Finite Group Databasebeta

← C3×#10588885 #13143497

Subgroups — C3×#10588885 #13143497

Order 1728.

486 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63716C211yesmaximal, characteristic
2C3×C9×#137C211yesmaximal, characteristic, complemented
3C3×#63506C211yesmaximal, characteristic, complemented
4C3×#63505C211yesmaximal, characteristic, complemented
5C3×#63396C211yesmaximal, characteristic, complemented
6C3×#63384C211yesmaximal, characteristic, complemented
7C3×#63382C211yesmaximal, characteristic, complemented
8C3×#5205—31nomaximal
9#10588885C311yesmaximal, characteristic, direct product factor
10C3×C9×#41C2211yescharacteristic
11C2×C3×Q8×C9C2211yescharacteristic
12C3×Q8×D9—21no—
13C3×#3699—21no—
14C3×#3699—22no—
15C3×#3697—21no—
16C3×#3697C2211yescharacteristic
17C2×C3×Dic18—21no—
18C2×C3×Dic18C2211yescharacteristic
19C3×C9×Q16—21no—
20C3×C9×SD16—21no—
21C3×C9×M16C2211yescharacteristic
22C3×#3675—21no—
23C3×#3674—21no—
24C3×Dic9⋊C4—22no—
25C3×C4×Dic9C2211yescharacteristic
26C3×#3667C2211yescharacteristic
27C3×#1010—31no—
28C32×#137D311yescharacteristic
29C3×#944—31no—
30C3×#942—31no—
31#63716C2×C311yescharacteristic
32C3×#837—31no—
33C3×#825—31no—
34C3×#824—31no—
35C9×#137—21no—
36C9×#137C2×C311yescharacteristic, complemented
37#63506C2×C311yescharacteristic, complemented
38#63505C2×C311yescharacteristic, complemented
39#63396C2×C311yescharacteristic, complemented
40#63384C2×C311yescharacteristic, complemented
41#63382C2×C311yescharacteristic, complemented
42C3×Q8×C9—21no—
43C3×Q8×C9—21no—
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—22no—
53C2×C3×Dic9—21no—
54C2×C3×Dic9D412yes—
55C3×D36D411yescharacteristic
56C3×C4×D9—22no—
57C3×C4×D9—21no—
58C3×Dic18—22no—
59C3×Dic18—21no—
60C3×Dic18—21no—
61C3×Dic18D411yescharacteristic
62C3×C8×C9—21no—
63C3×C9⋊C8—21no—
64C3×#456—91no—
65#5205—31no—
66C32×#41C2×D311yescharacteristic
67C2×C32×Q8C2×D311yescharacteristic
68C3×D3×Q8—61no—
69C3×#237—62no—
70C3×#237—61no—
71C3×#235—31no—
72C3×#235—61no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—61no—
75C32×Q16—21no—
76C32×SD16—21no—
77C32×M16C2×D311yescharacteristic
78C3×#216—61no—
79C3×#215—61no—
80C3×Dic3⋊C4—62no—
81C3×C4×Dic3—31no—
82C3×#208—31no—
83C9×#41—21no—
84C9×#41C22×C311yescharacteristic
85C2×Q8×C9—21no—
86C2×Q8×C9C22×C311yescharacteristic
87Q8×D9—21no—
88#3699—21no—
89#3699—22no—
90#3697—21no—
91#3697C22×C311yescharacteristic
92C2×Dic18—21no—
93C2×Dic18C22×C311yescharacteristic
94C9×Q16—22no—
95C9×Q16—21no—
96C9×SD16—22no—
97C9×SD16—21no—
98C9×M16—21no—
99C9×M16C22×C311yescharacteristic
100#3675—21no—
101#3674—21no—
102Dic9⋊C4—22no—
103C4×Dic9C22×C311yescharacteristic
104#3667C22×C311yescharacteristic
105C22×C3×C9—21no—
106C22×C3×C9C2×D411yescharacteristic
107C2×C3×D9—21no—
108C3×C4×C9—21no—
109C3×C4×C9—41no—
110C3×C4×C9C2×D411yescharacteristic
111C3×C4×C9C2×D411yescharacteristic
112C3×Dic9—22no—
113C3×Dic9—22no—
114C3×Dic9—21no—
115C3×2−1+4—91no—
116#1010—31no—
117C3×#137—21no—
118C3×#137—91no—
119C3×#137D911yescharacteristic, complemented
120C3×#137C3×D311yescharacteristic
121C3×C4⋊Q8—91no—
122#944—31no—
123#942—31no—
124C3×#104—91no—
125C3×#104—91no—
126C3×#101—91no—
127#837—31no—
128#825—31no—
