Finite Group Databasebeta

← C3×#10590331 #13157346

Subgroups — C3×#10590331 #13157346

Order 1728.

766 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C42×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3710C212yesmaximal
3C32×C4⋊(C3⋊C8)C212yesmaximal, complemented
4C3×#63572C212yesmaximal, complemented
5C3×D3×C4⋊C8—32nomaximal
6#10590331C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C411yescharacteristic
8C2×C3×C4×C3⋊D3C411yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C33×C42C2211yescharacteristic
11C3×C4×C3⋊Dic3C411yescharacteristic
12C3×C4×C3⋊Dic3C411yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C32×C3⋊C8C2214yes—
15C3×#3710—24no—
16C42×C3⋊D3C2×C311yescharacteristic
17C2×C3×D3×C8—34no—
18C3×C42×D3—32no—
19C3×C42×D3—61no—
20C2×#3710C2×C312yes—
21C32×C4⋊C8—32no—
22C3×Dic3⋊C8—34no—
23C3×C4⋊(C3⋊C8)—22no—
24C3×C4⋊(C3⋊C8)D312yescomplemented
25C3×C4⋊(C3⋊C8)C2×C312yescomplemented
26#63572C2×C312yescomplemented
27C22×C3×C3⋊D3C2×C411yescharacteristic
28C2×C33×C4C2×C411yescharacteristic
29C2×C33×C4C2×C411yescharacteristic
30C2×C33×C4C2311yescharacteristic
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C3×C4×C3⋊D3—22no—
35C3×C4×C3⋊D3C814yescomplemented
36C3×C4×C3⋊D3D412yes—
37C3×C4×C3⋊D3Q812yes—
38C32×C3⋊C8—24no—
39C2×C3×C4⋊C8—91no—
40D3×C4⋊C8—32no—
41C2×C4×C3⋊D3C3×C411yescharacteristic
42C2×C4×C3⋊D3C3×C411yescharacteristic
43C2×C4×C3⋊D3C22×C311yescharacteristic
44C2×C3×C4×D3—32no—
45C2×C3×C4×D3—32no—
46C2×C3×C4×D3—32no—
47C2×C3×C4×D3—61no—
48C2×C3×C4×D3—61no—
49C2×C3×C4×D3—61no—
50C2×C32×C8—34no—
51C32×C42—21no—
52C32×C42—22no—
53C32×C42—41no—
54C32×C42C2×D312yes—
55C32×C42C22×C311yescharacteristic
56C4×C3⋊Dic3C3×C411yescharacteristic
57C4×C3⋊Dic3C3×C411yescharacteristic
58C4×C3⋊Dic3C22×C311yescharacteristic
59C3×C4×Dic3—32no—
60C3×C4×Dic3—32no—
61C3×C4×Dic3—32no—
62C3×C4×Dic3—61no—
63C3×C4×Dic3—61no—
64C3×C4×Dic3—61no—
65C2×C3×C3⋊C8—24no—
66C2×C3×C3⋊C8C2×D314yes—
67C2×C3×C3⋊C8C22×C314yes—
68C3×D3×C8—68no—
69#3710—24no—
70C22×C33C22×C411yescharacteristic
71C2×C3×C3⋊D3C4⋊C412yes—
72C2×C3×C3⋊D3C2×C812yes—
73C2×C3×C3⋊D3M1612yes—
74C33×C4—21no—
75C33×C4C2×C812yescomplemented
76C33×C4C2×D411yescharacteristic
77C33×C4C2×Q811yescharacteristic
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3C4⋊C412yes—
80C3×C3⋊Dic3C2×C812yescomplemented
81C22×C3×C8—92no—
82C2×C3×C42—91no—
83C2×D3×C8—34no—
84C42×D3—32no—
85C42×D3—61no—
86C3×C4⋊C8—32no—
87C3×C4⋊C8—32no—
88C3×C4⋊C8—62no—
89C3×C4⋊C8—92no—
90Dic3⋊C8—34no—
91C4⋊(C3⋊C8)C3×D312yescomplemented
92C22×C3⋊D3C2×C3×C411yescharacteristic
93C22×C3×D3—32no—
94C22×C3×D3—61no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4—22no—
101C2×C32×C4—41no—
102C2×C32×C4—41no—
103C2×C32×C4—41no—
104C2×C32×C4C4×D312yes—
105C2×C32×C4C4×D312yes—
106C2×C32×C4C2×C3×C411yescharacteristic
107C2×C32×C4C2×C3×C411yescharacteristic
108C2×C32×C4C22×D312yes—
109C2×C32×C4C23×C311yescharacteristic
110C2×C3⋊Dic3C2×C3×C411yescharacteristic
111C2×C3⋊Dic3C2×C3×C411yescharacteristic
112C2×C3⋊Dic3C2×C3×C411yescharacteristic
113C4×C3⋊D3—22no—
114C4×C3⋊D3C3×C814yescomplemented
115C4×C3⋊D3C3×D412yes—
116C4×C3⋊D3C3×Q812yes—
117C2×C3×Dic3—32no—
