Finite Group Databasebeta

← C3×#69511 #11785341

Subgroups — C3×#69511 #11785341

Order 1152.

789 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊C8C211yesmaximal, characteristic
2C32×#380C211yesmaximal, characteristic
3C3×#5212C211yesmaximal, characteristic
4C2×C3×#808C212yesmaximal, complemented
5C3×#4529C212yesmaximal, complemented
6C3×#1807—31nomaximal
7#69511C311yesmaximal, characteristic, direct product factor
8C23×C32×C4C2211yescharacteristic
9C2×C32×C4⋊C4C2212yes—
10C2×C32×C22⋊C4—21no—
11C2×C3×#217—21no—
12C2×C3×C4⋊Dic3C2212yes—
13C22×C3×C3⋊C8—24no—
14C22×C3×C3⋊C8C414yescomplemented
15C22×C3×C3⋊C8C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C32×#95—21no—
18C3×C3⋊#95—21no—
19C3×#808—24no—
20C23×C3×C8—31no—
21C23×C3⋊C8C2×C311yescharacteristic
22C2×C3×#106—32no—
23C3×#380—21no—
24C3×#380—31no—
25C3×#380D311yescharacteristic
26C3×#380C2×C311yescharacteristic
27#5212C2×C311yescharacteristic
28C2×#808C2×C312yescomplemented
29C3×#340—32no—
30#4529C2×C312yescomplemented
31C24×C32D411yescharacteristic
32C22×C32×C4—22no—
33C22×C32×C4—22no—
34C22×C32×C4—22no—
35C22×C32×C4C2×C412yes—
36C22×C32×C4D411yescharacteristic
37C22×C32×C4Q812yes—
38C22×C32×C4C2311yescharacteristic
39C22×C3×Dic3—22no—
40C32×C4⋊C4—22no—
41C32×C4⋊C4—22no—
42C32×C22⋊C4—42no—
43C3×#217—42no—
44C3×C4⋊Dic3—22no—
45C3×C4⋊Dic3—22no—
46C2×C3×C3⋊C8—22no—
47C2×C3×C3⋊C8—28no—
48C2×C3×C3⋊C8C2×C412yes—
49C2×C3×C3⋊C8C2×C412yes—
50C2×C3×C3⋊C8D414yes—
51#1807—31noSylow
52C23×C3×C4—21no—
53C23×C3×C4C2×D311yescharacteristic
54C23×C3×C4C22×C311yescharacteristic
55C22×C3×C8—31no—
56C22×C3×C8—34no—
57C22×C3×C8—31no—
58C22×C3×C8—64no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—32no—
61C2×C3×C4⋊C4C2×D312yes—
62C2×C3×C4⋊C4C22×C312yes—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—61no—
67C2×#217—21no—
68C2×C4⋊Dic3C22×C312yes—
69C22×C3⋊C8—24no—
70C22×C3⋊C8C3×C414yescomplemented
71C22×C3⋊C8C22×C311yescharacteristic
72C22×C3⋊C8C22×C311yescharacteristic
73C3×#106—64no—
74C3×#95—22no—
75C3×#95—21no—
76C3×#95—21no—
77C3×#95—61no—
78C3⋊#95—21no—
79#808—24no—
80C23×C32—22no—
81C23×C32—22no—
82C23×C32C4⋊C412yes—
83C23×C32SD1612yes—
84C23×C32SD1612yes—
85C23×C32C2×D411yescharacteristic
86C2×C32×C4—22no—
87C2×C32×C4—22no—
88C2×C32×C4—22no—
89C2×C32×C4—24no—
90C2×C32×C4—42no—
91C2×C32×C4—42no—
92C2×C32×C4—42no—
93C2×C32×C4C4⋊C412yes—
94C2×C32×C4C22×C411yescharacteristic
95C2×C32×C4C2×D412yes—
96C2×C32×C4C2×Q811yescharacteristic
97C2×C32×C4#4112yes—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—42no—
100C2×C3×Dic3—42no—
101C3×C3⋊C8—22no—
102C3×C3⋊C8C22⋊C414yescomplemented
103C23×C8—31no—
104C2×#106—32no—
105#380—31no—
106#380C3×D311yescharacteristic
107#340—32no—
108C24×C3—21no—
109C24×C3#6711yescharacteristic
110C24×C3C3×D411yescharacteristic
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—24no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—42no—
123C22×C3×C4—42no—
124C22×C3×C4—62no—
125C22×C3×C4Dic612yes—
126C22×C3×C4C4×D312yes—
127C22×C3×C4#6711yescharacteristic
128C22×C3×C4C2×C3×C412yes—
129C22×C3×C4C3×D411yescharacteristic
130C22×C3×C4C3×Q812yes—
131C22×C3×C4C22×D311yescharacteristic
