Finite Group Databasebeta

← C2×C3×#63870 #13174401

Subgroups — C2×C3×#63870 #13174401

Order 1728.

1250 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C32×Dic3⋊C4C212yesmaximal
3C2×C3×#3724C211yesmaximal, characteristic
4C2×C3×#3724C211yesmaximal, characteristic
5C2×C3×Dic3⋊Dic3C212yesmaximal
6C3×#63870C218yesmaximal, direct product factor
7C2×C3×#889—32nomaximal
8C2×#63870C311yesmaximal, characteristic, direct product factor
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2212yes—
14C3×C4×C3⋊Dic3C2214yes—
15C32×Dic3⋊C4C2218yes—
16C3×#3724C2214yes—
17C3×#3724C2214yes—
18C3×Dic3⋊Dic3C2218yes—
19C2×C32×C4⋊C4—32no—
20C2×C4×C3⋊Dic3C2×C311yescharacteristic
21C2×C3×C4⋊Dic3—32no—
22C2×C3×Dic3⋊C4—22no—
23C2×C3×Dic3⋊C4—32no—
24C2×C3×Dic3⋊C4—32no—
25C2×C3×Dic3⋊C4—32no—
26C2×C3×Dic3⋊C4D312yes—
27C2×C3×Dic3⋊C4C2×C312yes—
28C2×C3×C4×Dic3—32no—
29C2×C3×C4×Dic3—61no—
30C3×#889—316no—
31C2×#3724C2×C311yescharacteristic
32C2×#3724C2×C311yescharacteristic
33C2×Dic3⋊Dic3C2×C312yes—
34#63870C2×C318yesdirect product factor
35C23×C33C2311yescharacteristic
36C2×C33×C4—22no—
37C2×C33×C4C2312yes—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3Q814yes—
40C2×C3×C3⋊Dic3C2312yes—
41C2×C3×C3⋊Dic3C2312yes—
42C2×C32×Dic3—24no—
43C2×C32×Dic3—24no—
44C2×C32×Dic3C2314yes—
45C2×C32×Dic3C2314yes—
46C2×C3×#125—91no—
47C2×#889—32no—
48C22×C32×C4—21no—
49C22×C32×C4—22no—
50C22×C32×C4—32no—
51C22×C32×C4—32no—
52C22×C32×C4—41no—
53C22×C32×C4C2×D312yes—
54C22×C32×C4C22×C311yescharacteristic
55C22×C3⋊Dic3C22×C311yescharacteristic
56C22×C3⋊Dic3C22×C311yescharacteristic
57C22×C3×Dic3—22no—
58C22×C3×Dic3—22no—
59C22×C3×Dic3—32no—
60C22×C3×Dic3—32no—
61C22×C3×Dic3—61no—
62C22×C3×Dic3—61no—
63C22×C3×Dic3C2×D312yes—
64C22×C3×Dic3C2×D312yes—
65C22×C3×Dic3C22×C312yes—
66C22×C3×Dic3C22×C312yes—
67C32×C4⋊C4—38no—
68C4×C3⋊Dic3C22×C314yes—
69C3×C4⋊Dic3—38no—
70C3×Dic3⋊C4—28no—
71C3×Dic3⋊C4—38no—
72C3×Dic3⋊C4—38no—
73C3×Dic3⋊C4—38no—
74C3×Dic3⋊C4C2×D318yes—
75C3×Dic3⋊C4C22×C318yes—
76C3×C4×Dic3—38no—
77C3×C4×Dic3—64no—
78#3724C22×C314yes—
79#3724C22×C314yes—
80Dic3⋊Dic3C22×C318yes—
81C22×C33C2×Q812yes—
82C22×C33#4112yes—
83C22×C33#4112yes—
84C22×C33C2411yescharacteristic
85C33×C4—22no—
86C3×C3⋊Dic3—22no—
87C3×C3⋊Dic3C2×Q814yes—
88C32×Dic3—24no—
89C32×Dic3—24no—
90C3×#125—98no—
91C2×C3×C4⋊C4—32no—
92C2×C3×C4⋊C4—32no—
93C2×C3×C4⋊C4—62no—
94C2×C3×C4⋊C4—91no—
95C2×C3×C4⋊C4—91no—
96C2×C3×C4⋊C4—92no—
97C2×C3×C42—91no—
98C2×C4⋊Dic3—32no—
99C2×Dic3⋊C4—32no—
100C2×Dic3⋊C4—32no—
101C2×Dic3⋊C4—32no—
102C2×Dic3⋊C4C3×D312yes—
103C2×C4×Dic3—32no—
104C2×C4×Dic3—61no—
105#889—316no—
106C23×C32—21no—
107C23×C32—22no—
108C23×C32—41no—
109C23×C32C22×D312yes—
110C23×C32C23×C311yescharacteristic
111C2×C32×C4—22no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—22no—
115C2×C32×C4—24no—
116C2×C32×C4—34no—
117C2×C32×C4—34no—
118C2×C32×C4—42no—
119C2×C32×C4—44no—
120C2×C32×C4—44no—
121C2×C32×C4—64no—
122C2×C32×C4—64no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C23×C312yes—
125C2×C3⋊Dic3—22no—
126C2×C3⋊Dic3C3×Q814yes—
