Finite Group Databasebeta

← C3×#71711 #11788168

Subgroups — C3×#71711 #11788168

Order 1152.

1049 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#525C211yesmaximal, characteristic
2C22×C3×C4×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#837C212yesmaximal, complemented
4C3×#5207C211yesmaximal, characteristic, complemented
5C3×#4532C212yesmaximal, complemented
6C3×#1896—31nomaximal
7#71711C311yesmaximal, characteristic, direct product factor
8C2×C32×#41C2211yescharacteristic
9C23×C32×C4C2211yescharacteristic
10C23×C3×Dic3—21no—
11C32×#123—21no—
12C32×C22⋊Q8C412yescomplemented
13C32×#121C412yescomplemented
14C32×#120—21no—
15C32×C4×D4—22no—
16C32×#117C2211yescharacteristic
17C2×C3×C4×Dic3—21no—
18C2×C3×C4×Dic3—24no—
19C2×C3×C4×Dic3C2212yes—
20C2×C3×#208C2212yes—
21C3×#837—24no—
22C3×#830—22no—
23C3×#525—21no—
24C3×#525D311yescharacteristic
25C3×#525C2×C311yescharacteristic
26C22×C3×C42—31no—
27C22×C4×Dic3C2×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, complemented
38C2×C32×D4—22no—
39C2×C32×D4—21no—
40C2×C32×D4C2×C411yescharacteristic, complemented
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—24no—
49C22×C3×Dic3—28no—
50C22×C3×Dic3—22no—
51C32×C4⋊C4—22no—
52C32×C4⋊C4C2×C412yescomplemented
53C32×C22⋊C4—22no—
54C32×C22⋊C4—21no—
55C32×C22⋊C4—22no—
56C32×C42—21no—
57C3×C4×Dic3—24no—
58C3×C4×Dic3—22no—
59C3×C4×Dic3D414yescomplemented
60C3×#208—22no—
61C3×#208—22no—
62C2×C3×C3⋊C8—22no—
63#1896—31noSylow
64C2×C3×#41—21no—
65C2×C3×#41C2×D311yescharacteristic
66C2×C3×#41C22×C311yescharacteristic
67C23×C3×C4—21no—
68C23×C3×C4—61no—
69C23×C3×C4C2×D311yescharacteristic
70C23×C3×C4C22×C311yescharacteristic
71C23×Dic3—21no—
72C2×C3×M16—32no—
73C3×#123—22no—
74C3×#123—21no—
75C3×#123—21no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8Dic312yescomplemented
78C3×C22⋊Q8C3×C412yescomplemented
79C3×#121—22no—
80C3×#121Dic312yescomplemented
81C3×#121C3×C412yescomplemented
82C3×#120—22no—
83C3×#120—21no—
84C3×#120—21no—
85C3×C4×D4—24no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×#117—21no—
89C3×#117C2×D311yescharacteristic
90C3×#117C22×C311yescharacteristic
91C2×C3×C42—32no—
92C2×C3×C42—64no—
93C2×C3×C42—61no—
94C2×C4×Dic3—21no—
95C2×C4×Dic3—24no—
96C2×C4×Dic3C22×C312yes—
97C2×#208C22×C312yes—
98C3×#104—64no—
99C3×C22⋊C8—62no—
100#837—24no—
101#830—22no—
102C23×C32—21no—
103C23×C32—24no—
104C23×C32—21no—
105C23×C32—21no—
106C23×C32C22⋊C412yes—
107C23×C32C2×D411yescharacteristic
108C32×Q8—22no—
109C32×D4—22no—
110C32×D4—44no—
111C32×D4—41no—
112C32×D4—41no—
113C2×C32×C4—22no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—42no—
120C2×C32×C4C22⋊C412yes—
121C2×C32×C4C22×C411yescharacteristic
122C2×C32×C4C2×D412yes—
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4#4112yes—
125C2×C3×Dic3—216no—
126C2×C3×Dic3—28no—
127C2×C3×Dic3—24no—
128C3×C3⋊C8—42no—
129#525C3×D311yescharacteristic
130C22×C42—31no—
131C2×#104—32no—
132#407—31no—
133#339—32no—
134C24×C3—21no—
135C24×C3#6711yescharacteristic
136C24×C3C3×D411yescharacteristic
137C3×#41—22no—
138C3×#41—22no—
139C3×#41—22no—
140C3×#41—22no—
141C3×#41—42no—
142C3×#41—42no—
143C2×C3×Q8—21no—
144C2×C3×Q8C2×Dic311yescharacteristic, complemented
145C2×C3×Q8C2×C3×C411yescharacteristic, complemented
146C2×C3×D4—24no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4C2×Dic311yescharacteristic, complemented
154C2×C3×D4C2×C3×C411yescharacteristic, complemented
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4—64no—
169C22×C3×C4—68no—
170C22×C3×C4—62no—
171C22×C3×C4C2×Dic312yes—
172C22×C3×C4#6711yescharacteristic
