Finite Group Databasebeta

← C3×#10591219 #13158241

Subgroups — C3×#10591219 #13158241

Order 1728.

1314 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3722C212yesmaximal
3C32×C3⋊#95C212yesmaximal, complemented
4C3×#63579C212yesmaximal, complemented
5C3×#4664—32nomaximal
6#10591219C311yesmaximal, characteristic, direct product factor
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C3×C4×C3⋊D3—24no—
11C2×C3×C4×C3⋊D3C414yescomplemented
12C22×C32×Dic3C2214yes—
13C3×#3722—24no—
14C22×C4×C3⋊D3C2×C311yescharacteristic
15C22×C3×C4×D3—32no—
16C22×C3×C4×D3—61no—
17C2×C3×(C2×D3)⋊C4—34no—
18C2×#3722C2×C312yes—
19C32×#95—32no—
20C3×C3⋊#95—22no—
21C3×C3⋊#95—34no—
22C3×C3⋊#95D312yescomplemented
23C3×C3⋊#95C2×C312yescomplemented
24#63579C2×C312yescomplemented
25C23×C33C2311yescharacteristic
26C22×C3×C3⋊D3—22no—
27C22×C3×C3⋊D3—22no—
28C22×C3×C3⋊D3C2×C412yes—
29C22×C3×C3⋊D3D414yes—
30C2×C33×C4—22no—
31C2×C33×C4C2×C412yes—
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3C2×C412yes—
34C3×C4×C3⋊D3—28no—
35C2×C32×Dic3—24no—
36C2×C32×Dic3—24no—
37C2×C32×Dic3—24no—
38C3×#381—91no—
39#4664—32no—
40C23×C3⋊D3C22×C311yescharacteristic
41C23×C3×D3—32no—
42C23×C3×D3—61no—
43C22×C32×C4—21no—
44C22×C32×C4—22no—
45C22×C32×C4—34no—
46C22×C32×C4—41no—
47C22×C32×C4C2×D312yes—
48C22×C32×C4C22×C311yescharacteristic
49C22×C3⋊Dic3C22×C311yescharacteristic
50C2×C4×C3⋊D3—24no—
51C2×C4×C3⋊D3C3×C414yescomplemented
52C22×C3×Dic3—24no—
53C22×C3×Dic3—32no—
54C22×C3×Dic3—61no—
55C22×C3×Dic3C2×D314yes—
56C22×C3×Dic3C22×C314yes—
57C2×C3×C4×D3—38no—
58C2×C3×C4×D3—68no—
59C2×C3×C4×D3—64no—
60C2×C3×C4×D3—68no—
61C3×(C2×D3)⋊C4—68no—
62#3722—24no—
63C22×C33C22×C411yescharacteristic
64C22×C33C2×D412yes—
65C22×C33#4112yes—
66C22×C33#4112yes—
67C2×C3×C3⋊D3—22no—
68C2×C3×C3⋊D3—22no—
69C2×C3×C3⋊D3—28no—
70C2×C3×C3⋊D3C22⋊C414yes—
71C33×C4—22no—
72C3×C3⋊Dic3—22no—
73C32×Dic3—44no—
74C23×C3×C4—91no—
75C22×C4×D3—32no—
76C22×C4×D3—61no—
77C2×C3×C22⋊C4—92no—
78C2×(C2×D3)⋊C4—34no—
79C3×#95—32no—
80C3×#95—32no—
81C3×#95—62no—
82C3×#95—92no—
83C3⋊#95—34no—
84C3⋊#95C3×D312yescomplemented
85C23×C32—21no—
86C23×C32—22no—
87C23×C32—41no—
88C23×C32C22×D312yes—
89C23×C32C23×C311yescharacteristic
90C22×C3⋊D3—22no—
91C22×C3⋊D3—22no—
92C22×C3⋊D3C2×C3×C412yes—
93C22×C3⋊D3C3×D414yes—
94C22×C3×D3—34no—
95C22×C3×D3—38no—
96C22×C3×D3—62no—
97C22×C3×D3—64no—
98C22×C3×D3—64no—
99C22×C3×D3—64no—
100C22×C3×D3—64no—
101C22×C3×D3—62no—
102C22×C3×D3—62no—
103C2×C32×C4—22no—
104C2×C32×C4—24no—
105C2×C32×C4—24no—
106C2×C32×C4—22no—
107C2×C32×C4—24no—
108C2×C32×C4—42no—
109C2×C32×C4—44no—
110C2×C32×C4—44no—
111C2×C32×C4—64no—
112C2×C32×C4—64no—
113C2×C32×C4—64no—
114C2×C32×C4C4×D314yes—
115C2×C32×C4C2×C3×C412yes—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3C2×C3×C412yes—
118C4×C3⋊D3—28no—
119C2×C3×Dic3—28no—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—24no—
122C2×C3×Dic3—24no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—24no—
125C2×C3×Dic3—24no—
