Finite Group Databasebeta

← C3×C4×#3718 #13157393

Subgroups — C3×C4×#3718 #13157393

Order 1728.

654 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×Dic6C212yesmaximal
2C2×C3×#3718C211yesmaximal, characteristic
3C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic
4C3×#63576C211yesmaximal, characteristic
5C3×#63565C212yesmaximal
6C3×C4×#216—32nomaximal
7C4×#3718C311yesmaximal, characteristic, direct product factor
8C2×C32×Dic6C2212yes—
9C33×C42C2211yescharacteristic
10C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
12C32×C4⋊Dic3C2212yes—
13C32×Dic3⋊C4—22no—
14C32×C4×Dic3—22no—
15C3×#3718C414yesdirect product factor
16C32×C4×Q8—32no—
17C2×C3×#216—32no—
18C3×C4×Dic6—22no—
19C3×C4×Dic6D312yes—
20C3×C4×Dic6C2×C312yes—
21C2×#3718C2×C311yescharacteristic
22C4×C3⋊(C3⋊C8)C2×C311yescharacteristic
23C3×#835—32no—
24C3×#810—32no—
25C3×#807—32no—
26C3×C4×C3⋊C8—32no—
27C3×C4×C3⋊C8—61no—
28#63576C2×C311yescharacteristic
29#63565C2×C312yes—
30C2×C33×C4D411yescharacteristic
31C2×C33×C4D411yescharacteristic
32C2×C33×C4C2311yescharacteristic
33C2×C32×Dic3—22no—
34C2×C32×Dic3—22no—
35C32×Dic6—22no—
36C32×Dic6C2×C414yes—
37C3×C3⋊(C3⋊C8)—21no—
38C3×C3⋊(C3⋊C8)C2×C412yes—
39C3×C4×Q16—91no—
40C4×#216—32no—
41C2×C32×Q8—32no—
42C2×C3×Dic6—22no—
43C2×C3×Dic6C2×D312yes—
44C2×C3×Dic6C22×C312yes—
45C32×C4⋊C4—32no—
46C32×C4⋊C4—62no—
47C32×C42—21no—
48C32×C42—22no—
49C32×C42—41no—
50C32×C42—62no—
51C32×C42C2×D312yes—
52C32×C42C22×C311yescharacteristic
53C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
54C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
55C3×#216—38no—
56C3×C4⋊Dic3—22no—
57C3×C4⋊Dic3C2×D312yes—
58C3×C4⋊Dic3C22×C312yes—
59C3×Dic3⋊C4—24no—
60C3×Dic3⋊C4—22no—
61C3×Dic3⋊C4—22no—
62C3×C4×Dic3—24no—
63C3×C4×Dic3—22no—
64C3×C4×Dic3—22no—
65C2×C3×C3⋊C8—32no—
66C2×C3×C3⋊C8—32no—
67C2×C3×C3⋊C8—61no—
68C2×C3×C3⋊C8—61no—
69#3718C3×C414yesdirect product factor
70C22×C33C2×D411yescharacteristic
71C33×C4—21no—
72C33×C4Q1612yescomplemented
73C33×C4C22×C411yescharacteristic
74C33×C4#4111yescharacteristic
75C32×Dic3—24no—
76C32×Dic3—42no—
77C2×C3×Q16—91no—
78C3×C4×Q8—32no—
79C3×C4×Q8—32no—
80C3×C4×Q8—62no—
81C2×#216—32no—
82C4×Dic6C3×D312yes—
83C3×#107—91no—
84C3×#103—92no—
85C3×C4×C8—91no—
86#835—32no—
87#810—32no—
88#807—32no—
89C4×C3⋊C8—32no—
90C4×C3⋊C8—61no—
91C32×Q8—34no—
92C32×Q8—62no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—22no—
96C2×C32×C4—22no—
97C2×C32×C4—21no—
98C2×C32×C4—22no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4—41no—
102C2×C32×C4—62no—
103C2×C32×C4—62no—
104C2×C32×C4#6712yes—
105C2×C32×C4#6712yes—
106C2×C32×C4C3×D411yescharacteristic
107C2×C32×C4C3×D411yescharacteristic
108C2×C32×C4C22×D312yes—