173C22×C3×C4#6712yes—
174C22×C3×C4C2×C3×C412yes—
175C22×C3×C4C3×D411yescharacteristic
176C22×C3×C4C3×D412yes—
177C22×C3×C4C22×D311yescharacteristic
178C22×C3×C4C23×C311yescharacteristic
179C22×Dic3—24no—
180C22×Dic3—28no—
181C22×Dic3—22no—
182C3×M16—62no—
183C3×M16—62no—
184C2×C3×C8—62no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4C2×Dic312yescomplemented
190C3×C4⋊C4C2×C3×C412yescomplemented
191C3×C22⋊C4—22no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—22no—
195C3×C22⋊C4—22no—
196C3×C22⋊C4—21no—
197C3×C22⋊C4—21no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C42—22no—
201C3×C42—21no—
202C3×C42—21no—
203C3×C42—34no—
204C3×C42—64no—
205C3×C42—62no—
206C4×Dic3—24no—
207C4×Dic3—22no—
208C4×Dic3C3×D414yescomplemented
209#208—22no—
210#208—22no—
211C2×C3⋊C8—22no—
212C22×C32—24no—
213C22×C32—24no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—41no—
218C22×C32—42no—
219C22×C32#10414yescomplemented
220C22×C32C2×C22⋊C411yescharacteristic
221C22×C32#12011yescharacteristic
222C22×C32#12311yescharacteristic
223C32×C4—22no—
224C32×C4—41no—
225C32×C4—42no—
226C32×C4C4×D412yes—
227C32×C4#12112yes—
228C3×Dic3—28no—
229C2×#41C2×C3×D311yescharacteristic
230C23×C4—61no—
231C23×C4C2×C3×D311yescharacteristic
232C2×M16—32no—
233#123—21no—
234C22⋊Q8C3×Dic312yescomplemented
235#121C3×Dic312yescomplemented
236#120—21no—
237C4×D4—22no—
238#117C2×C3×D311yescharacteristic
239C2×C42—32no—
240C2×C42—64no—
241C2×C42—61no—
242#104—64no—
243C22⋊C8—62no—
244C23×C3—24no—
245C23×C3—24no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—22no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—22no—
254C23×C3—21no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—44no—
258C23×C3#21712yes—
259C23×C3C3×C22⋊C412yes—
260C23×C3C2×#6711yescharacteristic
261C23×C3C2×C3×D411yescharacteristic
262C3×Q8—22no—
263C3×Q8—22no—
264C3×Q8—42no—
265C3×D4—22no—
266C3×D4—22no—
267C3×D4—42no—
268C3×D4—48no—
269C3×D4—42no—
270C3×D4—42no—
271C3×D4—44no—
272C3×D4—44no—
273C3×D4—41no—
274C3×D4—41no—
275C3×D4—41no—
276C3×D4—41no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—24no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—21no—
289C2×C3×C4—22no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—44no—
297C2×C3×C4—44no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4—44no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—616no—
304C2×C3×C4—68no—
305C2×C3×C4—64no—
306C2×C3×C4#21712yes—
307C2×C3×C4C3×C22⋊C412yes—
308C2×C3×C4D3×D412yes—
309C2×C3×C4#23712yes—
310C2×C3×C4C22×Dic311yescharacteristic
311C2×C3×C4C2×#6711yescharacteristic
312C2×C3×C4C22×C3×C411yescharacteristic
313C2×C3×C4C2×C3×D412yes—
314C2×C3×C4C2×C3×D411yescharacteristic
315C2×C3×C4C3×#4112yes—
316C2×Dic3—216no—
317C2×Dic3—28no—
318C2×Dic3—24no—
319C3×C8—122no—
320C3⋊C8—42no—
321C2×C32—21no—
322C2×C32—24no—
323C2×C32—21no—
324C2×C32—81no—
325C2×C32#38611yescharacteristic
326C2×C32C2×#10412yes—
327C24C3×#6711yescharacteristic
328#41—22no—
329#41—22no—
330C2×Q8C2×C3×Dic311yescharacteristic, complemented
331C2×D4—22no—
332C2×D4—21no—
333C2×D4C2×C3×Dic311yescharacteristic, complemented
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—64no—
338C22×C4—68no—
339C22×C4—62no—
340C22×C4C2×C3×Dic312yes—
341C22×C4C3×#6711yescharacteristic
342C22×C4C3×#6712yes—
343C22×C4C22×C3×D311yescharacteristic
344M16—62no—
345M16—62no—
346C2×C8—62no—
347C4⋊C4—22no—
348C4⋊C4C2×C3×Dic312yescomplemented
349C22⋊C4—22no—
350C22⋊C4—21no—
351C22⋊C4—22no—
352C42—21no—
353C42—34no—
354C42—64no—
355C42—62no—
356C22×C3—24no—
357C22×C3—24no—
