Finite Group Databasebeta

← C3×#10588506 #13141970

Subgroups — C3×#10588506 #13141970

Order 1728.

470 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×#63716C211yesmaximal, characteristic
2C3×#63511C211yesmaximal, characteristic, complemented
3C3×#63471C211yesmaximal, characteristic, complemented
4C3×#63431C211yesmaximal, characteristic, complemented
5C3×C9×#104C211yesmaximal, characteristic, complemented
6C3×#63382C211yesmaximal, characteristic, complemented
7C3×#63364C211yesmaximal, characteristic, complemented
8C3×#4826—31nomaximal
9#10588506C311yesmaximal, characteristic, direct product factor
10C3×C9×#41C2211yescharacteristic
11C3×Q8×D9—21no—
12C3×#3699—21no—
13C3×#3699—22no—
14C3×#3697—21no—
15C3×#3697C2211yescharacteristic
16C2×C3×Dic18—21no—
17C2×C3×Dic18C2211yescharacteristic
18C2×C3×Dic18C2211yescharacteristic
19C3×C9×M16C2211yescharacteristic
20C3×C42×C9C2211yescharacteristic
21C3×#3674—21no—
22C3×#3672—21no—
23C3×C4⋊Dic9—22no—
24C3×#3667C2211yescharacteristic
25C3×#3664—21no—
26C3×Dic36—21no—
27C3×#1010—31no—
28C3×#951—31no—
29C3×#909—31no—
30C3×#869—31no—
31#63716C2×C311yescharacteristic
32C32×#104D311yescharacteristic
33C3×#824—31no—
34C3×#805—31no—
35#63511C2×C311yescharacteristic, complemented
36#63471C2×C311yescharacteristic, complemented
37#63431C2×C311yescharacteristic, complemented
38C9×#104—21no—
39C9×#104C2×C311yescharacteristic, complemented
40#63382C2×C311yescharacteristic, complemented
41#63364C2×C311yescharacteristic, complemented
42C3×Q8×C9D411yescharacteristic
43C3×D4×C9—21no—
44C3×D4×C9D411yescharacteristic
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9C2311yescharacteristic
48C3×#611—22no—
49C3×#611—21no—
50C2×C3×Dic9—22no—
51C2×C3×Dic9—21no—
52C2×C3×Dic9D412yes—
53C3×D36D411yescharacteristic
54C3×C4×D9—22no—
55C3×C4×D9—21no—
56C3×Dic18—22no—
57C3×Dic18—21no—
58C3×Dic18—21no—
59C3×Dic18—21no—
60C3×Dic18—21no—
61C3×Dic18D411yescharacteristic
62C3×C8×C9—21no—
63C3×C9⋊C8—21no—
64C3×#456—91no—
65#4826—31no—
66C32×#41C2×D311yescharacteristic
67C3×D3×Q8—61no—
68C3×#237—62no—
69C3×#237—61no—
70C3×#235—31no—
71C3×#235—61no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—61no—
75C32×M16C2×D311yescharacteristic
76C32×C42C2×D311yescharacteristic
77C3×#216—61no—
78C3×#214—61no—
79C3×C4⋊Dic3—62no—
80C3×#208—31no—
81C3×Dic12—61no—
82C3×#204—61no—
83C9×#41—21no—
84C9×#41C22×C311yescharacteristic
85Q8×D9—21no—
86#3699—21no—
87#3699—22no—
88#3697—21no—
89#3697C22×C311yescharacteristic
90C2×Dic18—21no—
91C2×Dic18C22×C311yescharacteristic
92C2×Dic18C22×C311yescharacteristic
93C9×M16—21no—
94C9×M16C22×C311yescharacteristic
95C42×C9—21no—
96C42×C9C22×C311yescharacteristic
97#3674—21no—
98#3672—21no—
99C4⋊Dic9—22no—
100#3667C22×C311yescharacteristic
101#3664—21no—
102Dic36—21no—
103C22×C3×C9—21no—
104C22×C3×C9C2×D411yescharacteristic
105C2×C3×D9—21no—
106C3×C4×C9—22no—
107C3×C4×C9—21no—
108C3×C4×C9C2×D411yescharacteristic
109C3×C4×C9C2×D411yescharacteristic
110C3×Dic9—22no—
111C3×Dic9—21no—
112C3×Dic9—41no—
113C3×2−1+4—91no—
114#1010—31no—
115C3×#137—91no—
116C3×#137—91no—
117C3×C4⋊Q8—91no—
118#951—31no—
119#909—31no—
120#869—31no—
121C3×#104—21no—
122C3×#104—91no—