129#824—31no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8#6711yescharacteristic
133C32×D4—21no—
134C32×D4#6711yescharacteristic
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4C22×D311yescharacteristic
138C3×#67—62no—
139C3×#67—61no—
140C2×C3×Dic3—32no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—61no—
143C3×D12—31no—
144C3×C4×D3—62no—
145C3×C4×D3—61no—
146C3×Dic6—31no—
147C3×Dic6—62no—
148C3×Dic6—61no—
149C3×Dic6—61no—
150C32×C8—21no—
151C3×C3⋊C8—61no—
152Q8×C9—22no—
153Q8×C9—21no—
154Q8×C9—21no—
155Q8×C9—21no—
156Q8×C9—41no—
157Q8×C9C3×D411yescharacteristic
158D4×C9—21no—
159D4×C9—21no—
160D4×C9—41no—
161D4×C9C3×D411yescharacteristic
162C2×C4×C9—21no—
163C2×C4×C9—21no—
164C2×C4×C9—21no—
165C2×C4×C9—41no—
166C2×C4×C9—41no—
167C2×C4×C9C23×C311yescharacteristic
168#611—22no—
169#611—21no—
170C2×Dic9—22no—
171C2×Dic9—21no—
172C2×Dic9C3×D412yes—
173D36C3×D411yescharacteristic
174C4×D9—22no—
175C4×D9—21no—
176Dic18—22no—
177Dic18—21no—
178Dic18—21no—
179Dic18C3×D411yescharacteristic
180C8×C9—22no—
181C8×C9—21no—
182C9⋊C8—21no—
183#456—91noSylow
184C2×C3×C9—21no—
185C2×C3×C9—41no—
186C2×C3×C9#12011yescharacteristic
187C3×D9—41no—
188C3×#41—21no—
189C3×#41—91no—
190C3×#41—181no—
191C3×#41—182no—
192C3×#41—181no—
193C3×#41C2×D911yescharacteristic
194C3×#41C2×C3×D311yescharacteristic
195C2×C3×Q8—21no—
196C2×C3×Q8—91no—
197C2×C3×Q8—181no—
198C2×C3×Q8—181no—
199C2×C3×Q8C2×D911yescharacteristic
200C2×C3×Q8C2×C3×D311yescharacteristic
201D3×Q8—61no—
202#237—62no—
203#237—61no—
204#235—31no—
205#235—61no—
206C2×Dic6—31no—
207C2×Dic6—61no—
208C3×Q16—22no—
209C3×Q16—21no—
210C3×Q16—21no—
211C3×Q16—181no—
212C3×SD16—22no—
213C3×SD16—21no—
214C3×SD16—21no—
215C3×SD16—181no—
216C3×M16—21no—
217C3×M16—91no—
218C3×M16C2×D911yescharacteristic
219C3×M16C2×C3×D311yescharacteristic
220C3×C4⋊C4—182no—
221C3×C42—91no—
222#216—61no—
223#215—61no—
224Dic3⋊C4—62no—
225C4×Dic3—31no—
226#208—31no—
227C22×C32—21no—
228C22×C32D3×D411yescharacteristic
229C2×C3×D3—61no—
230C32×C4—21no—
231C32×C4—41no—
232C32×C4D3×D411yescharacteristic
233C32×C4C2×#6711yescharacteristic
234C3×Dic3—62no—
235C3×Dic3—61no—
236C3×Dic3—62no—
237C22×C9—21no—
238C22×C9—21no—
239C22×C9—41no—
240C22×C9C2×C3×D411yescharacteristic
241C2×D9—21no—
242C4×C9—21no—
243C4×C9—21no—
244C4×C9—21no—
245C4×C9—41no—
246C4×C9—42no—
247C4×C9—41no—
248C4×C9C2×C3×D411yescharacteristic
249C4×C9C2×C3×D411yescharacteristic
250Dic9—22no—
251Dic9—22no—
252Dic9—21no—
2532−1+4—91no—
254#137—91no—
255#137C3×D911yescharacteristic, complemented
256C4⋊Q8—91no—
257#104—91no—
258#104—91no—
259#101—91no—
260C3×C9#45611yescharacteristic, complemented, Sylow
261C3×Q8—22no—
262C3×Q8—21no—
263C3×Q8—21no—
264C3×Q8—21no—
265C3×Q8—21no—
266C3×Q8—21no—
267C3×Q8—41no—
268C3×Q8—91no—
269C3×Q8—182no—
270C3×Q8—181no—
271C3×Q8—181no—
272C3×Q8#61111yescharacteristic
273C3×Q8C3×#6711yescharacteristic
274C3×D4—21no—
275C3×D4—21no—
276C3×D4—21no—
277C3×D4—41no—
278C3×D4—91no—
279C3×D4—182no—
280C3×D4—181no—
281C3×D4#61111yescharacteristic
282C3×D4C3×#6711yescharacteristic
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—92no—
291C2×C3×C4—182no—
292C2×C3×C4—182no—
293C2×C3×C4—181no—
294C2×C3×C4—181no—
295C2×C3×C4C22×D911yescharacteristic
296C2×C3×C4C22×C3×D311yescharacteristic
297#67—62no—
