Finite Group Databasebeta

← C2×C3×#4528 #11773733

Subgroups — C2×C3×#4528 #11773733

Order 1152.

865 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C4⋊C4C211yesmaximal, characteristic
2C2×C3×#830C212yesmaximal
3C3×#4528C214yesmaximal, direct product factor
4C2×C3×#324—31nomaximal
5C2×#4528C311yesmaximal, characteristic, direct product factor
6C23×C32×C4C411yescharacteristic
7C23×C32×C4C411yescharacteristic
8C23×C32×C4C2211yescharacteristic
9C2×C32×C4⋊C4—22no—
10C2×C32×C4⋊C4—22no—
11C2×C32×C4⋊C4C412yes—
12C2×C32×C4⋊C4C2212yes—
13C22×C3×C3⋊C8—22no—
14C3×#830—22no—
15C3×#830C2214yes—
16C22×C3×C4⋊C4—21no—
17C22×C3×C4⋊C4D311yescharacteristic
18C22×C3×C4⋊C4C2×C311yescharacteristic
19C2×C3×C22⋊C8—32no—
20C2×#830C2×C312yes—
21C3×#324—34no—
22#4528C2×C314yesdirect product factor
23C24×C32C2×C411yescharacteristic
24C22×C32×C4—21no—
25C22×C32×C4—21no—
26C22×C32×C4—22no—
27C22×C32×C4—21no—
28C22×C32×C4—21no—
29C22×C32×C4—42no—
30C22×C32×C4—42no—
31C22×C32×C4C814yescomplemented
32C22×C32×C4C2×C412yes—
33C22×C32×C4C2×C412yes—
34C22×C32×C4C2×C411yescharacteristic
35C22×C32×C4D412yes—
36C22×C32×C4D412yes—
37C22×C32×C4C2311yescharacteristic
38C32×C4⋊C4—24no—
39C32×C4⋊C4—24no—
40C2×C3×C3⋊C8—24no—
41C2×C3×C3⋊C8—44no—
42C2×#324—31noSylow
43C23×C3×C4—21no—
44C23×C3×C4—21no—
45C23×C3×C4—21no—
46C23×C3×C4Dic311yescharacteristic
47C23×C3×C4Dic311yescharacteristic
48C23×C3×C4C3×C411yescharacteristic
49C23×C3×C4C3×C411yescharacteristic
50C23×C3×C4C2×D311yescharacteristic
51C23×C3×C4C22×C311yescharacteristic
52C22×C3×C8—62no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4Dic312yes—
62C2×C3×C4⋊C4C3×C412yes—
63C2×C3×C4⋊C4C2×D312yes—
64C2×C3×C4⋊C4C22×C312yes—
65C22×C3⋊C8—22no—
66C3×C22⋊C8—34no—
67C3×C22⋊C8—62no—
68#830—22no—
69#830C22×C314yes—
70C23×C32—22no—
71C23×C32—22no—
72C23×C32C22⋊C411yescharacteristic
73C23×C32C22⋊C411yescharacteristic
74C23×C32C2×C812yes—
75C23×C32M1612yes—
76C23×C32C22×C411yescharacteristic
77C2×C32×C4—24no—
78C2×C32×C4—24no—
79C2×C32×C4—22no—
80C2×C32×C4—22no—
81C2×C32×C4—22no—
82C2×C32×C4—24no—
83C2×C32×C4—41no—
84C2×C32×C4—42no—
85C2×C32×C4—41no—
86C2×C32×C4—42no—
87C2×C32×C4—42no—
88C2×C32×C4—42no—
89C2×C32×C4C22⋊C412yes—
90C2×C32×C4C2×C814yescomplemented
91C2×C32×C4C2×D412yes—
92C3×C3⋊C8—44no—
93C22×C4⋊C4C3×D311yescharacteristic
94C2×C22⋊C8—32no—
95#324—34no—
96C24×C3—21no—
97C24×C3C2×Dic311yescharacteristic
98C24×C3C2×C3×C411yescharacteristic
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—24no—
102C22×C3×C4—22no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—24no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—42no—
122C22×C3×C4—42no—
123C22×C3×C4—44no—
124C22×C3×C4—44no—
125C22×C3×C4—42no—
126C22×C3×C4—42no—
127C22×C3×C4C3⋊C814yescomplemented
128C22×C3×C4C3×C814yescomplemented
129C22×C3×C4C2×Dic312yes—
130C22×C3×C4C2×Dic312yes—
131C22×C3×C4C2×Dic311yescharacteristic
132C22×C3×C4#6712yes—
133C22×C3×C4#6712yes—
134C22×C3×C4C2×C3×C412yes—
135C22×C3×C4C2×C3×C412yes—
