Finite Group Databasebeta

← C2×C3×#63557 #13158098

Subgroups — C2×C3×#63557 #13158098

Order 1728.

1250 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C22×C32×C3⋊C8C212yesmaximal, complemented
3C3×#63557C214yesmaximal, direct product factor
4C2×C3×(C2×D3)⋊C8—32nomaximal
5C2×#63557C311yesmaximal, characteristic, direct product factor
6C23×C3×C3⋊D3C411yescharacteristic
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C411yescharacteristic
9C2×C3×C4×C3⋊D3—24no—
10C2×C3×C4×C3⋊D3C412yes—
11C2×C3×C4×C3⋊D3C2212yes—
12C2×C32×C3⋊C8—24no—
13C2×C32×C3⋊C8C2214yescomplemented
14C22×C4×C3⋊D3C2×C311yescharacteristic
15C22×C3×C4×D3—32no—
16C22×C3×C4×D3—61no—
17C22×C32×C8—32no—
18C22×C3×C3⋊C8—22no—
19C22×C3×C3⋊C8D312yescomplemented
20C22×C3×C3⋊C8C2×C312yescomplemented
21C3×(C2×D3)⋊C8—38no—
22#63557C2×C314yesdirect product factor
23C23×C33C2×C411yescharacteristic
24C22×C3×C3⋊D3—22no—
25C22×C3×C3⋊D3—22no—
26C22×C3×C3⋊D3C814yescomplemented
27C22×C3×C3⋊D3C2×C412yes—
28C2×C33×C4C2×C411yescharacteristic
29C2×C33×C4D414yes—
30C2×C33×C4C2311yescharacteristic
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3C2×C412yes—
33C3×C4×C3⋊D3—24no—
34C3×C4×C3⋊D3—24no—
35C32×C3⋊C8—24no—
36C2×C3×C22⋊C8—91no—
37C2×(C2×D3)⋊C8—32no—
38C23×C3⋊D3C3×C411yescharacteristic
39C23×C3×D3—32no—
40C23×C3×D3—61no—
41C22×C32×C4—21no—
42C22×C32×C4—22no—
43C22×C32×C4—41no—
44C22×C32×C4C2×D312yes—
45C22×C32×C4C22×C311yescharacteristic
46C22×C3⋊Dic3C3×C411yescharacteristic
47C2×C4×C3⋊D3—24no—
48C2×C4×C3⋊D3C3×C412yes—
49C2×C4×C3⋊D3C22×C312yes—
50C22×C3×Dic3—32no—
51C22×C3×Dic3—61no—
52C2×C3×C4×D3—34no—
53C2×C3×C4×D3—34no—
54C2×C3×C4×D3—62no—
55C2×C3×C4×D3—62no—
56C2×C3×C4×D3—68no—
57C2×C3×C4×D3—64no—
58C2×C3×C4×D3—64no—
59C2×C32×C8—34no—
60C2×C32×C8—64no—
61C2×C3×C3⋊C8—28no—
62C2×C3×C3⋊C8—24no—
63C2×C3×C3⋊C8—24no—
64C2×C3×C3⋊C8—24no—
65C2×C3×C3⋊C8C2×D314yescomplemented
66C2×C3×C3⋊C8C22×C314yescomplemented
67C22×C33C22⋊C412yes—
68C22×C33C2×C812yes—
69C22×C33M1612yes—
70C22×C33C22×C411yescharacteristic
71C2×C3×C3⋊D3—28no—
72C2×C3×C3⋊D3—22no—
73C2×C3×C3⋊D3—22no—
74C2×C3×C3⋊D3C2×C814yescomplemented
75C33×C4C22⋊C412yes—
76C33×C4C2×D412yes—
77C3×C3⋊Dic3—22no—
78C23×C3×C4—91no—
79C22×C4×D3—32no—
80C22×C4×D3—61no—
81C22×C3×C8—32no—
82C22×C3×C8—32no—
83C22×C3×C8—62no—
84C22×C3⋊C8C3×D312yescomplemented
85C3×C22⋊C8—94no—
86(C2×D3)⋊C8—38no—
87C23×C32—21no—
88C23×C32—22no—
89C23×C32—41no—
90C23×C32C4×D312yes—
91C23×C32C2×C3×C411yescharacteristic
92C22×C3⋊D3—22no—
93C22×C3⋊D3—22no—
94C22×C3⋊D3C3×C814yescomplemented
