Finite Group Databasebeta

← C3×#71686 #11790835

Subgroups — C3×#71686 #11790835

Order 1152.

1029 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#121C211yesmaximal, characteristic
2C2×C3×#834C214yesmaximal, complemented
3C3×#5212C211yesmaximal, characteristic, complemented
4C3×C4×#217C211yesmaximal, characteristic, complemented
5C3×#1964—31nomaximal
6#71686C311yesmaximal, characteristic, direct product factor
7C22×C32×D4C411yescharacteristic, complemented
8C22×C32×D4C2212yes—
9C23×C32×C4C2211yescharacteristic
10C32×#121—21no—
11C32×#121—21no—
12C32×#121—21no—
13C32×#121—21no—
14C2×C32×C4⋊C4C411yescharacteristic, complemented
15C2×C32×C22⋊C4C412yescomplemented
16C2×C3×#217C2212yes—
17C2×C3×#217C2212yes—
18C2×C3×C4⋊Dic3—21no—
19C2×C3×C4×Dic3—21no—
20C3×#834—28no—
21C3×C3⋊#95—21no—
22C3×C3⋊#95—21no—
23C2×C3×#121—21no—
24C2×C3×#121D311yescharacteristic
25C2×C3×#121C2×C311yescharacteristic
26C2×C3×#99—34no—
27C3×#380—31no—
28C3×C4×C22⋊C4—31no—
29C2×#834C2×C314yescomplemented
30#5212C2×C311yescharacteristic, complemented
31C4×#217C2×C311yescharacteristic, complemented
32C24×C32C2×C412yes—
33C24×C32C2311yescharacteristic
34C2×C32×D4—22no—
35C2×C32×D4—22no—
36C2×C32×D4—24no—
37C2×C32×D4—24no—
38C2×C32×D4—21no—
39C2×C32×D4—21no—
40C2×C32×D4—42no—
41C22×C32×C4—42no—
42C22×C32×C4C2×C412yes—
43C22×C32×C4C2×C411yescharacteristic
44C22×C32×C4C2×C411yescharacteristic
45C22×C32×C4D411yescharacteristic
46C22×C32×C4D411yescharacteristic
47C22×C32×C4D411yescharacteristic
48C22×C32×C4D411yescharacteristic
49C22×C3×Dic3—22no—
50C22×C3×Dic3—22no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4—21no—
53C32×C22⋊C4—22no—
54C32×C22⋊C4—22no—
55C3×#217—22no—
56C3×#217—24no—
57C3×#217D414yescomplemented
58C3×C4⋊Dic3—42no—
59C3×C4×Dic3—42no—
60#1964—31noSylow
61C22×C3×D4—21no—
62C22×C3×D4—22no—
63C22×C3×D4Dic311yescharacteristic, complemented
64C22×C3×D4C3×C411yescharacteristic, complemented
65C22×C3×D4C2×D312yes—
66C22×C3×D4C22×C312yes—
67C23×C3×C4—21no—
68C23×C3×C4C2×D311yescharacteristic
69C23×C3×C4C22×C311yescharacteristic
70C3×#121—22no—
71C3×#121—22no—
72C3×#121—22no—
73C3×#121—22no—
74C3×#121—21no—
75C3×#121—21no—
76C3×#121—21no—
77C3×#121—21no—
78C3×#121—21no—
79C3×#121—21no—
80C3×#121—21no—
81C3×#121—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—61no—
84C2×C3×C4⋊C4Dic311yescharacteristic, complemented
85C2×C3×C4⋊C4C3×C411yescharacteristic, complemented
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—32no—
88C2×C3×C22⋊C4—32no—
89C2×C3×C22⋊C4Dic312yescomplemented
90C2×C3×C22⋊C4C3×C412yescomplemented
91C2×C3×C42—61no—
92C2×#217C22×C312yes—
93C2×#217C22×C312yes—
94C2×C4⋊Dic3—21no—
95C2×C4×Dic3—21no—
96C3×#99—68no—
97C3×#95—61no—
98C3×#95—61no—
99#834—28no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102C23×C32—22no—
103C23×C32—24no—
104C23×C32—41no—
105C23×C32—41no—
106C23×C32—42no—
107C23×C32—42no—
108C23×C32C22×C411yescharacteristic
109C23×C32C2×D412yes—
110C23×C32C2×D411yescharacteristic
111C23×C32C2×D411yescharacteristic
112C23×C32#4112yes—
113C32×D4—42no—
114C32×D4—44no—
115C32×D4—42no—
116C32×D4—42no—
117C32×D4—42no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—24no—
123C2×C32×C4—41no—
124C2×C32×C4—41no—
125C2×C32×C4—41no—
126C2×C32×C4C22⋊C411yescharacteristic
127C2×C32×C4C22⋊C411yescharacteristic
128C2×C32×C4C22⋊C411yescharacteristic
129C2×C32×C4C22⋊C411yescharacteristic
130C2×C3×Dic3—24no—
131C2×C3×Dic3—42no—
132C2×C3×Dic3—42no—
133C2×C3×Dic3—42no—
134C2×C3×Dic3—42no—
135C2×C3×Dic3—42no—
136C2×#121C3×D311yescharacteristic
137C2×#99—34no—
138#380—31no—
139C4×C22⋊C4—31no—
140C24×C3—22no—
141C24×C3—21no—
142C24×C3C2×Dic312yes—
143C24×C3C2×C3×C412yes—
