Finite Group Databasebeta

← C3×#10590810 #13157784

Subgroups — C3×#10590810 #13157784

Order 1728.

590 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3746C211yesmaximal, characteristic
2C3×#63844C212yesmaximal
3C32×#817C211yesmaximal, characteristic
4C32×#815C211yesmaximal, characteristic
5C3×#63567C212yesmaximal
6C3×#4853—31nomaximal
7C3×#4716—31nomaximal
8#10590810C311yesmaximal, characteristic, direct product factor
9C2×C3×#634C2212yes—
10C2×C33×C8C2211yescharacteristic
11C3×#3746C414yescomplemented
12C32×C4⋊Dic3C2212yes—
13C32×C4×Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2211yescharacteristic
15C3×#3724—22no—
16C3×Dic32—22no—
17C2×#3746C2×C311yescharacteristic
18C2×C3×Dic12—31no—
19C2×C3×Dic12—31no—
20C2×C3×Dic12—61no—
21C3×#887—32no—
22C3×C4×Dic6—32no—
23#63844C2×C312yes—
24C32×#106—31no—
25C32×#97—31no—
26C3×#817—21no—
27C3×#817D311yescharacteristic
28C3×#817C2×C311yescharacteristic
29C3×#816—32no—
30C3×#815—21no—
31C3×#815D311yescharacteristic
32C3×#815C2×C311yescharacteristic
33C3×#810—32no—
34#63567C2×C312yes—
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—22no—
37C3×#634—22no—
38C3×#634C2×C414yes—
39C2×C32×Dic3—22no—
40C2×C32×Dic3D412yes—
41C33×C8C2×C412yes—
42C32×C3⋊C8—21no—
43C3×#441—91no—
44#4853—31no—
45#4716—31no—
46C2×#634C22×C312yes—
47C2×C3×Dic6—32no—
48C2×C3×Dic6—32no—
49C2×C3×Dic6—62no—
50C2×C32×C8—21no—
51C2×C32×C8—21no—
52C2×C32×C8—21no—
53C2×C32×C8—31no—
54C2×C32×C8—41no—
55C2×C32×C8C2×D311yescharacteristic
56C2×C32×C8C2×D311yescharacteristic
57C2×C32×C8C22×C311yescharacteristic
58C32×C4⋊C4—32no—
59C32×C42—31no—
60#3746C3×C414yescomplemented
61C3×C4⋊Dic3—22no—
62C3×C4⋊Dic3C2×D312yes—
63C3×C4⋊Dic3C22×C312yes—
64C3×Dic3⋊C4—62no—
65C3×Dic3⋊C4—62no—
66C3×C4×Dic3—21no—
67C3×C4×Dic3—62no—
68C3×C4×Dic3—62no—
69C3×C4×Dic3C2×D311yescharacteristic
70C3×C4×Dic3C22×C311yescharacteristic
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8C2×D311yescharacteristic
73C2×C3×C3⋊C8C22×C311yescharacteristic
74C3×Dic12—34no—
75C3×Dic12—34no—
76C3×Dic12—64no—
77#3724—22no—
78Dic32—22no—
79C22×C33C2×D411yescharacteristic
80C33×C4C22×C411yescharacteristic
81C33×C4#4111yescharacteristic
82C3×C3⋊Dic3—24no—
83C32×Dic3—22no—
84C32×Dic3—42no—
85C2×C3×Q16—91no—
86C3×C4×Q8—92no—
87C2×Dic12—31no—
88C2×Dic12—31no—
89C2×Dic12—61no—
90#887—32no—
91C4×Dic6—32no—
92C3×#106—31no—
93C3×#106—31no—
94C3×#106—61no—
95C3×#103—92no—
96C3×#97—31no—
97C3×#97—31no—
98C3×#97—61no—
99#817C3×D311yescharacteristic
100#816—32no—
101#815C3×D311yescharacteristic
102#810—32no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—32no—
107C2×C32×C4—41no—
108C2×C32×C4—62no—
109C2×C32×C4C22×D311yescharacteristic
110C2×C32×C4C22×D311yescharacteristic
