Finite Group Databasebeta

← C3×#10588505 #13141969

Subgroups — C3×#10588505 #13141969

Order 1728.

500 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63715C211yesmaximal, characteristic
2C3×#63511C211yesmaximal, characteristic, complemented
3C3×#63471C211yesmaximal, characteristic, complemented
4C3×#63437C211yesmaximal, characteristic, complemented
5C3×C9×#104C211yesmaximal, characteristic, complemented
6C3×#63383C211yesmaximal, characteristic, complemented
7C3×#63364C211yesmaximal, characteristic, complemented
8C3×#4825—31nomaximal
9#10588505C311yesmaximal, characteristic, direct product factor
10C3×C9×#41C2211yescharacteristic
11C3×#3701—21no—
12C3×D4×D9—22no—
13C3×D4×D9—21no—
14C3×#3697—21no—
15C3×#3697C2211yescharacteristic
16C2×C3×D36—21no—
17C2×C3×D36C2211yescharacteristic
18C2×C3×Dic18C2211yescharacteristic, complemented
19C3×C9×M16C2211yescharacteristic, complemented
20C3×C42×C9C2211yescharacteristic, complemented
21C3×#3674—21no—
22C3×#3672—21no—
23C3×(C2×D9)⋊C4—22no—
24C3×#3667C2211yescharacteristic, complemented
25C3×#3664—21no—
26C3×Dic36—21no—
27C3×#1009—31no—
28C3×#951—31no—
29C3×#909—31no—
30C3×#875—31no—
31#63715C2×C311yescharacteristic
32C32×#104D311yescharacteristic
33C3×#823—31no—
34C3×#805—31no—
35#63511C2×C311yescharacteristic, complemented
36#63471C2×C311yescharacteristic, complemented
37#63437C2×C311yescharacteristic, complemented
38C9×#104—21no—
39C9×#104C2×C311yescharacteristic, complemented
40#63383C2×C311yescharacteristic, complemented
41#63364C2×C311yescharacteristic, complemented
42C22×C3×D9—22no—
43C22×C3×D9D412yes—
44C3×Q8×C9D411yescharacteristic
45C3×D4×C9—21no—
46C3×D4×C9D411yescharacteristic
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3×C4×C9C2311yescharacteristic
50C3×#611—22no—
51C3×#611—21no—
52C2×C3×Dic9—21no—
53C3×D36—22no—
54C3×D36—21no—
55C3×D36—21no—
56C3×D36D411yescharacteristic
57C3×C4×D9—22no—
58C3×C4×D9—21no—
59C3×Dic18—21no—
60C3×Dic18—21no—
61C3×Dic18D411yescharacteristic
62C3×C8×C9—21no—
63C3×C9⋊C8—21no—
64C3×#455—91no—
65#4825—31no—
66C32×#41C2×D311yescharacteristic
67C3×#239—61no—
68C3×D3×D4—62no—
69C3×D3×D4—61no—
70C3×#235—31no—
71C3×#235—61no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×Dic6—31no—
75C32×M16C2×D311yescharacteristic
76C32×C42C2×D311yescharacteristic
77C3×#216—61no—
78C3×#214—61no—
79C3×(C2×D3)⋊C4—62no—
80C3×#208—31no—
81C3×Dic12—61no—
82C3×#204—61no—
83C9×#41—21no—
84C9×#41C22×C311yescharacteristic
85#3701—21no—
86D4×D9—22no—
87D4×D9—21no—
88#3697—21no—
89#3697C22×C311yescharacteristic
90C2×D36—21no—
91C2×D36C22×C311yescharacteristic
92C2×Dic18C22×C311yescharacteristic, complemented
93C9×M16—21no—
94C9×M16C22×C311yescharacteristic, complemented
95C42×C9—21no—
96C42×C9C22×C311yescharacteristic, complemented
97#3674—21no—
98#3672—21no—
99(C2×D9)⋊C4—22no—
100#3667C22×C311yescharacteristic, complemented
101#3664—21no—
102Dic36—21no—
103C22×C3×C9—21no—
104C22×C3×C9C2×D411yescharacteristic
105C2×C3×D9—22no—
106C2×C3×D9—21no—
107C2×C3×D9—42no—
108C2×C3×D9—82no—
109C3×C4×C9—21no—
110C3×C4×C9—41no—
111C3×C4×C9C2×D411yescharacteristic
112C3×C4×C9C2×D411yescharacteristic
113C3×Dic9—21no—
114C3×Dic9—41no—
115C3×H(5,𝔽2)—91no—
116#1009—31no—
117C3×#137—91no—
118C3×#137—91no—
119C3×#124—91no—
120#951—31no—
121#909—31no—
122#875—31no—
123C3×#104—21no—
124C3×#104—91no—
125C3×#104D911yescharacteristic, complemented
126C3×#104C3×D311yescharacteristic
127C3×#100—91no—
128#823—31no—
129#805—31no—
130C22×C3×D3—32no—
131C22×C3×D3—62no—
132C32×Q8D1211yescharacteristic
133C32×D4—21no—
134C32×D4D1211yescharacteristic
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4C22×D311yescharacteristic
138C3×#67—62no—
139C3×#67—61no—
140C2×C3×Dic3—61no—
141C3×D12—31no—
142C3×D12—62no—
143C3×D12—61no—
144C3×D12—61no—
145C3×C4×D3—62no—
146C3×C4×D3—61no—
147C3×Dic6—31no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C22×D9—22no—
151C22×D9C3×D412yes—
152C32×C8—21no—
153C3×C3⋊C8—61no—
154Q8×C9—21no—
155Q8×C9C3×D411yescharacteristic
156D4×C9—21no—
157D4×C9—21no—
158D4×C9—41no—
159D4×C9C3×D411yescharacteristic
160C2×C4×C9—21no—
161C2×C4×C9—21no—
162C2×C4×C9—21no—
163C2×C4×C9—41no—
164C2×C4×C9—41no—
165C2×C4×C9C23×C311yescharacteristic
166#611—22no—
167#611—21no—
168C2×Dic9—21no—
169D36—22no—
170D36—21no—
171D36—21no—
172D36C3×D411yescharacteristic
173C4×D9—22no—
174C4×D9—21no—
175Dic18—21no—
176Dic18—21no—
177Dic18C3×D411yescharacteristic
178C8×C9—21no—
179C8×C9—41no—
180C9⋊C8—21no—
181#455—91noSylow
182C2×C3×C9—21no—
183C2×C3×C9—41no—
184C2×C3×C9#12011yescharacteristic
185C3×D9—42no—
186C3×D9—41no—
187C3×#41—21no—
188C3×#41—91no—
189C3×#41—181no—
190C3×#41—181no—
191C3×#41C2×D911yescharacteristic
192C3×#41C2×C3×D311yescharacteristic
193C2×C3×Q8—91no—
194C2×C3×D4—91no—