127C2×C3⋊Dic3C23×C312yes—
128C2×C3⋊Dic3C23×C312yes—
129C2×C3×Dic3—28no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—24no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—38no—
137C2×C3×Dic3—34no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—44no—
140C2×C3×Dic3—62no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3C22×D314yes—
146C2×C3×Dic3C22×D314yes—
147C2×C3×Dic3C23×C314yes—
148C2×C3×Dic3C23×C314yes—
149C2×#125—91noSylow
150C2×C33#12514yescomplemented
151C2×C33C22×Q811yescharacteristic
152C2×C33C2×#4111yescharacteristic
153C2×C33C2×#4111yescharacteristic
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—32no—
157C22×C3×C4—32no—
158C22×C3×C4—32no—
159C22×C3×C4—32no—
160C22×C3×C4—41no—
161C22×C3×C4—62no—
162C22×C3×C4—62no—
163C22×C3×C4—91no—
164C22×C3×C4—91no—
165C22×C3×C4D3211yescharacteristic
166C22×C3×C4C2×C3×D312yes—
167C22×Dic3—32no—
168C22×Dic3—32no—
169C22×Dic3—61no—
170C22×Dic3—61no—
171C22×Dic3C2×C3×D312yes—
172C22×Dic3C2×C3×D312yes—
173C3×C4⋊C4—38no—
174C3×C4⋊C4—38no—
175C3×C4⋊C4—68no—
176C3×C4⋊C4—94no—
177C3×C4⋊C4—94no—
178C3×C4⋊C4—98no—
179C3×C42—94no—
180C4⋊Dic3—38no—
181Dic3⋊C4—38no—
182Dic3⋊C4—38no—
183Dic3⋊C4—38no—
184Dic3⋊C4C2×C3×D318yes—
185C4×Dic3—38no—
186C4×Dic3—64no—
187C22×C32—21no—
188C22×C32—22no—
189C22×C32—22no—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—24no—
193C22×C32—24no—
194C22×C32—24no—
195C22×C32—42no—
196C22×C32—42no—
197C22×C32—42no—
198C22×C32—41no—
199C22×C32#23514yes—
200C22×C32#23714yes—
201C22×C32D3×Q814yes—
202C22×C32C2×C3×Q812yes—
203C22×C32C3×#4112yes—
204C22×C32C3×#4112yes—
205C22×C32C23×D312yes—
206C22×C32C24×C311yescharacteristic
207C32×C4—22no—
208C32×C4—24no—
209C32×C4—24no—
210C32×C4—44no—
211C32×C4—44no—
212C32×C4—64no—
213C32×C4—64no—
214C3⋊Dic3—22no—
215C3⋊Dic3C2×C3×Q814yes—
216C3×Dic3—28no—
217C3×Dic3—24no—
218C3×Dic3—24no—
219C3×Dic3—24no—
220C3×Dic3—24no—
221C3×Dic3—38no—
222C3×Dic3—44no—
223C3×Dic3—64no—
224C3×Dic3—64no—
225C3×Dic3—64no—
226#125—98no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—91no—
229C2×C4⋊C4—91no—
230C2×C4⋊C4—92no—
231C2×C42—91no—
232C33C2×#12511yescharacteristic, complemented, Sylow
233C23×C3—22no—
234C23×C3—21no—
235C23×C3—41no—
236C23×C3C2×D3211yescharacteristic
237C23×C3C22×C3×D312yes—
238C2×C3×C4—24no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—34no—
244C2×C3×C4—34no—
245C2×C3×C4—34no—
246C2×C3×C4—34no—
247C2×C3×C4—44no—
248C2×C3×C4—44no—
249C2×C3×C4—42no—
250C2×C3×C4—64no—
251C2×C3×C4—64no—
252C2×C3×C4—68no—
253C2×C3×C4—64no—
254C2×C3×C4—64no—
255C2×C3×C4—64no—
256C2×C3×C4—64no—
257C2×C3×C4—92no—
258C2×C3×C4—94no—
259C2×C3×C4—92no—
260C2×C3×C4—124no—
261C2×C3×C4—182no—
262C2×C3×C4C2×D3212yes—
263C2×C3×C4C22×C3×D314yes—
264C2×Dic3—24no—
265C2×Dic3—24no—
266C2×Dic3—34no—
267C2×Dic3—38no—
268C2×Dic3—34no—
269C2×Dic3—64no—
270C2×Dic3—62no—
271C2×Dic3—62no—
272C2×Dic3—64no—
273C2×Dic3—64no—
274C2×Dic3C22×C3×D314yes—
275C2×Dic3C22×C3×D314yes—
276C2×C32—28no—
277C2×C32—21no—
278C2×C32—21no—
279C2×C32—21no—
280C2×C32—22no—