358C22×C3—24no—
359C22×C3—24no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—24no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—44no—
374C22×C3—44no—
375C22×C3—42no—
376C22×C3—44no—
377C22×C3—41no—
378C22×C3—42no—
379C22×C3—41no—
380C22×C3—42no—
381C22×C3#83714yescomplemented
382C22×C3C3×#10414yescomplemented
383C22×C3#93511yescharacteristic
384C22×C3#93711yescharacteristic
385C22×C3C2×#21711yescharacteristic
386C22×C3C2×C3×C22⋊C411yescharacteristic
387C22×C3C3×#12011yescharacteristic
388C22×C3C3×#12311yescharacteristic
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—22no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—44no—
396C3×C4—41no—
397C3×C4—42no—
398C3×C4—41no—
399C3×C4—42no—
400C3×C4—68no—
401C3×C4D4×Dic312yes—
402C3×C4#93912yes—
403C3×C4C3×C4×D412yes—
404C3×C4C3×#12112yes—
405Dic3—28no—
406C32#189611yescharacteristic, complemented, Sylow
407C23—21no—
408C23—21no—
409C23—24no—
410C23—21no—
411C23C3×#21712yes—
412C23C2×C3×#6711yescharacteristic
413Q8—22no—
414D4—22no—
415D4—41no—
416D4—41no—
417D4—44no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—42no—
425C2×C4—616no—
426C2×C4—68no—
427C2×C4—64no—
428C2×C4C3×#21712yes—
429C2×C4C3×D3×D412yes—
430C2×C4C3×#23712yes—
431C2×C4C22×C3×Dic311yescharacteristic
432C2×C4C2×C3×#6711yescharacteristic
433C8—122no—
434C2×C3—21no—
435C2×C3—24no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—24no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—22no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—44no—
445C2×C3—82no—
446C2×C3—81no—
447C2×C3—81no—
448C2×C3#522011yescharacteristic
449C2×C3C2×#83712yes—
450C2×C3C3×#38611yescharacteristic
451C2×C3C2×C3×#10412yes—
452C22—24no—
453C22—24no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—41no—
458C22—42no—
459C22C3×#83714yescomplemented
460C22C3×#93511yescharacteristic
461C22C3×#93711yescharacteristic
462C22C2×C3×#21711yescharacteristic
463C4—22no—
464C4—41no—
465C4—42no—
466C4—68no—
467C4C3×D4×Dic312yes—
468C4C3×#93912yes—
469C3—21no—
470C3#7171111yescharacteristic, direct product factor
471C3C3×#189611yescharacteristic, complemented
472C2—21no—
473C2—21no—
474C2—24no—
475C2—81no—
476C2C3×#522011yescharacteristic
477C2C2×C3×#83712yes—
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×#1896
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. C22×C3×C4×Dic3C2semidirect
3. C2×C3×#837C2semidirect
4. C3×#5207C2semidirect
5. C3×#4532C2semidirect
7. #71711C3direct (N × H)
12. C32×C22⋊Q8C4semidirect
13. C32×#121C4semidirect
27. C22×C4×Dic3C2×C3semidirect
30. C2×#837C2×C3semidirect
31. #5207C2×C3semidirect
33. #4532C2×C3semidirect
37. C2×C32×Q8C2×C4semidirect
40. C2×C32×D4C2×C4semidirect
52. C32×C4⋊C4C2×C4semidirect
59. C3×C4×Dic3D4semidirect
77. C3×C22⋊Q8Dic3semidirect
78. C3×C22⋊Q8C3×C4semidirect
80. C3×#121Dic3semidirect
81. C3×#121C3×C4semidirect
144. C2×C3×Q8C2×Dic3semidirect
145. C2×C3×Q8C2×C3×C4semidirect
153. C2×C3×D4C2×Dic3semidirect
154. C2×C3×D4C2×C3×C4semidirect
189. C3×C4⋊C4C2×Dic3semidirect
190. C3×C4⋊C4C2×C3×C4semidirect
208. C4×Dic3C3×D4semidirect
219. C22×C32#104semidirect
234. C22⋊Q8C3×Dic3semidirect
235. #121C3×Dic3semidirect
330. C2×Q8C2×C3×Dic3semidirect
333. C2×D4C2×C3×Dic3semidirect
348. C4⋊C4C2×C3×Dic3semidirect
381. C22×C3#837semidirect
382. C22×C3C3×#104semidirect
406. C32#1896semidirect
459. C22C3×#837semidirect
470. C3#71711direct (N × H)
471. C3C3×#1896semidirect

Maximal subgroups

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

subgroupindex
C32×#5252
C22×C3×C4×Dic32
C2×C3×#8372
C3×#52072
C3×#45322
C3×#18963
#717113