Finite Group Databasebeta

← C3×#60569 #10679176

Subgroups — C3×#60569 #10679176

Order 768.

2010 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic, complemented
2C3×#3428C211yesmaximal, characteristic
3C3×#3127C214yesmaximal, complemented
4C3×#3123C214yesmaximal, complemented
5C3×#3120C218yesmaximal, complemented
6C3×#3095C212yesmaximal, complemented
7C3×#3080C212yesmaximal, complemented
8C3×#3039C212yesmaximal, complemented
9C3×#3007C212yesmaximal
10C3×C4×#137C214yesmaximal, complemented
11C3×#2921C211yesmaximal, characteristic, complemented
12#60569C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#137C2214yes—
15C3×#541—22no—
16C3×#541—22no—
17C3×#541C2214yes—
18C3×#541C2214yescomplemented
19C2×C3×C4⋊Q8C2211yescharacteristic, complemented
20C2×C3×#126—22no—
21C2×C3×#125—22no—
22C2×C3×#125C2211yescharacteristic, complemented
23C2×C3×C22⋊Q8C2212yes—
24C3×#520—22no—
25C3×#520C2214yes—
26C3×#519C2212yes—
27C3×#518—22no—
28C3×#518C2212yes—
29C3×C4×#41—22no—
30C3×C4×#41C2212yes—
31C3×C4×#41C2212yes—
32C2×C3×C4×Q8C2212yescomplemented
33C2×C3×#117—22no—
34C2×C3×#117C2211yescharacteristic
35C2×C3×#117C2211yescharacteristic
36C3×#485C2218yes—
37C3×#484C2214yescomplemented
38C3×#483C2214yescomplemented
39C3×#479C2218yescomplemented
40C3×#478C2218yescomplemented
41C3×#462C2218yescomplemented
42C3×#461C2218yescomplemented
43C3×#452C2218yes—
44C3×#451C2214yes—
45C3×#450C2214yes—
46C3×#441C2218yescomplemented
47C3×#440C2218yescomplemented
48C3×C4×Q16C2218yescomplemented
49C3×C4×SD16C2218yescomplemented
50C3×#433C2214yes—
51C3×#433C2214yescomplemented
52C3×#428C2214yescomplemented
53C3×#423C2212yescomplemented
54C3×#419C2214yescomplemented
55C3×#417C2214yes—
56C3×#407C2212yes—
57C3×C4×M16C2212yescomplemented
58C2×#541C2×C311yescharacteristic, complemented
59#3428C2×C311yescharacteristic
60#3127C2×C314yescomplemented
61#3123C2×C314yescomplemented
62#3120C2×C318yescomplemented
63#3095C2×C312yescomplemented
64#3080C2×C312yescomplemented
65#3039C2×C312yescomplemented
66#3007C2×C312yes—
67C4×#137C2×C314yescomplemented
68#2921C2×C311yescharacteristic, complemented
69C2×C3×#41—28no—
70C2×C3×#41—21no—
71C2×C3×#41—21no—
72C2×C3×#41C2312yes—
73C2×C3×#41C2312yes—
74C22×C3×Q8C2312yes—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4—21no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C23×C3×C4C2311yescharacteristic
80C3×#137—216no—
81C2×C3×Q16C2318yes—
82C2×C3×SD16C2318yes—
83C2×C3×M16C2314yes—
84C3×C4⋊Q8—22no—
85C3×C4⋊Q8C2314yes—
86C3×#126—28no—
87C3×#126—28no—
88C3×#125—28no—
89C3×#125—28no—
90C3×#125—22no—
91C3×#125C2314yes—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—22no—
94C3×C22⋊Q8C2314yes—
95C3×C22⋊Q8C2314yes—
96C3×C4×Q8—24no—
97C3×C4×Q8—24no—
98C3×C4×Q8C2318yes—
99C3×C4×Q8C2314yes—
100C3×C4×Q8C2314yes—
101C3×C4×D4—24no—
102C3×C4×D4—24no—
103C3×C4×D4—24no—
104C3×C4×D4—24no—
105C3×C4×D4—24no—
106C3×C4×D4C2314yes—
107C3×C4×D4C2314yes—
108C3×#117—21no—
109C3×#117—21no—
110C3×#117—24no—
111C3×#117—24no—
112C3×#117—22no—
113C3×#117—22no—
114C3×#117D412yes—
115C3×#117D412yes—
116C3×#117C2312yes—
117C3×#117C2312yes—
118C3×#117C2312yes—
119C3×#117C2312yes—
120C2×C3×C4⋊C4—22no—
121C2×C3×C4⋊C4—22no—
122C2×C3×C4⋊C4—22no—
123C2×C3×C4⋊C4—22no—
124C2×C3×C4⋊C4—24no—
125C2×C3×C4⋊C4D412yes—
126C2×C3×C4⋊C4C2312yes—
