Finite Group Databasebeta

← C3×#71990 #11788132

Subgroups — C3×#71990 #11788132

Order 1152.

973 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×M16C211yesmaximal, characteristic, complemented
2C2×C3×#930C211yesmaximal, characteristic
3C3×#5212C211yesmaximal, characteristic, complemented
4C2×C3×#821C212yesmaximal, complemented
5C3×#4551C212yesmaximal, complemented
6C3×#1895—31nomaximal
7#71990C311yesmaximal, characteristic, direct product factor
8C23×C32×C4C2211yescharacteristic
9C22×C3×#67—21no—
10C22×C3×D12C2211yescharacteristic
11C2×C32×M16—22no—
12C2×C32×M16—24no—
13C22×C32×C8C2212yes—
14C2×C3×#217—21no—
15C3×#930—22no—
16C3×#930C414yescomplemented
17C2×C3×(C2×D3)⋊C4—21no—
18C2×C3×C4⋊Dic3C2212yes—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C3×C3⋊#95—21no—
21C3×#821—24no—
22C22×C3×M16—21no—
23C22×C3×M16D311yescharacteristic, complemented
24C22×C3×M16C2×C311yescharacteristic, complemented
25C2×C3×#121—31no—
26C2×#930C2×C311yescharacteristic
27C2×C3×#102—32no—
28C3×#380—31no—
29#5212C2×C311yescharacteristic, complemented
30C2×#821C2×C312yescomplemented
31C3×#340—32no—
32#4551C2×C312yescomplemented
33C24×C32D411yescharacteristic
34C23×C3×D3—21no—
35C22×C32×C4—22no—
36C22×C32×C4—22no—
37C22×C32×C4C2×C412yes—
38C22×C32×C4D411yescharacteristic
39C22×C32×C4D412yes—
40C22×C32×C4C2311yescharacteristic
41C2×C3×#67—24no—
42C2×C3×#67—44no—
43C22×C3×Dic3—21no—
44C22×C3×Dic3—22no—
45C2×C3×D12—22no—
46C2×C3×D12—22no—
47C2×C3×D12—21no—
48C2×C3×D12C2×C412yes—
49C32×M16—28no—
50C2×C32×C8—22no—
51C2×C32×C8—22no—
52C2×C32×C8D414yescomplemented
53C3×#217—42no—
54C3×(C2×D3)⋊C4—24no—
55C3×C4⋊Dic3—21no—
56C3×C4⋊Dic3—22no—
57C3×C4⋊Dic3—22no—
58C3×C4⋊Dic3C2×C412yes—
59#1895—31noSylow
60C22×C3×D4—31no—
61C22×C3×D4—61no—
62C23×C3×C4—21no—
63C23×C3×C4C2×D311yescharacteristic
64C23×C3×C4C22×C311yescharacteristic
65C22×#67—21no—
66C22×D12C22×C311yescharacteristic
67C2×C3×M16—24no—
68C2×C3×M16—28no—
69C2×C3×M16—22no—
70C2×C3×M16—22no—
71C2×C3×M16—24no—
72C2×C3×M16—24no—
73C22×C3×C8—22no—
74C22×C3×C8C2×D312yes—
75C22×C3×C8C22×C312yes—
76C3×#121—34no—
77C3×#121—62no—
78C2×C3×C4⋊C4—31no—
79C2×C3×C4⋊C4—32no—
80C2×C3×C22⋊C4—61no—
81C2×C3×C22⋊C4—61no—
82C2×#217—21no—
83#930—22no—
84#930C3×C414yescomplemented
85C2×(C2×D3)⋊C4—21no—
86C2×C4⋊Dic3C22×C312yes—
87C2×C4⋊Dic3C22×C311yescharacteristic
88C3×#102—64no—
89C3×#95—61no—
90C3⋊#95—21no—
91#821—24no—
92C23×C32—22no—
93C23×C32—21no—
94C23×C32—22no—
95C23×C32—21no—
96C23×C32C22⋊C412yes—
97C23×C32C2×D411yescharacteristic
98C22×C3×D3—22no—
99C22×C3×D3—42no—
100C22×C3×D3—41no—
101C22×C3×D3—42no—
102C22×C3×D3—41no—
103C2×C32×C4—22no—
104C2×C32×C4—24no—
105C2×C32×C4—22no—
106C2×C32×C4—22no—
107C2×C32×C4C22⋊C412yes—
108C2×C32×C4C22×C411yescharacteristic
109C2×C32×C4C2×D412yes—
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4#4112yes—
112C3×#67—48no—
113C2×C3×Dic3—22no—
114C2×C3×Dic3—42no—
115C2×C3×Dic3—42no—
116C2×C3×Dic3—42no—
117C2×C3×Dic3—42no—
118C3×D12—24no—
119C3×D12—41no—
120C3×D12—41no—
121C32×C8—24no—
122C22×M16C3×D311yescharacteristic, complemented
123C2×#121—31no—
124C2×#102—32no—
125#380—31no—
126#340—32no—
127C24×C3—21no—
128C24×C3—61no—
129C24×C3D1211yescharacteristic
130C24×C3C3×D411yescharacteristic
131C23×D3—21no—
132C2×C3×D4—32no—
133C2×C3×D4—64no—
134C2×C3×D4—62no—
135C2×C3×D4—62no—
136C2×C3×D4—61no—
137C2×C3×D4—124no—
138C22×C3×C4—24no—
139C22×C3×C4—24no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—61no—
149C22×C3×C4—62no—
150C22×C3×C4C4×D312yes—
151C22×C3×C4D1211yescharacteristic
152C22×C3×C4#6712yes—
153C22×C3×C4C2×C3×C412yes—
154C22×C3×C4C3×D411yescharacteristic
155C22×C3×C4C3×D412yes—
