
51C3×#235—32no—
52C3×#235—32no—
53C3×#235—62no—
54C32×M16—32no—
55C2×C32×C8—21no—
56C2×C32×C8—21no—
57C2×C32×C8—21no—
58C2×C32×C8—31no—
59C2×C32×C8—41no—
60C2×C32×C8C2×D311yescharacteristic
61C2×C32×C8C2×D311yescharacteristic
62C2×C32×C8C22×C311yescharacteristic
63C32×C42—31no—
64#3746C3×C411yescharacteristic, complemented
65#3745C3×C411yescharacteristic, complemented
66#3744C3×C412yescomplemented
67C3×C4×Dic3—21no—
68C3×C4×Dic3C2×D311yescharacteristic
69C3×C4×Dic3C22×C311yescharacteristic
70C3×#208—22no—
71C3×#208C2×D312yes—
72C3×#208C22×C312yes—
73C2×C3×C3⋊C8—21no—
74C2×C3×C3⋊C8C2×D311yescharacteristic
75C2×C3×C3⋊C8C22×C311yescharacteristic
76C3×Dic12—31no—
77C3×Dic12—31no—
78C3×Dic12—61no—
79C3×D24—31no—
80C3×D24—31no—
81C3×D24—61no—
82C3×#204—32no—
83C3×#204—32no—
84C3×#204—62no—
85C3×ΓL(1,𝔽25)—62no—
86C3×ΓL(1,𝔽25)—62no—
87C3×D3×C8—62no—
88C3×D3×C8—62no—
89#3712—22no—
90#3710—22no—
91C22×C33#4111yescharacteristic
92C2×C3×C3⋊D3—22no—
93C33×C4C22×C411yescharacteristic
94C33×C4C2×D411yescharacteristic
95C3×C3⋊Dic3—22no—
96C32×Dic3—41no—
97C3×#135—91no—
98C3×#131—92no—
99#907—32no—
100#904—31no—
101#904—31no—
102#904—61no—
103#901—32no—
104C3×#108—31no—
105C3×#108—31no—
106C3×#108—61no—
107C3×#104—92no—
108C3×#97—31no—
109C3×#97—31no—
110C3×#97—61no—
111#825—32no—
112#819C3×D311yescharacteristic, complemented
113#815C3×D311yescharacteristic, complemented
114#805—32no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—41no—
119C2×C32×C4—61no—
120C2×C32×C4C22×D311yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C23×C311yescharacteristic
123#638—22no—
124C4⋊(C3⋊D3)C2×C3×C412yes—
125C4×C3⋊D3—22no—
126#634C2×C3×C412yes—
127C3×#67—62no—
128C3×#67—62no—
129C3×#67—122no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C3×D12—32no—
134C3×D12—32no—
135C3×D12—62no—
136C3×C4×D3—62no—
137C3×C4×D3—62no—
138C3×C4×D3—122no—
139C3×Dic6—32no—
140C3×Dic6—32no—
141C3×Dic6—62no—
142C32×C8—21no—
143C32×C8—21no—
144C32×C8—21no—
145C32×C8—21no—
146C32×C8—21no—
147C32×C8—21no—
148C32×C8—32no—
149C32×C8—41no—
150C32×C8—41no—
151C32×C8—62no—
152C32×C8C4×D311yescharacteristic
153C32×C8C4×D311yescharacteristic
154C32×C8C4×D311yescharacteristic
155C32×C8C4×D311yescharacteristic
156C32×C8C2×C3×C411yescharacteristic
157C32×C8C2×C3×C411yescharacteristic
158C3×C3⋊C8—22no—
159C3×C3⋊C8—22no—
160C3×C3⋊C8—22no—
161C3×C3⋊C8—42no—
162C3×C3⋊C8D1212yes—
163C3×C3⋊C8C3×D412yes—
164#444—91noSylow
165C2×C33—21no—
166C2×C33C4×D411yescharacteristic
167C3×C3⋊D3—42no—
168C3×#41—92no—
169#235—32no—
170#235—32no—
171#235—62no—
172C3×Q16—91no—
173C3×SD16—92no—
174C3×D8—91no—
175C3×M16—32no—
176C3×M16—32no—
177C3×M16—62no—
178C3×M16—182no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—31no—
183C2×C3×C8—31no—
184C2×C3×C8—41no—
185C2×C3×C8—61no—
186C2×C3×C8—182no—
187C2×C3×C8D3211yescharacteristic
188C2×C3×C8C2×C3×D311yescharacteristic
189C2×C3×C8C2×C3×D311yescharacteristic
190C3×C42—31no—
191C3×C42—31no—
192C3×C42—61no—
193C4×Dic3C2×C3×D311yescharacteristic
194#208C2×C3×D312yes—
195C2×C3⋊C8C2×C3×D311yescharacteristic
196Dic12—31no—
197Dic12—31no—
198Dic12—61no—
199D24—31no—
200D24—31no—
201D24—61no—
202#204—32no—
203#204—32no—
204#204—62no—
205ΓL(1,𝔽25)—62no—
206ΓL(1,𝔽25)—62no—
207D3×C8—62no—
208D3×C8—62no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—41no—
213C22×C32#23511yescharacteristic
214C22×C32#23911yescharacteristic
215C22×C32C3×#4111yescharacteristic
216C2×C3⋊D3—22no—
217C2×C3×D3—62no—
218C2×C3×D3—62no—
219C2×C3×D3—122no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—41no—
227C32×C4—41no—
228C32×C4—121no—
229C32×C4C2×C4×D311yescharacteristic
230C32×C4C2×C4×D311yescharacteristic
231C32×C4C2×D1211yescharacteristic
232C32×C4D3×D411yescharacteristic
233C32×C4C22×C3×C411yescharacteristic
234C32×C4C2×C3×D411yescharacteristic
235C3⋊Dic3—22no—
236C3×Dic3—42no—
237C3×Dic3—41no—
238C3×Dic3—41no—
239C3×Dic3—62no—
240C3×Dic3—62no—
241C3×Dic3—122no—
242#135—91no—
243#131—92no—
244#108—31no—