95C22×C3⋊D3C2×C3×C412yes—
96C22×C3×D3—38no—
97C22×C3×D3—34no—
98C22×C3×D3—64no—
99C22×C3×D3—62no—
100C22×C3×D3—64no—
101C22×C3×D3—64no—
102C22×C3×D3—64no—
103C22×C3×D3—64no—
104C2×C32×C4—21no—
105C2×C32×C4—24no—
106C2×C32×C4—22no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—21no—
110C2×C32×C4—22no—
111C2×C32×C4—41no—
112C2×C32×C4—44no—
113C2×C32×C4—41no—
114C2×C32×C4C4×D312yes—
115C2×C32×C4D1214yes—
116C2×C32×C4#6714yes—
117C2×C32×C4C2×C3×C411yescharacteristic
118C2×C32×C4C3×D414yes—
119C2×C32×C4C22×D312yes—
120C2×C32×C4C23×C311yescharacteristic
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3C2×C3×C412yes—
123C4×C3⋊D3—24no—
124C4×C3⋊D3—24no—
125C2×C3×Dic3—34no—
126C2×C3×Dic3—62no—
127C2×C3×Dic3—64no—
128C2×C3×Dic3—64no—
129C3×C4×D3—68no—
130C3×C4×D3—68no—
131C3×C4×D3—68no—
132C3×C4×D3—64no—
133C3×C4×D3—64no—
134C32×C8—64no—
135C3×C3⋊C8—28no—
136C3×C3⋊C8—24no—
137C3×C3⋊C8—24no—
138C2×C22⋊C8—91noSylow
139C2×C33C22⋊C814yescomplemented
140C2×C33C2×C22⋊C411yescharacteristic
141C2×C33C22×C811yescharacteristic
142C2×C33C2×M1611yescharacteristic
143C3×C3⋊D3—24no—
144C24×C3—91no—
145C23×D3—32no—
146C23×D3—61no—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—41no—
150C22×C3×C4—91no—
151C22×C3×C4—92no—
152C22×C3×C4—92no—
153C22×C3×C4—184no—
154C22×C3×C4D3211yescharacteristic
155C22×C3×C4C2×C3×D312yes—
156C22×Dic3—32no—
157C22×Dic3—61no—
158C2×C4×D3—34no—
159C2×C4×D3—34no—
160C2×C4×D3—62no—
161C2×C4×D3—62no—
162C2×C4×D3—68no—
163C2×C4×D3—64no—
164C2×C4×D3—64no—
165C2×C3×C8—34no—
166C2×C3×C8—34no—
167C2×C3×C8—64no—
168C2×C3×C8—68no—
169C2×C3×C8—64no—
170C2×C3×C8—64no—
171C2×C3⋊C8—24no—
172C2×C3⋊C8C2×C3×D314yescomplemented
173C22×C32—22no—
174C22×C32—22no—
175C22×C32—22no—
176C22×C32—24no—
177C22×C32—24no—
178C22×C32—24no—
179C22×C32—21no—
180C22×C32—22no—
181C22×C32—42no—
182C22×C32—42no—
183C22×C32—42no—
184C22×C32—41no—
185C22×C32D3×C814yes—
186C22×C32ΓL(1,𝔽25)14yes—
187C22×C32(C2×D3)⋊C414yes—
188C22×C32C3×C22⋊C412yes—
189C22×C32C2×C3×C812yes—
190C22×C32C3×M1612yes—
191C22×C32C2×C4×D312yes—
192C22×C32C22×C3×C411yescharacteristic
193C2×C3⋊D3—28no—
194C2×C3⋊D3—22no—
195C2×C3⋊D3—22no—
196C2×C3⋊D3C2×C3×C814yescomplemented
197C2×C3×D3—38no—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C2×C3×D3—616no—
201C2×C3×D3—64no—
202C2×C3×D3—616no—
203C2×C3×D3—64no—
204C2×C3×D3—64no—
205C32×C4—22no—
206C32×C4—24no—
207C32×C4—22no—
208C32×C4—22no—
209C32×C4—22no—
210C32×C4—42no—
211C32×C4—42no—
212C32×C4(C2×D3)⋊C414yes—
213C32×C4C3×C22⋊C412yes—
214C32×C4C2×D1212yes—