136C22×C3×C4C2×C3×C411yescharacteristic
137C22×C3×C4C3×D412yes—
138C22×C3×C4C3×D412yes—
139C22×C3×C4C22×D311yescharacteristic
140C22×C3×C4C23×C311yescharacteristic
141C2×C3×C8—64no—
142C2×C3×C8—124no—
143C3×C4⋊C4—28no—
144C3×C4⋊C4—28no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4—24no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—24no—
149C2×C3⋊C8—24no—
150C2×C3⋊C8—44no—
151C22×C32—24no—
152C22×C32—24no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32C22⋊C812yes—
158C22×C32C22⋊C812yes—
159C22×C32#9912yes—
160C22×C32#10112yes—
161C22×C32C2×C22⋊C411yescharacteristic
162C22×C32C22×C811yescharacteristic
163C22×C32C2×M1611yescharacteristic
164C32×C4—24no—
165C32×C4—22no—
166C32×C4—22no—
167C32×C4—42no—
168C23×C4C3×Dic311yescharacteristic
169C23×C4C3×Dic311yescharacteristic
170C23×C4C2×C3×D311yescharacteristic
171C22×C8—62no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C3×Dic312yes—
175C2×C4⋊C4C2×C3×D312yes—
176C22⋊C8—34no—
177C22⋊C8—62no—
178C23×C3—24no—
179C23×C3—24no—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—22no—
183C23×C3—22no—
184C23×C3—22no—
185C23×C3—22no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3C2×C3⋊C812yes—
190C23×C3#20812yes—
191C23×C3#21711yescharacteristic
192C23×C3#21711yescharacteristic
193C23×C3C3×C22⋊C411yescharacteristic
194C23×C3C3×C22⋊C411yescharacteristic
195C23×C3C2×C3×C812yes—
196C23×C3C3×M1612yes—
197C23×C3C22×Dic311yescharacteristic
198C23×C3C22×C3×C411yescharacteristic
199C2×C3×C4—24no—
200C2×C3×C4—24no—
201C2×C3×C4—28no—
202C2×C3×C4—28no—
203C2×C3×C4—24no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
206C2×C3×C4—24no—
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—24no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—24no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—44no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—44no—
228C2×C3×C4—42no—
229C2×C3×C4—44no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—44no—
233C2×C3×C4—44no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4C2×C3⋊C814yescomplemented
239C2×C3×C4#21712yes—
240C2×C3×C4C3×C22⋊C412yes—
241C2×C3×C4C2×C3×C814yescomplemented
242C2×C3×C4C2×#6712yes—
243C2×C3×C4C2×C3×D412yes—
244C3×C8—124no—
245C3⋊C8—44no—
246C2×C32—22no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32#32414yescomplemented
250C2×C32C2×C22⋊C811yescharacteristic
251C2×C32C2×#9911yescharacteristic
252C2×C32C2×#10111yescharacteristic
253C24C2×C3×Dic311yescharacteristic
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4C3×C3⋊C814yescomplemented
262C22×C4C2×C3×Dic312yes—
263C22×C4C2×C3×Dic312yes—
264C22×C4C2×C3×Dic311yescharacteristic
265C22×C4C3×#6712yes—
266C22×C4C3×#6712yes—
267C22×C4C22×C3×D311yescharacteristic
268C2×C8—64no—
269C2×C8—124no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—28no—
277C22×C3—28no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—24no—
287C22×C3—24no—
288C22×C3—24no—
289C22×C3—24no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3#83012yes—
