Finite Group Databasebeta

← C3×#71712 #11788202

Subgroups — C3×#71712 #11788202

Order 1152.

881 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#525C211yesmaximal, characteristic
2C2×C3×#926C211yesmaximal, characteristic, complemented
3C2×C3×#837C212yesmaximal, complemented
4C3×#5207C211yesmaximal, characteristic, complemented
5C3×#4532C212yesmaximal, complemented
6C3×#1897—31nomaximal
7#71712C311yesmaximal, characteristic, direct product factor
8C2×C32×#41C2211yescharacteristic
9C23×C32×C4C2211yescharacteristic
10C32×#123—21no—
11C32×C22⋊Q8C412yescomplemented
12C32×#121C412yescomplemented
13C32×#120—21no—
14C32×C4×D4—22no—
15C32×#117C2211yescharacteristic
16C2×C3×#217—21no—
17C3×#926—24no—
18C2×C3×C4⋊Dic3—21no—
19C2×C3×C4×Dic3C2212yes—
20C2×C3×#208C2212yes—
21C3×#837—24no—
22C3×#830—22no—
23C3×#525—21no—
24C3×#525D311yescharacteristic
25C3×#525C2×C311yescharacteristic
26C2×C3×#117—31no—
27C2×#926C2×C311yescharacteristic, complemented
28C2×C3×#104—32no—
29C3×#407—31no—
30C2×#837C2×C312yescomplemented
31#5207C2×C311yescharacteristic, complemented
32C3×#339—32no—
33#4532C2×C312yescomplemented
34C24×C32D411yescharacteristic
35C32×#41—22no—
36C32×#41—22no—
37C2×C32×Q8C2×C411yescharacteristic
38C2×C32×D4—22no—
39C2×C32×D4—21no—
40C2×C32×D4C2×C411yescharacteristic
41C22×C32×C4—22no—
42C22×C32×C4—22no—
43C22×C32×C4—21no—
44C22×C32×C4C2×C412yes—
45C22×C32×C4D411yescharacteristic
46C22×C32×C4D412yes—
47C22×C32×C4C2311yescharacteristic
48C22×C3×Dic3—22no—
49C32×C4⋊C4—22no—
50C32×C4⋊C4C2×C412yes—
51C32×C22⋊C4—22no—
52C32×C22⋊C4—21no—
53C32×C22⋊C4—22no—
54C32×C42—21no—
55C3×#217—24no—
56C3×C4⋊Dic3—24no—
57C3×C4×Dic3—22no—
58C3×C4×Dic3D414yescomplemented
59C3×#208—22no—
60C3×#208—22no—
61C2×C3×C3⋊C8—22no—
62#1897—31noSylow
63C2×C3×#41—21no—
64C2×C3×#41C2×D311yescharacteristic
65C2×C3×#41C22×C311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4C2×D311yescharacteristic
68C23×C3×C4C22×C311yescharacteristic
69C2×C3×M16—32no—
70C3×#123—22no—
71C3×#123—21no—
72C3×#123—21no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8Dic312yescomplemented
75C3×C22⋊Q8C3×C412yescomplemented
76C3×#121—22no—
77C3×#121Dic312yescomplemented
78C3×#121C3×C412yescomplemented
79C3×#120—22no—
80C3×#120—21no—
81C3×#120—21no—
82C3×C4×D4—24no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×#117—21no—
86C3×#117—64no—
87C3×#117C2×D311yescharacteristic
88C3×#117C22×C311yescharacteristic
89C2×C3×C4⋊C4—61no—
90C2×C3×C22⋊C4—61no—
91C2×C3×C42—32no—
92C2×#217—21no—
93#926—24no—
94C2×C4⋊Dic3—21no—
95C2×C4×Dic3C22×C312yes—
96C2×#208C22×C312yes—
97C3×#104—64no—
98C3×C22⋊C8—62no—
99#837—24no—
100#830—22no—
101C23×C32—21no—
102C23×C32—21no—
103C23×C32—21no—
104C23×C32—42no—
105C23×C32C22⋊C412yes—
106C23×C32C2×D411yescharacteristic
107C32×Q8—22no—
108C32×D4—22no—
109C32×D4—44no—
110C32×D4—41no—
111C32×D4—41no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—42no—
119C2×C32×C4C22⋊C412yes—
120C2×C32×C4C22×C411yescharacteristic
121C2×C32×C4C2×D412yes—
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4#4112yes—
124C2×C3×Dic3—24no—
125C2×C3×Dic3—44no—
126C3×C3⋊C8—42no—
127#525C3×D311yescharacteristic
128C2×#117—31no—
129C2×#104—32no—
130#407—31no—
131#339—32no—
132C24×C3—21no—
133C24×C3#6711yescharacteristic
134C24×C3C3×D411yescharacteristic
135C3×#41—22no—
136C3×#41—22no—
137C3×#41—22no—
138C3×#41—22no—
139C3×#41—42no—