144C24×C3C22×D311yescharacteristic
145C24×C3C23×C311yescharacteristic
146C2×C3×D4—24no—
147C2×C3×D4—24no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—24no—
151C2×C3×D4—24no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—24no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4—24no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—44no—
163C2×C3×D4—44no—
164C2×C3×D4—44no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—42no—
175C22×C3×C4—42no—
176C22×C3×C4—44no—
177C22×C3×C4—62no—
178C22×C3×C4—62no—
179C22×C3×C4C2×Dic312yes—
180C22×C3×C4C2×Dic311yescharacteristic
181C22×C3×C4C2×Dic311yescharacteristic
182C22×C3×C4#6711yescharacteristic
183C22×C3×C4#6711yescharacteristic
184C22×C3×C4#6711yescharacteristic
185C22×C3×C4#6711yescharacteristic
186C22×C3×C4C2×C3×C412yes—
187C22×C3×C4C2×C3×C411yescharacteristic
188C22×C3×C4C2×C3×C411yescharacteristic
189C22×C3×C4C3×D411yescharacteristic
190C22×C3×C4C3×D411yescharacteristic
191C22×C3×C4C3×D411yescharacteristic
192C22×C3×C4C3×D411yescharacteristic
193C22×Dic3—22no—
194C22×Dic3—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—122no—
202C3×C22⋊C4—24no—
203C3×C22⋊C4—24no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—22no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—34no—
209C3×C22⋊C4—62no—
210C3×C22⋊C4—64no—
211C3×C42—122no—
212#217—22no—
213#217—24no—
214#217C3×D414yescomplemented
215C4⋊Dic3—42no—
216C4×Dic3—42no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—42no—
224C22×C32—42no—
225C22×C32—42no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32—84no—
229C22×C32C2×C22⋊C411yescharacteristic
230C22×C32C4×D412yes—
231C22×C32#12112yes—
232C22×C32#12411yescharacteristic
233C22×C32#12711yescharacteristic
234C32×C4—21no—
235C32×C4—21no—
236C32×C4—21no—
237C32×C4—42no—
238C32×C4#9912yescomplemented
239C3×Dic3—44no—
240C3×Dic3—82no—
241C22×D4C3×Dic311yescharacteristic, complemented
242C22×D4C2×C3×D312yes—
243C23×C4C2×C3×D311yescharacteristic
244#121—21no—
245#121—21no—
246#121—21no—
247#121—21no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4C3×Dic311yescharacteristic, complemented
250C2×C22⋊C4—32no—
251C2×C22⋊C4—32no—
252C2×C22⋊C4C3×Dic312yescomplemented
253C2×C42—61no—
254#99—68no—
255#95—61no—
256#95—61no—
257C23×C3—24no—
258C23×C3—22no—
259C23×C3—22no—
260C23×C3—24no—
261C23×C3—24no—
262C23×C3—22no—
263C23×C3—22no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—44no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3—44no—
275C23×C3—44no—
276C23×C3—42no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3—42no—
280C23×C3D3×D412yes—
281C23×C3#23712yes—
282C23×C3C22×Dic311yescharacteristic
283C23×C3C2×#6711yescharacteristic
284C23×C3C2×#6711yescharacteristic
285C23×C3C22×C3×C411yescharacteristic
286C23×C3C2×C3×D412yes—
287C23×C3C2×C3×D411yescharacteristic
288C23×C3C2×C3×D411yescharacteristic
289C23×C3C3×#4112yes—
290C3×D4—44no—
291C3×D4—42no—
292C3×D4—42no—
293C3×D4—44no—
294C3×D4—44no—
295C3×D4—48no—
296C3×D4—44no—
297C3×D4—42no—
298C3×D4—44no—
299C3×D4—42no—
300C3×D4—44no—
301C3×D4—42no—
302C3×D4—42no—
303C3×D4—42no—
304C3×D4—42no—
305C2×C3×C4—24no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—44no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—64no—
334C2×C3×C4—122no—
335C2×C3×C4—122no—
336C2×C3×C4—122no—
337C2×C3×C4—122no—
338C2×C3×C4—122no—
339C2×C3×C4#21711yescharacteristic
340C2×C3×C4#21711yescharacteristic
341C2×C3×C4#21711yescharacteristic
342C2×C3×C4#21711yescharacteristic
343C2×C3×C4C3×C22⋊C411yescharacteristic
344C2×C3×C4C3×C22⋊C411yescharacteristic
345C2×C3×C4C3×C22⋊C411yescharacteristic
346C2×C3×C4C3×C22⋊C411yescharacteristic
