Finite Group Databasebeta

← C3×#10590916 #13157882

Subgroups — C3×#10590916 #13157882

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
1C32×#904C211yesmaximal, characteristic
2C3×#63814C212yesmaximal, complemented
3C3×#63646C211yesmaximal, characteristic, complemented
4C32×#815C211yesmaximal, characteristic, complemented
5C3×#63569C212yesmaximal, complemented
6C3×#5197—31nomaximal
7C3×#4717—31nomaximal
8#10590916C311yesmaximal, characteristic, direct product factor
9C32×#235C2212yes—
10C2×C33×C8C2211yescharacteristic
11C3×#3748C2212yes—
12C32×C4×Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2211yescharacteristic
14C32×Dic12C411yescharacteristic, complemented
15C32×D24C411yescharacteristic, complemented
16C32×#204C412yescomplemented
17C3×#3711—22no—
18C3×D3×C3⋊C8—22no—
19C32×#135—31no—
20C3×#948—32no—
21C3×#904—21no—
22C3×#904D311yescharacteristic
23C3×#904C2×C311yescharacteristic
24C3×#901—32no—
25#63814C2×C312yescomplemented
26C32×#97—31no—
27#63646C2×C311yescharacteristic, complemented
28C3×#837—32no—
29C3×#819—31no—
30C3×#819—31no—
31C3×#819—61no—
32C3×#815—21no—
33C3×#815D311yescharacteristic, complemented
34C3×#815C2×C311yescharacteristic, complemented
35C3×#805—32no—
36#63569C2×C312yescomplemented
37C2×C33×C4C2311yescharacteristic
38C32×#67—22no—
39C2×C32×Dic3—21no—
40C32×D12C2×C412yes—
41C32×C4×D3—22no—
42C32×Dic6C2×C412yes—
43C33×C8C2×C411yescharacteristic
44C33×C8C2×C411yescharacteristic
45C3×C3⋊(C3⋊C8)—22no—
46C32×C3⋊C8D412yes—
47C3×#444—91no—
48#5197—31no—
49#4717—31no—
50C32×#41—32no—
51C3×#235—22no—
52C3×#235C2×D312yes—
53C3×#235C22×C312yes—
54C32×Q16—31no—
55C32×SD16—32no—
56C32×D8—31no—
57C2×C32×C8—21no—
58C2×C32×C8—21no—
59C2×C32×C8—21no—
60C2×C32×C8—31no—
61C2×C32×C8—41no—
62C2×C32×C8C2×D311yescharacteristic
63C2×C32×C8C2×D311yescharacteristic
64C2×C32×C8C22×C311yescharacteristic
65C32×C42—31no—
66#3748C22×C312yes—
67C3×C4×Dic3—21no—
68C3×C4×Dic3C2×D311yescharacteristic
69C3×C4×Dic3C22×C311yescharacteristic
70C3×#208—32no—
71C3×#208—32no—
72C3×#208—62no—
73C3×#208—62no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—62no—
76C2×C3×C3⋊C8C2×D311yescharacteristic
77C2×C3×C3⋊C8C22×C311yescharacteristic
78C3×Dic12—21no—
79C3×Dic12Dic311yescharacteristic, complemented
80C3×Dic12C3×C411yescharacteristic, complemented
81C3×D24—21no—
82C3×D24Dic311yescharacteristic, complemented
83C3×D24C3×C411yescharacteristic, complemented
84C3×#204—22no—
85C3×#204Dic312yescomplemented
86C3×#204C3×C412yescomplemented
87C3×ΓL(1,𝔽25)—62no—
88C3×D3×C8—62no—
89#3711—22no—
90D3×C3⋊C8—22no—
91C22×C33#4111yescharacteristic
92C2×C32×D3—22no—
93C33×C4C22×C411yescharacteristic
94C33×C4C2×D411yescharacteristic
95C32×Dic3—22no—
96C32×Dic3—41no—
97C3×#135—31no—
98C3×#135—31no—
99C3×#135—61no—
100C3×#131—92no—
101#948—32no—
102#904C3×D311yescharacteristic
103#901—32no—
104C3×#108—91no—
105C3×#104—92no—
106C3×#97—31no—
107C3×#97—31no—
108C3×#97—61no—
109#837—32no—
110#819—31no—
111#819—31no—
112#819—61no—
113#815C3×D311yescharacteristic, complemented
114#805—32no—
115C32×Q8—32no—
116C32×D4—32no—
117C32×D4—62no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—41no—
122C2×C32×C4—61no—