118C2×C3×Dic3—32no—
119C2×C3×Dic3—32no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C3×C4×D3—38no—
124C3×C4×D3—34no—
125C3×C4×D3—34no—
126C3×C4×D3—64no—
127C3×C4×D3—62no—
128C3×C4×D3—62no—
129C3×C4×D3—64no—
130C3×C4×D3—64no—
131C32×C8—64no—
132C3×C3⋊C8—24no—
133C3×C3⋊C8—24no—
134C3×C3⋊C8—44no—
135C2×C4⋊C8—91noSylow
136C2×C33C2×C4⋊C411yescharacteristic
137C2×C33C22×C811yescharacteristic
138C2×C33C2×M1611yescharacteristic
139C3×C3⋊D3C4⋊C814yescomplemented
140C22×C3×C4—91no—
141C22×C3×C4—91no—
142C22×C3×C4—91no—
143C2×C4×D3—32no—
144C2×C4×D3—32no—
145C2×C4×D3—32no—
146C2×C4×D3—61no—
147C2×C4×D3—61no—
148C2×C4×D3—61no—
149C2×C3×C8—34no—
150C2×C3×C8—34no—
151C2×C3×C8—64no—
152C2×C3×C8—184no—
153C3×C42—22no—
154C3×C42—21no—
155C3×C42—41no—
156C3×C42—91no—
157C3×C42—91no—
158C3×C42—91no—
159C3×C42D3211yescharacteristic
160C3×C42C2×C3×D312yes—
161C4×Dic3—32no—
162C4×Dic3—32no—
163C4×Dic3—32no—
164C4×Dic3—61no—
165C4×Dic3—61no—
166C4×Dic3—61no—
167C2×C3⋊C8C2×C3×D314yes—
168D3×C8—68no—
169C22×C32—21no—
170C22×C32—22no—
171C22×C32—41no—
172C22×C32C2×C4×D312yes—
173C22×C32C22×C3×C411yescharacteristic
174C2×C3⋊D3C3×C4⋊C412yes—
175C2×C3⋊D3C2×C3×C812yes—
176C2×C3⋊D3C3×M1612yes—
177C2×C3×D3—34no—
178C2×C3×D3—34no—
179C2×C3×D3—34no—
180C2×C3×D3—62no—
181C2×C3×D3—62no—
182C2×C3×D3—62no—
183C32×C4—21no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—24no—
188C32×C4—21no—
189C32×C4—21no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—42no—
193C32×C4—41no—
194C32×C4—41no—
195C32×C4—42no—
196C32×C4—42no—
197C32×C4D3×C814yescomplemented
198C32×C4C2×C3×C812yescomplemented
199C32×C4D3×D412yes—
200C32×C4D3×Q812yes—
201C32×C4C2×C3×D411yescharacteristic
202C32×C4C2×C3×Q811yescharacteristic
203C3⋊Dic3—21no—
204C3⋊Dic3C3×C4⋊C412yes—
205C3⋊Dic3C2×C3×C812yescomplemented
206C3×Dic3—34no—
207C3×Dic3—34no—
208C3×Dic3—62no—
209C3×Dic3—62no—
210C3×Dic3—62no—
211C3×Dic3—62no—
212C22×C8—92no—
213C2×C42—91no—
214C4⋊C8—32no—
215C4⋊C8—92no—
216C33C2×C4⋊C811yescharacteristic, complemented, Sylow
217C23×C3—91no—
218C22×D3—32no—
219C22×D3—61no—
220C2×C3×C4—22no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—22no—
224C2×C3×C4—22no—
225C2×C3×C4—21no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—91no—
230C2×C3×C4—91no—
231C2×C3×C4—94no—
232C2×C3×C4—92no—
233C2×C3×C4—92no—
234C2×C3×C4—91no—
235C2×C3×C4—182no—
236C2×C3×C4#62411yescharacteristic
237C2×C3×C4#62411yescharacteristic
238C2×C3×C4C3×C4×D312yes—
239C2×C3×C4C3×C4×D312yes—
240C2×C3×C4C2×D3211yescharacteristic
241C2×C3×C4C22×C3×D312yes—
242C2×Dic3—32no—
243C2×Dic3—32no—
244C2×Dic3—32no—
245C2×Dic3—61no—
246C2×Dic3—61no—
247C2×Dic3—61no—
248C4×D3—38no—
249C4×D3—34no—
250C4×D3—34no—
251C4×D3—64no—
252C4×D3—62no—
253C4×D3—62no—
254C4×D3—64no—
255C4×D3—64no—
256C3×C8—64no—
257C3×C8—64no—
258C3×C8—124no—
259C3⋊C8—24no—
260C2×C32—22no—
261C2×C32—22no—
262C2×C32—22no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—21no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32D3×C4⋊C412yes—
270C2×C32C2×D3×C812yes—
271C2×C32D3×M1612yes—
272C2×C32C2×C3×C4⋊C411yescharacteristic
273C2×C32C22×C3×C811yescharacteristic
274C2×C32C2×C3×M1611yescharacteristic
