Finite Group Databasebeta

← C3×#60370 #10678977

Subgroups — C3×#60370 #10678977

Order 768.

4746 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#585C211yesmaximal, characteristic
2C22×C3×#137C212yesmaximal, complemented
3C2×C3×#536C211yesmaximal, characteristic, complemented
4C3×#3011C218yesmaximal, complemented
5C3×#3009C218yesmaximal, complemented
6C3×#3007C218yesmaximal, complemented
7C3×#2896C212yesmaximal, complemented
8C3×#2879C211yesmaximal, characteristic, complemented
9#60370C311yesmaximal, characteristic, direct product factor, Sylow
10C3×#585—216no—
11C2×C3×2−1+4—21no—
12C2×C3×2−1+4C2214yes—
13C2×C3×H(5,𝔽2)—22no—
14C2×C3×H(5,𝔽2)—21no—
15C2×C3×H(5,𝔽2)C2214yes—
16C22×C3×#41—24no—
17C22×C3×#41C2214yes—
18C22×C3×#41C2212yes—
19C22×C3×#41C2211yescharacteristic
20C23×C3×Q8C2211yescharacteristic, complemented
21C2×C3×#137—216no—
22C2×C3×#137C2218yescomplemented
23C2×C3×#137C2218yescomplemented
24C22×C3×Q16C2214yescomplemented
25C22×C3×SD16C2214yescomplemented
26C22×C3×M16C2212yescomplemented
27C3×#536—24no—
28C3×#536C2218yescomplemented
29C2×C3×C4⋊Q8—21no—
30C2×C3×#124—21no—
31C2×C3×#123—22no—
32C2×C3×C22⋊Q8C2214yescomplemented
33C2×C3×#117C2211yescharacteristic, complemented
34C3×#452C22132yescomplemented
35C3×#451C22116yescomplemented
36C3×#450C22116yescomplemented
37C3×#419C2214yescomplemented
38C3×#418C2214yescomplemented
39C3×#417C2218yescomplemented
40C3×#416C2218yescomplemented
41C3×#408C2218yescomplemented
42C3×#407C2214yescomplemented
43C2×#585C2×C311yescharacteristic
44C22×#137C2×C312yescomplemented
45C2×#536C2×C311yescharacteristic, complemented
46#3011C2×C318yescomplemented
47#3009C2×C318yescomplemented
48#3007C2×C318yescomplemented
49#2896C2×C312yescomplemented
50#2879C2×C311yescharacteristic, complemented
51C3×2−1+4—216no—
52C3×2−1+4—216no—
53C3×2−1+4—216no—
54C3×H(5,𝔽2)—216no—
55C3×H(5,𝔽2)—232no—
56C3×H(5,𝔽2)—216no—
57C3×H(5,𝔽2)—216no—
58C2×C3×#41—264no—
59C2×C3×#41—232no—
60C2×C3×#41—24no—
61C2×C3×#41—28no—
62C2×C3×#41—24no—
63C2×C3×#41—24no—
64C2×C3×#41—22no—
65C2×C3×#41—24no—
66C2×C3×#41—22no—
67C2×C3×#41—28no—
68C2×C3×#41—28no—
69C2×C3×#41D4116yes—
70C2×C3×#41D414yes—
71C2×C3×#41C2318yes—
72C2×C3×#41C2314yes—
73C2×C3×#41C2314yes—
74C2×C3×#41C2314yes—
75C22×C3×Q8—24no—
76C22×C3×Q8—21no—
77C22×C3×Q8—22no—
78C22×C3×Q8—28no—
79C22×C3×Q8D411yescharacteristic
80C22×C3×Q8C2314yes—
81C22×C3×Q8C2312yes—
82C22×C3×Q8C2314yes—
83C22×C3×Q8C2312yes—
84C22×C3×D4—24no—
85C22×C3×D4—22no—
86C22×C3×D4—28no—