156C22×C3×C4C22×D311yescharacteristic
157C22×C3×C4C23×C311yescharacteristic
158C2×#67—24no—
159C2×#67—44no—
160C22×Dic3—21no—
161C22×Dic3—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—21no—
165C2×D12C2×C3×C412yes—
166C3×M16—216no—
167C3×M16—28no—
168C3×M16—28no—
169C2×C3×C8—24no—
170C2×C3×C8—24no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—24no—
176C2×C3×C8#6714yescomplemented
177C2×C3×C8C3×D414yescomplemented
178C3×C4⋊C4—32no—
179C3×C4⋊C4—61no—
180C3×C4⋊C4—62no—
181C3×C4⋊C4—62no—
182C3×C22⋊C4—64no—
183C3×C22⋊C4—122no—
184#217—42no—
185(C2×D3)⋊C4—24no—
186C4⋊Dic3—21no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3C2×C3×C412yes—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—22no—
193C22×C32—42no—
194C22×C32—42no—
195C22×C32C2×C22⋊C411yescharacteristic
196C22×C32#12112yes—
197C22×C32AGL(1,ℤ/8)11yescharacteristic
198C22×C32AGL(1,ℤ/8)12yes—
199C22×C32#13711yescharacteristic
200C2×C3×D3—42no—
201C2×C3×D3—42no—
202C2×C3×D3—42no—
203C2×C3×D3—82no—
204C2×C3×D3—82no—
205C32×C4—22no—
206C32×C4C4×D412yes—
207C32×C4#12011yescharacteristic
208C32×C4#12311yescharacteristic
209C3×Dic3—42no—
210C3×Dic3—82no—
211C22×D4—31no—
212C22×D4—61no—
213C23×C4C2×C3×D311yescharacteristic
214C2×M16—22no—
215C2×M16—24no—
216C22×C8C2×C3×D312yes—
217#121—34no—
218#121—62no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—32no—
221C2×C22⋊C4—61no—
222C2×C22⋊C4—61no—
223#102—64no—
224#95—61no—
225C23×C3—24no—
226C23×C3—21no—
227C23×C3—22no—
228C23×C3—21no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—21no—
232C23×C3—22no—
233C23×C3—21no—
234C23×C3—22no—
235C23×C3—21no—
236C23×C3—41no—
237C23×C3—42no—
238C23×C3—41no—
239C23×C3—62no—
240C23×C3—122no—
241C23×C3—121no—
242C23×C3—121no—
243C23×C3—122no—
244C23×C3(C2×D3)⋊C412yes—
245C23×C3C3×C22⋊C412yes—
246C23×C3C2×D1211yescharacteristic
247C23×C3C2×C3×D411yescharacteristic
248C22×D3—22no—
249C22×D3—42no—
250C22×D3—42no—
251C22×D3—41no—
252C22×D3—41no—
253C3×D4—64no—
254C3×D4—128no—
255C3×D4—121no—
256C3×D4—121no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—28no—
260C2×C3×C4—24no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—24no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—24no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—21no—
274C2×C3×C4—62no—
275C2×C3×C4—122no—
276C2×C3×C4—122no—
277C2×C3×C4—122no—
278C2×C3×C4—122no—
279C2×C3×C4(C2×D3)⋊C412yes—
280C2×C3×C4C3×C22⋊C412yes—
281C2×C3×C4C2×C4×D311yescharacteristic
282C2×C3×C4#23512yes—
283C2×C3×C4C2×#6712yes—
284C2×C3×C4C2×#6711yescharacteristic
285C2×C3×C4C22×C3×C411yescharacteristic
286C2×C3×C4C2×C3×D412yes—
287C2×C3×C4C2×C3×D411yescharacteristic
288C2×C3×C4C3×#4112yes—
289#67—48no—
290C2×Dic3—22no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295D12—24no—
296D12—41no—
297D12—41no—
298C3×C8—28no—
299C3×C8—24no—
300C3×C8—24no—
301C2×C32—42no—
302C2×C32#38611yescharacteristic
303C2×C32#41711yescharacteristic
304C2×C32#41811yescharacteristic
305C2×C32#46812yes—
306C2×C32#46912yes—
307C3×D3—82no—
308C24—61no—
309C24C3×D1211yescharacteristic
310C2×D4—32no—
311C2×D4—64no—
312C2×D4—62no—
313C2×D4—62no—
314C2×D4—61no—
315C2×D4—124no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—61no—
319C22×C4—62no—
320C22×C4C3×C4×D312yes—
321C22×C4C3×D1211yescharacteristic
322C22×C4C3×#6712yes—
323C22×C4C22×C3×D311yescharacteristic
324M16—28no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8C3×#6714yescomplemented
328C4⋊C4—32no—
329C4⋊C4—61no—
330C4⋊C4—62no—
331C4⋊C4—62no—
332C22⋊C4—64no—
333C22⋊C4—122no—
334C22×C3—24no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—42no—