127C2×C3×C4⋊C4C2312yes—
128C2×C3×C22⋊C4—21no—
129C2×C3×C22⋊C4—22no—
130C2×C3×C22⋊C4—24no—
131C2×C3×C22⋊C4D412yes—
132C2×C3×C42—21no—
133C2×C3×C42—22no—
134C2×C3×C42—22no—
135C2×C3×C42—22no—
136C2×C3×C42C2311yescharacteristic
137C2×C3×C42C2311yescharacteristic
138C3×#107C2318yes—
139C3×#106C2318yes—
140C3×C4⋊C8C2318yes—
141C3×#103C2318yes—
142C3×#103C2318yes—
143C3×#103C2318yes—
144C3×#102C2318yes—
145C3×C22⋊C8C2314yes—
146C3×C22⋊C8C2314yes—
147C3×#97C2314yes—
148C3×C4×C8C2314yes—
149C22×#41C22×C311yescharacteristic
150C2×#137C22×C314yes—
151#541—22no—
152#541—22no—
153#541C22×C314yes—
154#541C22×C314yescomplemented
155C2×C4⋊Q8C22×C311yescharacteristic, complemented
156C2×#126—22no—
157C2×#125—22no—
158C2×#125C22×C311yescharacteristic, complemented
159C2×C22⋊Q8C22×C312yes—
160#520—22no—
161#520C22×C314yes—
162#519C22×C312yes—
163#518—22no—
164#518C22×C312yes—
165C4×#41—22no—
166C4×#41C22×C312yes—
167C4×#41C22×C312yes—
168C2×C4×Q8C22×C312yescomplemented
169C2×#117—22no—
170C2×#117C22×C311yescharacteristic
171C2×#117C22×C311yescharacteristic
172#485C22×C318yes—
173#484C22×C314yescomplemented
174#483C22×C314yescomplemented
175#479C22×C318yescomplemented
176#478C22×C318yescomplemented
177#462C22×C318yescomplemented
178#461C22×C318yescomplemented
179#452C22×C318yes—
180#451C22×C314yes—
181#450C22×C314yes—
182#441C22×C318yescomplemented
183#440C22×C318yescomplemented
184C4×Q16C22×C318yescomplemented
185C4×SD16C22×C318yescomplemented
186#433C22×C314yes—
187#433C22×C314yescomplemented
188#428C22×C314yescomplemented
189#423C22×C312yescomplemented
190#419C22×C314yescomplemented
191#417C22×C314yes—
192#407C22×C312yes—
193C4×M16C22×C312yescomplemented
194C24×C3—21no—
195C24×C3C2×D411yescharacteristic
196C3×#41—28no—
197C3×#41—28no—
198C3×#41—24no—
199C3×#41—24no—
200C3×#41—24no—
201C3×#41#4118yes—
202C2×C3×Q8—24no—
203C2×C3×Q8—21no—
204C2×C3×Q8—22no—
205C2×C3×Q8—28no—
206C2×C3×Q8C2414yes—
207C2×C3×Q8C2412yes—
208C2×C3×D4—28no—
209C2×C3×D4—22no—
210C2×C3×D4—22no—
211C2×C3×D4—24no—
212C2×C3×D4—21no—
213C2×C3×D4C2412yes—
214C22×C3×C4—28no—
215C22×C3×C4—21no—
216C22×C3×C4—22no—
217C22×C3×C4—22no—
218C22×C3×C4—24no—
219C22×C3×C4—22no—
220C22×C3×C4—22no—
221C22×C3×C4—22no—
222C22×C3×C4—24no—
223C22×C3×C4—24no—
224C22×C3×C4C2×D411yescharacteristic
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D412yes—
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2411yescharacteristic
232C3×Q16—216no—
233C3×SD16—216no—
234C3×M16—28no—
235C2×C3×C8C2418yes—
236C3×C4⋊C4—22no—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—24no—
239C3×C4⋊C4—28no—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—28no—
243C3×C4⋊C4—28no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4C2×D414yes—
246C3×C4⋊C4C2×D414yes—
247C3×C4⋊C4C2414yes—
248C3×C4⋊C4C2414yes—
249C3×C22⋊C4—22no—
250C3×C22⋊C4—22no—
251C3×C22⋊C4—24no—
252C3×C22⋊C4—24no—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—24no—
255C3×C22⋊C4C2×D414yes—
256C3×C22⋊C4C2×D414yes—
257C3×C42—22no—
258C3×C42—24no—
259C3×C42—24no—
260C3×C42—22no—
261C3×C42—28no—
262C3×C42C2×D412yes—
263C3×C42C2×D412yes—
264C3×C42C2412yes—
265C3×C42C2412yes—
266C2×#41—28no—
267C2×#41—21no—
268C2×#41—21no—
269C2×#41C23×C312yes—
270C2×#41C23×C312yes—
271C22×Q8C23×C312yes—
272C22×Q8C23×C311yescharacteristic
