Finite Group Databasebeta

← C3×#71585 #11787276

Subgroups — C3×#71585 #11787276

Order 1152.

885 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#117C211yesmaximal, characteristic
2C2×C3×#924C211yesmaximal, characteristic
3C2×C3×#830C211yesmaximal, characteristic
4C2×C3×#810C212yesmaximal, complemented
5C3×#4529C212yesmaximal, complemented
6C3×#1873—31nomaximal
7#71585C311yesmaximal, characteristic, direct product factor
8C23×C32×C4C2211yescharacteristic
9C22×C3×Dic6C2211yescharacteristic
10C32×#117—24no—
11C2×C32×C4⋊C4C2212yes—
12C2×C32×C22⋊C4—21no—
13C2×C32×C42—21no—
14C2×C3×#217—21no—
15C2×C3×C4⋊Dic3C2211yescharacteristic
16C3×#924—22no—
17C3×#924C414yescomplemented
18C2×C3×Dic3⋊C4—21no—
19C22×C3×C3⋊C8C2212yes—
20C3×#830—22no—
21C3×#810—24no—
22C2×C3×C22⋊Q8—31no—
23C2×C3×#117—21no—
24C2×C3×#117D311yescharacteristic
25C2×C3×#117C2×C311yescharacteristic
26C2×#924C2×C311yescharacteristic
27C2×C3×#103—32no—
28C2×C3×C22⋊C8—31no—
29C2×#830C2×C311yescharacteristic
30C2×#810C2×C312yescomplemented
31C3×#340—32no—
32#4529C2×C312yescomplemented
33C24×C32D411yescharacteristic
34C22×C32×C4—22no—
35C22×C32×C4—22no—
36C22×C32×C4—22no—
37C22×C32×C4C2×C412yes—
38C22×C32×C4D411yescharacteristic
39C22×C32×C4D412yes—
40C22×C32×C4C2311yescharacteristic
41C22×C3×Dic3—21no—
42C22×C3×Dic3—21no—
43C2×C3×Dic6—22no—
44C2×C3×Dic6—22no—
45C2×C3×Dic6—21no—
46C2×C3×Dic6C2×C412yes—
47C32×C4⋊C4—22no—
48C32×C4⋊C4D414yes—
49C32×C22⋊C4—24no—
50C32×C42—24no—
51C3×#217—24no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3C2×C412yes—
54C3×Dic3⋊C4—24no—
55C2×C3×C3⋊C8—22no—
56C2×C3×C3⋊C8—22no—
57C2×C3×C3⋊C8—22no—
58#1873—31noSylow
59C22×C3×Q8—31no—
60C23×C3×C4—21no—
61C23×C3×C4C2×D311yescharacteristic
62C23×C3×C4C22×C311yescharacteristic
63C22×Dic6C22×C311yescharacteristic
64C22×C3×C8—32no—
65C3×C22⋊Q8—34no—
66C3×C22⋊Q8—62no—
67C3×#117—28no—
68C3×#117—24no—
69C3×#117—24no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—31no—
72C2×C3×C4⋊C4—61no—
73C2×C3×C4⋊C4C2×D312yes—
74C2×C3×C4⋊C4C22×C312yes—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—61no—
79C2×C3×C42—22no—
80C2×C3×C42—21no—
81C2×C3×C42—21no—
82C2×#217—21no—
83C2×C4⋊Dic3C22×C311yescharacteristic
84#924—22no—
85#924C3×C414yescomplemented
86C2×Dic3⋊C4—21no—
87C22×C3⋊C8C22×C312yes—
88C3×#103—64no—
89C3×C22⋊C8—62no—
90#830—22no—
91#810—24no—
92C23×C32—21no—
93C23×C32—21no—
94C23×C32—22no—
95C23×C32—22no—
96C23×C32C22⋊C412yes—
97C23×C32C2×D411yescharacteristic
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—24no—
101C2×C32×C4—24no—
102C2×C32×C4—24no—
103C2×C32×C4—42no—
104C2×C32×C4—42no—
105C2×C32×C4C22⋊C412yes—
106C2×C32×C4C22×C411yescharacteristic
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4C2×D412yes—
109C2×C32×C4#4112yes—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—22no—
112C2×C3×Dic3—41no—
113C2×C3×Dic3—41no—
114C2×C3×Dic3—42no—
115C3×Dic6—24no—
116C3×Dic6—42no—
117C3×C3⋊C8—42no—
118C2×C22⋊Q8—31no—
119C2×#117C3×D311yescharacteristic
120C2×#103—32no—
121C2×C22⋊C8—31no—
122#340—32no—
123C24×C3—21no—
124C24×C3#6711yescharacteristic
125C24×C3C3×D411yescharacteristic
126C2×C3×Q8—32no—
127C2×C3×Q8—61no—
128C2×C3×Q8—62no—