275C3⋊D3C3×C4⋊C814yescomplemented
276C3×D3—38no—
277C3×D3—64no—
278C22×C4—91no—
279C22×C4—91no—
280C22×C4—91no—
281C2×C8—34no—
282C2×C8—184no—
283C42—91no—
284C42—91no—
285C42—91no—
286C42C3×D3211yescharacteristic
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—41no—
290C22×C3—92no—
291C22×C3—92no—
292C22×C3—92no—
293C22×C3C2×#62411yescharacteristic
294C22×C3C2×C3×C4×D312yes—
295C2×D3—34no—
296C2×D3—34no—
297C2×D3—34no—
298C2×D3—62no—
299C2×D3—62no—
300C2×D3—62no—
301C3×C4—22no—
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—22no—
306C3×C4—22no—
307C3×C4—21no—
308C3×C4—22no—
309C3×C4—24no—
310C3×C4—42no—
311C3×C4—42no—
312C3×C4—41no—
313C3×C4—41no—
314C3×C4—42no—
315C3×C4—92no—
316C3×C4—92no—
317C3×C4—181no—
318C3×C4#371012yescomplemented
319C3×C4C3×D3×C814yescomplemented
320C3×C4#379711yescharacteristic
321C3×C4#380211yescharacteristic
322C3×C4C3×D3×D412yes—
323C3×C4C3×D3×Q812yes—
324Dic3—34no—
325Dic3—34no—
326Dic3—62no—
327Dic3—62no—
328Dic3—62no—
329Dic3—62no—
330C32—21no—
331C32—22no—
332C32—41no—
333C32D3×C4⋊C812yescomplemented
334C32C2×C3×C4⋊C811yescharacteristic, complemented
335C23—91no—
336C2×C4—91no—
337C2×C4—91no—
338C2×C4—92no—
339C2×C4—92no—
340C2×C4—94no—
341C2×C4—91no—
342C2×C4—182no—
343C2×C4C3×#62411yescharacteristic
344C2×C4C3×#62411yescharacteristic
345C2×C4C2×C3×D3211yescharacteristic
346C8—64no—
347C2×C3—21no—
348C2×C3—22no—
349C2×C3—21no—
350C2×C3—22no—
351C2×C3—21no—
352C2×C3—22no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—94no—
357C2×C3C2×#371011yescharacteristic
358C2×C3#6381811yescharacteristic
359C2×C3#6390011yescharacteristic
360C2×C3C3×D3×C4⋊C412yes—
361C2×C3C2×C3×D3×C812yes—
362C2×C3C3×D3×M1612yes—
363D3—38no—
364D3—64no—
365C22—92no—
366C22—92no—
367C22—92no—
368C22C2×C3×#62411yescharacteristic
369C4—21no—
370C4—92no—
371C4—92no—
372C4—181no—
373C4C3×#371012yescomplemented
374C4C3×#379711yescharacteristic
375C4C3×#380211yescharacteristic
376C3—21no—
377C3—22no—
378C3—41no—
379C3#1059033111yescharacteristic, direct product factor
380C3C3×D3×C4⋊C812yescomplemented
381C2—94no—
382C2C2×C3×#371011yescharacteristic
383C2C3×#6381811yescharacteristic
384C2C3×#6390011yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC2×C32C22×C3×C8
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C2×C3×C4⋊C8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
3. C32×C4⋊(C3⋊C8)C2semidirect
4. C3×#63572C2semidirect
6. #10590331C3direct (N × H)
24. C3×C4⋊(C3⋊C8)D3semidirect
25. C3×C4⋊(C3⋊C8)C2×C3semidirect
26. #63572C2×C3semidirect
35. C3×C4×C3⋊D3C8semidirect
75. C33×C4C2×C8semidirect
80. C3×C3⋊Dic3C2×C8semidirect
91. C4⋊(C3⋊C8)C3×D3semidirect
114. C4×C3⋊D3C3×C8semidirect
139. C3×C3⋊D3C4⋊C8semidirect
197. C32×C4D3×C8semidirect
198. C32×C4C2×C3×C8semidirect
205. C3⋊Dic3C2×C3×C8semidirect
216. C33C2×C4⋊C8semidirect
275. C3⋊D3C3×C4⋊C8semidirect
318. C3×C4#3710semidirect
319. C3×C4C3×D3×C8semidirect
333. C32D3×C4⋊C8semidirect
334. C32C2×C3×C4⋊C8semidirect
373. C4C3×#3710semidirect
379. C3#10590331direct (N × H)
380. C3C3×D3×C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C3×C42×C3⋊D32
C2×C3×#37102
C32×C4⋊(C3⋊C8)2
C3×#635722
C3×D3×C4⋊C83
#105903313