111C2×C32×C4C23×C311yescharacteristic
112C2×C3⋊Dic3—22no—
113#634—22no—
114#634C2×C3×C414yes—
115C2×C3×Dic3—22no—
116C2×C3×Dic3—22no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—42no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—122no—
122C2×C3×Dic3D1212yes—
123C2×C3×Dic3C3×D412yes—
124C3×Dic6—34no—
125C3×Dic6—34no—
126C3×Dic6—64no—
127C3×Dic6—62no—
128C3×Dic6—62no—
129C3×Dic6—122no—
130C32×C8—22no—
131C32×C8—22no—
132C32×C8—22no—
133C32×C8—42no—
134C32×C8—61no—
135C32×C8C4×D312yes—
136C32×C8C4×D312yes—
137C32×C8C2×C3×C412yes—
138C3×C3⋊C8—22no—
139C3×C3⋊C8—21no—
140C3×C3⋊C8—21no—
141#441—91noSylow
142C2×C33C4×D411yescharacteristic
143C2×C33#13712yes—
144C2×C3×Q8—92no—
145C2×Dic6—32no—
146C2×Dic6—32no—
147C2×Dic6—62no—
148C3×Q16—94no—
149C2×C3×C8—21no—
150C2×C3×C8—21no—
151C2×C3×C8—21no—
152C2×C3×C8—31no—
153C2×C3×C8—31no—
154C2×C3×C8—41no—
155C2×C3×C8—61no—
156C2×C3×C8D3211yescharacteristic
157C2×C3×C8C2×C3×D311yescharacteristic
158C2×C3×C8C2×C3×D311yescharacteristic
159C3×C4⋊C4—32no—
160C3×C4⋊C4—32no—
161C3×C4⋊C4—62no—
162C3×C4⋊C4—182no—
163C3×C42—31no—
164C3×C42—31no—
165C3×C42—61no—
166C3×C42—182no—
167C4⋊Dic3C2×C3×D312yes—
168Dic3⋊C4—62no—
169Dic3⋊C4—62no—
170C4×Dic3—62no—
171C4×Dic3—62no—
172C4×Dic3C2×C3×D311yescharacteristic
173C2×C3⋊C8C2×C3×D311yescharacteristic
174Dic12—34no—
175Dic12—34no—
176Dic12—64no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—41no—
181C22×C32C2×D1211yescharacteristic
182C22×C32D3×D411yescharacteristic
183C22×C32C2×C3×D411yescharacteristic
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—21no—
187C32×C4—21no—
188C32×C4—21no—
189C32×C4—21no—
190C32×C4—41no—
191C32×C4—41no—
192C32×C4—62no—
193C32×C4—122no—
194C32×C4C2×C4×D311yescharacteristic
195C32×C4C2×C4×D311yescharacteristic
196C32×C4#23511yescharacteristic
197C32×C4#23911yescharacteristic
198C32×C4C22×C3×C411yescharacteristic
199C32×C4C3×#4111yescharacteristic
200C3⋊Dic3—24no—
201C3×Dic3—22no—
202C3×Dic3—22no—
203C3×Dic3—42no—
204C3×Dic3—42no—
205C3×Dic3—42no—
206C3×Dic3—64no—
207C3×Dic3—64no—
208C3×Dic3—82no—
209C3×Dic3—124no—
210C2×Q16—91no—
211C4×Q8—92no—
212#106—31no—
213#103—92no—
214#97—31no—
215C33#44111yescharacteristic, complemented, Sylow
216C3×Q8—94no—
217C3×Q8—182no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—32no—
222C2×C3×C4—32no—
223C2×C3×C4—41no—
224C2×C3×C4—62no—
225C2×C3×C4—62no—
226C2×C3×C4—62no—
227C2×C3×C4—122no—
228C2×C3×C4—182no—
229C2×C3×C4C2×D3211yescharacteristic
230C2×C3×C4C22×C3×D311yescharacteristic
231C2×C3×C4C22×C3×D311yescharacteristic
232C2×Dic3—22no—
233C2×Dic3—62no—
234C2×Dic3—62no—
235C2×Dic3—122no—
236C2×Dic3C3×D1212yes—
237Dic6—34no—
238Dic6—34no—
239Dic6—64no—
240Dic6—62no—
241Dic6—62no—