123C2×C32×C4—62no—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C23×C311yescharacteristic
127C3×#67—22no—
128C3×#67—22no—
129C3×#67—42no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C3×D12—22no—
134C3×D12C2×Dic312yes—
135C3×D12C2×C3×C412yes—
136C3×C4×D3—22no—
137C3×C4×D3—22no—
138C3×C4×D3—42no—
139C3×Dic6—22no—
140C3×Dic6C2×Dic312yes—
141C3×Dic6C2×C3×C412yes—
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×C8C4×D311yescharacteristic
152C32×C8C4×D311yescharacteristic
153C32×C8C2×Dic311yescharacteristic
154C32×C8C2×Dic311yescharacteristic
155C32×C8C2×C3×C411yescharacteristic
156C32×C8C2×C3×C411yescharacteristic
157C3⋊(C3⋊C8)—22no—
158C3×C3⋊C8—22no—
159C3×C3⋊C8—62no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—122no—
162C3×C3⋊C8D1212yes—
163C3×C3⋊C8C3×D412yes—
164#444—91noSylow
165C2×C33—21no—
166C2×C33C4×D411yescharacteristic
167C32×D3—42no—
168C3×#41—32no—
169C3×#41—32no—
170C3×#41—62no—
171#235C2×C3×D312yes—
172C3×Q16—31no—
173C3×Q16—31no—
174C3×Q16—61no—
175C3×SD16—32no—
176C3×SD16—32no—
177C3×SD16—62no—
178C3×D8—31no—
179C3×D8—31no—
180C3×D8—61no—
181C3×M16—92no—
182C3×M16—182no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—31no—
187C2×C3×C8—31no—
188C2×C3×C8—41no—
189C2×C3×C8—61no—
190C2×C3×C8—182no—
191C2×C3×C8D3211yescharacteristic
192C2×C3×C8C2×C3×D311yescharacteristic
193C2×C3×C8C2×C3×D311yescharacteristic
194C3×C42—31no—
195C3×C42—31no—
196C3×C42—61no—
197C4×Dic3C2×C3×D311yescharacteristic
198#208—32no—
199#208—32no—
200#208—62no—
201#208—62no—
202C2×C3⋊C8—62no—
203C2×C3⋊C8C2×C3×D311yescharacteristic
204Dic12C3×Dic311yescharacteristic, complemented
205D24C3×Dic311yescharacteristic, complemented
206#204C3×Dic312yescomplemented
207ΓL(1,𝔽25)—62no—
208D3×C8—62no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—41no—
213C22×C32—62no—
214C22×C32#23511yescharacteristic
215C22×C32#23711yescharacteristic
216C22×C32C3×#4111yescharacteristic
217C2×C3×D3—22no—
218C2×C3×D3—22no—
219C2×C3×D3—42no—
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—62no—
229C32×C4—121no—
230C32×C4C2×C4×D311yescharacteristic
231C32×C4C2×D1211yescharacteristic
232C32×C4D3×D411yescharacteristic
233C32×C4C22×Dic311yescharacteristic
234C32×C4C22×C3×C411yescharacteristic
235C32×C4C2×C3×D411yescharacteristic
236C3×Dic3—22no—
237C3×Dic3—22no—
238C3×Dic3—42no—
239C3×Dic3—42no—
240C3×Dic3—41no—
241C3×Dic3—41no—
242#135—31no—
243#131—92no—
244#108—91no—
245#104—92no—
246#97—31no—
247C33#44411yescharacteristic, complemented, Sylow
248C3×Q8—32no—
249C3×Q8—32no—
250C3×Q8—62no—
251C3×D4—32no—
252C3×D4—32no—
253C3×D4—62no—
254C3×D4—62no—
255C3×D4—62no—
256C3×D4—122no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—41no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—62no—
264C2×C3×C4—61no—
265C2×C3×C4—61no—
266C2×C3×C4—122no—
267C2×C3×C4C2×D3211yescharacteristic
268C2×C3×C4C22×C3×D311yescharacteristic
269C2×C3×C4C22×C3×D311yescharacteristic
270#67—22no—
271C2×Dic3—21no—
272D12C2×C3×Dic312yes—
273C4×D3—22no—
274Dic6C2×C3×Dic312yes—
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—182no—
287C3×C8D3×Dic311yescharacteristic
288C3×C8D3×Dic311yescharacteristic
289C3×C8C3×C4×D311yescharacteristic
