Finite Group Databasebeta

← C3×#10591274 #13158192

Subgroups — C3×#10591274 #13158192

Order 1728.

1122 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C32×C4⋊Dic3C212yesmaximal
3C2×C3×Dic32C212yesmaximal
4C3×#63579C212yesmaximal
5C3×Dic3×C4⋊C4—32nomaximal
6#10591274C311yesmaximal, characteristic, direct product factor
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C32×Dic3C2214yes—
11C3×C4×C3⋊Dic3C414yescomplemented
12C32×C4⋊Dic3C418yescomplemented
13C3×Dic32—24no—
14C2×C32×C4⋊C4—32no—
15C2×C4×C3⋊Dic3C2×C311yescharacteristic
16C2×C3×C4⋊Dic3—22no—
17C2×C3×C4⋊Dic3D312yes—
18C2×C3×C4⋊Dic3C2×C312yes—
19C2×C3×C4×Dic3—32no—
20C2×C3×C4×Dic3—34no—
21C2×C3×C4×Dic3—61no—
22C2×Dic32C2×C312yes—
23C3×C3⋊#95—34no—
24#63579C2×C312yes—
25C23×C33C2311yescharacteristic
26C2×C33×C4C2×C412yes—
27C2×C33×C4C2×C414yes—
28C2×C3×C3⋊Dic3—22no—
29C2×C3×C3⋊Dic3C2×C412yes—
30C2×C3×C3⋊Dic3C2×C412yes—
31C2×C3×C3⋊Dic3D412yes—
32C2×C3×C3⋊Dic3Q812yes—
33C2×C32×Dic3—24no—
34C2×C32×Dic3—24no—
35C2×C32×Dic3C2×C418yes—
36C3×C4×C4⋊C4—91no—
37Dic3×C4⋊C4—32no—
38C22×C32×C4—21no—
39C22×C32×C4—22no—
40C22×C32×C4—34no—
41C22×C32×C4—41no—
42C22×C32×C4C2×D312yes—
43C22×C32×C4C22×C311yescharacteristic
44C22×C3⋊Dic3C22×C311yescharacteristic
45C22×C3⋊Dic3C22×C311yescharacteristic
46C22×C3×Dic3—24no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3C2×D314yes—
52C22×C3×Dic3C22×C314yes—
53C32×C4⋊C4—38no—
54C4×C3⋊Dic3C3×C414yescomplemented
55C3×C4⋊Dic3—28no—
56C3×C4⋊Dic3Dic318yescomplemented
57C3×C4⋊Dic3C3×C418yescomplemented
58C3×C4×Dic3—38no—
59C3×C4×Dic3—68no—
60C3×C4×Dic3—64no—
61Dic32—24no—
62C22×C33C22×C411yescharacteristic
63C22×C33C22×C412yes—
64C22×C33C2×D411yescharacteristic
65C22×C33C2×Q811yescharacteristic
66C22×C33#4112yes—
67C33×C4C4214yescomplemented
68C3×C3⋊Dic3—22no—
69C3×C3⋊Dic3C4⋊C414yescomplemented
70C32×Dic3—28no—
71C2×C3×C4⋊C4—32no—
72C2×C3×C4⋊C4—32no—
73C2×C3×C4⋊C4—62no—
74C2×C3×C42—91no—
75C2×C3×C42—92no—
76C2×C4⋊Dic3C3×D312yes—
77C2×C4×Dic3—32no—
78C2×C4×Dic3—34no—
79C2×C4×Dic3—61no—
80C3×#95—92no—
81C3⋊#95—34no—
82C23×C32—21no—
83C23×C32—22no—
84C23×C32—41no—
85C23×C32C22×D312yes—
86C23×C32C23×C311yescharacteristic
87C2×C32×C4—24no—
88C2×C32×C4—22no—
89C2×C32×C4—24no—
90C2×C32×C4—24no—
91C2×C32×C4—24no—
92C2×C32×C4—38no—
93C2×C32×C4—44no—
94C2×C32×C4—42no—
95C2×C32×C4—64no—
96C2×C32×C4—64no—
97C2×C32×C4C4×D314yes—
98C2×C32×C4C4×D314yes—
99C2×C32×C4C2×Dic314yes—
100C2×C32×C4C2×C3×C412yes—
101C2×C32×C4C2×C3×C414yes—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3C2×C3×C412yes—
104C2×C3⋊Dic3C2×C3×C412yes—
105C2×C3⋊Dic3C3×D412yes—
106C2×C3⋊Dic3C3×Q812yes—
107C2×C3×Dic3—24no—
108C2×C3×Dic3—24no—
109C2×C3×Dic3—24no—
110C2×C3×Dic3—24no—
111C2×C3×Dic3—28no—
112C2×C3×Dic3—34no—
113C2×C3×Dic3—34no—
114C2×C3×Dic3—34no—
115C2×C3×Dic3—34no—
116C2×C3×Dic3—44no—
117C2×C3×Dic3—44no—
118C2×C3×Dic3—62no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—62no—
125C2×C3×Dic3C2×Dic318yes—
126C2×C3×Dic3C2×C3×C418yes—
127C4×C4⋊C4—91noSylow
128C2×C33C2×C4211yescharacteristic
129C2×C33C2×C4⋊C411yescharacteristic
130C2×C33#11711yescharacteristic
131C2×C33C4×D412yes—