347C22×C3—42no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—42no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—122no—
355C22×C3—122no—
356C22×C3—122no—
357C22×C3—242no—
358C22×C3—242no—
359C22×C3#90812yes—
360C22×C3#90811yescharacteristic
361C22×C3#90911yescharacteristic
362C22×C3C2×(C2×D3)⋊C411yescharacteristic
363C22×C3#93012yes—
364C22×C3C2×C3×C22⋊C411yescharacteristic
365C22×C3C3×#12112yes—
366C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
367C22×C3C3×AGL(1,ℤ/8)12yes—
368C22×C3C3×#13711yescharacteristic
369C2×D3—42no—
370C2×D3—42no—
371C2×D3—42no—
372C2×D3—82no—
373C2×D3—82no—
374C3×C4—24no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—122no—
381C3×C4—242no—
382C3×C4C4×#6712yes—
383C3×C4#92911yescharacteristic
384C3×C4#95311yescharacteristic
385C3×C4C3×C4×D412yes—
386C3×C4C3×#12011yescharacteristic
387C3×C4C3×#12311yescharacteristic
388Dic3—42no—
389Dic3—82no—
390C32#189511yescharacteristic, complemented, Sylow
391C23—21no—
392C23—22no—
393C23—21no—
394C23—22no—
395C23—62no—
396C23—122no—
397C23—121no—
398C23—121no—
399C23—122no—
400C23C3×(C2×D3)⋊C412yes—
401C23C2×C3×D1211yescharacteristic
402D4—64no—
403D4—128no—
404D4—121no—
405D4—121no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—62no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4C3×(C2×D3)⋊C412yes—
416C2×C4C2×C3×C4×D311yescharacteristic
417C2×C4C3×#23512yes—
418C2×C4C2×C3×#6712yes—
419C2×C4C2×C3×#6711yescharacteristic
420C8—24no—
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—22no—
425C2×C3—22no—
426C2×C3—44no—
427C2×C3—42no—
428C2×C3—42no—
429C2×C3—242no—
430C2×C3#513111yescharacteristic
431C2×C3#513311yescharacteristic
432C2×C3#513412yes—
433C2×C3#513512yes—
434C2×C3#521311yescharacteristic
435C2×C3C3×#38611yescharacteristic
436C2×C3C3×#41711yescharacteristic
437C2×C3C3×#41811yescharacteristic
438C2×C3C3×#46812yes—
439C2×C3C3×#46912yes—
440D3—82no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—42no—
445C22—42no—
446C22—122no—
447C22—122no—
448C22—122no—
449C22—242no—
450C22—242no—
451C22C3×#90812yes—
452C22C3×#90811yescharacteristic
453C22C3×#90911yescharacteristic
454C22C2×C3×(C2×D3)⋊C411yescharacteristic
455C22C3×#93012yes—
456C4—22no—
457C4—122no—
458C4—242no—
459C4C3×C4×#6712yes—
460C4C3×#92911yescharacteristic
461C4C3×#95311yescharacteristic
462C3—21no—
463C3#7199011yescharacteristic, direct product factor
464C3C3×#189511yescharacteristic, complemented
465C2—42no—
466C2—242no—
467C2C3×#513111yescharacteristic
468C2C3×#513311yescharacteristic
469C2C3×#513412yes—
470C2C3×#513512yes—
471C2C3×#521311yescharacteristic
subgroupG/H
centerC23×C3C2×D12
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC22×C32×M16C2
socleC23×C32C2×D4
prime radicalC2×C3×#930C2
cosocleC22×C3×C4C23×C3
hypercenterC22×C3×M16D3
hypocenterC3C3×#1895
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C22×C3×M16

Complements

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

NH ≅ G/Ntype
1. C22×C32×M16C2semidirect
3. C3×#5212C2semidirect
4. C2×C3×#821C2semidirect
5. C3×#4551C2semidirect
7. #71990C3direct (N × H)
16. C3×#930C4semidirect
23. C22×C3×M16D3semidirect
24. C22×C3×M16C2×C3semidirect
29. #5212C2×C3semidirect
30. C2×#821C2×C3semidirect
32. #4551C2×C3semidirect
52. C2×C32×C8D4semidirect
84. #930C3×C4semidirect
122. C22×M16C3×D3semidirect
176. C2×C3×C8#67semidirect
177. C2×C3×C8C3×D4semidirect
327. C2×C8C3×#67semidirect
390. C32#1895semidirect
463. C3#71990direct (N × H)
464. C3C3×#1895semidirect

Maximal subgroups

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

subgroupindex
C22×C32×M162
C2×C3×#9302
C3×#52122
C2×C3×#8212
C3×#45512
C3×#18953
#719903