Finite Group Databasebeta

← C3×#10590873 #13157835

Subgroups — C3×#10590873 #13157835

Order 1728.

482 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63869C212yesmaximal
2C3×#63645C211yesmaximal, characteristic
3C32×#818C211yesmaximal, characteristic
4C32×C8×Dic3C211yesmaximal, characteristic
5C3×#63573C212yesmaximal
6C3×#4878—31nomaximal
7C3×#4683—31nomaximal
8#10590873C311yesmaximal, characteristic, direct product factor
9C2×C33×C8C2211yescharacteristic
10C3×C4⋊(C3⋊Dic3)C2212yes—
11C32×C4⋊Dic3C2212yes—
12C32×C4×Dic3C2211yescharacteristic
13C2×C32×C3⋊C8C2211yescharacteristic
14C3×#3724—22no—
15C3×Dic3⋊Dic3—22no—
16C3×#890—32no—
17C3×#870—32no—
18#63869C2×C312yes—
19C32×#107—31no—
20C32×C4×C8—31no—
21#63645C2×C311yescharacteristic
22C3×#818—21no—
23C3×#818—31no—
24C3×#818—31no—
25C3×#818—61no—
26C3×#818D311yescharacteristic
27C3×#818C2×C311yescharacteristic
28C3×#817—32no—
29C3×C8×Dic3—21no—
30C3×C8×Dic3D311yescharacteristic
31C3×C8×Dic3C2×C311yescharacteristic
32C3×#808—32no—
33#63573C2×C312yes—
34C2×C33×C4C2311yescharacteristic
35C2×C3×C3⋊Dic3—22no—
36C2×C32×Dic3—22no—
37C2×C32×Dic3D411yescharacteristic
38C2×C32×Dic3D411yescharacteristic
39C33×C8Q812yes—
40C32×C3⋊C8Q812yes—
41C3×#499—91no—
42#4878—31no—
43#4683—31no—
44C2×C32×C8—21no—
45C2×C32×C8—21no—
46C2×C32×C8—21no—
47C2×C32×C8—31no—
48C2×C32×C8—41no—
49C2×C32×C8C2×D311yescharacteristic
50C2×C32×C8C2×D311yescharacteristic
51C2×C32×C8C22×C311yescharacteristic
52C32×C4⋊C4—32no—
53C32×C42—31no—
54C4⋊(C3⋊Dic3)C22×C312yes—
55C3×C4⋊Dic3—22no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—32no—
58C3×C4⋊Dic3—62no—
59C3×C4⋊Dic3—62no—
60C3×C4⋊Dic3C2×D312yes—
61C3×C4⋊Dic3C22×C312yes—
62C3×Dic3⋊C4—62no—
63C3×Dic3⋊C4—62no—
64C3×Dic3⋊C4—62no—
65C3×C4×Dic3—21no—
66C3×C4×Dic3C2×D311yescharacteristic
67C3×C4×Dic3C22×C311yescharacteristic
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8C2×D311yescharacteristic
70C2×C3×C3⋊C8C22×C311yescharacteristic
71#3724—22no—
72Dic3⋊Dic3—22no—
73C22×C33C2×D411yescharacteristic
74C33×C4C2×Q811yescharacteristic
75C33×C4C2×Q811yescharacteristic
76C3×C3⋊Dic3—42no—
77C32×Dic3—21no—
78C32×Dic3—21no—
79C32×Dic3—42no—
80C3×#125—92no—
81#890—32no—
82#870—32no—
83C3×#107—31no—
84C3×#107—31no—
85C3×#107—61no—
86C3×#107—91no—
87C3×#106—92no—
88C3×C4×C8—31no—
89C3×C4×C8—31no—
90C3×C4×C8—61no—
91#818—31no—
92#818—31no—
93#818—61no—
94#818C3×D311yescharacteristic
95#817—32no—
96C8×Dic3C3×D311yescharacteristic
97#808—32no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—31no—
102C2×C32×C4—31no—
103C2×C32×C4—41no—
104C2×C32×C4—62no—
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C22×D311yescharacteristic
107C2×C32×C4C23×C311yescharacteristic
108C2×C3⋊Dic3—22no—
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—42no—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—62no—