132C2×C33C4×Q812yes—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—34no—
136C22×C3×C4—34no—
137C22×C3×C4—41no—
138C22×C3×C4—64no—
139C22×C3×C4—91no—
140C22×C3×C4—91no—
141C22×C3×C4D3211yescharacteristic
142C22×C3×C4C2×C3×D312yes—
143C22×Dic3—32no—
144C22×Dic3—32no—
145C22×Dic3—61no—
146C22×Dic3—61no—
147C22×Dic3C2×C3×D314yes—
148C3×C4⋊C4—38no—
149C3×C4⋊C4—38no—
150C3×C4⋊C4—68no—
151C3×C42—94no—
152C3×C42—184no—
153C4⋊Dic3C3×Dic318yescomplemented
154C4×Dic3—38no—
155C4×Dic3—64no—
156C4×Dic3—68no—
157C22×C32—22no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32—22no—
161C22×C32—21no—
162C22×C32—22no—
163C22×C32—22no—
164C22×C32—22no—
165C22×C32—22no—
166C22×C32—22no—
167C22×C32—22no—
168C22×C32—22no—
169C22×C32—42no—
170C22×C32—41no—
171C22×C32—41no—
172C22×C32—41no—
173C22×C32—42no—
174C22×C32C2×C4×D312yes—
175C22×C32C2×C4×D312yes—
176C22×C32D3×D412yes—
177C22×C32#23712yes—
178C22×C32D3×Q812yes—
179C22×C32#23912yes—
180C22×C32C22×Dic312yes—
181C22×C32C22×C3×C411yescharacteristic
182C22×C32C22×C3×C412yes—
183C22×C32C2×C3×D411yescharacteristic
184C22×C32C2×C3×Q811yescharacteristic
185C22×C32C3×#4112yes—
186C32×C4—24no—
187C32×C4—28no—
188C32×C4—44no—
189C32×C4—68no—
190C32×C4C4×Dic318yescomplemented
191C32×C4C3×C4214yescomplemented
192C3⋊Dic3—22no—
193C3⋊Dic3C3×C4⋊C414yescomplemented
194C3×Dic3—28no—
195C3×Dic3—28no—
196C3×Dic3—38no—
197C3×Dic3—48no—
198C3×Dic3—64no—
199C3×Dic3—64no—
200C3×Dic3—64no—
201C2×C4⋊C4—32no—
202C2×C42—91no—
203C2×C42—92no—
204#95—92no—
205C33C4×C4⋊C411yescharacteristic, complemented, Sylow
206C23×C3—22no—
207C23×C3—21no—
208C23×C3—41no—
209C23×C3C2×D3211yescharacteristic
210C23×C3C22×C3×D312yes—
211C2×C3×C4—24no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—24no—
215C2×C3×C4—22no—
216C2×C3×C4—38no—
217C2×C3×C4—38no—
218C2×C3×C4—44no—
219C2×C3×C4—42no—
220C2×C3×C4—68no—
221C2×C3×C4—64no—
222C2×C3×C4—64no—
223C2×C3×C4—64no—
224C2×C3×C4—64no—
225C2×C3×C4—92no—
226C2×C3×C4—92no—
227C2×C3×C4—92no—
228C2×C3×C4—92no—
229C2×C3×C4—124no—
230C2×C3×C4—124no—
231C2×C3×C4—182no—
232C2×C3×C4D3×Dic314yes—
233C2×C3×C4#62412yes—
234C2×C3×C4C3×C4×D314yes—
235C2×C3×C4C3×C4×D314yes—
236C2×C3×C4C2×C3×Dic314yes—
237C2×Dic3—24no—
238C2×Dic3—24no—
239C2×Dic3—34no—
240C2×Dic3—34no—
241C2×Dic3—34no—
242C2×Dic3—34no—
243C2×Dic3—62no—
244C2×Dic3—62no—
245C2×Dic3—62no—
246C2×Dic3—62no—
247C2×Dic3—64no—
248C2×Dic3—62no—
249C2×Dic3—62no—
250C2×Dic3C2×C3×Dic318yes—
251C2×C32—22no—
252C2×C32—22no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—22no—
256C2×C32—22no—
257C2×C32—22no—
258C2×C32—22no—
259C2×C32—21no—
260C2×C32—22no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—41no—
264C2×C32—41no—
265C2×C32—41no—
266C2×C32—42no—
267C2×C32—42no—
268C2×C32#88712yes—
269C2×C32D3×C4⋊C412yes—
270C2×C32#89212yes—
271C2×C32#89312yes—
272C2×C32C2×C4×Dic312yes—
273C2×C32D4×Dic312yes—
274C2×C32Q8×Dic312yes—
275C2×C32C2×C3×C4211yescharacteristic
276C2×C32C2×C3×C4⋊C411yescharacteristic
277C2×C32C3×#11711yescharacteristic
278C2×C32C3×C4×D412yes—
279C2×C32C3×C4×Q812yes—
280C22×C4—34no—
281C22×C4—91no—
282C22×C4—91no—
283C22×C4C3×D3211yescharacteristic
284C4⋊C4—38no—
