Finite Group Databasebeta

← C3×#71886 #11778504

Subgroups — C3×#71886 #11778504

Order 1152.

541 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#539C211yesmaximal, characteristic
2C3×#5043C211yesmaximal, characteristic, complemented
3C3×#5033C211yesmaximal, characteristic
4C3×#4981C211yesmaximal, characteristic, complemented
5C3×#4538C211yesmaximal, characteristic, complemented
6C3×#4537C211yesmaximal, characteristic
7C3×#4528C211yesmaximal, characteristic, complemented
8C3×#1609—31nomaximal
9#71886C311yesmaximal, characteristic, direct product factor
10C32×#126—21no—
11C32×#125—21no—
12C32×#123—21no—
13C32×#123—21no—
14C32×C22⋊Q8—21no—
15C32×C22⋊Q8C2211yescharacteristic
16C32×#121C2211yescharacteristic
17C32×C4×D4—21no—
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×Dic3⋊C4—21no—
21C3×#835—21no—
22C3×#832—21no—
23C3×C3⋊#95—21no—
24C3×C3⋊#95C2211yescharacteristic
25C3×#830C2211yescharacteristic
26C3×#830C2211yescharacteristic
27C3×#808—21no—
28C3×#807—21no—
29C3×#539—21no—
30C3×#539D311yescharacteristic
31C3×#539C2×C311yescharacteristic
32C3×#483—31no—
33C3×#480—31no—
34C3×#398—31no—
35#5043C2×C311yescharacteristic, complemented
36#5033C2×C311yescharacteristic
37#4981C2×C311yescharacteristic, complemented
38C3×#328—31no—
39C3×#327—31no—
40C3×#324—31no—
41#4538C2×C311yescharacteristic, complemented
42#4537C2×C311yescharacteristic
43#4528C2×C311yescharacteristic, complemented
44C2×C32×Q8D411yescharacteristic
45C2×C32×D4—21no—
46C2×C32×D4D411yescharacteristic
47C22×C32×C4—21no—
48C22×C32×C4D411yescharacteristic
49C22×C32×C4D411yescharacteristic
50C22×C32×C4C2311yescharacteristic
51C22×C3×Dic3—21no—
52C22×C3×Dic3—21no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4D411yescharacteristic
60C32×C4⋊C4D411yescharacteristic
61C32×C22⋊C4—21no—
62C32×C22⋊C4—21no—
63C32×C22⋊C4—21no—
64C32×C22⋊C4—21no—
65C32×C22⋊C4—21no—
66C32×C42—21no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3—21no—
69C3×Dic3⋊C4—42no—
70C2×C3×C3⋊C8—21no—
71C2×C3×C3⋊C8—21no—
72#1609—31noSylow
73C3×#126—22no—
74C3×#126—21no—
75C3×#126—21no—
76C3×#125—22no—
77C3×#125—21no—
78C3×#125—21no—
79C3×#123—22no—
80C3×#123—22no—
81C3×#123—21no—
82C3×#123—21no—
83C3×#123—21no—
84C3×#123—21no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—21no—
87C3×C22⋊Q8—21no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8C2×D311yescharacteristic
90C3×C22⋊Q8C22×C311yescharacteristic
91C3×#121—21no—
92C3×#121C2×D311yescharacteristic
93C3×#121C22×C311yescharacteristic
94C3×C4×D4—22no—
95C3×C4×D4—21no—
96C3×C4×D4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—31no—
99C2×C3×C4⋊C4—61no—
100C2×C3×C4⋊C4C2×D311yescharacteristic
101C2×C3×C4⋊C4C22×C311yescharacteristic
102C2×C4⋊Dic3C22×C311yescharacteristic
103C2×Dic3⋊C4—21no—
104C3×#107—61no—
105C3×#106—61no—
106C3×#103—61no—
107C3×#102—61no—
108C3×C22⋊C8—31no—
109C3×C22⋊C8—31no—
110C3×#95—31no—
111C3×#95—61no—
112#835—21no—
113#832—21no—
114C3⋊#95—21no—
115C3⋊#95C22×C311yescharacteristic
116#830C22×C311yescharacteristic
117#830C22×C311yescharacteristic
118#808—21no—
119#807—21no—
120C23×C32—21no—
121C23×C32C2×D411yescharacteristic
122C32×Q8—41no—
123C32×D4—41no—
124C32×D4—41no—
125C32×D4—41no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—41no—
134C2×C32×C4—41no—