195C2×C3×D4—181no—
196C2×C3×D4—182no—
197C2×C3×D4—181no—
198#239—61no—
199D3×D4—62no—
200D3×D4—61no—
201#235—31no—
202#235—61no—
203C2×D12—31no—
204C2×D12—61no—
205C2×Dic6—31no—
206C3×Q16—181no—
207C3×Q16—181no—
208C3×SD16—181no—
209C3×SD16—181no—
210C3×M16—21no—
211C3×M16—91no—
212C3×M16C2×D911yescharacteristic, complemented
213C3×M16C2×C3×D311yescharacteristic
214C3×C22⋊C4—182no—
215C3×C42—21no—
216C3×C42C2×D911yescharacteristic, complemented
217C3×C42C2×C3×D311yescharacteristic
218#216—61no—
219#214—61no—
220(C2×D3)⋊C4—62no—
221#208—31no—
222Dic12—61no—
223#204—61no—
224C22×C32—21no—
225C22×C32D3×D411yescharacteristic
226C2×C3×D3—61no—
227C2×C3×D3—62no—
228C2×C3×D3—122no—
229C2×C3×D3—242no—
230C32×C4—21no—
231C32×C4—41no—
232C32×C4C2×D1211yescharacteristic
233C32×C4D3×D411yescharacteristic
234C3×Dic3—61no—
235C3×Dic3—121no—
236C22×C9—21no—
237C22×C9—21no—
238C22×C9—41no—
239C22×C9C2×C3×D411yescharacteristic
240C2×D9—22no—
241C2×D9—21no—
242C2×D9—42no—
243C2×D9—82no—
244C4×C9—21no—
245C4×C9—21no—
246C4×C9—21no—
247C4×C9—41no—
248C4×C9—41no—
249C4×C9—42no—
250C4×C9C2×C3×D411yescharacteristic
251C4×C9C2×C3×D411yescharacteristic
252Dic9—21no—
253Dic9—41no—
254H(5,𝔽2)—91no—
255#137—91no—
256#137—91no—
257#124—91no—
258#104—91no—
259#104C3×D911yescharacteristic, complemented
260#100—91no—
261C3×C9#45511yescharacteristic, complemented, Sylow
262C23×C3—92no—
263C23×C3—182no—
264C22×D3—32no—
265C22×D3—62no—
266C3×Q8—21no—
267C3×Q8—91no—
268C3×Q8—181no—
269C3×Q8—181no—
270C3×Q8D3611yescharacteristic
271C3×Q8C3×D1211yescharacteristic
272C3×D4—21no—
273C3×D4—21no—
274C3×D4—21no—
275C3×D4—41no—
276C3×D4—91no—
277C3×D4—182no—
278C3×D4—182no—
279C3×D4—181no—
280C3×D4—181no—
281C3×D4—181no—
282C3×D4D3611yescharacteristic
283C3×D4C3×D1211yescharacteristic
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—182no—
292C2×C3×C4—181no—
293C2×C3×C4—181no—
294C2×C3×C4C22×D911yescharacteristic
295C2×C3×C4C22×C3×D311yescharacteristic
296#67—62no—
297#67—61no—
298C2×Dic3—61no—
299D12—31no—
300D12—62no—
301D12—61no—
302D12—61no—
303C4×D3—62no—
304C4×D3—61no—
305Dic6—31no—
306Dic6—61no—
307Dic6—61no—
308C3×C8—21no—
309C3×C8—21no—
310C3×C8—41no—
311C3×C8—181no—
312C3⋊C8—61no—
313C2×C32—21no—
314C2×C32—41no—
315C2×C32#88211yescharacteristic
316C3×D3—122no—
317C3×D3—121no—
318C2×C9—21no—
319C2×C9—21no—
320C2×C9—41no—
321C2×C9—41no—
322C2×C9—81no—
323C2×C9C3×#12011yescharacteristic
324D9—42no—
325D9—41no—
326#41—91no—
327#41—181no—
328#41—181no—
329#41C2×C3×D911yescharacteristic
330C2×Q8—91no—
331C2×D4—91no—
332C2×D4—181no—
333C2×D4—182no—
334C2×D4—181no—
335Q16—181no—
336Q16—181no—
337SD16—181no—
338SD16—181no—
339M16—91no—
340M16C2×C3×D911yescharacteristic, complemented
341C22⋊C4—182no—
342C42C2×C3×D911yescharacteristic, complemented
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—21no—
346C22×C3—41no—
347C22×C3—181no—
348C22×C3—182no—
349C22×C3—362no—
350C22×C3—722no—
351C22×C3D4×D911yescharacteristic
352C22×C3C3×D3×D411yescharacteristic
353C2×D3—61no—
354C2×D3—62no—
355C2×D3—122no—
356C2×D3—242no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—21no—
360C3×C4—21no—
361C3×C4—41no—
362C3×C4—41no—
363C3×C4—41no—
364C3×C4—42no—
365C3×C4—181no—
366C3×C4—361no—
367C3×C4C2×D3611yescharacteristic
368C3×C4D4×D911yescharacteristic
369C3×C4C2×C3×D1211yescharacteristic
370C3×C4C3×D3×D411yescharacteristic
371Dic3—61no—
372Dic3—121no—
373C32#482511yescharacteristic
374C9—21no—
375C9C3×#45511yescharacteristic, complemented
376C23—92no—
377C23—182no—
378Q8—91no—
379Q8—181no—
380Q8—181no—
381Q8C3×D3611yescharacteristic
382D4—21no—
383D4—91no—
384D4—182no—
385D4—182no—
386D4—181no—
387D4—181no—
388D4—181no—
389D4C3×D3611yescharacteristic
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—182no—
393C2×C4—181no—
394C2×C4—181no—
395C2×C4C22×C3×D911yescharacteristic
396C8—21no—
397C8—181no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—41no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3—81no—
405C2×C3—362no—
406C2×C3—361no—
407C2×C3#6344411yescharacteristic
408C2×C3C3×#88211yescharacteristic
409D3—122no—
410D3—121no—
411C22—21no—
412C22—181no—
413C22—182no—
414C22—362no—
415C22—722no—
416C22C3×D4×D911yescharacteristic
417C4—21no—
418C4—41no—
419C4—181no—
420C4—361no—
421C4C2×C3×D3611yescharacteristic
422C4C3×D4×D911yescharacteristic
423C3—21no—
424C3#1058850511yescharacteristic, direct product factor
425C3C3×#482511yescharacteristic
426C2—21no—
427C2—41no—
428C2—362no—
429C2—361no—
430C2C3×#6344411yescharacteristic
subgroupG/H
centerC2×C3#63444
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#104C2
socleC2×C32#882
prime radicalC3×#63715C2
hypercenterC3×#104D9
hypocenterC9C3×#455
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#104