87C22×C3×D4—24no—
88C22×C3×D4D412yes—
89C22×C3×D4D411yescharacteristic
90C22×C3×D4C2314yes—
91C22×C3×D4C2312yes—
92C23×C3×C4—24no—
93C23×C3×C4—22no—
94C23×C3×C4—21no—
95C23×C3×C4C2312yes—
96C23×C3×C4C2311yescharacteristic
97C3×#137—232no—
98C3×#137—232no—
99C2×C3×Q16—216no—
100C2×C3×Q16C23116yes—
101C2×C3×SD16—216no—
102C2×C3×SD16C23116yes—
103C2×C3×M16—28no—
104C2×C3×M16C2314yes—
105C2×C3×M16C2314yes—
106C22×C3×C8C2314yes—
107C3×C4⋊Q8—28no—
108C3×#124—28no—
109C3×#123—216no—
110C3×C22⋊Q8—28no—
111C3×C22⋊Q8C23116yes—
112C3×#117—22no—
113C3×#117C2314yes—
114C2×C3×C4⋊C4—24no—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C22⋊C4—21no—
117C2×C3×C22⋊C4—24no—
118C2×C3×C42—21no—
119C3×#103C23116yes—
120C3×#102C23116yes—
121C3×C22⋊C8C23116yes—
122#585—216no—
123C2×2−1+4—21no—
124C2×2−1+4C22×C314yes—
125C2×H(5,𝔽2)—22no—
126C2×H(5,𝔽2)—21no—
127C2×H(5,𝔽2)C22×C314yes—
128C22×#41—24no—
129C22×#41C22×C314yes—
130C22×#41C22×C312yes—
131C22×#41C22×C311yescharacteristic
132C23×Q8C22×C311yescharacteristic, complemented
133C2×#137—216no—
134C2×#137C22×C318yescomplemented
135C2×#137C22×C318yescomplemented
136C22×Q16C22×C314yescomplemented
137C22×SD16C22×C314yescomplemented
138C22×M16C22×C312yescomplemented
139#536—24no—
140#536C22×C318yescomplemented
141C2×C4⋊Q8—21no—
142C2×#124—21no—
143C2×#123—22no—
144C2×C22⋊Q8C22×C314yescomplemented
145C2×#117C22×C311yescharacteristic, complemented
146#452C22×C3132yescomplemented
147#451C22×C3116yescomplemented
148#450C22×C3116yescomplemented
149#419C22×C314yescomplemented
150#418C22×C314yescomplemented
151#417C22×C318yescomplemented
152#416C22×C318yescomplemented
153#408C22×C318yescomplemented
154#407C22×C314yescomplemented
155C24×C3—24no—
156C24×C3—22no—
157C24×C3C2×D412yes—
158C24×C3C2×D411yescharacteristic
159C3×#41—232no—
160C3×#41—264no—
161C3×#41—232no—
162C3×#41—232no—
163C3×#41—232no—
164C3×#41—216no—
165C3×#41—216no—
166C3×#41—216no—
167C3×#41—216no—
168C3×#41—216no—
169C3×#41—28no—
170C3×#41—28no—
171C3×#41—28no—
172C3×#41—28no—
173C3×#41C2×D4116yes—
174C3×#41C2×D4116yes—
175C2×C3×Q8—232no—
176C2×C3×Q8—216no—
177C2×C3×Q8—24no—
178C2×C3×Q8—24no—
179C2×C3×Q8—22no—
180C2×C3×Q8—24no—
181C2×C3×Q8—22no—
182C2×C3×Q8—24no—
183C2×C3×Q8—28no—
184C2×C3×Q8—28no—
185C2×C3×Q8—28no—
186C2×C3×Q8—48no—
187C2×C3×Q8—44no—
188C2×C3×Q8C2×D418yes—
189C2×C3×Q8C2×D414yes—
190C2×C3×Q8C2414yes—
191C2×C3×Q8C2414yes—
192C2×C3×D4—232no—
193C2×C3×D4—264no—
194C2×C3×D4—232no—
195C2×C3×D4—216no—
196C2×C3×D4—28no—
