Finite Group Databasebeta

← C3×#71611 #11787173

Subgroups — C3×#71611 #11787173

Order 1152.

1029 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic
2C22×C3×#208C211yesmaximal, characteristic, complemented
3C2×C3×#830C211yesmaximal, characteristic, complemented
4C2×C3×#830C212yesmaximal, complemented
5C3×#4524C212yesmaximal, complemented
6C3×#1870—31nomaximal
7#71611C311yesmaximal, characteristic, direct product factor
8C22×C32×D4C411yescharacteristic
9C23×C32×C4C2211yescharacteristic
10C23×C32×C4C2211yescharacteristic
11C32×C4×D4—22no—
12C32×C4×D4—22no—
13C2×C32×C4⋊C4C411yescharacteristic
14C2×C32×C22⋊C4C411yescharacteristic
15C2×C32×C22⋊C4C411yescharacteristic
16C2×C32×C42C2211yescharacteristic
17C2×C3×#208—22no—
18C2×C3×#208—24no—
19C22×C3×C3⋊C8C2212yes—
20C22×C3×C3⋊C8C2212yes—
21C3×#830—22no—
22C3×#830—24no—
23C22×C3×M16—31no—
24C2×C3×C4×D4—21no—
25C2×C3×C4×D4D311yescharacteristic
26C2×C3×C4×D4C2×C311yescharacteristic
27C22×#208C2×C311yescharacteristic, complemented
28C2×C3×C22⋊C8—32no—
29C2×C3×C22⋊C8—31no—
30C2×#830C2×C311yescharacteristic, complemented
31C2×#830C2×C312yescomplemented
32C3×#336—32no—
33#4524C2×C312yescomplemented
34C24×C32C2×C411yescharacteristic
35C24×C32C2×C411yescharacteristic
36C2×C32×D4—22no—
37C2×C32×D4—22no—
38C2×C32×D4—22no—
39C22×C32×C4—22no—
40C22×C32×C4—22no—
41C22×C32×C4—21no—
42C22×C32×C4—21no—
43C22×C32×C4—22no—
44C22×C32×C4—22no—
45C22×C32×C4C2×C411yescharacteristic
46C22×C32×C4C2×C411yescharacteristic
47C22×C32×C4C2×C411yescharacteristic
48C22×C32×C4C2×C411yescharacteristic
49C22×C32×C4D412yes—
50C22×C32×C4D412yes—
51C22×C32×C4C2311yescharacteristic
52C32×C4⋊C4—22no—
53C32×C22⋊C4—22no—
54C32×C22⋊C4—22no—
55C32×C42—21no—
56C32×C42—21no—
57C3×#208—28no—
58C2×C3×C3⋊C8—22no—
59C2×C3×C3⋊C8—22no—
60C2×C3×C3⋊C8—22no—
61C2×C3×C3⋊C8—22no—
62C2×C3×C3⋊C8—22no—
63C2×C3×C3⋊C8D414yescomplemented
64#1870—31noSylow
65C22×C3×D4—21no—
66C22×C3×D4Dic311yescharacteristic
67C22×C3×D4C3×C411yescharacteristic
68C23×C3×C4—21no—
69C23×C3×C4—21no—
70C23×C3×C4C2×D311yescharacteristic
71C23×C3×C4C2×D311yescharacteristic
72C23×C3×C4C22×C311yescharacteristic
73C23×C3×C4C22×C311yescharacteristic
74C2×C3×M16—64no—
75C2×C3×M16—62no—
76C22×C3×C8—32no—
77C22×C3×C8—32no—
78C3×C4×D4—24no—
79C3×C4×D4—24no—
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4Dic311yescharacteristic
86C2×C3×C4⋊C4C3×C411yescharacteristic
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4Dic311yescharacteristic
90C2×C3×C22⋊C4Dic311yescharacteristic
91C2×C3×C22⋊C4C3×C411yescharacteristic
92C2×C3×C22⋊C4C3×C411yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42C2×D311yescharacteristic
95C2×C3×C42C22×C311yescharacteristic
96C2×#208—22no—
97C2×#208—24no—
98C22×C3⋊C8C22×C312yes—
99C22×C3⋊C8C22×C312yes—
100C3×C22⋊C8—64no—
101C3×C22⋊C8—62no—
102#830—22no—
103#830—24no—
104C23×C32—21no—
105C23×C32—21no—
106C23×C32—21no—
107C23×C32—21no—
108C23×C32—22no—
109C23×C32—22no—
110C23×C32—22no—
111C23×C32—22no—
112C23×C32C22⋊C412yes—
113C23×C32M1612yescomplemented
114C23×C32C22×C411yescharacteristic
115C32×D4—28no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—24no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—41no—
127C2×C32×C4—42no—
128C2×C32×C4—41no—
129C2×C32×C4C22⋊C412yes—
130C2×C32×C4M1612yescomplemented
131C2×C32×C4C2×D412yes—
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4#4112yes—
135C3×C3⋊C8—24no—
136C3×C3⋊C8—42no—
137C22×M16—31no—
138C2×C4×D4C3×D311yescharacteristic
139C2×C22⋊C8—32no—
140C2×C22⋊C8—31no—
141#336—32no—
142C24×C3—21no—
143C24×C3—21no—
144C24×C3C2×Dic311yescharacteristic
145C24×C3C2×Dic311yescharacteristic
146C24×C3C2×C3×C411yescharacteristic
147C24×C3C2×C3×C411yescharacteristic
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—24no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—42no—