123C3×#104D911yescharacteristic, complemented
124C3×#104C3×D311yescharacteristic
125C3×#101—91no—
126#824—31no—
127#805—31no—
128C32×Q8D1211yescharacteristic
129C32×D4—21no—
130C32×D4D1211yescharacteristic
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4C22×D311yescharacteristic
134C3×#67—62no—
135C3×#67—61no—
136C2×C3×Dic3—32no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—61no—
139C3×D12—31no—
140C3×C4×D3—62no—
141C3×C4×D3—61no—
142C3×Dic6—31no—
143C3×Dic6—62no—
144C3×Dic6—61no—
145C3×Dic6—61no—
146C3×Dic6—61no—
147C3×Dic6—61no—
148C32×C8—21no—
149C3×C3⋊C8—61no—
150Q8×C9—21no—
151Q8×C9C3×D411yescharacteristic
152D4×C9—21no—
153D4×C9—21no—
154D4×C9—41no—
155D4×C9C3×D411yescharacteristic
156C2×C4×C9—21no—
157C2×C4×C9—21no—
158C2×C4×C9—21no—
159C2×C4×C9—41no—
160C2×C4×C9—41no—
161C2×C4×C9C23×C311yescharacteristic
162#611—22no—
163#611—21no—
164C2×Dic9—22no—
165C2×Dic9—21no—
166C2×Dic9C3×D412yes—
167D36C3×D411yescharacteristic
168C4×D9—22no—
169C4×D9—21no—
170Dic18—22no—
171Dic18—21no—
172Dic18—21no—
173Dic18—21no—
174Dic18—21no—
175Dic18C3×D411yescharacteristic
176C8×C9—21no—
177C8×C9—41no—
178C9⋊C8—21no—
179#456—91noSylow
180C2×C3×C9—21no—
181C2×C3×C9—41no—
182C2×C3×C9#12011yescharacteristic
183C3×D9—41no—
184C3×#41—21no—
185C3×#41—91no—
186C3×#41—181no—
187C3×#41—182no—
188C3×#41—181no—
189C3×#41C2×D911yescharacteristic
190C3×#41C2×C3×D311yescharacteristic
191C2×C3×Q8—91no—
192C2×C3×Q8—91no—
193C2×C3×Q8—181no—
194C2×C3×Q8—181no—
195D3×Q8—61no—
196#237—62no—
197#237—61no—
198#235—31no—
199#235—61no—
200C2×Dic6—31no—
201C2×Dic6—31no—
202C2×Dic6—61no—
203C3×Q16—181no—
204C3×Q16—181no—
205C3×SD16—181no—
206C3×SD16—181no—
207C3×M16—21no—
208C3×M16—91no—
209C3×M16C2×D911yescharacteristic
210C3×M16C2×C3×D311yescharacteristic
211C3×C4⋊C4—182no—
212C3×C42—21no—
213C3×C42C2×D911yescharacteristic
214C3×C42C2×C3×D311yescharacteristic
215#216—61no—
216#214—61no—
217C4⋊Dic3—62no—
218#208—31no—
219Dic12—61no—
220#204—61no—
221C22×C32—21no—
222C22×C32D3×D411yescharacteristic
223C2×C3×D3—61no—
224C32×C4—22no—
225C32×C4—21no—
226C32×C4C2×D1211yescharacteristic
227C32×C4D3×D411yescharacteristic
228C3×Dic3—61no—
229C3×Dic3—62no—
230C3×Dic3—121no—
231C22×C9—21no—
232C22×C9—21no—
233C22×C9—41no—
234C22×C9C2×C3×D411yescharacteristic
235C2×D9—21no—
236C4×C9—22no—
237C4×C9—21no—
238C4×C9—21no—
239C4×C9—21no—
240C4×C9—41no—
241C4×C9—42no—
242C4×C9C2×C3×D411yescharacteristic
243C4×C9C2×C3×D411yescharacteristic
244Dic9—22no—
245Dic9—21no—
246Dic9—41no—
2472−1+4—91no—
248#137—91no—
249#137—91no—
250C4⋊Q8—91no—
251#104—91no—
252#104C3×D911yescharacteristic, complemented
253#101—91no—
254C3×C9#45611yescharacteristic, complemented, Sylow
255C3×Q8—21no—
256C3×Q8—91no—
257C3×Q8—182no—
258C3×Q8—181no—
259C3×Q8—181no—
260C3×Q8—181no—
261C3×Q8—181no—
262C3×Q8D3611yescharacteristic
263C3×Q8C3×D1211yescharacteristic
264C3×D4—21no—
265C3×D4—21no—
266C3×D4—21no—
267C3×D4—41no—
268C3×D4—91no—
269C3×D4—182no—
270C3×D4—181no—
271C3×D4D3611yescharacteristic
272C3×D4C3×D1211yescharacteristic
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—92no—
281C2×C3×C4—182no—
282C2×C3×C4—182no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4C22×D911yescharacteristic