109C2×C32×C4C23×C311yescharacteristic
110C2×C3×Dic3—24no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—22no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—22no—
115C2×C3×Dic3—22no—
116C3×Dic6—24no—
117C3×Dic6—22no—
118C3×Dic6—22no—
119C3×Dic6—24no—
120C3×Dic6C4×D314yes—
121C3×Dic6C2×C3×C414yes—
122C3⋊(C3⋊C8)—21no—
123C3⋊(C3⋊C8)C2×C3×C412yes—
124C3×C3⋊C8—34no—
125C3×C3⋊C8—62no—
126C3×C3⋊C8—62no—
127C3×C3⋊C8—62no—
128C4×Q16—91noSylow
129C2×C33C4×D411yescharacteristic
130C2×C33C2×Q1611yescharacteristic
131C2×C33#13511yescharacteristic
132C2×C3×Q8—32no—
133C2×C3×Q8—32no—
134C2×C3×Q8—62no—
135C2×Dic6C2×C3×D312yes—
136C3×Q16—94no—
137C2×C3×C8—91no—
138C2×C3×C8—91no—
139C3×C4⋊C4—32no—
140C3×C4⋊C4—32no—
141C3×C4⋊C4—62no—
142C3×C4⋊C4—64no—
143C3×C4⋊C4—62no—
144C3×C4⋊C4—62no—
145C3×C42—21no—
146C3×C42—22no—
147C3×C42—41no—
148C3×C42—64no—
149C3×C42—62no—
150C3×C42—62no—
151C3×C42D3211yescharacteristic
152C3×C42C2×C3×D312yes—
153#216—38no—
154C4⋊Dic3C2×C3×D312yes—
155Dic3⋊C4—22no—
156C4×Dic3—22no—
157C2×C3⋊C8—32no—
158C2×C3⋊C8—32no—
159C2×C3⋊C8—61no—
160C2×C3⋊C8—61no—
161C22×C32—21no—
162C22×C32—22no—
163C22×C32—41no—
164C22×C32C2×#6712yes—
165C22×C32C2×C3×D411yescharacteristic
166C32×C4—21no—
167C32×C4—22no—
168C32×C4—22no—
169C32×C4—22no—
170C32×C4—24no—
171C32×C4—21no—
172C32×C4—21no—
173C32×C4—22no—
174C32×C4—22no—
175C32×C4—42no—
176C32×C4—41no—
177C32×C4—41no—
178C32×C4—42no—
179C32×C4—42no—
180C32×C4—64no—
181C32×C4—122no—
182C32×C4#21614yescomplemented
183C32×C4C3×Q1612yescomplemented
184C32×C4C2×C4×D312yes—
185C32×C4#23512yes—
186C32×C4C22×C3×C411yescharacteristic
187C32×C4C3×#4111yescharacteristic
188C3×Dic3—28no—
189C3×Dic3—24no—
190C3×Dic3—24no—
191C3×Dic3—44no—
192C3×Dic3—42no—
193C3×Dic3—42no—
194C2×Q16—91no—
195C4×Q8—32no—
196#107—91no—
197#103—92no—
198C4×C8—91no—
199C33C4×Q1611yescharacteristic, complemented, Sylow
200C3×Q8—34no—
201C3×Q8—34no—
202C3×Q8—64no—
203C3×Q8—64no—
204C3×Q8—62no—
205C3×Q8—62no—
206C2×C3×C4—22no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—21no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—64no—
216C2×C3×C4—62no—
217C2×C3×C4—62no—
218C2×C3×C4—64no—
219C2×C3×C4—62no—
220C2×C3×C4—62no—
221C2×C3×C4#62511yescharacteristic
222C2×C3×C4#62511yescharacteristic
223C2×C3×C4C3×#6712yes—
224C2×C3×C4C3×#6712yes—
225C2×C3×C4C2×D3211yescharacteristic
226C2×C3×C4C22×C3×D312yes—
227C2×Dic3—22no—
228C2×Dic3—22no—
229Dic6—22no—
230Dic6C3×C4×D314yes—
231C3×C8—92no—
232C3×C8—181no—
233C3⋊C8—34no—
234C3⋊C8—62no—
235C3⋊C8—62no—
236C3⋊C8—62no—
237C2×C32—22no—
238C2×C32—22no—
239C2×C32—22no—
240C2×C32—21no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—41no—
244C2×C32—41no—
245C2×C32—41no—
