Finite Group Databasebeta

← C3×#10590298 #13157314

Subgroups — C3×#10590298 #13157314

Order 1728.

574 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64081C211yesmaximal, characteristic
2C32×#803C211yesmaximal, characteristic, complemented
3C32×C4×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×#63572C212yesmaximal, complemented
5C3×#63557C212yesmaximal
6C3×#4709—31nomaximal
7C3×#4690—31nomaximal
8#10590298C311yesmaximal, characteristic, direct product factor
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C33×C42C2211yescharacteristic
11C3×#3752C412yes—
12C3×#3750C412yes—
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2212yes—
15C2×C32×C3⋊C8C2212yes—
16#64081C2×C311yescharacteristic
17C3×#872—31no—
18C3×#872—31no—
19C3×#872—61no—
20C32×#97—31no—
21C32×C4×C8—31no—
22C3×(C2×D3)⋊C8—32no—
23C3×(C2×D3)⋊C8—32no—
24C3×Dic3⋊C8—32no—
25C3×Dic3⋊C8—32no—
26C3×#803—21no—
27C3×#803D311yescharacteristic, complemented
28C3×#803C2×C311yescharacteristic, complemented
29C3×C4×C3⋊C8—21no—
30C3×C4×C3⋊C8D311yescharacteristic, complemented
31C3×C4×C3⋊C8C2×C311yescharacteristic, complemented
32#63572C2×C312yescomplemented
33#63557C2×C312yes—
34C22×C3×C3⋊D3C2×C411yescharacteristic
35C2×C33×C4C2×C412yes—
36C2×C33×C4C2311yescharacteristic
37C2×C3×C3⋊Dic3C2×C412yes—
38C2×C3×C3⋊Dic3C2×C411yescharacteristic
39C3×C4×C3⋊D3—22no—
40C32×C3⋊C8—22no—
41C32×C3⋊C8—22no—
42C3×#433—91no—
43#4709—31no—
44#4690—31no—
45C2×C4×C3⋊D3C22×C311yescharacteristic
46C2×C3×C4×D3—31no—
47C2×C3×C4×D3—31no—
48C2×C3×C4×D3—61no—
49C2×C32×C8—32no—
50C2×C32×C8—32no—
51C32×C42—21no—
52C32×C42—21no—
53C32×C42—21no—
54C32×C42—41no—
55C32×C42C2×D311yescharacteristic
56C32×C42C2×D311yescharacteristic
57C32×C42C22×C311yescharacteristic
58#3752C3×C412yes—
59#3750C3×C412yes—
60C4×C3⋊Dic3C22×C311yescharacteristic
61C3×(C2×D3)⋊C4—32no—
62C3×(C2×D3)⋊C4—32no—
63C3×(C2×D3)⋊C4—62no—
64C3×Dic3⋊C4—32no—
65C3×Dic3⋊C4—32no—
66C3×Dic3⋊C4—62no—
67C3×C4×Dic3—31no—
68C3×C4×Dic3—31no—
69C3×C4×Dic3—61no—
70C2×C3×C3⋊C8—22no—
71C2×C3×C3⋊C8—22no—
72C2×C3×C3⋊C8C2×D312yes—
73C2×C3×C3⋊C8C2×D312yes—
74C2×C3×C3⋊C8C22×C312yes—
75C2×C3×C3⋊C8C22×C312yes—
76C22×C33C22×C411yescharacteristic
77C2×C3×C3⋊D3—21no—
78C2×C3×C3⋊D3—22no—
79C33×C4—22no—
80C33×C4#4112yes—
81C3×C3⋊Dic3—21no—
82C3×C3⋊Dic3—22no—
83C3×#117—91no—
84#872—31no—
85#872—31no—
86#872—61no—
87C3×C4⋊C8—92no—
88C3×C22⋊C8—92no—
89C3×#97—31no—
90C3×#97—31no—
91C3×#97—61no—
92C3×C4×C8—31no—
93C3×C4×C8—31no—
94C3×C4×C8—61no—
95(C2×D3)⋊C8—32no—
96(C2×D3)⋊C8—32no—
97Dic3⋊C8—32no—
98Dic3⋊C8—32no—
99#803C3×D311yescharacteristic, complemented
100C4×C3⋊C8C3×D311yescharacteristic, complemented
101C22×C3⋊D3C2×C3×C411yescharacteristic
102C22×C3×D3—31no—
103C22×C3×D3—31no—
104C22×C3×D3—61no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—42no—
112C2×C32×C4—41no—