286C2×C3×C4C22×C3×D311yescharacteristic
287#67—62no—
288#67—61no—
289C2×Dic3—32no—
290C2×Dic3—62no—
291C2×Dic3—61no—
292D12—31no—
293C4×D3—62no—
294C4×D3—61no—
295Dic6—31no—
296Dic6—62no—
297Dic6—61no—
298Dic6—61no—
299Dic6—61no—
300Dic6—61no—
301C3×C8—21no—
302C3×C8—21no—
303C3×C8—41no—
304C3×C8—181no—
305C3⋊C8—61no—
306C2×C32—21no—
307C2×C32—41no—
308C2×C32#88211yescharacteristic
309C3×D3—121no—
310C2×C9—21no—
311C2×C9—21no—
312C2×C9—41no—
313C2×C9—41no—
314C2×C9—81no—
315C2×C9C3×#12011yescharacteristic
316D9—41no—
317#41—91no—
318#41—181no—
319#41—182no—
320#41—181no—
321#41C2×C3×D911yescharacteristic
322C2×Q8—91no—
323C2×Q8—91no—
324C2×Q8—181no—
325C2×Q8—181no—
326Q16—181no—
327Q16—181no—
328SD16—181no—
329SD16—181no—
330M16—91no—
331M16C2×C3×D911yescharacteristic
332C4⋊C4—182no—
333C42C2×C3×D911yescharacteristic
334C22×C3—21no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—41no—
338C22×C3—181no—
339C22×C3D4×D911yescharacteristic
340C22×C3C3×D3×D411yescharacteristic
341C2×D3—61no—
342C3×C4—22no—
343C3×C4—22no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—41no—
349C3×C4—42no—
350C3×C4—182no—
351C3×C4—181no—
352C3×C4—361no—
353C3×C4C2×D3611yescharacteristic
354C3×C4D4×D911yescharacteristic
355C3×C4C2×C3×D1211yescharacteristic
356C3×C4C3×D3×D411yescharacteristic
357Dic3—61no—
358Dic3—62no—
359Dic3—121no—
360C32#482611yescharacteristic
361C9—21no—
362C9C3×#45611yescharacteristic, complemented
363Q8—91no—
364Q8—182no—
365Q8—181no—
366Q8—181no—
367Q8—181no—
368Q8—181no—
369Q8C3×D3611yescharacteristic
370D4—21no—
371D4—91no—
372D4—182no—
373D4—181no—
374D4C3×D3611yescharacteristic
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—92no—
378C2×C4—182no—
379C2×C4—182no—
380C2×C4—181no—
381C2×C4—181no—
382C2×C4C22×C3×D911yescharacteristic
383C8—21no—
384C8—181no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—81no—
392C2×C3—361no—
393C2×C3#6344411yescharacteristic
394C2×C3C3×#88211yescharacteristic
395D3—121no—
396C22—21no—
397C22—181no—
398C22C3×D4×D911yescharacteristic
399C4—22no—
400C4—21no—
401C4—182no—
402C4—181no—
403C4—361no—
404C4C2×C3×D3611yescharacteristic
405C4C3×D4×D911yescharacteristic
406C3—21no—
407C3#1058850611yescharacteristic, direct product factor
408C3C3×#482611yescharacteristic
409C2—21no—
410C2—41no—
411C2—361no—
412C2C3×#6344411yescharacteristic
subgroupG/H
centerC2×C3#63444
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#104C2
socleC2×C32#882
prime radicalC3×#63716C2
hypercenterC3×#104D9
hypocenterC9C3×#456
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×#63431C2semidirect
5. C3×C9×#104C2semidirect
6. C3×#63382C2semidirect
7. C3×#63364C2semidirect
9. #10588506C3direct (N × H)
35. #63511C2×C3semidirect
36. #63471C2×C3semidirect
37. #63431C2×C3semidirect
39. C9×#104C2×C3semidirect
40. #63382C2×C3semidirect
41. #63364C2×C3semidirect
123. C3×#104D9semidirect
252. #104C3×D9semidirect
254. C3×C9#456semidirect
362. C9C3×#456semidirect
407. C3#10588506direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#637162
C3×#635112
C3×#634712
C3×#634312
C3×C9×#1042
C3×#633822
C3×#633642
C3×#48263
#105885063