129C2×C3×Q8—62no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—24no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—61no—
144C22×C3×C4—61no—
145C22×C3×C4C4×D312yes—
146C22×C3×C4D1212yes—
147C22×C3×C4#6711yescharacteristic
148C22×C3×C4C2×C3×C412yes—
149C22×C3×C4C3×D412yes—
150C22×C3×C4C3×D411yescharacteristic
151C22×C3×C4C22×D311yescharacteristic
152C22×C3×C4C23×C311yescharacteristic
153C22×Dic3—21no—
154C22×Dic3—21no—
155C2×Dic6—22no—
156C2×Dic6—22no—
157C2×Dic6—21no—
158C2×Dic6C2×C3×C412yes—
159C2×C3×C8—62no—
160C2×C3×C8—62no—
161C2×C3×C8—62no—
162C3×C4⋊C4—24no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—32no—
167C3×C4⋊C4—64no—
168C3×C4⋊C4—61no—
169C3×C4⋊C4#6714yes—
170C3×C4⋊C4C3×D414yes—
171C3×C22⋊C4—28no—
172C3×C22⋊C4—24no—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—64no—
175C3×C42—28no—
176C3×C42—24no—
177C3×C42—24no—
178#217—24no—
179C4⋊Dic3—21no—
180C4⋊Dic3C2×C3×C412yes—
181Dic3⋊C4—24no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—22no—
185C22×C32—22no—
186C22×C32—22no—
187C22×C32—22no—
188C22×C32—42no—
189C22×C32—42no—
190C22×C32C2×C22⋊C411yescharacteristic
191C22×C32#12011yescharacteristic
192C22×C32#12311yescharacteristic
193C22×C32#13512yes—
194C22×C32#13712yes—
195C32×C4—22no—
196C32×C4—44no—
197C32×C4C4×D412yes—
198C32×C4#12112yes—
199C3×Dic3—42no—
200C3×Dic3—42no—
201C22×Q8—31no—
202C23×C4C2×C3×D311yescharacteristic
203C22×C8—32no—
204C22⋊Q8—34no—
205C22⋊Q8—62no—
206#117—24no—
207C2×C4⋊C4—31no—
208C2×C4⋊C4—61no—
209C2×C4⋊C4C2×C3×D312yes—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—61no—
212C2×C42—21no—
213#103—64no—
214C22⋊C8—62no—
215C23×C3—22no—
216C23×C3—22no—
217C23×C3—21no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—22no—
224C23×C3—22no—
225C23×C3—22no—
226C23×C3—21no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3(C2×D3)⋊C412yes—
230C23×C3C3×C22⋊C412yes—
231C23×C3C2×#6711yescharacteristic
232C23×C3C2×C3×D411yescharacteristic
233C3×Q8—64no—
234C3×Q8—122no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—28no—
238C2×C3×C4—28no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—28no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—44no—
256C2×C3×C4—44no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—62no—
262C2×C3×C4—62no—
263C2×C3×C4—121no—
264C2×C3×C4—122no—
265C2×C3×C4—121no—
266C2×C3×C4(C2×D3)⋊C412yes—
267C2×C3×C4C3×C22⋊C412yes—
268C2×C3×C4C2×C4×D311yescharacteristic
269C2×C3×C4C2×D1211yescharacteristic
270C2×C3×C4#23512yes—
271C2×C3×C4C2×#6712yes—
272C2×C3×C4C22×C3×C411yescharacteristic
273C2×C3×C4C2×C3×D411yescharacteristic
274C2×C3×C4C2×C3×D412yes—
275C2×C3×C4C3×#4112yes—
276C2×Dic3—22no—
277C2×Dic3—22no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—42no—
281Dic6—24no—
282Dic6—42no—
283C3×C8—122no—
284C3⋊C8—42no—
285C2×C32—42no—
286C2×C32#38611yescharacteristic
287C2×C32#41611yescharacteristic
288C2×C32#41911yescharacteristic
289C2×C32#45212yes—
290C2×C32#48512yes—
291C24C3×#6711yescharacteristic
292C2×Q8—32no—
293C2×Q8—61no—
294C2×Q8—62no—
295C2×Q8—62no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—61no—
300C22×C4—61no—
301C22×C4C3×C4×D312yes—
302C22×C4C3×D1212yes—
303C22×C4C3×#6711yescharacteristic