285C42—94no—
286C42—184no—
287C22×C3—22no—
288C22×C3—22no—
289C22×C3—22no—
290C22×C3—21no—
291C22×C3—22no—
292C22×C3—21no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—21no—
298C22×C3—22no—
299C22×C3—42no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—42no—
303C22×C3—41no—
304C22×C3#379612yes—
305C22×C3#379711yescharacteristic
306C22×C3#380211yescharacteristic
307C22×C3C2×D3×Dic312yes—
308C22×C3C2×#62411yescharacteristic
309C22×C3C2×C3×C4×D312yes—
310C22×C3C2×C3×C4×D312yes—
311C22×C3C3×D3×D412yes—
312C22×C3C3×#23712yes—
313C22×C3C3×D3×Q812yes—
314C22×C3C3×#23912yes—
315C22×C3C22×C3×Dic312yes—
316C3×C4—28no—
317C3×C4—24no—
318C3×C4—44no—
319C3×C4—68no—
320C3×C4—68no—
321C3×C4—94no—
322C3×C4—128no—
323C3×C4—182no—
324C3×C4Dic3214yescomplemented
325C3×C4C3×C4×Dic318yescomplemented
326Dic3—28no—
327Dic3—38no—
328Dic3—64no—
329Dic3—64no—
330Dic3—64no—
331C32—21no—
332C32—22no—
333C32—41no—
334C32Dic3×C4⋊C412yescomplemented
335C32C3×C4×C4⋊C411yescharacteristic, complemented
336C23C2×C3×D3211yescharacteristic
337C2×C4—38no—
338C2×C4—64no—
339C2×C4—64no—
340C2×C4—92no—
341C2×C4—92no—
342C2×C4—92no—
343C2×C4—92no—
344C2×C4—182no—
345C2×C4C3×D3×Dic314yes—
346C2×C4C3×#62412yes—
347C2×C3—22no—
348C2×C3—21no—
349C2×C3—22no—
350C2×C3—22no—
351C2×C3—22no—
352C2×C3—21no—
353C2×C3—22no—
354C2×C3—22no—
355C2×C3—22no—
356C2×C3—22no—
357C2×C3—21no—
358C2×C3—22no—
359C2×C3—41no—
360C2×C3—42no—
361C2×C3—41no—
362C2×C3—42no—
363C2×C3—41no—
364C2×C3#6384312yes—
365C2×C3#6384911yescharacteristic
366C2×C3#6389812yes—
367C2×C3#6390011yescharacteristic
368C2×C3C2×Dic3211yescharacteristic
369C2×C3C3×#88712yes—
370C2×C3C3×D3×C4⋊C412yes—
371C2×C3C3×#89212yes—
372C2×C3C3×#89312yes—
373C2×C3C2×C3×C4×Dic312yes—
374C2×C3C3×D4×Dic312yes—
375C2×C3C3×Q8×Dic312yes—
376C22C3×#379612yes—
377C22C3×#379711yescharacteristic
378C22C3×#380211yescharacteristic
379C22C2×C3×D3×Dic312yes—
380C22C2×C3×#62411yescharacteristic
381C4—68no—
382C4—94no—
383C4—182no—
384C4C3×Dic3214yescomplemented
385C3—22no—
386C3—21no—
387C3—41no—
388C3#1059127411yescharacteristic, direct product factor
389C3C3×Dic3×C4⋊C412yescomplemented
390C2C3×#6384312yes—
391C2C3×#6384911yescharacteristic
392C2C3×#6389812yes—
393C2C3×#6390011yescharacteristic
394C2C2×C3×Dic3211yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC2×C32C2×C3×C42
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×C4×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
6. #10591274C3direct (N × H)
11. C3×C4×C3⋊Dic3C4semidirect
12. C32×C4⋊Dic3C4semidirect
54. C4×C3⋊Dic3C3×C4semidirect
56. C3×C4⋊Dic3Dic3semidirect
57. C3×C4⋊Dic3C3×C4semidirect
67. C33×C4C42semidirect
69. C3×C3⋊Dic3C4⋊C4semidirect
153. C4⋊Dic3C3×Dic3semidirect
190. C32×C4C4×Dic3semidirect
191. C32×C4C3×C42semidirect
193. C3⋊Dic3C3×C4⋊C4semidirect
205. C33C4×C4⋊C4semidirect
324. C3×C4Dic32semidirect
325. C3×C4C3×C4×Dic3semidirect
334. C32Dic3×C4⋊C4semidirect
335. C32C3×C4×C4⋊C4semidirect
384. C4C3×Dic32semidirect
388. C3#10591274direct (N × H)
389. C3C3×Dic3×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
C2×C32×C4⋊Dic32
C2×C3×Dic322
C3×#635792
C3×Dic3×C4⋊C43
#105912743