273C22×D4—21no—
274C22×D4C23×C311yescharacteristic
275C23×C4—21no—
276C23×C4C23×C311yescharacteristic
277#137—216no—
278C2×Q16C23×C318yes—
279C2×SD16C23×C318yes—
280C2×M16C23×C314yes—
281C4⋊Q8—22no—
282C4⋊Q8C23×C314yes—
283#126—28no—
284#126—28no—
285#125—28no—
286#125—28no—
287#125—22no—
288#125C23×C314yes—
289C22⋊Q8—22no—
290C22⋊Q8—22no—
291C22⋊Q8C23×C314yes—
292C22⋊Q8C23×C314yes—
293C4×Q8—24no—
294C4×Q8—24no—
295C4×Q8C23×C318yes—
296C4×Q8C23×C314yes—
297C4×Q8C23×C314yes—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4C23×C314yes—
304C4×D4C23×C314yes—
305#117—21no—
306#117—21no—
307#117—24no—
308#117—24no—
309#117—22no—
310#117—22no—
311#117C3×D412yes—
312#117C3×D412yes—
313#117C23×C312yes—
314#117C23×C312yes—
315#117C23×C312yes—
316#117C23×C312yes—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4C3×D412yes—
323C2×C4⋊C4C23×C312yes—
324C2×C4⋊C4C23×C312yes—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4C3×D412yes—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42C23×C311yescharacteristic
334C2×C42C23×C311yescharacteristic
335#107C23×C318yes—
336#106C23×C318yes—
337C4⋊C8C23×C318yes—
338#103C23×C318yes—
339#103C23×C318yes—
340#103C23×C318yes—
341#102C23×C318yes—
342C22⋊C8C23×C314yes—
343C22⋊C8C23×C314yes—
344#97C23×C314yes—
345C4×C8C23×C314yes—
346C23×C3—24no—
347C23×C3—22no—
348C23×C3—22no—
349C23×C3—42no—
350C23×C3—82no—
351C23×C3C22×D411yescharacteristic
352C23×C3C22×D411yescharacteristic
353C23×C3C22×D411yescharacteristic
354C3×Q8—24no—
355C3×Q8—22no—
356C3×Q8—28no—
357C3×Q8—44no—
358C3×Q8C2×#4114yes—
359C3×D4—28no—
360C3×D4—24no—
361C3×D4—28no—
362C3×D4—22no—
363C3×D4C2×#4114yes—
364C2×C3×C4—24no—
365C2×C3×C4—28no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—28no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—24no—
374C2×C3×C4—24no—
375C2×C3×C4—24no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—44no—
379C2×C3×C4—44no—
380C2×C3×C4—84no—
381C2×C3×C4—84no—
382C2×C3×C4—82no—
383C2×C3×C4C22×D414yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D414yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4114yes—
389C2×C3×C4C2511yescharacteristic
390C3×C8—28no—
391C24—21no—
392C24C2×C3×D411yescharacteristic
393#41—28no—
394#41—28no—
395#41—24no—
396#41—24no—
397#41—24no—
398#41C3×#4118yes—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—28no—
403C2×Q8C24×C314yes—
404C2×Q8C24×C312yes—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4C24×C312yes—
411C22×C4—28no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4C2×C3×D411yescharacteristic
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C2×C3×D412yes—
425C22×C4C2×C3×D412yes—
426C22×C4C24×C311yescharacteristic
427C22×C4C24×C311yescharacteristic
428C22×C4C24×C311yescharacteristic
429Q16—216no—
430SD16—216no—
431M16—28no—
432C2×C8C24×C318yes—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—28no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—22no—
442C4⋊C4C2×C3×D414yes—
443C4⋊C4C2×C3×D414yes—
444C4⋊C4C24×C314yes—
445C4⋊C4C24×C314yes—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4C2×C3×D414yes—
453C22⋊C4C2×C3×D414yes—
454C42—22no—
455C42—24no—
456C42—24no—
457C42—22no—
458C42—28no—
459C42C2×C3×D412yes—
460C42C2×C3×D412yes—
461C42C24×C312yes—
462C42C24×C312yes—
463C22×C3—22no—
464C22×C3—22no—