215C32×C4C2×#6712yes—
216C32×C4C2×C3×D412yes—
217C3⋊Dic3—22no—
218C3×Dic3—64no—
219C3×Dic3—64no—
220C23×C4—91no—
221C22×C8—32no—
222C22⋊C8—94no—
223C33C2×C22⋊C811yescharacteristic, complemented, Sylow
224C23×C3—22no—
225C23×C3—21no—
226C23×C3—41no—
227C23×C3—94no—
228C23×C3—92no—
229C23×C3—182no—
230C23×C3—182no—
231C23×C3#62411yescharacteristic
232C23×C3C3×C4×D312yes—
233C22×D3—38no—
234C22×D3—34no—
235C22×D3—64no—
236C22×D3—62no—
237C22×D3—64no—
238C22×D3—64no—
239C22×D3—64no—
240C22×D3—64no—
241C2×C3×C4—22no—
242C2×C3×C4—21no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—22no—
247C2×C3×C4—21no—
248C2×C3×C4—41no—
249C2×C3×C4—44no—
250C2×C3×C4—41no—
251C2×C3×C4—92no—
252C2×C3×C4—182no—
253C2×C3×C4—184no—
254C2×C3×C4—184no—
255C2×C3×C4#62411yescharacteristic
256C2×C3×C4#62614yes—
257C2×C3×C4C3×C4×D312yes—
258C2×C3×C4C3×D1214yes—
259C2×C3×C4C3×#6714yes—
260C2×C3×C4C2×D3211yescharacteristic
261C2×C3×C4C22×C3×D312yes—
262C2×Dic3—34no—
263C2×Dic3—62no—
264C2×Dic3—64no—
265C2×Dic3—64no—
266C4×D3—68no—
267C4×D3—68no—
268C4×D3—68no—
269C4×D3—64no—
270C4×D3—64no—
271C3×C8—68no—
272C3×C8—64no—
273C3×C8—64no—
274C3⋊C8—24no—
275C2×C32—22no—
276C2×C32—22no—
277C2×C32—22no—
278C2×C32—24no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—28no—
282C2×C32—21no—
283C2×C32—44no—
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—41no—
287C2×C32(C2×D3)⋊C818yescomplemented
288C2×C32C3×C22⋊C814yescomplemented
289C2×C32C2×D3×C812yes—
290C2×C32C2×ΓL(1,𝔽25)12yes—
291C2×C32C2×(C2×D3)⋊C412yes—
292C2×C32C2×C3×C22⋊C411yescharacteristic
293C2×C32C22×C3×C811yescharacteristic
294C2×C32C2×C3×M1611yescharacteristic
295C3⋊D3—24no—
296C3×D3—68no—
297C3×D3—68no—
298C24—91no—
299C22×C4—91no—
300C22×C4—92no—
301C22×C4—92no—
302C22×C4—184no—
303C22×C4C3×D3211yescharacteristic
304C2×C8—34no—
305C2×C8—64no—
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—94no—
319C22×C3—188no—
320C22×C3—182no—
321C22×C3—182no—
322C22×C3#371012yes—
323C22×C3#371212yes—
324C22×C3#372212yes—
325C22×C3C3×D3×C814yes—
326C22×C3C3×ΓL(1,𝔽25)14yes—
327C22×C3C3×(C2×D3)⋊C414yes—
328C22×C3C2×#62411yescharacteristic
329C22×C3C2×C3×C4×D312yes—
330C2×D3—38no—
331C2×D3—64no—
332C2×D3—64no—
333C2×D3—64no—
334C2×D3—616no—
335C2×D3—64no—
336C2×D3—64no—
337C2×D3—616no—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—24no—
343C3×C4—42no—
344C3×C4—42no—
345C3×C4—182no—
346C3×C4#372212yes—
347C3×C4C3×(C2×D3)⋊C414yes—
348C3×C4C2×#62612yes—
349C3×C4C2×C3×D1212yes—
350C3×C4C2×C3×#6712yes—
351Dic3—64no—
352Dic3—64no—
