Finite Group Databasebeta

← C3×#71888 #11778052

Subgroups — C3×#71888 #11778052

Order 1152.

653 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#539C211yesmaximal, characteristic, complemented
2C3×#5046C211yesmaximal, characteristic, complemented
3C3×#5036C211yesmaximal, characteristic
4C3×#4991C211yesmaximal, characteristic, complemented
5C3×#4538C211yesmaximal, characteristic, complemented
6C3×#4537C211yesmaximal, characteristic, complemented
7C3×#4528C211yesmaximal, characteristic, complemented
8C3×#1597—31nomaximal
9#71888C311yesmaximal, characteristic, direct product factor
10C22×C3×D12C2211yescharacteristic
11C32×#126—21no—
12C32×#125—21no—
13C32×#123—21no—
14C32×#123—21no—
15C32×C22⋊Q8—21no—
16C32×C22⋊Q8C2211yescharacteristic
17C32×#121C2211yescharacteristic
18C32×C4×D4—21no—
19C2×C32×C4⋊C4C2211yescharacteristic
20C2×C3×#215—21no—
21C2×C3×#213—21no—
22C2×C3×(C2×D3)⋊C4—21no—
23C2×C3×(C2×D3)⋊C4—21no—
24C3×C3⋊#95C2211yescharacteristic
25C3×#830C2211yescharacteristic
26C3×#830C2211yescharacteristic
27C3×#809—21no—
28C3×#809—21no—
29C3×#539—21no—
30C3×#539D311yescharacteristic, complemented
31C3×#539C2×C311yescharacteristic, complemented
32C3×#450—31no—
33C3×#447—31no—
34C3×#394—31no—
35#5046C2×C311yescharacteristic, complemented
36#5036C2×C311yescharacteristic
37#4991C2×C311yescharacteristic, complemented
38C3×#328—31no—
39C3×#327—31no—
40C3×#324—31no—
41#4538C2×C311yescharacteristic, complemented
42#4537C2×C311yescharacteristic, complemented
43#4528C2×C311yescharacteristic, complemented
44C23×C3×D3—21no—
45C2×C32×Q8D411yescharacteristic
46C2×C32×D4—21no—
47C2×C32×D4D411yescharacteristic
48C22×C32×C4—21no—
49C22×C32×C4D411yescharacteristic
50C22×C32×C4D411yescharacteristic
51C22×C32×C4C2311yescharacteristic
52C22×C3×Dic3—21no—
53C2×C3×D12—21no—
54C2×C3×D12—21no—
55C2×C3×D12—42no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4D411yescharacteristic
63C32×C4⋊C4D411yescharacteristic
64C32×C22⋊C4—21no—
65C32×C22⋊C4—21no—
66C32×C22⋊C4—21no—
67C32×C22⋊C4—21no—
68C32×C22⋊C4—21no—
69C32×C42—21no—
70C3×#215—42no—
71C3×#213—42no—
72C3×(C2×D3)⋊C4—42no—
73C3×(C2×D3)⋊C4—42no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76#1597—31noSylow
77C22×C3×D4—31no—
78C22×D12C22×C311yescharacteristic
79C2×C3×SD16—61no—
80C2×C3×D8—61no—
81C3×#126—22no—
82C3×#126—21no—
83C3×#126—21no—
84C3×#125—22no—
85C3×#125—21no—
86C3×#125—21no—
87C3×#123—22no—
88C3×#123—22no—
89C3×#123—21no—
90C3×#123—21no—
91C3×#123—21no—
92C3×#123—21no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8—21no—
95C3×C22⋊Q8—21no—
96C3×C22⋊Q8—21no—
97C3×C22⋊Q8C2×D311yescharacteristic
98C3×C22⋊Q8C22×C311yescharacteristic
99C3×#121—21no—
100C3×#121C2×D311yescharacteristic
101C3×#121C22×C311yescharacteristic
102C3×C4×D4—22no—
103C3×C4×D4—21no—
104C3×C4×D4—21no—
105C2×C3×C4⋊C4—21no—
106C2×C3×C4⋊C4C2×D311yescharacteristic
107C2×C3×C4⋊C4C22×C311yescharacteristic
108C2×C3×C22⋊C4—61no—
109C2×C3×C22⋊C4—61no—
110C2×#215—21no—
111C2×#213—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C3×#102—61no—
115C3×#102—61no—
116C3×C22⋊C8—31no—
117C3×C22⋊C8—31no—
118C3×#95—31no—
119C3⋊#95C22×C311yescharacteristic
120#830C22×C311yescharacteristic
121#830C22×C311yescharacteristic
122#809—21no—
123#809—21no—
124C23×C32—21no—
125C23×C32C2×D411yescharacteristic
126C22×C3×D3—22no—
127C22×C3×D3—41no—
128C22×C3×D3—41no—
129C22×C3×D3—41no—
130C22×C3×D3—41no—
131C22×C3×D3—41no—
132C22×C3×D3—41no—
133C32×Q8—41no—
134C32×D4—41no—
135C32×D4—41no—
136C32×D4—41no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—41no—
145C2×C32×C4—41no—
146C2×C32×C4—81no—
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2×D411yescharacteristic
149C2×C3×Dic3—41no—
