Finite Group Databasebeta

← C3×#10588420 #13141644

Subgroups — C3×#10588420 #13141644

Order 1728.

526 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3699C211yesmaximal, characteristic
2C3×#63439C212yesmaximal, complemented
3C3×C9×#99C211yesmaximal, characteristic, complemented
4C3×#63393C211yesmaximal, characteristic, complemented
5C3×#63365C212yesmaximal, complemented
6C3×#4740—31nomaximal
7#10588420C311yesmaximal, characteristic, direct product factor
8C2×C3×D4×C9C2211yescharacteristic
9C2×C3×#611C411yescharacteristic, complemented
10C2×C3×#611C2211yescharacteristic
11C22×C3×Dic9C411yescharacteristic, complemented
12C22×C3×Dic9C2211yescharacteristic
13C3×#3699—22no—
14C3×#3699—22no—
15C2×C3×C4×D9C411yescharacteristic, complemented
16C2×C3×Dic18C411yescharacteristic, complemented
17C3×C9×C22⋊C4C2212yes—
18C3×#3676C2212yes—
19C3×Dic9⋊C4—22no—
20C3×C4×Dic9—22no—
21C2×C3×#237—31no—
22C3×#877—32no—
23C2×#3699C2×C311yescharacteristic
24C32×#99D311yescharacteristic
25C3×#834—31no—
26C3×#806—32no—
27#63439C2×C312yescomplemented
28C9×#99—21no—
29C9×#99C2×C311yescharacteristic, complemented
30#63393C2×C311yescharacteristic, complemented
31#63365C2×C312yescomplemented
32C23×C3×C9C2×C411yescharacteristic
33C23×C3×C9C2311yescharacteristic
34C22×C3×D9C2×C411yescharacteristic
35C3×D4×C9—22no—
36C2×C3×C4×C9—22no—
37C2×C3×C4×C9C2×C411yescharacteristic
38C3×#611—22no—
39C3×#611—22no—
40C2×C3×Dic9—21no—
41C2×C3×Dic9—21no—
42C2×C3×Dic9—22no—
43C2×C3×Dic9C2×C411yescharacteristic
44C2×C3×Dic9C2×C411yescharacteristic
45C2×C3×Dic9C2×C411yescharacteristic
46C2×C3×Dic9D412yes—
47C2×C3×Dic9D412yes—
48C3×C4×D9—21no—
49C3×C4×D9—21no—
50C3×Dic18—22no—
51C3×#410—91no—
52#4740—31no—
53C2×C32×D4C2×D311yescharacteristic
54C2×C3×#67—31no—
55C2×C3×#67—31no—
56C22×C3×Dic3—31no—
57C22×C3×Dic3—31no—
58C3×#237—62no—
59C3×#237—62no—
60C2×C3×C4×D3—31no—
61C2×C3×Dic6—31no—
62C32×C22⋊C4C2×D312yes—
63C3×#217—32no—
64C3×Dic3⋊C4—62no—
65C3×C4×Dic3—62no—
66C2×D4×C9—21no—
67C2×D4×C9C22×C311yescharacteristic
68C2×#611C3×C411yescharacteristic, complemented
69C2×#611C22×C311yescharacteristic
70C22×Dic9C3×C411yescharacteristic, complemented
71C22×Dic9C22×C311yescharacteristic
72#3699—22no—
73#3699—22no—
74C2×C4×D9C3×C411yescharacteristic, complemented
75C2×Dic18C3×C411yescharacteristic, complemented
76C9×C22⋊C4—22no—
77C9×C22⋊C4C22×C312yes—
78#3676C22×C312yes—
79Dic9⋊C4—22no—
80C4×Dic9—22no—
81C22×C3×C9—21no—
82C22×C3×C9—41no—
83C22×C3×C9—41no—
84C22×C3×C9C22×C411yescharacteristic
85C22×C3×C9C2×D412yes—
86C2×C3×D9—21no—
87C2×C3×D9—41no—
88C3×C4×C9—21no—
89C3×C4×C9—42no—
90C3×Dic9—21no—
91C3×Dic9—42no—
92C3×Dic9C22⋊C412yes—
93C3×Dic9C22⋊C411yescharacteristic
94C3×Dic9C22⋊C411yescharacteristic
95C2×C3×#41—91no—
96C2×#237—31no—
97C3×#117—92no—
98#877—32no—
99C3×#99—21no—
100C3×#99—91no—
101C3×#99—92no—
102C3×#99D911yescharacteristic, complemented
103C3×#99C3×D311yescharacteristic
104#834—31no—
105#806—32no—
106C23×C32C4×D311yescharacteristic
107C23×C32C22×D311yescharacteristic
108C22×C3×D3—31no—
109C32×D4—22no—
110C2×C32×C4—22no—
111C2×C32×C4C4×D311yescharacteristic
112C3×#67—62no—
113C3×#67—62no—