290C3×C8C3×C4×D311yescharacteristic
291C3×C8C2×C3×Dic311yescharacteristic
292C3×C8C2×C3×Dic311yescharacteristic
293C3⋊C8—62no—
294C3⋊C8—62no—
295C3⋊C8—122no—
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×C32—122no—
309C2×C32C4×D1211yescharacteristic
310C2×C32D4×Dic311yescharacteristic
311C2×C32C3×C4×D411yescharacteristic
312C3×D3—42no—
313C3×D3—42no—
314C3×D3—82no—
315#41—32no—
316Q16—31no—
317SD16—32no—
318D8—31no—
319M16—92no—
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—62no—
330C22×C3—62no—
331C22×C3—122no—
332C22×C3#379511yescharacteristic
333C22×C3C3×#23511yescharacteristic
334C22×C3C3×#23711yescharacteristic
335C2×D3—22no—
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—62no—
345C3×C4—62no—
346C3×C4—122no—
347C3×C4—122no—
348C3×C4—121no—
349C3×C4—121no—
350C3×C4D3×D1211yescharacteristic
351C3×C4C2×D3×Dic311yescharacteristic
352C3×C4C2×C3×C4×D311yescharacteristic
353C3×C4C2×C3×D1211yescharacteristic
354C3×C4C3×D3×D411yescharacteristic
355C3×C4C22×C3×Dic311yescharacteristic
356Dic3—22no—
357Dic3—41no—
358C32—21no—
359C32—21no—
360C32—21no—
361C32—41no—
362C32#471711yescharacteristic, complemented
363C32#519711yescharacteristic, complemented
364C32C3×#44411yescharacteristic, complemented
365Q8—32no—
366D4—32no—
367D4—62no—
368C2×C4—62no—
369C2×C4—61no—
370C2×C4C2×C3×D3211yescharacteristic
371C8—32no—
372C8—182no—
373C8C3×D3×Dic311yescharacteristic
374C8C3×D3×Dic311yescharacteristic
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—122no—
387C2×C3—122no—
388C2×C3—242no—
389C2×C3Dic3×D1211yescharacteristic
390C2×C3C3×C4×D1211yescharacteristic
391C2×C3C3×D4×Dic311yescharacteristic
392D3—42no—
393C22—62no—
394C22C3×#379511yescharacteristic
395C4—62no—
396C4—121no—
397C4C3×D3×D1211yescharacteristic
398C4C2×C3×D3×Dic311yescharacteristic
399C3—21no—
400C3—21no—
401C3—21no—
402C3—41no—
403C3#1059091611yescharacteristic, direct product factor
404C3C3×#471711yescharacteristic, complemented
405C3C3×#519711yescharacteristic, complemented
406C2—21no—
407C2—122no—
408C2C3×Dic3×D1211yescharacteristic
subgroupG/H
centerC3×C4D3×D12
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC2×C33C4×D4
prime radicalC32×#904C2
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×#63814C2semidirect
3. C3×#63646C2semidirect
4. C32×#815C2semidirect
5. C3×#63569C2semidirect
8. #10590916C3direct (N × H)
14. C32×Dic12C4semidirect
15. C32×D24C4semidirect
16. C32×#204C4semidirect
25. #63814C2×C3semidirect
27. #63646C2×C3semidirect
33. C3×#815D3semidirect
34. C3×#815C2×C3semidirect
36. #63569C2×C3semidirect
79. C3×Dic12Dic3semidirect
80. C3×Dic12C3×C4semidirect
82. C3×D24Dic3semidirect
83. C3×D24C3×C4semidirect
85. C3×#204Dic3semidirect
86. C3×#204C3×C4semidirect
113. #815C3×D3semidirect
204. Dic12C3×Dic3semidirect
205. D24C3×Dic3semidirect
206. #204C3×Dic3semidirect
247. C33#444semidirect
362. C32#4717semidirect
363. C32#5197semidirect
364. C32C3×#444semidirect
403. C3#10590916direct (N × H)
404. C3C3×#4717semidirect
405. C3C3×#5197semidirect

Maximal subgroups

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

subgroupindex
C32×#9042
C3×#638142
C3×#636462
C32×#8152
C3×#635692
C3×#51973
C3×#47173
#105909163