465C22×C3—22no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—24no—
469C22×C3—44no—
470C22×C3—82no—
471C22×C3—84no—
472C22×C3#52512yes—
473C22×C3C23×D411yescharacteristic
474C3×C4—24no—
475C3×C4—24no—
476C3×C4—21no—
477C3×C4—24no—
478C3×C4—21no—
479C3×C4—44no—
480C3×C4—44no—
481C3×C4—44no—
482C3×C4#52512yes—
483C3×C4C22×#4112yes—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—42no—
488C23—82no—
489C23C22×C3×D411yescharacteristic
490C23C22×C3×D411yescharacteristic
491C23C22×C3×D411yescharacteristic
492Q8—24no—
493Q8—22no—
494Q8—28no—
495Q8—44no—
496Q8C2×C3×#4114yes—
497D4—28no—
498D4—24no—
499D4—28no—
500D4—22no—
501D4C2×C3×#4114yes—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4C22×C3×D414yes—
522C2×C4C22×C3×D411yescharacteristic
523C2×C4C22×C3×D414yes—
524C2×C4C22×C3×D411yescharacteristic
525C2×C4C22×C3×D411yescharacteristic
526C2×C4C2×C3×#4114yes—
527C2×C4C25×C311yescharacteristic
528C8—28no—
529C2×C3—22no—
530C2×C3—41no—
531C2×C3—41no—
532C2×C3—44no—
533C2×C3C2×#52511yescharacteristic
534C2×C3#358612yes—
535C22—22no—
536C22—21no—
537C22—21no—
538C22—24no—
539C22—22no—
540C22—22no—
541C22—44no—
542C22—82no—
543C22—84no—
544C22C3×#52512yes—
545C22C23×C3×D411yescharacteristic
546C4—21no—
547C4—24no—
548C4—21no—
549C4—24no—
550C4—24no—
551C4—44no—
552C4—44no—
553C4—44no—
554C4C3×#52512yes—
555C4C22×C3×#4112yes—
556C3#6056911yescharacteristic, direct product factor, Sylow
557C2—22no—
558C2—41no—
559C2—41no—
560C2—44no—
561C2C2×C3×#52511yescharacteristic
562C2C3×#358612yes—
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC22×C3×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#541C2semidirect
3. C3×#3127C2semidirect
4. C3×#3123C2semidirect
5. C3×#3120C2semidirect
6. C3×#3095C2semidirect
7. C3×#3080C2semidirect
8. C3×#3039C2semidirect
10. C3×C4×#137C2semidirect
11. C3×#2921C2semidirect
12. #60569C3direct (N × H)
18. C3×#541C22semidirect
19. C2×C3×C4⋊Q8C22semidirect
22. C2×C3×#125C22semidirect
32. C2×C3×C4×Q8C22semidirect
37. C3×#484C22semidirect
38. C3×#483C22semidirect
39. C3×#479C22semidirect
40. C3×#478C22semidirect
41. C3×#462C22semidirect
42. C3×#461C22semidirect
46. C3×#441C22semidirect
47. C3×#440C22semidirect
48. C3×C4×Q16C22semidirect
49. C3×C4×SD16C22semidirect
51. C3×#433C22semidirect
52. C3×#428C22semidirect
53. C3×#423C22semidirect
54. C3×#419C22semidirect
57. C3×C4×M16C22semidirect
58. C2×#541C2×C3semidirect
60. #3127C2×C3semidirect
61. #3123C2×C3semidirect
62. #3120C2×C3semidirect
63. #3095C2×C3semidirect
64. #3080C2×C3semidirect
65. #3039C2×C3semidirect
67. C4×#137C2×C3semidirect
68. #2921C2×C3semidirect
154. #541C22×C3semidirect
155. C2×C4⋊Q8C22×C3semidirect
158. C2×#125C22×C3semidirect
168. C2×C4×Q8C22×C3semidirect
173. #484C22×C3semidirect
174. #483C22×C3semidirect
175. #479C22×C3semidirect
176. #478C22×C3semidirect
177. #462C22×C3semidirect
178. #461C22×C3semidirect
182. #441C22×C3semidirect
183. #440C22×C3semidirect
184. C4×Q16C22×C3semidirect
185. C4×SD16C22×C3semidirect
187. #433C22×C3semidirect
188. #428C22×C3semidirect
189. #423C22×C3semidirect
190. #419C22×C3semidirect
193. C4×M16C22×C3semidirect
556. C3#60569direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C3×#34282
C3×#31272
C3×#31232
C3×#31202
C3×#30952
C3×#30802
C3×#30392
C3×#30072
C3×C4×#1372
C3×#29212
#605693