114C2×C3×Dic3—31no—
115C2×C3×Dic3—31no—
116C2×C3×Dic3—32no—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—31no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—62no—
122C3×C4×D3—61no—
123C3×C4×D3—61no—
124C3×Dic6—62no—
125C23×C9—21no—
126C23×C9—21no—
127C23×C9C2×C3×C411yescharacteristic
128C23×C9C23×C311yescharacteristic
129C22×D9C2×C3×C411yescharacteristic
130D4×C9—22no—
131D4×C9—42no—
132C2×C4×C9—22no—
133C2×C4×C9—21no—
134C2×C4×C9—42no—
135C2×C4×C9C2×C3×C411yescharacteristic
136#611—22no—
137#611—22no—
138C2×Dic9—21no—
139C2×Dic9—21no—
140C2×Dic9—22no—
141C2×Dic9C2×C3×C411yescharacteristic
142C2×Dic9C2×C3×C411yescharacteristic
143C2×Dic9C2×C3×C411yescharacteristic
144C2×Dic9C3×D412yes—
145C2×Dic9C3×D412yes—
146C4×D9—21no—
147C4×D9—21no—
148Dic18—22no—
149#410—91noSylow
150C2×C3×C9—22no—
151C2×C3×C9—21no—
152C2×C3×C9—41no—
153C2×C3×C9C2×C22⋊C411yescharacteristic
154C3×D9—41no—
155C3×#41—182no—
156C3×#41—182no—
157C2×C3×Q8—91no—
158C2×C3×D4—21no—
159C2×C3×D4—91no—
160C2×C3×D4—91no—
161C2×C3×D4C2×D911yescharacteristic
162C2×C3×D4C2×C3×D311yescharacteristic
163C22×C3×C4—91no—
164C22×C3×C4—91no—
165C22×C3×C4—91no—
166C2×#67—31no—
167C2×#67—31no—
168C22×Dic3—31no—
169C22×Dic3—31no—
170#237—62no—
171#237—62no—
172C2×C4×D3—31no—
173C2×Dic6—31no—
174C3×C4⋊C4—182no—
175C3×C22⋊C4—22no—
176C3×C22⋊C4—92no—
177C3×C22⋊C4C2×D912yes—
178C3×C22⋊C4C2×C3×D312yes—
179C3×C42—182no—
180#217—32no—
181Dic3⋊C4—62no—
182C4×Dic3—62no—
183C22×C32—21no—
184C22×C32—41no—
185C22×C32—41no—
186C22×C32C2×C4×D311yescharacteristic
187C22×C32D3×D412yes—
188C2×C3×D3—61no—
189C2×C3×D3—121no—
190C32×C4—21no—
191C32×C4—42no—
192C3×Dic3—32no—
193C3×Dic3—31no—
194C3×Dic3—31no—
195C3×Dic3—61no—
196C3×Dic3—122no—
197C22×C9—21no—
198C22×C9—21no—
199C22×C9—22no—
200C22×C9—41no—
201C22×C9—41no—
202C22×C9—41no—
203C22×C9—81no—
204C22×C9—81no—
205C22×C9C22×C3×C411yescharacteristic
206C22×C9C2×C3×D412yes—
207C2×D9—21no—
208C2×D9—41no—
209C4×C9—21no—
210C4×C9—41no—
211C4×C9—42no—
212C4×C9—82no—
213Dic9—21no—
214Dic9—42no—
215Dic9C3×C22⋊C412yes—
216Dic9C3×C22⋊C411yescharacteristic
217Dic9C3×C22⋊C411yescharacteristic
218C2×#41—91no—
219#117—92no—
220#99—91no—
221#99—92no—
222#99C3×D911yescharacteristic, complemented
223C3×C9#41011yescharacteristic, complemented, Sylow
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—91no—
227C23×C3C4×D911yescharacteristic
228C23×C3C22×D911yescharacteristic
229C23×C3C3×C4×D311yescharacteristic
230C23×C3C22×C3×D311yescharacteristic
231C22×D3—31no—
232C3×Q8—182no—
233C3×D4—22no—
234C3×D4—22no—
235C3×D4—42no—
236C3×D4—182no—
237C3×D4—182no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—42no—
242C2×C3×C4—91no—
243C2×C3×C4—91no—
244C2×C3×C4—92no—
245C2×C3×C4—92no—
246C2×C3×C4—91no—
247C2×C3×C4—181no—
248C2×C3×C4—181no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4—182no—
252C2×C3×C4C4×D911yescharacteristic
253C2×C3×C4C3×C4×D311yescharacteristic
254#67—62no—
255#67—62no—
256C2×Dic3—31no—
257C2×Dic3—31no—
258C2×Dic3—32no—
259C2×Dic3—32no—
260C2×Dic3—31no—
261C2×Dic3—61no—
262C2×Dic3—61no—
263C2×Dic3—62no—
264C4×D3—61no—
265C4×D3—61no—
266Dic6—62no—