304C22×C4C22×C3×D311yescharacteristic
305C2×C8—62no—
306C2×C8—62no—
307C2×C8—62no—
308C4⋊C4—22no—
309C4⋊C4—32no—
310C4⋊C4—64no—
311C4⋊C4—61no—
312C4⋊C4C3×#6714yes—
313C22⋊C4—24no—
314C22⋊C4—64no—
315C42—24no—
316C22×C3—24no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—21no—
328C22×C3—42no—
329C22×C3—42no—
330C22×C3—44no—
331C22×C3—44no—
332C22×C3—42no—
333C22×C3—42no—
334C22×C3—42no—
335C22×C3—42no—
336C22×C3C2×(C2×D3)⋊C411yescharacteristic
337C22×C3#92911yescharacteristic
338C22×C3#95012yes—
339C22×C3#95112yes—
340C22×C3#95311yescharacteristic
341C22×C3C2×C3×C22⋊C411yescharacteristic
342C22×C3C3×#12011yescharacteristic
343C22×C3C3×#12311yescharacteristic
344C22×C3C3×#13512yes—
345C22×C3C3×#13712yes—
346C3×C4—24no—
347C3×C4—22no—
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—48no—
352C3×C4—44no—
353C3×C4—44no—
354C3×C4—122no—
355C3×C4—122no—
356C3×C4C4×#6712yes—
357C3×C4#93012yes—
358C3×C4C3×C4×D412yes—
359C3×C4C3×#12112yes—
360Dic3—42no—
361Dic3—42no—
362C32#187311yescharacteristic, complemented, Sylow
363C23—21no—
364C23—22no—
365C23—22no—
366C23—21no—
367C23C3×(C2×D3)⋊C412yes—
368C23C2×C3×#6711yescharacteristic
369Q8—64no—
370Q8—122no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—121no—
381C2×C4—122no—
382C2×C4—121no—
383C2×C4C3×(C2×D3)⋊C412yes—
384C2×C4C2×C3×C4×D311yescharacteristic
385C2×C4C2×C3×D1211yescharacteristic
386C2×C4C3×#23512yes—
387C2×C4C2×C3×#6712yes—
388C8—122no—
389C2×C3—21no—
390C2×C3—22no—
391C2×C3—22no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—44no—
395C2×C3—42no—
396C2×C3—42no—
397C2×C3#499612yes—
398C2×C3#500211yescharacteristic
399C2×C3#500411yescharacteristic
400C2×C3#521311yescharacteristic
401C2×C3#523912yes—
402C2×C3C3×#38611yescharacteristic
403C2×C3C3×#41611yescharacteristic
404C2×C3C3×#41911yescharacteristic
405C2×C3C3×#45212yes—
406C2×C3C3×#48512yes—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—42no—
411C22—42no—
412C22C2×C3×(C2×D3)⋊C411yescharacteristic
413C22C3×#92911yescharacteristic
414C22C3×#95012yes—
415C22C3×#95112yes—
416C22C3×#95311yescharacteristic
417C4—22no—
418C4—44no—
419C4—122no—
420C4—122no—
421C4C3×C4×#6712yes—
422C4C3×#93012yes—
423C3—21no—
424C3#7158511yescharacteristic, direct product factor
425C3C3×#187311yescharacteristic, complemented
426C2—42no—
427C2C3×#499612yes—
428C2C3×#500211yescharacteristic
429C2C3×#500411yescharacteristic
430C2C3×#521311yescharacteristic
431C2C3×#523912yes—
subgroupG/H
centerC23×C3C2×#67
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#117C2
socleC23×C32C2×D4
prime radicalC24×C32D4
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#117D3
hypocenterC3C3×#1873
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×#117

Complements

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

NH ≅ G/Ntype
4. C2×C3×#810C2semidirect
5. C3×#4529C2semidirect
7. #71585C3direct (N × H)
17. C3×#924C4semidirect
30. C2×#810C2×C3semidirect
32. #4529C2×C3semidirect
85. #924C3×C4semidirect
362. C32#1873semidirect
424. C3#71585direct (N × H)
425. C3C3×#1873semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1172
C2×C3×#9242
C2×C3×#8302
C2×C3×#8102
C3×#45292
C3×#18733
#715853