281C2×C32—22no—
282C2×C32—22no—
283C2×C32—24no—
284C2×C32—44no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32#88918yescomplemented
289C2×C32C3×#12514yescomplemented
290C2×C32C2×#23512yes—
291C2×C32C2×#23712yes—
292C2×C32C2×D3×Q812yes—
293C2×C32C22×C3×Q811yescharacteristic
294C2×C32C2×C3×#4111yescharacteristic
295C2×C32C2×C3×#4111yescharacteristic
296C22×C4—32no—
297C22×C4—32no—
298C22×C4—91no—
299C22×C4—91no—
300C22×C4C3×D3211yescharacteristic
301C4⋊C4—38no—
302C4⋊C4—94no—
303C4⋊C4—94no—
304C4⋊C4—98no—
305C42—94no—
306C22×C3—21no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—24no—
310C22×C3—22no—
311C22×C3—24no—
312C22×C3—22no—
313C22×C3—24no—
314C22×C3—41no—
315C22×C3—42no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3#379712yes—
319C22×C3#379812yes—
320C22×C3#380512yes—
321C22×C3C3×#23514yes—
322C22×C3C3×#23714yes—
323C22×C3C3×D3×Q814yes—
324C22×C3C22×D3211yescharacteristic
325C22×C3C23×C3×D312yes—
326C3×C4—24no—
327C3×C4—22no—
328C3×C4—24no—
329C3×C4—44no—
330C3×C4—44no—
331C3×C4—68no—
332C3×C4—64no—
333C3×C4—64no—
334C3×C4—64no—
335C3×C4—64no—
336C3×C4—94no—
337C3×C4—124no—
338C3×C4—182no—
339Dic3—24no—
340Dic3—24no—
341Dic3—38no—
342Dic3—64no—
343Dic3—64no—
344Dic3—64no—
345C32—22no—
346C32—21no—
347C32—41no—
348C32C2×#88912yescomplemented
349C32C2×C3×#12511yescharacteristic, complemented
350C23C2×C3×D3211yescharacteristic
351C2×C4—22no—
352C2×C4—34no—
353C2×C4—34no—
354C2×C4—64no—
355C2×C4—64no—
356C2×C4—92no—
357C2×C4—94no—
358C2×C4—92no—
359C2×C4—182no—
360C2×C4C2×C3×D3212yes—
361C2×C3—21no—
362C2×C3—22no—
363C2×C3—28no—
364C2×C3—24no—
365C2×C3—21no—
366C2×C3—22no—
367C2×C3—21no—
368C2×C3—22no—
369C2×C3—41no—
370C2×C3—44no—
371C2×C3—41no—
372C2×C3—41no—
373C2×C3#6387014yesdirect product factor
374C2×C3C3×#88918yescomplemented
375C2×C3C2×#379711yescharacteristic
376C2×C3C2×#379811yescharacteristic
377C2×C3C2×#380511yescharacteristic
378C2×C3C2×C3×#23512yes—
379C2×C3C2×C3×#23712yes—
380C2×C3C2×C3×D3×Q812yes—
381C22C3×#379712yes—
382C22C3×#379812yes—
383C22C3×#380512yes—
384C22C22×C3×D3211yescharacteristic
385C4—22no—
386C4—64no—
387C4—64no—
388C4—94no—
389C4—182no—
390C3—22no—
391C3—21no—
392C3—41no—
393C3C2×#6387011yescharacteristic, direct product factor
394C3C2×C3×#88912yescomplemented
395C2C3×#6387014yesdirect product factor
396C2C2×C3×#379711yescharacteristic
397C2C2×C3×#379811yescharacteristic
398C2C2×C3×#380511yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×#125
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
6. C3×#63870C2direct (N × H)
8. C2×#63870C3direct (N × H)
34. #63870C2×C3direct (N × H)
150. C2×C33#125semidirect
232. C33C2×#125semidirect
288. C2×C32#889semidirect
289. C2×C32C3×#125semidirect
348. C32C2×#889semidirect
349. C32C2×C3×#125semidirect
373. C2×C3#63870direct (N × H)
374. C2×C3C3×#889semidirect
393. C3C2×#63870direct (N × H)
394. C3C2×C3×#889semidirect
395. C2C3×#63870direct (N × H)

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×Dic3⋊C42
C2×C3×#37242
C2×C3×Dic3⋊Dic32
C3×#638702
C2×C3×#8893
C2×#638703