150C2×C3×Dic3—41no—
151C2×C3×Dic3—41no—
152C3×D12—22no—
153C3×D12—22no—
154C3×D12—41no—
155C3×D12—41no—
156C3×C3⋊C8—41no—
157C3×C3⋊C8—41no—
158#539C3×D311yescharacteristic, complemented
159#450—31no—
160#447—31no—
161#394—31no—
162#328—31no—
163#327—31no—
164#324—31no—
165C24×C3—61no—
166C23×D3—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8#6711yescharacteristic
169C2×C3×Q8C3×D411yescharacteristic
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C2×C3×D4—61no—
175C2×C3×D4—61no—
176C2×C3×D4—122no—
177C2×C3×D4#6711yescharacteristic
178C2×C3×D4C3×D411yescharacteristic
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—61no—
186C22×C3×C4#6711yescharacteristic
187C22×C3×C4#6711yescharacteristic
188C22×C3×C4C3×D411yescharacteristic
189C22×C3×C4C3×D411yescharacteristic
190C22×C3×C4C22×D311yescharacteristic
191C22×C3×C4C23×C311yescharacteristic
192C22×Dic3—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—42no—
196C3×SD16—122no—
197C3×D8—122no—
198C2×C3×C8—61no—
199C2×C3×C8—61no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4#6711yescharacteristic
221C3×C4⋊C4#6711yescharacteristic
222C3×C4⋊C4C3×D411yescharacteristic
223C3×C4⋊C4C3×D411yescharacteristic
224C3×C22⋊C4—22no—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—22no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—21no—
231C3×C22⋊C4—21no—
232C3×C22⋊C4—22no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—21no—
237C3×C22⋊C4—21no—
238C3×C22⋊C4—21no—
239C3×C22⋊C4—122no—
240C3×C22⋊C4—122no—
241C3×C42—22no—
242C3×C42—21no—
243C3×C42—21no—
244#215—42no—
245#213—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—41no—
253C22×C32—41no—
254C22×C32—41no—
255C22×C32#12011yescharacteristic
256C22×C32#13511yescharacteristic
257C22×C32AGL(1,ℤ/8)11yescharacteristic
258C2×C3×D3—42no—
259C2×C3×D3—42no—
260C2×C3×D3—42no—
261C2×C3×D3—81no—
262C2×C3×D3—81no—
263C2×C3×D3—81no—
264C2×C3×D3—81no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—41no—
268C32×C4—41no—
269C32×C4—41no—
270C32×C4—41no—
271C32×C4—41no—
272C3×Dic3—81no—
273C22×D4—31no—
274C2×SD16—61no—
275C2×D8—61no—
276#126—21no—
277#125—21no—
278#123—21no—
279#123—21no—
280C22⋊Q8—21no—
281C22⋊Q8C2×C3×D311yescharacteristic
282#121C2×C3×D311yescharacteristic
283C4×D4—21no—
284C2×C4⋊C4C2×C3×D311yescharacteristic
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287#102—61no—
288#102—61no—
289C22⋊C8—31no—
290C22⋊C8—31no—
291#95—31no—
292C23×C3—22no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—62no—
297C23×C3—121no—
298C23×C3—121no—
299C23×C3—121no—
300C23×C3—121no—
301C23×C3—121no—
302C23×C3—121no—
303C23×C3C2×#6711yescharacteristic
304C23×C3C2×C3×D411yescharacteristic
305C22×D3—22no—
306C22×D3—41no—
307C22×D3—41no—
308C22×D3—41no—
309C22×D3—41no—
310C22×D3—41no—
311C22×D3—41no—
312C3×Q8—41no—
313C3×Q8—41no—
314C3×Q8—81no—
315C3×D4—41no—
316C3×D4—41no—
317C3×D4—42no—
318C3×D4—42no—
319C3×D4—41no—
320C3×D4—41no—
321C3×D4—41no—
322C3×D4—41no—
323C3×D4—62no—
324C3×D4—62no—
325C3×D4—81no—
326C3×D4—121no—
327C3×D4—121no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—81no—
356C2×C3×C4—81no—
357C2×C3×C4—82no—
358C2×C3×C4—81no—
359C2×C3×C4—81no—
360C2×C3×C4—121no—
361C2×C3×C4—121no—
362C2×C3×C4—121no—
363C2×C3×C4C2×#6711yescharacteristic
364C2×C3×C4C2×#6711yescharacteristic
365C2×C3×C4C2×C3×D411yescharacteristic
366C2×C3×C4C2×C3×D411yescharacteristic
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370D12—22no—
371D12—22no—
372D12—41no—
373D12—41no—
374C3×C8—121no—
375C3×C8—121no—
376C3⋊C8—41no—
377C3⋊C8—41no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—81no—
381C2×C32#44911yescharacteristic
382C2×C32#45411yescharacteristic