135C2×C32×C4—81no—
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4C2×D411yescharacteristic
138C2×C3×Dic3—22no—
139C2×C3×Dic3—41no—
140C2×C3×Dic3—41no—
141C2×C3×Dic3—41no—
142C2×C3×Dic3—41no—
143C2×C3×Dic3—41no—
144C3×C3⋊C8—41no—
145C3×C3⋊C8—41no—
146#539C3×D311yescharacteristic
147#483—31no—
148#480—31no—
149#398—31no—
150#328—31no—
151#327—31no—
152#324—31no—
153C2×C3×Q8—21no—
154C2×C3×Q8#6711yescharacteristic
155C2×C3×Q8C3×D411yescharacteristic
156C2×C3×D4—22no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4#6711yescharacteristic
161C2×C3×D4C3×D411yescharacteristic
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—61no—
169C22×C3×C4—61no—
170C22×C3×C4#6711yescharacteristic
171C22×C3×C4#6711yescharacteristic
172C22×C3×C4C3×D411yescharacteristic
173C22×C3×C4C3×D411yescharacteristic
174C22×C3×C4C22×D311yescharacteristic
175C22×C3×C4C23×C311yescharacteristic
176C22×Dic3—21no—
177C22×Dic3—21no—
178C2×C3×C8—61no—
179C2×C3×C8—61no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—61no—
201C3×C4⋊C4—61no—
202C3×C4⋊C4—122no—
203C3×C4⋊C4#6711yescharacteristic
204C3×C4⋊C4#6711yescharacteristic
205C3×C4⋊C4C3×D411yescharacteristic
206C3×C4⋊C4C3×D411yescharacteristic
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—21no—
210C3×C22⋊C4—21no—
211C3×C22⋊C4—22no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—21no—
214C3×C22⋊C4—21no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—21no—
217C3×C22⋊C4—21no—
218C3×C22⋊C4—21no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—21no—
221C3×C22⋊C4—21no—
222C3×C42—22no—
223C3×C42—21no—
224C3×C42—21no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227Dic3⋊C4—42no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—21no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—41no—
233C22×C32—41no—
234C22×C32—41no—
235C22×C32#12011yescharacteristic
236C22×C32#13511yescharacteristic
237C22×C32AGL(1,ℤ/8)11yescharacteristic
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—41no—
241C32×C4—41no—
242C32×C4—41no—
243C32×C4—41no—
244C32×C4—41no—
245C3×Dic3—42no—
246C3×Dic3—81no—
247#126—21no—
248#125—21no—
249#123—21no—
250#123—21no—
251C22⋊Q8—21no—
252C22⋊Q8C2×C3×D311yescharacteristic
253#121C2×C3×D311yescharacteristic
254C4×D4—21no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—61no—
257C2×C4⋊C4C2×C3×D311yescharacteristic
258#107—61no—
259#106—61no—
260#103—61no—
261#102—61no—
262C22⋊C8—31no—
263C22⋊C8—31no—
264#95—31no—
265#95—61no—
266C23×C3—22no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3C2×#6711yescharacteristic
271C23×C3C2×C3×D411yescharacteristic
272C3×Q8—41no—
273C3×Q8—41no—
274C3×Q8—81no—
275C3×D4—41no—
276C3×D4—41no—
277C3×D4—42no—
278C3×D4—42no—
279C3×D4—41no—
280C3×D4—41no—
281C3×D4—41no—
282C3×D4—41no—
283C3×D4—81no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—62no—
312C2×C3×C4—81no—
313C2×C3×C4—81no—
314C2×C3×C4—82no—
315C2×C3×C4—81no—
316C2×C3×C4—81no—
317C2×C3×C4—121no—
318C2×C3×C4—121no—
319C2×C3×C4—121no—
320C2×C3×C4—121no—
321C2×C3×C4—121no—
322C2×C3×C4C2×#6711yescharacteristic
323C2×C3×C4C2×#6711yescharacteristic
324C2×C3×C4C2×C3×D411yescharacteristic
325C2×C3×C4C2×C3×D411yescharacteristic
326C2×Dic3—22no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C3×C8—121no—