132C22×C3×C4C23×C311yescharacteristic
133C22×Dic3—22no—
134C2×C3×C8—32no—
135C2×C3×C8—34no—
136C2×C3×C8—32no—
137C2×C3×C8—68no—
138C2×C3×C8—62no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—42no—
144C3×C4⋊C4—42no—
145C3×C4⋊C4—62no—
146C3×C4⋊C4—62no—
147C3×C22⋊C4—44no—
148C3×C22⋊C4—42no—
149C3×C22⋊C4—42no—
150C3×C22⋊C4—122no—
151#217—42no—
152C4⋊Dic3—22no—
153C4⋊Dic3—22no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—28no—
156C2×C3⋊C8C2×C3×C412yes—
157C2×C3⋊C8C2×C3×C412yes—
158C2×C3⋊C8C3×D414yes—
159C22×C32—22no—
160C22×C32—22no—
161C22×C32—24no—
162C22×C32—24no—
163C22×C32#10614yescomplemented
164C22×C32C2×C4⋊C411yescharacteristic
165C22×C32#12112yes—
166C22×C32C2×SD1611yescharacteristic
167C22×C32C2×SD1611yescharacteristic
168C22×C32#13512yes—
169C32×C4—22no—
170C32×C4—82no—
171C32×C4C2×C22⋊C411yescharacteristic
172C32×C4#11711yescharacteristic
173C32×C4C22⋊Q812yes—
174C3×Dic3—82no—
175C23×C4C2×C3×D311yescharacteristic
176C22×C8—31no—
177C22×C8—34no—
178C22×C8—31no—
179C22×C8—64no—
180C2×C4⋊C4—32no—
181C2×C4⋊C4C2×C3×D312yes—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—61no—
184#106—64no—
185#95—21no—
186#95—61no—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—22no—
190C23×C3—22no—
191C23×C3—22no—
192C23×C3—22no—
193C23×C3—22no—
194C23×C3—21no—
195C23×C3—42no—
196C23×C3—42no—
197C23×C3Dic3⋊C412yes—
198C23×C3#21412yes—
199C23×C3#21512yes—
200C23×C3C3×C4⋊C412yes—
201C23×C3C3×SD1612yes—
202C23×C3C3×SD1612yes—
203C23×C3C2×#6711yescharacteristic
204C23×C3C2×C3×D411yescharacteristic
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—24no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—24no—
213C2×C3×C4—22no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—42no—
219C2×C3×C4—44no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—44no—
223C2×C3×C4—44no—
224C2×C3×C4—44no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—42no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—122no—
232C2×C3×C4—122no—
233C2×C3×C4—122no—
234C2×C3×C4Dic3⋊C412yes—
235C2×C3×C4C3×C4⋊C412yes—
236C2×C3×C4C2×Dic611yescharacteristic
237C2×C3×C4C2×C4×D311yescharacteristic
238C2×C3×C4D3×D412yes—
239C2×C3×C4#23712yes—
240C2×C3×C4C22×C3×C411yescharacteristic
241C2×C3×C4C2×C3×D412yes—
242C2×C3×C4C2×C3×Q811yescharacteristic
243C2×C3×C4C3×#4112yes—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C3×C8—34no—
248C3×C8—62no—
249C3⋊C8—22no—
250C3⋊C8C3×C22⋊C414yescomplemented
251C2×C32—24no—
252C2×C32#38011yescharacteristic
253C2×C32C2×#10611yescharacteristic
254C2×C32#42711yescharacteristic
255C2×C32#46512yes—
256C2×C32#46512yes—
257C24C3×#6711yescharacteristic
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—62no—
262C22×C4C3×Dic612yes—
263C22×C4C3×C4×D312yes—
264C22×C4C3×#6711yescharacteristic
265C22×C4C22×C3×D311yescharacteristic
266C2×C8—32no—
267C2×C8—34no—
268C2×C8—32no—
269C2×C8—68no—
270C2×C8—62no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—62no—
274C4⋊C4—62no—
275C22⋊C4—42no—
276C22⋊C4—122no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—24no—
281C22×C3—22no—
282C22×C3—24no—
283C22×C3—24no—
284C22×C3—24no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—22no—
288C22×C3—21no—