242Dic6—122no—
243C3×C8—22no—
244C3×C8—22no—
245C3×C8—22no—
246C3×C8—42no—
247C3×C8—62no—
248C3×C8—61no—
249C3×C8—61no—
250C3×C8#62412yes—
251C3×C8C3×C4×D312yes—
252C3×C8C3×C4×D312yes—
253C3⋊C8—21no—
254C2×C32—21no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—41no—
261C2×C32—42no—
262C2×C32C4×D1211yescharacteristic
263C2×C32#89311yescharacteristic
264C2×C32#90912yes—
265C2×C32#91512yes—
266C2×C32C3×C4×D411yescharacteristic
267C2×C32C3×#13712yes—
268C2×Q8—92no—
269Q16—94no—
270C2×C8—31no—
271C2×C8C3×D3211yescharacteristic
272C4⋊C4—32no—
273C4⋊C4—182no—
274C42—31no—
275C42—182no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—41no—
280C22×C3D3×D1211yescharacteristic
281C22×C3C2×C3×D1211yescharacteristic
282C22×C3C3×D3×D411yescharacteristic
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—62no—
292C3×C4—62no—
293C3×C4—122no—
294C3×C4—122no—
295C3×C4—122no—
296C3×C4—184no—
297C3×C4—242no—
298C3×C4#379911yescharacteristic
299C3×C4C2×#62411yescharacteristic
300C3×C4C2×C3×C4×D311yescharacteristic
301C3×C4C2×C3×C4×D311yescharacteristic
302C3×C4C3×#23511yescharacteristic
303C3×C4C3×#23911yescharacteristic
304Dic3—22no—
305Dic3—42no—
306Dic3—64no—
307Dic3—64no—
308Dic3—124no—
309C32—21no—
310C32—21no—
311C32—21no—
312C32—41no—
313C32#471611yescharacteristic, complemented
314C32#485311yescharacteristic, complemented
315C32C3×#44111yescharacteristic, complemented
316Q8—94no—
317Q8—182no—
318C2×C4—32no—
319C2×C4—62no—
320C2×C4—182no—
321C2×C4C2×C3×D3211yescharacteristic
322C8—61no—
323C8C3×#62412yes—
324C2×C3—22no—
325C2×C3—22no—
326C2×C3—22no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—42no—
331C2×C3—41no—
332C2×C3#6380012yes—
333C2×C3#6389411yescharacteristic
334C2×C3C3×C4×D1211yescharacteristic
335C2×C3C3×#89311yescharacteristic
336C2×C3C3×#90912yes—
337C2×C3C3×#91512yes—
338C22C3×D3×D1211yescharacteristic
339C4—62no—
340C4—122no—
341C4—184no—
342C4C3×#379911yescharacteristic
343C4C2×C3×#62411yescharacteristic
344C3—21no—
345C3—21no—
346C3—21no—
347C3—41no—
348C3#1059081011yescharacteristic, direct product factor
349C3C3×#471611yescharacteristic, complemented
350C3C3×#485311yescharacteristic, complemented
351C2C3×#6380012yes—
352C2C3×#6389411yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×#441
LCSG ▷ 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. #10590810C3direct (N × H)
11. C3×#3746C4semidirect
60. #3746C3×C4semidirect
215. C33#441semidirect
313. C32#4716semidirect
314. C32#4853semidirect
315. C32C3×#441semidirect
348. C3#10590810direct (N × H)
349. C3C3×#4716semidirect
350. C3C3×#4853semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37462
C3×#638442
C32×#8172
C32×#8152
C3×#635672
C3×#48533
C3×#47163
#105908103