298C22×C3#83012yes—
299C22×C3#83412yes—
300C22×C3#83612yes—
301C22×C3C3×C22⋊C812yes—
302C22×C3C3×C22⋊C812yes—
303C22×C3C3×#9912yes—
304C22×C3C3×#10112yes—
305C22×C3C22×C3⋊C811yescharacteristic
306C22×C3C2×#20811yescharacteristic
307C22×C3C2×#21711yescharacteristic
308C22×C3C2×C3×C22⋊C411yescharacteristic
309C22×C3C22×C3×C811yescharacteristic
310C22×C3C2×C3×M1611yescharacteristic
311C3×C4—28no—
312C3×C4—24no—
313C3×C4—22no—
314C3×C4—22no—
315C3×C4—24no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—42no—
319C3×C4—42no—
320C3×C4—44no—
321C3×C4—42no—
322C3×C4—42no—
323C32C2×#32411yescharacteristic, complemented, Sylow
324C23—22no—
325C23—22no—
326C23C2×C3×C3⋊C812yes—
327C23C3×#20812yes—
328C23C3×#21711yescharacteristic
329C23C3×#21711yescharacteristic
330C23C22×C3×Dic311yescharacteristic
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4C2×C3×C3⋊C814yescomplemented
344C2×C4C3×#21712yes—
345C2×C4C2×C3×#6712yes—
346C8—124no—
347C2×C3—22no—
348C2×C3—22no—
349C2×C3—24no—
350C2×C3—22no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—22no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—24no—
360C2×C3#452814yesdirect product factor
361C2×C3C3×#32414yescomplemented
362C2×C3C2×#83011yescharacteristic
363C2×C3C2×#83411yescharacteristic
364C2×C3C2×#83611yescharacteristic
365C2×C3C2×C3×C22⋊C811yescharacteristic
366C2×C3C2×C3×#9911yescharacteristic
367C2×C3C2×C3×#10111yescharacteristic
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—24no—
373C22—24no—
374C22C3×#83012yes—
375C22C3×#83012yes—
376C22C3×#83412yes—
377C22C3×#83612yes—
378C22C22×C3×C3⋊C811yescharacteristic
379C22C2×C3×#20811yescharacteristic
380C22C2×C3×#21711yescharacteristic
381C4—24no—
382C4—22no—
383C4—22no—
384C4—42no—
385C3—21no—
386C3C2×#452811yescharacteristic, direct product factor
387C3C2×C3×#32411yescharacteristic, complemented
388C2—21no—
389C2—21no—
390C2—22no—
391C2C3×#452814yesdirect product factor
392C2C2×C3×#83011yescharacteristic
393C2C2×C3×#83411yescharacteristic
394C2C2×C3×#83611yescharacteristic
subgroupG/H
centerC23×C3#217
commutatorC22×C3C22×C3×C8
FrattiniC22×C4C22×C3×D3
FittingC22×C32×C4⋊C4C2
socleC23×C32C22⋊C4
prime radicalC24×C32C2×C4
cosocleC22×C3×C4C23×C3
hypercenterC22×C3×C4⋊C4D3
hypocenterC3C2×C3×#324
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C22×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#4528C2direct (N × H)
5. C2×#4528C3direct (N × H)
22. #4528C2×C3direct (N × H)
31. C22×C32×C4C8semidirect
90. C2×C32×C4C2×C8semidirect
127. C22×C3×C4C3⋊C8semidirect
128. C22×C3×C4C3×C8semidirect
238. C2×C3×C4C2×C3⋊C8semidirect
241. C2×C3×C4C2×C3×C8semidirect
249. C2×C32#324semidirect
261. C22×C4C3×C3⋊C8semidirect
323. C32C2×#324semidirect
343. C2×C4C2×C3×C3⋊C8semidirect
360. C2×C3#4528direct (N × H)
361. C2×C3C3×#324semidirect
386. C3C2×#4528direct (N × H)
387. C3C2×C3×#324semidirect
391. C2C3×#4528direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C4⋊C42
C2×C3×#8302
C3×#45282
C2×C3×#3243
C2×#45283