140C3×#41—42no—
141C2×C3×Q8—21no—
142C2×C3×Q8C2×Dic311yescharacteristic
143C2×C3×Q8C2×C3×C411yescharacteristic
144C2×C3×D4—24no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4C2×Dic311yescharacteristic
152C2×C3×D4C2×C3×C411yescharacteristic
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—62no—
167C22×C3×C4C2×Dic312yes—
168C22×C3×C4#6711yescharacteristic
169C22×C3×C4#6712yes—
170C22×C3×C4C2×C3×C412yes—
171C22×C3×C4C3×D411yescharacteristic
172C22×C3×C4C3×D412yes—
173C22×C3×C4C22×D311yescharacteristic
174C22×C3×C4C23×C311yescharacteristic
175C22×Dic3—22no—
176C3×M16—62no—
177C3×M16—62no—
178C2×C3×C8—62no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—64no—
184C3×C4⋊C4C2×Dic312yes—
185C3×C4⋊C4C2×C3×C412yes—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—24no—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—21no—
192C3×C22⋊C4—21no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—22no—
195C3×C22⋊C4—64no—
196C3×C42—22no—
197C3×C42—21no—
198C3×C42—21no—
199C3×C42—34no—
200C3×C42—62no—
201#217—24no—
202C4⋊Dic3—24no—
203C4×Dic3—22no—
204C4×Dic3C3×D414yescomplemented
205#208—22no—
206#208—22no—
207C2×C3⋊C8—22no—
208C22×C32—22no—
209C22×C32—22no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—42no—
213C22×C32—42no—
214C22×C32—41no—
215C22×C32—42no—
216C22×C32C2×C22⋊C411yescharacteristic
217C22×C32#12011yescharacteristic
218C22×C32#12311yescharacteristic
219C32×C4—22no—
220C32×C4—41no—
221C32×C4—42no—
222C32×C4C4×D412yes—
223C32×C4#12112yes—
224C3×Dic3—44no—
225C2×#41C2×C3×D311yescharacteristic
226C23×C4C2×C3×D311yescharacteristic
227C2×M16—32no—
228#123—21no—
229C22⋊Q8C3×Dic312yescomplemented
230#121C3×Dic312yescomplemented
231#120—21no—
232C4×D4—22no—
233#117—64no—
234#117C2×C3×D311yescharacteristic
235C2×C4⋊C4—61no—
236C2×C22⋊C4—61no—
237C2×C42—32no—
238#104—64no—
239C22⋊C8—62no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—22no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—22no—
248C23×C3—21no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—44no—
254C23×C3#21712yes—
255C23×C3C3×C22⋊C412yes—
256C23×C3C2×#6711yescharacteristic
257C23×C3C2×C3×D411yescharacteristic
258C3×Q8—22no—
259C3×Q8—22no—
260C3×Q8—42no—
261C3×D4—22no—
262C3×D4—22no—
263C3×D4—42no—
264C3×D4—48no—
265C3×D4—44no—
266C3×D4—44no—
267C3×D4—42no—
268C3×D4—42no—
269C3×D4—41no—
270C3×D4—41no—
271C3×D4—41no—
272C3×D4—41no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—44no—
293C2×C3×C4—44no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—44no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—64no—
300C2×C3×C4—124no—
301C2×C3×C4#21712yes—
302C2×C3×C4C3×C22⋊C412yes—
303C2×C3×C4D3×D412yes—
304C2×C3×C4#23712yes—
305C2×C3×C4C22×Dic311yescharacteristic
306C2×C3×C4C2×#6711yescharacteristic
307C2×C3×C4C22×C3×C411yescharacteristic
308C2×C3×C4C2×C3×D412yes—
309C2×C3×C4C2×C3×D411yescharacteristic
310C2×C3×C4C3×#4112yes—
311C2×Dic3—24no—
312C2×Dic3—44no—
313C3×C8—122no—
314C3⋊C8—42no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—42no—
318C2×C32—81no—
319C2×C32#38611yescharacteristic
320C2×C32#42112yes—
321C24C3×#6711yescharacteristic
322#41—22no—
323#41—22no—
324C2×Q8C2×C3×Dic311yescharacteristic
325C2×D4—22no—
326C2×D4—21no—
327C2×D4C2×C3×Dic311yescharacteristic
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—62no—
332C22×C4C2×C3×Dic312yes—
333C22×C4C3×#6711yescharacteristic