126C2×C3×Dic3—34no—
127C2×C3×Dic3—44no—
128C2×C3×Dic3—44no—
129C2×C3×Dic3—62no—
130C2×C3×Dic3—64no—
131C2×C3×Dic3—64no—
132C3×C4×D3—616no—
133C3×C4×D3—616no—
134#381—91noSylow
135C2×C33C2×C22⋊C411yescharacteristic
136C2×C33#11711yescharacteristic
137C2×C33#11711yescharacteristic
138C2×C33#12314yes—
139C3×C3⋊D3—24no—
140C24×C3—91no—
141C23×D3—32no—
142C23×D3—61no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—34no—
146C22×C3×C4—34no—
147C22×C3×C4—41no—
148C22×C3×C4—64no—
149C22×C3×C4—91no—
150C22×C3×C4—94no—
151C22×C3×C4—184no—
152C22×C3×C4D3211yescharacteristic
153C22×C3×C4C2×C3×D312yes—
154C22×Dic3—32no—
155C22×Dic3—61no—
156C22×Dic3C2×C3×D314yes—
157C2×C4×D3—38no—
158C2×C4×D3—64no—
159C2×C4×D3—68no—
160C2×C4×D3—68no—
161C3×C22⋊C4—184no—
162(C2×D3)⋊C4—68no—
163C22×C32—22no—
164C22×C32—22no—
165C22×C32—21no—
166C22×C32—22no—
167C22×C32—22no—
168C22×C32—24no—
169C22×C32—24no—
170C22×C32—22no—
171C22×C32—22no—
172C22×C32—41no—
173C22×C32—42no—
174C22×C32—42no—
175C22×C32—42no—
176C22×C32C2×C4×D312yes—
177C22×C32#23514yes—
178C22×C32D3×D414yes—
179C22×C32#23712yes—
180C22×C32#23912yes—
181C22×C32C22×C3×C411yescharacteristic
182C22×C32C2×C3×D412yes—
183C22×C32C3×#4112yes—
184C22×C32C3×#4112yes—
185C2×C3⋊D3—22no—
186C2×C3⋊D3—22no—
187C2×C3⋊D3—28no—
188C2×C3⋊D3C3×C22⋊C414yes—
189C2×C3×D3—38no—
190C2×C3×D3—64no—
191C2×C3×D3—64no—
192C2×C3×D3—616no—
193C2×C3×D3—64no—
194C2×C3×D3—64no—
195C2×C3×D3—68no—
196C2×C3×D3—64no—
197C2×C3×D3—68no—
198C32×C4—22no—
199C32×C4—24no—
200C32×C4—24no—
201C32×C4—44no—
202C32×C4—44no—
203C32×C4—124no—
204C3⋊Dic3—22no—
205C3×Dic3—48no—
206C3×Dic3—44no—
207C3×Dic3—44no—
208C3×Dic3—64no—
209C3×Dic3—64no—
210C23×C4—91no—
211C2×C22⋊C4—92no—
212#95—32no—
213#95—92no—
214C33#38111yescharacteristic, complemented, Sylow
215C23×C3—22no—
216C23×C3—21no—
217C23×C3—41no—
218C23×C3—92no—
219C23×C3—94no—
220C23×C3—182no—
221C23×C3—182no—
222C23×C3C2×D3211yescharacteristic
223C23×C3C22×C3×D312yes—
224C22×D3—34no—
225C22×D3—38no—
226C22×D3—62no—
227C22×D3—64no—
228C22×D3—64no—
229C22×D3—64no—
230C22×D3—64no—
231C22×D3—62no—
232C22×D3—62no—
233C2×C3×C4—24no—
234C2×C3×C4—22no—
235C2×C3×C4—24no—
236C2×C3×C4—22no—
237C2×C3×C4—24no—
238C2×C3×C4—44no—
239C2×C3×C4—44no—
240C2×C3×C4—42no—
241C2×C3×C4—68no—
242C2×C3×C4—64no—
243C2×C3×C4—64no—
244C2×C3×C4—64no—
245C2×C3×C4—64no—
246C2×C3×C4—64no—
247C2×C3×C4—64no—
248C2×C3×C4—92no—
249C2×C3×C4—124no—
250C2×C3×C4—124no—
251C2×C3×C4—188no—
252C2×C3×C4—182no—
253C2×C3×C4#62412yes—
254C2×C3×C4C3×C4×D314yes—
255C2×Dic3—24no—
256C2×Dic3—24no—
257C2×Dic3—24no—
258C2×Dic3—34no—
259C2×Dic3—62no—
260C2×Dic3—64no—
261C2×Dic3—64no—
262C4×D3—616no—
263C4×D3—616no—
264C2×C32—22no—
265C2×C32—24no—
266C2×C32—22no—
267C2×C32—24no—
268C2×C32—21no—
269C2×C32—24no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—22no—
273C2×C32—41no—
274C2×C32—41no—
275C2×C32—41no—
276C2×C32—44no—
277C2×C32#87212yes—
278C2×C32D3×C22⋊C412yes—
279C2×C32#88314yes—
280C2×C32#89212yes—