179C22×C3×C4—42no—
180C22×C3×C4—42no—
181C22×C3×C4—42no—
182C22×C3×C4C2×Dic311yescharacteristic
183C22×C3×C4C2×Dic311yescharacteristic
184C22×C3×C4C2×Dic311yescharacteristic
185C22×C3×C4C2×Dic311yescharacteristic
186C22×C3×C4#6712yes—
187C22×C3×C4#6712yes—
188C22×C3×C4C2×C3×C411yescharacteristic
189C22×C3×C4C2×C3×C411yescharacteristic
190C22×C3×C4C2×C3×C411yescharacteristic
191C22×C3×C4C2×C3×C411yescharacteristic
192C22×C3×C4C3×D412yes—
193C22×C3×C4C3×D412yes—
194C22×C3×C4C22×D311yescharacteristic
195C22×C3×C4C23×C311yescharacteristic
196C3×M16—68no—
197C2×C3×C8—34no—
198C2×C3×C8—62no—
199C2×C3×C8—62no—
200C2×C3×C8—62no—
201C2×C3×C8—62no—
202C2×C3×C8—62no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—24no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—22no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—22no—
212C3×C42—22no—
213C3×C42—22no—
214C3×C42—21no—
215C3×C42—21no—
216C3×C42—21no—
217C3×C42—21no—
218#208—28no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C2×C3⋊C8C3×D414yescomplemented
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—22no—
229C22×C32—24no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—42no—
233C22×C32—42no—
234C22×C32—42no—
235C22×C32C2×C22⋊C411yescharacteristic
236C22×C32C4×D412yes—
237C22×C32C2×M1611yescharacteristic
238C22×C32C2×M1611yescharacteristic
239C22×C32#13112yes—
240C32×C4—22no—
241C32×C4—22no—
242C32×C4—41no—
243C32×C4—41no—
244C32×C4#12011yescharacteristic
245C32×C4#12112yes—
246C32×C4#12311yescharacteristic
247C22×D4C3×Dic311yescharacteristic
248C23×C4C2×C3×D311yescharacteristic
249C23×C4C2×C3×D311yescharacteristic
250C2×M16—64no—
251C2×M16—62no—
252C22×C8—32no—
253C22×C8—32no—
254C4×D4—22no—
255C4×D4—22no—
256C2×C4⋊C4C3×Dic311yescharacteristic
257C2×C22⋊C4C3×Dic311yescharacteristic
258C2×C22⋊C4C3×Dic311yescharacteristic
259C2×C42C2×C3×D311yescharacteristic
260C22⋊C8—64no—
261C22⋊C8—62no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—22no—
265C23×C3—22no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—21no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3#20812yescomplemented
290C23×C3#21712yes—
291C23×C3C3×C22⋊C412yes—
292C23×C3C3×M1612yescomplemented
293C23×C3C22×Dic311yescharacteristic
294C23×C3C22×C3×C411yescharacteristic
295C3×D4—216no—
296C3×D4—28no—
297C3×D4—28no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—21no—
301C2×C3×C4—22no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—21no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—24no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—44no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—44no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4—41no—
343C2×C3×C4#20812yescomplemented
344C2×C3×C4#21712yes—
345C2×C3×C4C3×C22⋊C412yes—
346C2×C3×C4C3×M1612yescomplemented
347C2×C3×C4D3×D412yes—
348C2×C3×C4#23712yes—
349C2×C3×C4C2×#6711yescharacteristic
350C2×C3×C4C2×#6711yescharacteristic
351C2×C3×C4C2×C3×D412yes—
352C2×C3×C4C2×C3×D411yescharacteristic
353C2×C3×C4C2×C3×D411yescharacteristic
354C2×C3×C4C3×#4112yes—
355C3×C8—64no—
356C3×C8—122no—
357C3⋊C8—24no—
358C3⋊C8—42no—
359C2×C32—42no—
360C2×C32—42no—
361C2×C32#38611yescharacteristic
362C2×C32#40711yescharacteristic
363C2×C32#40811yescharacteristic
364C2×C32#43512yes—
365C2×C32#43612yes—
366C24C2×C3×Dic311yescharacteristic
367C24C2×C3×Dic311yescharacteristic
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4C2×C3×Dic311yescharacteristic
378C22×C4C2×C3×Dic311yescharacteristic
379C22×C4C2×C3×Dic311yescharacteristic
380C22×C4C2×C3×Dic311yescharacteristic
381C22×C4C3×#6712yes—
382C22×C4C3×#6712yes—
383C22×C4C22×C3×D311yescharacteristic
384M16—68no—
385C2×C8—34no—
386C2×C8—62no—
387C2×C8—62no—
388C2×C8—62no—
389C2×C8—62no—