113C2×C32×C4C4×D312yes—
114C2×C32×C4C4×D312yes—
115C2×C32×C4C2×C3×C412yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C23×C311yescharacteristic
119C2×C3⋊Dic3C2×C3×C412yes—
120C2×C3⋊Dic3C2×C3×C411yescharacteristic
121C4×C3⋊D3—22no—
122C2×C3×Dic3—32no—
123C2×C3×Dic3—32no—
124C2×C3×Dic3—31no—
125C2×C3×Dic3—31no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—62no—
128C3×C4×D3—64no—
129C3×C4×D3—62no—
130C3×C4×D3—62no—
131C32×C8—62no—
132C32×C8—62no—
133C3×C3⋊C8—24no—
134C3×C3⋊C8—24no—
135C3×C3⋊C8—22no—
136C3×C3⋊C8—22no—
137C3×C3⋊C8—22no—
138C3×C3⋊C8—22no—
139#433—91noSylow
140C2×C33#11711yescharacteristic
141C2×C33#13112yes—
142C3×C3⋊D3—41no—
143C22×C3×C4—91no—
144C2×C4×D3—31no—
145C2×C4×D3—31no—
146C2×C4×D3—61no—
147C2×C3×C8—32no—
148C2×C3×C8—32no—
149C2×C3×C8—32no—
150C2×C3×C8—32no—
151C2×C3×C8—62no—
152C2×C3×C8—62no—
153C3×C4⋊C4—92no—
154C3×C22⋊C4—92no—
155C3×C42—21no—
156C3×C42—21no—
157C3×C42—21no—
158C3×C42—41no—
159C3×C42—91no—
160C3×C42D3211yescharacteristic
161C3×C42C2×C3×D311yescharacteristic
162C3×C42C2×C3×D311yescharacteristic
163(C2×D3)⋊C4—32no—
164(C2×D3)⋊C4—32no—
165(C2×D3)⋊C4—62no—
166Dic3⋊C4—32no—
167Dic3⋊C4—32no—
168Dic3⋊C4—62no—
169C4×Dic3—31no—
170C4×Dic3—31no—
171C4×Dic3—61no—
172C2×C3⋊C8C2×C3×D312yes—
173C2×C3⋊C8C2×C3×D312yes—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—41no—
178C22×C32C2×C4×D311yescharacteristic
179C22×C32C2×C4×D311yescharacteristic
180C22×C32C22×C3×C411yescharacteristic
181C2×C3⋊D3—21no—
182C2×C3⋊D3—22no—
183C2×C3×D3—62no—
184C2×C3×D3—61no—
185C2×C3×D3—61no—
186C2×C3×D3—62no—
187C2×C3×D3—62no—
188C2×C3×D3—122no—
189C32×C4—22no—
190C32×C4—24no—
191C32×C4—22no—
192C32×C4—22no—
193C32×C4—22no—
194C32×C4—22no—
195C32×C4—22no—
196C32×C4—42no—
197C32×C4—42no—
198C32×C4—42no—
199C32×C4—44no—
200C32×C4#23512yes—
201C32×C4#23512yes—
202C32×C4C3×#4112yes—
203C3⋊Dic3—21no—
204C3⋊Dic3—22no—
205C3×Dic3—62no—
206C3×Dic3—61no—
207C3×Dic3—61no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210C3×Dic3—122no—
211#117—91no—
212C4⋊C8—92no—
213C22⋊C8—92no—
214#97—31no—
215C4×C8—31no—
216C33#43311yescharacteristic, complemented, Sylow
217C23×C3—91no—
218C22×D3—31no—
219C22×D3—31no—
220C22×D3—61no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—92no—
230C2×C3×C4—91no—
231C2×C3×C4—182no—
232C2×C3×C4#62412yes—
233C2×C3×C4C3×C4×D312yes—
234C2×C3×C4C3×C4×D312yes—
235C2×C3×C4C2×D3211yescharacteristic
236C2×C3×C4C22×C3×D311yescharacteristic
237C2×C3×C4C22×C3×D311yescharacteristic
238C2×Dic3—32no—
239C2×Dic3—32no—
240C2×Dic3—31no—
241C2×Dic3—31no—
242C2×Dic3—61no—
243C2×Dic3—62no—
244C4×D3—64no—
245C4×D3—62no—
246C4×D3—62no—
247C3×C8—64no—
248C3×C8—64no—
249C3×C8—62no—
250C3×C8—62no—
251C3×C8—62no—
252C3×C8—62no—
253C3⋊C8—22no—
254C3⋊C8—22no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—21no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—41no—