Complements

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

NH ≅ G/Ntype
2. C3×#63511C2semidirect
3. C3×#63471C2semidirect
4. C3×#63437C2semidirect
5. C3×C9×#104C2semidirect
6. C3×#63383C2semidirect
7. C3×#63364C2semidirect
9. #10588505C3direct (N × H)
18. C2×C3×Dic18C22semidirect
19. C3×C9×M16C22semidirect
20. C3×C42×C9C22semidirect
24. C3×#3667C22semidirect
35. #63511C2×C3semidirect
36. #63471C2×C3semidirect
37. #63437C2×C3semidirect
39. C9×#104C2×C3semidirect
40. #63383C2×C3semidirect
41. #63364C2×C3semidirect
92. C2×Dic18C22×C3semidirect
94. C9×M16C22×C3semidirect
96. C42×C9C22×C3semidirect
100. #3667C22×C3semidirect
125. C3×#104D9semidirect
212. C3×M16C2×D9semidirect
216. C3×C42C2×D9semidirect
259. #104C3×D9semidirect
261. C3×C9#455semidirect
340. M16C2×C3×D9semidirect
342. C42C2×C3×D9semidirect
375. C9C3×#455semidirect
424. C3#10588505direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#637152
C3×#635112
C3×#634712
C3×#634372
C3×C9×#1042
C3×#633832
C3×#633642
C3×#48253
#105885053