347C2×Dic3—24no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×C32—22no—
354C2×C32—22no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—82no—
358C2×C32#39011yescharacteristic
359C2×C32C2×#9911yescharacteristic
360C2×C32#41011yescharacteristic
361C24C2×C3×Dic312yes—
362C24C22×C3×D311yescharacteristic
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—42no—
370C22×C4—42no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4C2×C3×Dic312yes—
374C22×C4C2×C3×Dic311yescharacteristic
375C22×C4C2×C3×Dic311yescharacteristic
376C22×C4C3×#6711yescharacteristic
377C22×C4C3×#6711yescharacteristic
378C22×C4C3×#6711yescharacteristic
379C22×C4C3×#6711yescharacteristic
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—122no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—34no—
386C22⋊C4—62no—
387C22⋊C4—64no—
388C42—122no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—22no—
404C22×C3—21no—
405C22×C3—22no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—41no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—44no—
417C22×C3—44no—
418C22×C3—44no—
419C22×C3—44no—
420C22×C3—44no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—84no—
428C22×C3—84no—
429C22×C3—88no—
430C22×C3D4×Dic312yes—
431C22×C3#93611yescharacteristic
432C22×C3#93812yes—
433C22×C3#94011yescharacteristic
434C22×C3C2×#21711yescharacteristic
435C22×C3C2×C3×C22⋊C411yescharacteristic
436C22×C3C3×C4×D412yes—
437C22×C3C3×#12112yes—
438C22×C3C3×#12411yescharacteristic
439C22×C3C3×#12711yescharacteristic
440C3×C4—22no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—22no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—44no—
451C3×C4—42no—
452C3×C4—42no—
453C3×C4—124no—
454C3×C4—242no—
455C3×C4#83412yescomplemented
456C3×C4C3×#9912yescomplemented
457Dic3—44no—
458Dic3—82no—
459C32#196411yescharacteristic, complemented, Sylow
460C23—22no—
461C23—24no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23C3×D3×D412yes—
467C23C3×#23712yes—
468C23C22×C3×Dic311yescharacteristic
469C23C2×C3×#6711yescharacteristic
470C23C2×C3×#6711yescharacteristic
471D4—42no—
472D4—42no—
473D4—44no—
474D4—42no—
475D4—42no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—64no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4C3×#21711yescharacteristic
491C2×C4C3×#21711yescharacteristic
492C2×C4C3×#21711yescharacteristic
493C2×C4C3×#21711yescharacteristic
494C2×C3—24no—
495C2×C3—22no—
496C2×C3—21no—
497C2×C3—22no—
498C2×C3—21no—
499C2×C3—22no—
500C2×C3—21no—
501C2×C3—22no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—42no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3—84no—
510C2×C3—82no—
511C2×C3—82no—
512C2×C3C2×#83411yescharacteristic
513C2×C3#522111yescharacteristic
514C2×C3#524011yescharacteristic
515C2×C3C3×#39011yescharacteristic
516C2×C3C2×C3×#9911yescharacteristic
517C2×C3C3×#41011yescharacteristic
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—84no—
530C22C3×D4×Dic312yes—
531C22C3×#93611yescharacteristic
532C22C3×#93812yes—
533C22C3×#94011yescharacteristic
534C22C2×C3×#21711yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—42no—
539C4—124no—
540C4—242no—
541C4C3×#83412yescomplemented
542C3—21no—
543C3#7168611yescharacteristic, direct product factor
544C3C3×#196411yescharacteristic, complemented
545C2—22no—
546C2—21no—
547C2—22no—
548C2—21no—
549C2—82no—
550C2C2×C3×#83411yescharacteristic
551C2C3×#522111yescharacteristic
552C2C3×#524011yescharacteristic
subgroupG/H
centerC22×C3C2×#217
commutatorC23×C3C22×C3×C4
FrattiniC24C22×C3×D3
Fitting, prime radicalC2×C32×#121C2
socleC22×C32C2×C22⋊C4
cosocleC24×C3C23×C3
hypercenterC2×C3×#121D3
hypocenterC3C3×#1964
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C2×C3×#121