281C2×C32#89414yes—
282C2×C32C2×C3×C22⋊C411yescharacteristic
283C2×C32C3×#11711yescharacteristic
284C2×C32C3×#11711yescharacteristic
285C2×C32C3×#12314yes—
286C3⋊D3—24no—
287C3×D3—68no—
288C3×D3—68no—
289C24—91no—
290C22×C4—34no—
291C22×C4—91no—
292C22×C4—94no—
293C22×C4—184no—
294C22×C4C3×D3211yescharacteristic
295C22⋊C4—184no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—24no—
300C22×C3—22no—
301C22×C3—21no—
302C22×C3—22no—
303C22×C3—22no—
304C22×C3—24no—
305C22×C3—42no—
306C22×C3—42no—
307C22×C3—41no—
308C22×C3—42no—
309C22×C3—94no—
310C22×C3—182no—
311C22×C3—188no—
312C22×C3—182no—
313C22×C3#379612yes—
314C22×C3#379812yes—
315C22×C3C2×#62411yescharacteristic
316C22×C3#381112yes—
317C22×C3C2×C3×C4×D312yes—
318C22×C3C3×#23514yes—
319C22×C3C3×D3×D414yes—
320C22×C3C3×#23712yes—
321C22×C3C3×#23912yes—
322C2×D3—38no—
323C2×D3—64no—
324C2×D3—64no—
325C2×D3—616no—
326C2×D3—64no—
327C2×D3—68no—
328C2×D3—64no—
329C2×D3—68no—
330C2×D3—64no—
331C3×C4—24no—
332C3×C4—24no—
333C3×C4—22no—
334C3×C4—44no—
335C3×C4—44no—
336C3×C4—128no—
337C3×C4—124no—
338C3×C4—124no—
339C3×C4—182no—
340Dic3—44no—
341Dic3—64no—
342Dic3—64no—
343C32—22no—
344C32—21no—
345C32—41no—
346C32#466412yescomplemented
347C32C3×#38111yescharacteristic, complemented
348C23—92no—
349C23—94no—
350C23—182no—
351C23—182no—
352C23C2×C3×D3211yescharacteristic
353C2×C4—22no—
354C2×C4—64no—
355C2×C4—64no—
356C2×C4—64no—
357C2×C4—92no—
358C2×C4—188no—
359C2×C4—182no—
360C2×C4C3×#62412yes—
361C2×C3—21no—
362C2×C3—22no—
363C2×C3—21no—
364C2×C3—22no—
365C2×C3—24no—
366C2×C3—24no—
367C2×C3—24no—
368C2×C3—21no—
369C2×C3—22no—
370C2×C3—41no—
371C2×C3—41no—
372C2×C3—44no—
373C2×C3—41no—
374C2×C3—184no—
375C2×C3#6384911yescharacteristic
376C2×C3#6385314yes—
377C2×C3#6387411yescharacteristic
378C2×C3#6397411yescharacteristic
379C2×C3C3×#87212yes—
380C2×C3C3×D3×C22⋊C412yes—
381C2×C3C3×#88314yes—
382C2×C3C3×#89212yes—
383C2×C3C3×#89414yes—
384D3—68no—
385D3—68no—
386C22—94no—
387C22—182no—
388C22—188no—
389C22—182no—
390C22C3×#379612yes—
391C22C3×#379812yes—
392C22C2×C3×#62411yescharacteristic
393C22C3×#381112yes—
394C4—22no—
395C4—124no—
396C4—182no—
397C3—21no—
398C3—22no—
399C3—41no—
400C3#1059121911yescharacteristic, direct product factor
401C3C3×#466412yescomplemented
402C2—184no—
403C2C3×#6384911yescharacteristic
404C2C3×#6385314yes—
405C2C3×#6387411yescharacteristic
406C2C3×#6397411yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#381
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
3. C32×C3⋊#95C2semidirect
4. C3×#63579C2semidirect
6. #10591219C3direct (N × H)
11. C2×C3×C4×C3⋊D3C4semidirect
22. C3×C3⋊#95D3semidirect
23. C3×C3⋊#95C2×C3semidirect
24. #63579C2×C3semidirect
51. C2×C4×C3⋊D3C3×C4semidirect
84. C3⋊#95C3×D3semidirect
214. C33#381semidirect
346. C32#4664semidirect
347. C32C3×#381semidirect
400. C3#10591219direct (N × H)
401. C3C3×#4664semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×C3×#37222
C32×C3⋊#952
C3×#635792
C3×#46643
#105912193