390C2×C8—62no—
391C4⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C42—21no—
395C42—21no—
396C22×C3—24no—
397C22×C3—24no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—22no—
403C22×C3—24no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—24no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—21no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—44no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—41no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3C2×#20811yescharacteristic
432C22×C3C2×#20811yescharacteristic
433C22×C3D4×Dic312yes—
434C22×C3#94812yes—
435C22×C3C2×#21711yescharacteristic
436C22×C3C2×C3×C22⋊C411yescharacteristic
437C22×C3C3×C4×D412yes—
438C22×C3C2×C3×M1611yescharacteristic
439C22×C3C2×C3×M1611yescharacteristic
440C22×C3C3×#13112yes—
441C3×C4—24no—
442C3×C4—22no—
443C3×C4—22no—
444C3×C4—22no—
445C3×C4—22no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—22no—
449C3×C4—42no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4#93511yescharacteristic
457C3×C4#93711yescharacteristic
458C3×C4#93912yes—
459C3×C4C3×#12011yescharacteristic
460C3×C4C3×#12112yes—
461C3×C4C3×#12311yescharacteristic
462C32#187011yescharacteristic, complemented, Sylow
463C23—21no—
464C23—21no—
465C23—21no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—21no—
471C23C3×#20812yescomplemented
472C23C3×#21712yes—
473C23C22×C3×Dic311yescharacteristic
474D4—28no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4C3×#20812yescomplemented
489C2×C4C3×#21712yes—
490C2×C4C3×D3×D412yes—
491C2×C4C3×#23712yes—
492C2×C4C2×C3×#6711yescharacteristic
493C2×C4C2×C3×#6711yescharacteristic
494C8—64no—
495C8—122no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—22no—
500C2×C3—22no—
501C2×C3—44no—
502C2×C3—44no—
503C2×C3—42no—
504C2×C3—42no—
505C2×C3—42no—
506C2×C3—42no—
507C2×C3#501112yes—
508C2×C3#501212yes—
509C2×C3#520711yescharacteristic
510C2×C3#522011yescharacteristic
511C2×C3#523411yescharacteristic
512C2×C3C3×#38611yescharacteristic
513C2×C3C3×#40711yescharacteristic
514C2×C3C3×#40811yescharacteristic
515C2×C3C3×#43512yes—
516C2×C3C3×#43612yes—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—24no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22C2×C3×#20811yescharacteristic
528C22C2×C3×#20811yescharacteristic
529C22C3×D4×Dic312yes—
530C22C3×#94812yes—
531C22C2×C3×#21711yescharacteristic
532C4—22no—
533C4—22no—
534C4—41no—
535C4—41no—
536C4C3×#93511yescharacteristic
537C4C3×#93711yescharacteristic
538C4C3×#93912yes—
539C3—21no—
540C3#7161111yescharacteristic, direct product factor
541C3C3×#187011yescharacteristic, complemented
542C2—42no—
543C2—42no—
544C2C3×#501112yes—
545C2C3×#501212yes—
546C2C3×#520711yescharacteristic
547C2C3×#522011yescharacteristic
548C2C3×#523411yescharacteristic
subgroupG/H
centerC22×C3×C4C22×D3
commutatorC23×C3C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×C4×D4C2
socleC23×C32C22×C4
prime radicalC22×C32×D4C4
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×C4×D4D3
hypocenterC3C3×#1870
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
2. C22×C3×#208C2semidirect
3. C2×C3×#830C2semidirect
4. C2×C3×#830C2semidirect
5. C3×#4524C2semidirect
7. #71611C3direct (N × H)
27. C22×#208C2×C3semidirect
30. C2×#830C2×C3semidirect
31. C2×#830C2×C3semidirect
33. #4524C2×C3semidirect
63. C2×C3×C3⋊C8D4semidirect
113. C23×C32M16semidirect
130. C2×C32×C4M16semidirect
224. C2×C3⋊C8C3×D4semidirect
289. C23×C3#208semidirect
292. C23×C3C3×M16semidirect
343. C2×C3×C4#208semidirect
346. C2×C3×C4C3×M16semidirect
462. C32#1870semidirect
471. C23C3×#208semidirect
488. C2×C4C3×#208semidirect
540. C3#71611direct (N × H)
541. C3C3×#1870semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×D42
C22×C3×#2082
C2×C3×#8302
C3×#45242
C3×#18703
#716113