245#104—92no—
246#97—31no—
247C33#44411yescharacteristic, complemented, Sylow
248C3×Q8—92no—
249C3×D4—92no—
250C3×D4—182no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—41no—
255C2×C3×C4—62no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—182no—
259C2×C3×C4C2×D3211yescharacteristic
260C2×C3×C4C22×C3×D311yescharacteristic
261C2×C3×C4C22×C3×D311yescharacteristic
262#67—62no—
263#67—62no—
264#67—122no—
265C2×Dic3—21no—
266D12—32no—
267D12—32no—
268D12—62no—
269C4×D3—62no—
270C4×D3—62no—
271C4×D3—122no—
272Dic6—32no—
273Dic6—32no—
274Dic6—62no—
275C3×C8—21no—
276C3×C8—21no—
277C3×C8—21no—
278C3×C8—21no—
279C3×C8—21no—
280C3×C8—21no—
281C3×C8—32no—
282C3×C8—32no—
283C3×C8—41no—
284C3×C8—41no—
285C3×C8—62no—
286C3×C8—62no—
287C3×C8—62no—
288C3×C8—122no—
289C3×C8#62411yescharacteristic
290C3×C8#62411yescharacteristic
291C3×C8C3×C4×D311yescharacteristic
292C3×C8C3×C4×D311yescharacteristic
293C3×C8C3×C4×D311yescharacteristic
294C3×C8C3×C4×D311yescharacteristic
295C3⋊C8—22no—
296C3⋊C8C3×D1212yes—
297C2×C32—21no—
298C2×C32—22no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—41no—
305C2×C32—41no—
306C2×C32—41no—
307C2×C32—42no—
308C2×C32C4×D1211yescharacteristic
309C2×C32#89311yescharacteristic
310C2×C32C3×C4×D411yescharacteristic
311C3⋊D3—42no—
312C3×D3—122no—
313C3×D3—122no—
314C3×D3—242no—
315#41—92no—
316Q16—91no—
317SD16—92no—
318D8—91no—
319M16—32no—
320M16—182no—
321C2×C8—31no—
322C2×C8—182no—
323C2×C8C3×D3211yescharacteristic
324C42—31no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—41no—
329C22×C3—182no—
330C22×C3#379911yescharacteristic
331C22×C3C3×#23511yescharacteristic
332C22×C3C3×#23911yescharacteristic
333C2×D3—62no—
334C2×D3—62no—
335C2×D3—122no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—122no—
345C3×C4—121no—
346C3×C4—121no—
347C3×C4—182no—
348C3×C4D3×D1211yescharacteristic
349C3×C4C2×#62411yescharacteristic
350C3×C4C2×C3×C4×D311yescharacteristic
351C3×C4C2×C3×C4×D311yescharacteristic
352C3×C4C2×C3×D1211yescharacteristic
353C3×C4C3×D3×D411yescharacteristic
354Dic3—41no—
355Dic3—62no—
356Dic3—62no—
357Dic3—122no—
358C32—21no—
359C32—21no—
360C32—21no—
361C32—41no—
362C32#471711yescharacteristic, complemented
363C32#489411yescharacteristic, complemented
364C32C3×#44411yescharacteristic, complemented
365Q8—92no—
366D4—92no—
367D4—182no—
368C2×C4—61no—
369C2×C4—182no—
370C2×C4C2×C3×D3211yescharacteristic
371C8—32no—
372C8—62no—
373C8C3×#62411yescharacteristic
374C8C3×#62411yescharacteristic
375C2×C3—22no—
376C2×C3—21no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—42no—
385C2×C3—41no—
386C2×C3—362no—
387C2×C3#6389411yescharacteristic
388C2×C3C3×C4×D1211yescharacteristic
389C2×C3C3×#89311yescharacteristic
390D3—122no—
391D3—122no—
392D3—242no—
393C22—182no—
394C22C3×#379911yescharacteristic
395C4—121no—
396C4—182no—
397C4C3×D3×D1211yescharacteristic
398C4C2×C3×#62411yescharacteristic
399C3—21no—
400C3—21no—
401C3—21no—
402C3—41no—
403C3#1059091811yescharacteristic, direct product factor
404C3C3×#471711yescharacteristic, complemented
405C3C3×#489411yescharacteristic, complemented
406C2—21no—
407C2—362no—
408C2C3×#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×#444
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×#815C2semidirect
5. C3×#63570C2semidirect
8. #10590918C3direct (N × H)
11. C3×#3746C4semidirect
12. C3×#3745C4semidirect
13. C3×#3744C4semidirect
25. #63817C2×C3semidirect
30. C3×#819D3semidirect
31. C3×#819C2×C3semidirect
33. C3×#815D3semidirect
34. C3×#815C2×C3semidirect
36. #63570C2×C3semidirect
64. #3746C3×C4semidirect
65. #3745C3×C4semidirect
66. #3744C3×C4semidirect
112. #819C3×D3semidirect
113. #815C3×D3semidirect
247. C33#444semidirect
362. C32#4717semidirect
363. C32#4894semidirect
364. C32C3×#444semidirect
403. C3#10590918direct (N × H)
404. C3C3×#4717semidirect
405. C3C3×#4894semidirect

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×#8152
C3×#635702
C3×#48943
C3×#47173
#105909183