383C2×C32B(4,𝔽2)11yescharacteristic
384C3×D3—82no—
385C24—61no—
386C2×Q8C3×#6711yescharacteristic
387C2×D4—21no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—122no—
391C2×D4C3×#6711yescharacteristic
392C22×C4—21no—
393C22×C4—61no—
394C22×C4C3×#6711yescharacteristic
395C22×C4C3×#6711yescharacteristic
396C22×C4C22×C3×D311yescharacteristic
397SD16—122no—
398D8—122no—
399C2×C8—61no—
400C2×C8—61no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4C3×#6711yescharacteristic
408C4⋊C4C3×#6711yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—122no—
415C22⋊C4—122no—
416C42—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—41no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—122no—
436C22×C3—122no—
437C22×C3—122no—
438C22×C3—241no—
439C22×C3—241no—
440C22×C3—241no—
441C22×C3—241no—
442C22×C3#94911yescharacteristic
443C22×C3#95011yescharacteristic
444C22×C3#95311yescharacteristic
445C22×C3C3×#12011yescharacteristic
446C22×C3C3×#13511yescharacteristic
447C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
448C2×D3—42no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—81no—
452C2×D3—81no—
453C2×D3—81no—
454C2×D3—81no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—41no—
460C3×C4—41no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—42no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—241no—
477Dic3—81no—
478C32#159711yescharacteristic, complemented, Sylow
479C23—21no—
480C23—62no—
481C23—121no—
482C23—121no—
483C23—121no—
484C23—121no—
485C23—121no—
486C23—121no—
487C23C2×C3×#6711yescharacteristic
488Q8—41no—
489D4—41no—
490D4—41no—
491D4—41no—
492D4—62no—
493D4—62no—
494D4—121no—
495D4—121no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—81no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4C2×C3×#6711yescharacteristic
510C2×C4C2×C3×#6711yescharacteristic
511C8—121no—
512C8—121no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—41no—
521C2×C3—41no—
522C2×C3—82no—
523C2×C3—81no—
524C2×C3—81no—
525C2×C3—242no—
526C2×C3#523811yescharacteristic
527C2×C3#524411yescharacteristic
528C2×C3#524511yescharacteristic
529C2×C3C3×#44911yescharacteristic
530C2×C3C3×#45411yescharacteristic
531C2×C3C3×B(4,𝔽2)11yescharacteristic
532D3—82no—
533C22—21no—
534C22—21no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—122no—
539C22—122no—
540C22—122no—
541C22—241no—
542C22—241no—
543C22—241no—
544C22—241no—
545C22C3×#94911yescharacteristic
546C22C3×#95011yescharacteristic
547C22C3×#95311yescharacteristic
548C4—21no—
549C4—21no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—241no—
556C3—21no—
557C3#7188811yescharacteristic, direct product factor
558C3C3×#159711yescharacteristic, complemented
559C2—21no—
560C2—21no—
561C2—81no—
562C2—242no—
563C2C3×#523811yescharacteristic
564C2C3×#524411yescharacteristic
565C2C3×#524511yescharacteristic
subgroupG/H
centerC22×C3#953
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#539C2
socleC22×C32#120
prime radicalC3×#5036C2
hypercenterC3×#539D3
hypocenterC3C3×#1597
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
1. C32×#539C2semidirect
2. C3×#5046C2semidirect
4. C3×#4991C2semidirect
5. C3×#4538C2semidirect
6. C3×#4537C2semidirect
7. C3×#4528C2semidirect
9. #71888C3direct (N × H)
30. C3×#539D3semidirect
31. C3×#539C2×C3semidirect
35. #5046C2×C3semidirect
37. #4991C2×C3semidirect
41. #4538C2×C3semidirect
42. #4537C2×C3semidirect
43. #4528C2×C3semidirect
158. #539C3×D3semidirect
478. C32#1597semidirect
557. C3#71888direct (N × H)
558. C3C3×#1597semidirect

Maximal subgroups

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

subgroupindex
C32×#5392
C3×#50462
C3×#50362
C3×#49912
C3×#45382
C3×#45372
C3×#45282
C3×#15973
#718883