334C22×C4C3×#6712yes—
335C22×C4C22×C3×D311yescharacteristic
336M16—62no—
337M16—62no—
338C2×C8—62no—
339C4⋊C4—22no—
340C4⋊C4—64no—
341C4⋊C4C2×C3×Dic312yes—
342C22⋊C4—22no—
343C22⋊C4—21no—
344C22⋊C4—22no—
345C22⋊C4—64no—
346C42—21no—
347C42—34no—
348C42—62no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—24no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3—44no—
370C22×C3—44no—
371C22×C3—44no—
372C22×C3—41no—
373C22×C3—42no—
374C22×C3—41no—
375C22×C3—42no—
376C22×C3#93511yescharacteristic
377C22×C3#93711yescharacteristic
378C22×C3C2×#21711yescharacteristic
379C22×C3C2×C3×C22⋊C411yescharacteristic
380C22×C3C3×#12011yescharacteristic
381C22×C3C3×#12311yescharacteristic
382C3×C4—22no—
383C3×C4—22no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—42no—
387C3×C4—42no—
388C3×C4—44no—
389C3×C4—41no—
390C3×C4—42no—
391C3×C4—41no—
392C3×C4—42no—
393C3×C4—124no—
394C3×C4D4×Dic312yes—
395C3×C4#93912yes—
396C3×C4C3×C4×D412yes—
397C3×C4C3×#12112yes—
398Dic3—44no—
399C32#189711yescharacteristic, complemented, Sylow
400C23—21no—
401C23—21no—
402C23—21no—
403C23—42no—
404C23C3×#21712yes—
405C23C2×C3×#6711yescharacteristic
406Q8—22no—
407D4—22no—
408D4—44no—
409D4—41no—
410D4—41no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—42no—
418C2×C4—64no—
419C2×C4—124no—
420C2×C4C3×#21712yes—
421C2×C4C3×D3×D412yes—
422C2×C4C3×#23712yes—
423C2×C4C22×C3×Dic311yescharacteristic
424C2×C4C2×C3×#6711yescharacteristic
425C8—122no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—22no—
432C2×C3—41no—
433C2×C3—41no—
434C2×C3—44no—
435C2×C3—42no—
436C2×C3—42no—
437C2×C3—82no—
438C2×C3—81no—
439C2×C3—81no—
440C2×C3#522011yescharacteristic
441C2×C3#523612yes—
442C2×C3C3×#38611yescharacteristic
443C2×C3C3×#42112yes—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—42no—
449C22—42no—
450C22—41no—
451C22—42no—
452C22C3×#93511yescharacteristic
453C22C3×#93711yescharacteristic
454C22C2×C3×#21711yescharacteristic
455C4—22no—
456C4—41no—
457C4—42no—
458C4—124no—
459C4C3×D4×Dic312yes—
460C4C3×#93912yes—
461C3—21no—
462C3#7171211yescharacteristic, direct product factor
463C3C3×#189711yescharacteristic, complemented
464C2—21no—
465C2—21no—
466C2—42no—
467C2—81no—
468C2C3×#522011yescharacteristic
469C2C3×#523612yes—
subgroupG/H
centerC2×C3×C4C2×#67
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
Fitting, prime radicalC32×#525C2
socleC22×C32C2×C22⋊C4
cosocleC22×C3×C4C23×C3
hypercenterC3×#525D3
hypocenterC3C3×#1897
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ C3×#525

Complements

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

NH ≅ G/Ntype
2. C2×C3×#926C2semidirect
3. C2×C3×#837C2semidirect
4. C3×#5207C2semidirect
5. C3×#4532C2semidirect
7. #71712C3direct (N × H)
11. C32×C22⋊Q8C4semidirect
12. C32×#121C4semidirect
27. C2×#926C2×C3semidirect
30. C2×#837C2×C3semidirect
31. #5207C2×C3semidirect
33. #4532C2×C3semidirect
58. C3×C4×Dic3D4semidirect
74. C3×C22⋊Q8Dic3semidirect
75. C3×C22⋊Q8C3×C4semidirect
77. C3×#121Dic3semidirect
78. C3×#121C3×C4semidirect
204. C4×Dic3C3×D4semidirect
229. C22⋊Q8C3×Dic3semidirect
230. #121C3×Dic3semidirect
399. C32#1897semidirect
462. C3#71712direct (N × H)
463. C3C3×#1897semidirect

Maximal subgroups

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

subgroupindex
C32×#5252
C2×C3×#9262
C2×C3×#8372
C3×#52072
C3×#45322
C3×#18973
#717123