333C3×C8—121no—
334C3⋊C8—41no—
335C3⋊C8—41no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—81no—
339C2×C32#44911yescharacteristic
340C2×C32#45611yescharacteristic
341C2×C32#45811yescharacteristic
342C2×Q8C3×#6711yescharacteristic
343C2×D4—21no—
344C2×D4C3×#6711yescharacteristic
345C22×C4—21no—
346C22×C4—61no—
347C22×C4—61no—
348C22×C4C3×#6711yescharacteristic
349C22×C4C3×#6711yescharacteristic
350C22×C4C22×C3×D311yescharacteristic
351C2×C8—61no—
352C2×C8—61no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—61no—
360C4⋊C4—61no—
361C4⋊C4—122no—
362C4⋊C4C3×#6711yescharacteristic
363C4⋊C4C3×#6711yescharacteristic
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C42—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—41no—
388C22×C3#94911yescharacteristic
389C22×C3#95011yescharacteristic
390C22×C3#95311yescharacteristic
391C22×C3C3×#12011yescharacteristic
392C22×C3C3×#13511yescharacteristic
393C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—41no—
399C3×C4—41no—
400C3×C4—42no—
401C3×C4—42no—
402C3×C4—42no—
403C3×C4—42no—
404C3×C4—42no—
405C3×C4—41no—
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—41no—
409C3×C4—41no—
410C3×C4—41no—
411C3×C4—41no—
412C3×C4—41no—
413C3×C4—41no—
414C3×C4—41no—
415C3×C4—122no—
416C3×C4—241no—
417Dic3—42no—
418Dic3—81no—
419C32#160911yescharacteristic, complemented, Sylow
420C23—21no—
421C23C2×C3×#6711yescharacteristic
422Q8—41no—
423D4—41no—
424D4—41no—
425D4—41no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—62no—
436C2×C4—81no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4C2×C3×#6711yescharacteristic
443C2×C4C2×C3×#6711yescharacteristic
444C8—121no—
445C8—121no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—82no—
456C2×C3—81no—
457C2×C3—81no—
458C2×C3#523811yescharacteristic
459C2×C3#524311yescharacteristic
460C2×C3#524611yescharacteristic
461C2×C3C3×#44911yescharacteristic
462C2×C3C3×#45611yescharacteristic
463C2×C3C3×#45811yescharacteristic
464C22—21no—
465C22—21no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22C3×#94911yescharacteristic
470C22C3×#95011yescharacteristic
471C22C3×#95311yescharacteristic
472C4—21no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—122no—
480C4—241no—
481C3—21no—
482C3#7188611yescharacteristic, direct product factor
483C3C3×#160911yescharacteristic, complemented
484C2—21no—
485C2—21no—
486C2—81no—
487C2C3×#523811yescharacteristic
488C2C3×#524311yescharacteristic
489C2C3×#524611yescharacteristic
subgroupG/H
centerC22×C3#953
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#539C2
socleC22×C32#120
prime radicalC32×#121C22
hypercenterC3×#539D3
hypocenterC3C3×#1609
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#539

Complements

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

NH ≅ G/Ntype
2. C3×#5043C2semidirect
4. C3×#4981C2semidirect
5. C3×#4538C2semidirect
7. C3×#4528C2semidirect
9. #71886C3direct (N × H)
35. #5043C2×C3semidirect
37. #4981C2×C3semidirect
41. #4538C2×C3semidirect
43. #4528C2×C3semidirect
419. C32#1609semidirect
482. C3#71886direct (N × H)
483. C3C3×#1609semidirect

Maximal subgroups

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

subgroupindex
C32×#5392
C3×#50432
C3×#50332
C3×#49812
C3×#45382
C3×#45372
C3×#45282
C3×#16093
#718863