353C32—21no—
354C32—22no—
355C32—41no—
356C32C2×(C2×D3)⋊C812yescomplemented
357C32C2×C3×C22⋊C811yescharacteristic, complemented
358C23—92no—
359C23—94no—
360C23—182no—
361C23—182no—
362C23C3×#62411yescharacteristic
363C2×C4—92no—
364C2×C4—182no—
365C2×C4—184no—
366C2×C4—184no—
367C2×C4C3×#62411yescharacteristic
368C2×C4C3×#62614yes—
369C2×C4C2×C3×D3211yescharacteristic
370C8—64no—
371C2×C3—21no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—22no—
377C2×C3—24no—
378C2×C3—28no—
379C2×C3—41no—
380C2×C3—41no—
381C2×C3—41no—
382C2×C3—44no—
383C2×C3—184no—
384C2×C3#6355714yesdirect product factor
385C2×C3C3×(C2×D3)⋊C818yescomplemented
386C2×C3C2×#371011yescharacteristic
387C2×C3C2×#371211yescharacteristic
388C2×C3C2×#372211yescharacteristic
389C2×C3C2×C3×D3×C812yes—
390C2×C3C2×C3×ΓL(1,𝔽25)12yes—
391C2×C3C2×C3×(C2×D3)⋊C412yes—
392D3—68no—
393D3—68no—
394C22—94no—
395C22—182no—
396C22—182no—
397C22—188no—
398C22C3×#371012yes—
399C22C3×#371212yes—
400C22C3×#372212yes—
401C22C2×C3×#62411yescharacteristic
402C4—182no—
403C4C3×#372212yes—
404C4C2×C3×#62612yes—
405C3—21no—
406C3—22no—
407C3—41no—
408C3C2×#6355711yescharacteristic, direct product factor
409C3C2×C3×(C2×D3)⋊C812yescomplemented
410C2—184no—
411C2C3×#6355714yesdirect product factor
412C2C2×C3×#371011yescharacteristic
413C2C2×C3×#371211yescharacteristic
414C2C2×C3×#372211yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutatorC2×C32C22×C3×C8
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C2×C4
prime radicalC23×C3×C3⋊D3C4
cosocleC2×C32×C4C23×C3
hypocenterC32C2×C3×C22⋊C8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C22×C32×C3⋊C8C2semidirect
3. C3×#63557C2direct (N × H)
5. C2×#63557C3direct (N × H)
13. C2×C32×C3⋊C8C22semidirect
19. C22×C3×C3⋊C8D3semidirect
20. C22×C3×C3⋊C8C2×C3semidirect
22. #63557C2×C3direct (N × H)
26. C22×C3×C3⋊D3C8semidirect
65. C2×C3×C3⋊C8C2×D3semidirect
66. C2×C3×C3⋊C8C22×C3semidirect
74. C2×C3×C3⋊D3C2×C8semidirect
84. C22×C3⋊C8C3×D3semidirect
94. C22×C3⋊D3C3×C8semidirect
139. C2×C33C22⋊C8semidirect
172. C2×C3⋊C8C2×C3×D3semidirect
196. C2×C3⋊D3C2×C3×C8semidirect
223. C33C2×C22⋊C8semidirect
287. C2×C32(C2×D3)⋊C8semidirect
288. C2×C32C3×C22⋊C8semidirect
356. C32C2×(C2×D3)⋊C8semidirect
357. C32C2×C3×C22⋊C8semidirect
384. C2×C3#63557direct (N × H)
385. C2×C3C3×(C2×D3)⋊C8semidirect
408. C3C2×#63557direct (N × H)
409. C3C2×C3×(C2×D3)⋊C8semidirect
411. C2C3×#63557direct (N × H)

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
C22×C32×C3⋊C82
C3×#635572
C2×C3×(C2×D3)⋊C83
C2×#635573