116C2×C3×Dic3—122no—
117C2×C3×Dic3D1211yescharacteristic
118C2×C3×Dic3D1211yescharacteristic
119C2×C3×Dic3C3×D411yescharacteristic
120C2×C3×Dic3C3×D411yescharacteristic
121C32×C8—22no—
122C32×C8—22no—
123C32×C8—22no—
124C32×C8—32no—
125C32×C8—42no—
126C32×C8Dic612yes—
127C32×C8Dic612yes—
128C32×C8C3×Q812yes—
129C3×C3⋊C8—22no—
130C3×C3⋊C8Dic612yes—
131C3×C3⋊C8C3×Q812yes—
132#499—91noSylow
133C2×C33C4⋊Q811yescharacteristic
134C2×C33#13511yescharacteristic
135C2×C33#13511yescharacteristic
136C2×C3×C8—21no—
137C2×C3×C8—21no—
138C2×C3×C8—21no—
139C2×C3×C8—31no—
140C2×C3×C8—31no—
141C2×C3×C8—41no—
142C2×C3×C8—61no—
143C2×C3×C8D3211yescharacteristic
144C2×C3×C8C2×C3×D311yescharacteristic
145C2×C3×C8C2×C3×D311yescharacteristic
146C3×C4⋊C4—32no—
147C3×C4⋊C4—32no—
148C3×C4⋊C4—62no—
149C3×C4⋊C4—92no—
150C3×C4⋊C4—182no—
151C3×C4⋊C4—182no—
152C3×C42—31no—
153C3×C42—31no—
154C3×C42—61no—
155C4⋊Dic3—32no—
156C4⋊Dic3—32no—
157C4⋊Dic3—62no—
158C4⋊Dic3—62no—
159C4⋊Dic3C2×C3×D312yes—
160Dic3⋊C4—62no—
161Dic3⋊C4—62no—
162Dic3⋊C4—62no—
163C4×Dic3C2×C3×D311yescharacteristic
164C2×C3⋊C8C2×C3×D311yescharacteristic
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—21no—
168C22×C32—41no—
169C22×C32C2×D1211yescharacteristic
170C22×C32D3×D411yescharacteristic
171C22×C32C2×C3×D411yescharacteristic
172C32×C4—21no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4—21no—
176C32×C4—21no—
177C32×C4—21no—
178C32×C4—41no—
179C32×C4—41no—
180C32×C4—61no—
181C32×C4—61no—
182C32×C4—122no—
183C32×C4C2×Dic611yescharacteristic
184C32×C4C2×Dic611yescharacteristic
185C32×C4C2×Dic611yescharacteristic
186C32×C4D3×Q811yescharacteristic
187C32×C4C2×C3×Q811yescharacteristic
188C32×C4C2×C3×Q811yescharacteristic
189C3⋊Dic3—42no—
190C3×Dic3—22no—
191C3×Dic3—22no—
192C3×Dic3—21no—
193C3×Dic3—21no—
194C3×Dic3—21no—
195C3×Dic3—21no—
196C3×Dic3—42no—
197C3×Dic3—42no—
198C3×Dic3—82no—
199C3×Dic3—122no—
200C3×Dic3—122no—
201C3×Dic3—242no—
202#125—92no—
203#107—31no—
204#107—91no—
205#106—92no—
206C4×C8—31no—
207C33#49911yescharacteristic, complemented, Sylow
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—31no—
215C2×C3×C4—41no—
216C2×C3×C4—61no—
217C2×C3×C4—61no—
218C2×C3×C4—62no—
219C2×C3×C4—62no—
220C2×C3×C4—122no—
221C2×C3×C4—182no—
222C2×C3×C4C2×D3211yescharacteristic
223C2×C3×C4C22×C3×D311yescharacteristic
224C2×C3×C4C22×C3×D311yescharacteristic
225C2×Dic3—22no—
226C2×Dic3—62no—
227C2×Dic3—62no—
228C2×Dic3—122no—
229C2×Dic3C3×D1211yescharacteristic
230C2×Dic3C3×D1211yescharacteristic
231C3×C8—22no—
232C3×C8—22no—
233C3×C8—22no—
234C3×C8—32no—
235C3×C8—32no—
236C3×C8—42no—
237C3×C8—62no—
238C3×C8#62712yes—
239C3×C8C3×Dic612yes—
240C3×C8C3×Dic612yes—
241C3⋊C8C3×Dic612yes—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32—21no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—41no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32#86911yescharacteristic