262C2×C32—42no—
263C2×C32#87211yescharacteristic
264C2×C32#87211yescharacteristic
265C2×C32#90112yes—
266C2×C32#90712yes—
267C2×C32C3×#11711yescharacteristic
268C2×C32C3×#13112yes—
269C3⋊D3—41no—
270C3×D3—122no—
271C3×D3—121no—
272C3×D3—121no—
273C22×C4—91no—
274C2×C8—32no—
275C2×C8—32no—
276C4⋊C4—92no—
277C22⋊C4—92no—
278C42—91no—
279C42C3×D3211yescharacteristic
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—41no—
284C22×C3—182no—
285C22×C3—181no—
286C22×C3C2×#62411yescharacteristic
287C22×C3C2×C3×C4×D311yescharacteristic
288C22×C3C2×C3×C4×D311yescharacteristic
289C2×D3—62no—
290C2×D3—62no—
291C2×D3—62no—
292C2×D3—61no—
293C2×D3—61no—
294C2×D3—122no—
295C3×C4—24no—
296C3×C4—22no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—42no—
303C3×C4—42no—
304C3×C4—44no—
305C3×C4—42no—
306C3×C4—182no—
307C3×C4—181no—
308C3×C4#379812yes—
309C3×C4C3×#23512yes—
310C3×C4C3×#23512yes—
311Dic3—62no—
312Dic3—62no—
313Dic3—62no—
314Dic3—61no—
315Dic3—61no—
316Dic3—122no—
317C32—21no—
318C32—21no—
319C32—21no—
320C32—41no—
321C32#469011yescharacteristic, complemented
322C32#470911yescharacteristic, complemented
323C32C3×#43311yescharacteristic, complemented
324C23—91no—
325C2×C4—92no—
326C2×C4—91no—
327C2×C4—182no—
328C2×C4C3×#62412yes—
329C2×C4C2×C3×D3211yescharacteristic
330C8—62no—
331C8—62no—
332C2×C3—22no—
333C2×C3—22no—
334C2×C3—22no—
335C2×C3—21no—
336C2×C3—21no—
337C2×C3—21no—
338C2×C3—42no—
339C2×C3—41no—
340C2×C3—361no—
341C2×C3#6381712yes—
342C2×C3#6387411yescharacteristic
343C2×C3C3×#87211yescharacteristic
344C2×C3C3×#87211yescharacteristic
345C2×C3C3×#90112yes—
346C2×C3C3×#90712yes—
347D3—122no—
348D3—121no—
349D3—121no—
350C22—182no—
351C22—181no—
352C22C2×C3×#62411yescharacteristic
353C4—22no—
354C4—182no—
355C4—181no—
356C4C3×#379812yes—
357C3—21no—
358C3—21no—
359C3—21no—
360C3—41no—
361C3#1059029811yescharacteristic, direct product factor
362C3C3×#469011yescharacteristic, complemented
363C3C3×#470911yescharacteristic, complemented
364C2—361no—
365C2C3×#6381712yes—
366C2C3×#6387411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C32×#803C2semidirect
3. C32×C4×C3⋊C8C2semidirect
4. C3×#63572C2semidirect
8. #10590298C3direct (N × H)
27. C3×#803D3semidirect
28. C3×#803C2×C3semidirect
30. C3×C4×C3⋊C8D3semidirect
31. C3×C4×C3⋊C8C2×C3semidirect
32. #63572C2×C3semidirect
99. #803C3×D3semidirect
100. C4×C3⋊C8C3×D3semidirect
216. C33#433semidirect
321. C32#4690semidirect
322. C32#4709semidirect
323. C32C3×#433semidirect
361. C3#10590298direct (N × H)
362. C3C3×#4690semidirect
363. C3C3×#4709semidirect

Maximal subgroups

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

subgroupindex
C3×#640812
C32×#8032
C32×C4×C3⋊C82
C3×#635722
C3×#635572
C3×#47093
C3×#46903
#105902983