246C2×C32C4×#6712yes—
247C2×C32C2×#21612yes—
248C2×C32#95012yes—
249C2×C32C3×C4×D411yescharacteristic
250C2×C32C2×C3×Q1611yescharacteristic
251C2×C32C3×#13511yescharacteristic
252C2×Q8—32no—
253Q16—94no—
254C2×C8—91no—
255C2×C8—91no—
256C4⋊C4—32no—
257C4⋊C4—62no—
258C42—62no—
259C42C3×D3211yescharacteristic
260C22×C3—22no—
261C22×C3—21no—
262C22×C3—41no—
263C22×C3C2×#62511yescharacteristic
264C22×C3C2×C3×#6712yes—
265C3×C4—22no—
266C3×C4—22no—
267C3×C4—21no—
268C3×C4—21no—
269C3×C4—22no—
270C3×C4—21no—
271C3×C4—22no—
272C3×C4—24no—
273C3×C4—22no—
274C3×C4—42no—
275C3×C4—42no—
276C3×C4—41no—
277C3×C4—41no—
278C3×C4—42no—
279C3×C4—68no—
280C3×C4—64no—
281C3×C4—64no—
282C3×C4—124no—
283C3×C4—122no—
284C3×C4—122no—
285C3×C4#371812yesdirect product factor
286C3×C4C3×#21614yescomplemented
287C3×C4#379811yescharacteristic
288C3×C4C4×D3211yescharacteristic
289C3×C4C2×C3×C4×D312yes—
290C3×C4C3×#23512yes—
291Dic3—24no—
292Dic3—42no—
293C32—22no—
294C32—21no—
295C32—41no—
296C32C4×#21612yescomplemented
297C32C3×C4×Q1611yescharacteristic, complemented
298Q8—34no—
299Q8—62no—
300C2×C4—62no—
301C2×C4—62no—
302C2×C4C3×#62511yescharacteristic
303C2×C4C3×#62511yescharacteristic
304C2×C4C2×C3×D3211yescharacteristic
305C8—92no—
306C8—181no—
307C2×C3—21no—
308C2×C3—22no—
309C2×C3—22no—
310C2×C3—21no—
311C2×C3—22no—
312C2×C3—21no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3#6382211yescharacteristic
317C2×C3C2×#371811yescharacteristic
318C2×C3C4×#62511yescharacteristic
319C2×C3C3×C4×#6712yes—
320C2×C3C2×C3×#21612yes—
321C2×C3C3×#95012yes—
322C22C2×C3×#62511yescharacteristic
323C4—21no—
324C4—64no—
325C4—122no—
326C4C3×#371812yesdirect product factor
327C4C3×#379811yescharacteristic
328C4C3×C4×D3211yescharacteristic
329C3—22no—
330C3—21no—
331C3—41no—
332C3C4×#371811yescharacteristic, direct product factor
333C3C3×C4×#21612yescomplemented
334C2C3×#6382211yescharacteristic
335C2C2×C3×#371811yescharacteristic
336C2C3×C4×#62511yescharacteristic
subgroupG/H
centerC2×C3×C4#625
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×C4×Q16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
7. C4×#3718C3direct (N × H)
15. C3×#3718C4direct (N × H)
69. #3718C3×C4direct (N × H)
72. C33×C4Q16semidirect
182. C32×C4#216semidirect
183. C32×C4C3×Q16semidirect
199. C33C4×Q16semidirect
285. C3×C4#3718direct (N × H)
286. C3×C4C3×#216semidirect
296. C32C4×#216semidirect
297. C32C3×C4×Q16semidirect
326. C4C3×#3718direct (N × H)
332. C3C4×#3718direct (N × H)
333. C3C3×C4×#216semidirect

Maximal subgroups

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

subgroupindex
C32×C4×Dic62
C2×C3×#37182
C3×C4×C3⋊(C3⋊C8)2
C3×#635762
C3×#635652
C3×C4×#2163
C4×#37183