255C2×C32#88811yescharacteristic
256C2×C32#90411yescharacteristic
257C2×C32#90411yescharacteristic
258C2×C32#91211yescharacteristic
259C2×C32#91911yescharacteristic
260C2×C32C3×C4⋊Q811yescharacteristic
261C2×C32C3×#13511yescharacteristic
262C2×C32C3×#13511yescharacteristic
263C2×C8—31no—
264C2×C8C3×D3211yescharacteristic
265C4⋊C4—32no—
266C4⋊C4—92no—
267C4⋊C4—182no—
268C4⋊C4—182no—
269C42—31no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—41no—
274C22×C3D3×D1211yescharacteristic
275C22×C3C2×C3×D1211yescharacteristic
276C22×C3C3×D3×D411yescharacteristic
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—62no—
286C3×C4—62no—
287C3×C4—61no—
288C3×C4—61no—
289C3×C4—61no—
290C3×C4—61no—
291C3×C4—122no—
292C3×C4—122no—
293C3×C4—242no—
294C3×C4—362no—
295C3×C4D3×Dic611yescharacteristic
296C3×C4C2×#62711yescharacteristic
297C3×C4C2×C3×Dic611yescharacteristic
298C3×C4C2×C3×Dic611yescharacteristic
299C3×C4C2×C3×Dic611yescharacteristic
300C3×C4C3×D3×Q811yescharacteristic
301Dic3—21no—
302Dic3—21no—
303Dic3—42no—
304Dic3—122no—
305Dic3—122no—
306Dic3—242no—
307C32—21no—
308C32—21no—
309C32—21no—
310C32—41no—
311C32#468311yescharacteristic, complemented
312C32#487811yescharacteristic, complemented
313C32C3×#49911yescharacteristic, complemented
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—62no—
317C2×C4—182no—
318C2×C4C2×C3×D3211yescharacteristic
319C8—32no—
320C8C3×#62712yes—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—41no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3#6380711yescharacteristic
334C2×C3#6380811yescharacteristic
335C2×C3#6391811yescharacteristic
336C2×C3C3×#86911yescharacteristic
337C2×C3C3×#88811yescharacteristic
338C2×C3C3×#90411yescharacteristic
339C2×C3C3×#90411yescharacteristic
340C2×C3C3×#91211yescharacteristic
341C2×C3C3×#91911yescharacteristic
342C22C3×D3×D1211yescharacteristic
343C4—61no—
344C4—61no—
345C4—122no—
346C4—362no—
347C4C3×D3×Dic611yescharacteristic
348C4C2×C3×#62711yescharacteristic
349C3—21no—
350C3—21no—
351C3—21no—
352C3—41no—
353C3#1059087311yescharacteristic, direct product factor
354C3C3×#468311yescharacteristic, complemented
355C3C3×#487811yescharacteristic, complemented
356C2C3×#6380711yescharacteristic
357C2C3×#6380811yescharacteristic
358C2C3×#6391811yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#499
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
8. #10590873C3direct (N × H)
207. C33#499semidirect
311. C32#4683semidirect
312. C32#4878semidirect
313. C32C3×#499semidirect
353. C3#10590873direct (N × H)
354. C3C3×#4683semidirect
355. C3C3×#4878semidirect

Maximal subgroups

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

subgroupindex
C3×#638692
C3×#636452
C32×#8182
C32×C8×Dic32
C3×#635732
C3×#48783
C3×#46833
#105908733