298#67—61no—
299C2×Dic3—32no—
300C2×Dic3—62no—
301C2×Dic3—61no—
302D12—31no—
303C4×D3—62no—
304C4×D3—61no—
305Dic6—31no—
306Dic6—62no—
307Dic6—61no—
308Dic6—61no—
309C3×C8—22no—
310C3×C8—21no—
311C3×C8—21no—
312C3×C8—181no—
313C3⋊C8—61no—
314C2×C32—21no—
315C2×C32—41no—
316C2×C32#93711yescharacteristic
317C3×D3—121no—
318C2×C9—21no—
319C2×C9—21no—
320C2×C9—41no—
321C2×C9—41no—
322C2×C9—81no—
323C2×C9C3×#12011yescharacteristic
324D9—41no—
325#41—91no—
326#41—181no—
327#41—182no—
328#41—181no—
329#41C2×C3×D911yescharacteristic
330C2×Q8—91no—
331C2×Q8—181no—
332C2×Q8—181no—
333C2×Q8C2×C3×D911yescharacteristic
334Q16—21no—
335Q16—181no—
336SD16—21no—
337SD16—181no—
338M16—91no—
339M16C2×C3×D911yescharacteristic
340C4⋊C4—182no—
341C42—91no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—41no—
346C22×C3—181no—
347C22×C3D4×D911yescharacteristic
348C22×C3C3×D3×D411yescharacteristic
349C2×D3—61no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—41no—
355C3×C4—42no—
356C3×C4—41no—
357C3×C4—41no—
358C3×C4—182no—
359C3×C4—182no—
360C3×C4—181no—
361C3×C4D4×D911yescharacteristic
362C3×C4C2×#61111yescharacteristic
363C3×C4C3×D3×D411yescharacteristic
364C3×C4C2×C3×#6711yescharacteristic
365Dic3—62no—
366Dic3—61no—
367Dic3—62no—
368C32#520511yescharacteristic
369C9—21no—
370C9C3×#45611yescharacteristic, complemented
371Q8—21no—
372Q8—21no—
373Q8—91no—
374Q8—182no—
375Q8—181no—
376Q8—181no—
377Q8C3×#61111yescharacteristic
378D4—21no—
379D4—91no—
380D4—182no—
381D4—181no—
382D4C3×#61111yescharacteristic
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—92no—
386C2×C4—182no—
387C2×C4—182no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4C22×C3×D911yescharacteristic
391C8—21no—
392C8—181no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—41no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—81no—
400C2×C3—361no—
401C2×C3#6349911yescharacteristic
402C2×C3C3×#93711yescharacteristic
403D3—121no—
404C22—21no—
405C22—181no—
406C22C3×D4×D911yescharacteristic
407C4—21no—
408C4—41no—
409C4—182no—
410C4—182no—
411C4—181no—
412C4C3×D4×D911yescharacteristic
413C4C2×C3×#61111yescharacteristic
414C3—21no—
415C3#1058888511yescharacteristic, direct product factor
416C3C3×#520511yescharacteristic
417C2—21no—
418C2—41no—
419C2—361no—
420C2C3×#6349911yescharacteristic
subgroupG/H
centerC2×C3#63499
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC3×C9×#137C2
socleC2×C32#937
prime radicalC3×#63716C2
hypercenterC3×#137D9
hypocenterC9C3×#456
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#137

Complements

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

NH ≅ G/Ntype
2. C3×C9×#137C2semidirect
3. C3×#63506C2semidirect
4. C3×#63505C2semidirect
5. C3×#63396C2semidirect
6. C3×#63384C2semidirect
7. C3×#63382C2semidirect
9. #10588885C3direct (N × H)
36. C9×#137C2×C3semidirect
37. #63506C2×C3semidirect
38. #63505C2×C3semidirect
39. #63396C2×C3semidirect
40. #63384C2×C3semidirect
41. #63382C2×C3semidirect
119. C3×#137D9semidirect
255. #137C3×D9semidirect
260. C3×C9#456semidirect
370. C9C3×#456semidirect
415. C3#10588885direct (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×C9×#1372
C3×#635062
C3×#635052
C3×#633962
C3×#633842
C3×#633822
C3×#52053
#105888853