Complements

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

NH ≅ G/Ntype
2. C2×C3×#834C2semidirect
3. C3×#5212C2semidirect
4. C3×C4×#217C2semidirect
6. #71686C3direct (N × H)
7. C22×C32×D4C4semidirect
14. C2×C32×C4⋊C4C4semidirect
15. C2×C32×C22⋊C4C4semidirect
29. C2×#834C2×C3semidirect
30. #5212C2×C3semidirect
31. C4×#217C2×C3semidirect
57. C3×#217D4semidirect
63. C22×C3×D4Dic3semidirect
64. C22×C3×D4C3×C4semidirect
84. C2×C3×C4⋊C4Dic3semidirect
85. C2×C3×C4⋊C4C3×C4semidirect
89. C2×C3×C22⋊C4Dic3semidirect
90. C2×C3×C22⋊C4C3×C4semidirect
214. #217C3×D4semidirect
238. C32×C4#99semidirect
241. C22×D4C3×Dic3semidirect
249. C2×C4⋊C4C3×Dic3semidirect
252. C2×C22⋊C4C3×Dic3semidirect
455. C3×C4#834semidirect
456. C3×C4C3×#99semidirect
459. C32#1964semidirect
541. C4C3×#834semidirect
543. C3#71686direct (N × H)
544. C3C3×#1964semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1212
C2×C3×#8342
C3×#52122
C3×C4×#2172
C3×#19643
#716863