267C2×C32—22no—
268C2×C32—21no—
269C2×C32—41no—
270C2×C32D3×C22⋊C411yescharacteristic
271C3×D3—121no—
272C2×C9—22no—
273C2×C9—21no—
274C2×C9—21no—
275C2×C9—42no—
276C2×C9—41no—
277C2×C9—41no—
278C2×C9—81no—
279C2×C9C2×C3×C22⋊C411yescharacteristic
280D9—41no—
281#41—182no—
282#41—182no—
283C2×Q8—91no—
284C2×D4—91no—
285C2×D4—91no—
286C2×D4C2×C3×D911yescharacteristic
287C22×C4—91no—
288C22×C4—91no—
289C22×C4—91no—
290C4⋊C4—182no—
291C22⋊C4—92no—
292C22⋊C4C2×C3×D912yes—
293C42—182no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—22no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3—181no—
306C22×C3—361no—
307C22×C3C2×C4×D911yescharacteristic
308C22×C3D4×D912yes—
309C22×C3C2×C3×C4×D311yescharacteristic
310C22×C3C3×D3×D412yes—
311C2×D3—61no—
312C2×D3—121no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—41no—
316C3×C4—42no—
317C3×C4—42no—
318C3×C4—82no—
319C3×C4—91no—
320C3×C4—91no—
321C3×C4—92no—
322C3×C4—181no—
323C3×C4—362no—
324Dic3—32no—
325Dic3—31no—
326Dic3—31no—
327Dic3—61no—
328Dic3—122no—
329C32#474011yescharacteristic
330C9—21no—
331C9C3×#41011yescharacteristic, complemented
332C23—91no—
333C23C3×C4×D911yescharacteristic
334C23C22×C3×D911yescharacteristic
335Q8—182no—
336D4—22no—
337D4—182no—
338D4—182no—
339C2×C4—22no—
340C2×C4—91no—
341C2×C4—91no—
342C2×C4—92no—
343C2×C4—92no—
344C2×C4—91no—
345C2×C4—181no—
346C2×C4—181no—
347C2×C4—181no—
348C2×C4—181no—
349C2×C4—182no—
350C2×C4C3×C4×D911yescharacteristic
351C2×C3—22no—
352C2×C3—21no—
353C2×C3—22no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—42no—
357C2×C3—41no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—81no—
361C2×C3—361no—
362C2×C3C22⋊C4×D911yescharacteristic
363C2×C3C3×D3×C22⋊C411yescharacteristic
364D3—121no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—181no—
369C22—361no—
370C22C2×C3×C4×D911yescharacteristic
371C22C3×D4×D912yes—
372C4—21no—
373C4—42no—
374C4—91no—
375C4—91no—
376C4—92no—
377C4—181no—
378C4—362no—
379C3—21no—
380C3#1058842011yescharacteristic, direct product factor
381C3C3×#474011yescharacteristic
382C2—22no—
383C2—21no—
384C2—41no—
385C2—361no—
386C2C3×C22⋊C4×D911yescharacteristic
subgroupG/H
centerC2×C3C22⋊C4×D9
commutatorC22×C9C22×C3×C4
FrattiniC23×C3C22×C3×D3
FittingC3×C9×#99C2
socleC2×C32D3×C22⋊C4
prime radicalC2×C3×#3699C2
cosocleC23×C9C23×C3
hypercenterC3×#99D9
hypocenterC9C3×#410
LCSG ▷ C22×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
2. C3×#63439C2semidirect
3. C3×C9×#99C2semidirect
4. C3×#63393C2semidirect
5. C3×#63365C2semidirect
7. #10588420C3direct (N × H)
9. C2×C3×#611C4semidirect
11. C22×C3×Dic9C4semidirect
15. C2×C3×C4×D9C4semidirect
16. C2×C3×Dic18C4semidirect
27. #63439C2×C3semidirect
29. C9×#99C2×C3semidirect
30. #63393C2×C3semidirect
31. #63365C2×C3semidirect
68. C2×#611C3×C4semidirect
70. C22×Dic9C3×C4semidirect
74. C2×C4×D9C3×C4semidirect
75. C2×Dic18C3×C4semidirect
102. C3×#99D9semidirect
222. #99C3×D9semidirect
223. C3×C9#410semidirect
331. C9C3×#410semidirect
380. C3#10588420direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#36992
C3×#634392
C3×C9×#992
C3×#633932
C3×#633652
C3×#47403
#105884203