289C22×C3—22no—
290C22×C3—24no—
291C22×C3—42no—
292C22×C3—42no—
293C22×C3—44no—
294C22×C3—44no—
295C22×C3#80814yescomplemented
296C22×C3C3×#10614yescomplemented
297C22×C3C2×Dic3⋊C411yescharacteristic
298C22×C3C2×#21411yescharacteristic
299C22×C3#93912yes—
300C22×C3C2×#21511yescharacteristic
301C22×C3#95012yes—
302C22×C3C2×C3×C4⋊C411yescharacteristic
303C22×C3C3×#12112yes—
304C22×C3C2×C3×SD1611yescharacteristic
305C22×C3C2×C3×SD1611yescharacteristic
306C22×C3C3×#13512yes—
307C3×C4—22no—
308C3×C4—22no—
309C3×C4—21no—
310C3×C4—22no—
311C3×C4—21no—
312C3×C4—42no—
313C3×C4—84no—
314C3×C4—82no—
315C3×C4—82no—
316C3×C4—242no—
317C3×C4#87711yescharacteristic
318C3×C4#87812yes—
319C3×C4D3×C22⋊C411yescharacteristic
320C3×C4C2×C3×C22⋊C411yescharacteristic
321C3×C4C3×#11711yescharacteristic
322C3×C4C3×C22⋊Q812yes—
323Dic3—82no—
324C32#180711yescharacteristic, complemented, Sylow
325C23—22no—
326C23—22no—
327C23C3×Dic3⋊C412yes—
328C23C3×#21412yes—
329C23C3×#21512yes—
330C23C2×C3×#6711yescharacteristic
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4C3×Dic3⋊C412yes—
342C2×C4C2×C3×Dic611yescharacteristic
343C2×C4C2×C3×C4×D311yescharacteristic
344C2×C4C3×D3×D412yes—
345C2×C4C3×#23712yes—
346C8—34no—
347C8—62no—
348C2×C3—24no—
349C2×C3—24no—
350C2×C3—21no—
351C2×C3—21no—
352C2×C3—21no—
353C2×C3—22no—
354C2×C3—22no—
355C2×C3—44no—
356C2×C3#494211yescharacteristic
357C2×C3C2×#80811yescharacteristic
358C2×C3#499811yescharacteristic
359C2×C3#504112yes—
360C2×C3#504812yes—
361C2×C3C3×#38011yescharacteristic
362C2×C3C2×C3×#10611yescharacteristic
363C2×C3C3×#42711yescharacteristic
364C2×C3C3×#46512yes—
365C2×C3C3×#46512yes—
366C22—22no—
367C22—22no—
368C22—24no—
369C22—24no—
370C22C3×#80814yescomplemented
371C22C2×C3×Dic3⋊C411yescharacteristic
372C22C2×C3×#21411yescharacteristic
373C22C3×#93912yes—
374C22C2×C3×#21511yescharacteristic
375C22C3×#95012yes—
376C4—22no—
377C4—82no—
378C4—242no—
379C4C3×#87711yescharacteristic
380C4C3×#87812yes—
381C4C3×D3×C22⋊C411yescharacteristic
382C3—21no—
383C3#6951111yescharacteristic, direct product factor
384C3C3×#180711yescharacteristic, complemented
385C2—24no—
386C2C3×#494211yescharacteristic
387C2C2×C3×#80811yescharacteristic
388C2C3×#499811yescharacteristic
389C2C3×#504112yes—
390C2C3×#504812yes—
subgroupG/H
centerC23×C3C2×#67
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#380C2
socleC23×C32C2×D4
prime radicalC24×C32D4
cosocleC22×C3×C4C23×C3
hypercenterC3×#380D3
hypocenterC3C3×#1807
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C3×#380

Complements

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

NH ≅ G/Ntype
4. C2×C3×#808C2semidirect
5. C3×#4529C2semidirect
7. #69511C3direct (N × H)
14. C22×C3×C3⋊C8C4semidirect
28. C2×#808C2×C3semidirect
30. #4529C2×C3semidirect
70. C22×C3⋊C8C3×C4semidirect
102. C3×C3⋊C8C22⋊C4semidirect
163. C22×C32#106semidirect
250. C3⋊C8C3×C22⋊C4semidirect
295. C22×C3#808semidirect
296. C22×C3C3×#106semidirect
324. C32#1807semidirect
370. C22C3×#808semidirect
383. C3#69511direct (N × H)
384. C3C3×#1807semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊C82
C32×#3802
C3×#52122
C2×C3×#8082
C3×#45292
C3×#18073
#695113