197C2×C3×D4—28no—
198C2×C3×D4—24no—
199C2×C3×D4—24no—
200C2×C3×D4—216no—
201C2×C3×D4—24no—
202C2×C3×D4—24no—
203C2×C3×D4—22no—
204C2×C3×D4—24no—
205C2×C3×D4C2×D418yes—
206C2×C3×D4C2×D414yes—
207C2×C3×D4C2×D418yes—
208C2×C3×D4C2414yes—
209C22×C3×C4—232no—
210C22×C3×C4—24no—
211C22×C3×C4—22no—
212C22×C3×C4—216no—
213C22×C3×C4—24no—
214C22×C3×C4—28no—
215C22×C3×C4—24no—
216C22×C3×C4—28no—
217C22×C3×C4—22no—
218C22×C3×C4—22no—
219C22×C3×C4—24no—
220C22×C3×C4—44no—
221C22×C3×C4C2×D412yes—
222C22×C3×C4C2×D414yes—
223C22×C3×C4C2×D411yescharacteristic
224C22×C3×C4C2×D412yes—
225C22×C3×C4C2×D418yes—
226C22×C3×C4C2414yes—
227C22×C3×C4C2411yescharacteristic
228C22×C3×C4C2412yes—
229C3×Q16—232no—
230C3×SD16—232no—
231C3×M16—28no—
232C3×M16—28no—
233C2×C3×C8—28no—
234C2×C3×C8C2418yes—
235C3×C4⋊C4—216no—
236C3×C4⋊C4—22no—
237C3×C4⋊C4C2414yes—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—216no—
240C3×C42—24no—
2412−1+4—216no—
2422−1+4—216no—
2432−1+4—216no—
244H(5,𝔽2)—216no—
245H(5,𝔽2)—232no—
246H(5,𝔽2)—216no—
247H(5,𝔽2)—216no—
248C2×#41—264no—
249C2×#41—232no—
250C2×#41—24no—
251C2×#41—28no—
252C2×#41—24no—
253C2×#41—24no—
254C2×#41—22no—
255C2×#41—24no—
256C2×#41—22no—
257C2×#41—28no—
258C2×#41—28no—
259C2×#41C3×D4116yes—
260C2×#41C3×D414yes—
261C2×#41C23×C318yes—
262C2×#41C23×C314yes—
263C2×#41C23×C314yes—
264C2×#41C23×C314yes—
265C22×Q8—24no—
266C22×Q8—21no—
267C22×Q8—22no—
268C22×Q8—28no—
269C22×Q8C3×D411yescharacteristic
270C22×Q8C23×C312yes—
271C22×Q8C23×C314yes—
272C22×Q8C23×C314yes—
273C22×Q8C23×C312yes—
274C22×D4—24no—
275C22×D4—22no—
276C22×D4—28no—
277C22×D4—24no—
278C22×D4C3×D412yes—
279C22×D4C3×D411yescharacteristic
280C22×D4C23×C314yes—
281C22×D4C23×C312yes—
282C23×C4—24no—
283C23×C4—22no—
284C23×C4—21no—
285C23×C4C23×C312yes—
286C23×C4C23×C311yescharacteristic
287#137—232no—
288#137—232no—
289C2×Q16—216no—
290C2×Q16C23×C3116yes—
291C2×SD16—216no—
292C2×SD16C23×C3116yes—
293C2×M16—28no—
294C2×M16C23×C314yes—
295C2×M16C23×C314yes—
296C22×C8C23×C314yes—
297C4⋊Q8—28no—
298#124—28no—
299#123—216no—
300C22⋊Q8—28no—
301C22⋊Q8C23×C3116yes—
302#117—22no—
303#117C23×C314yes—
304C2×C4⋊C4—24no—
305C2×C4⋊C4C23×C312yes—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—24no—
308C2×C42—21no—
309#103C23×C3116yes—
310#102C23×C3116yes—
311C22⋊C8C23×C3116yes—
312C23×C3—216no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—28no—
316C23×C3—42no—
317C23×C3—44no—
318C23×C3—84no—
319C23×C3—88no—
320C23×C3#12014yes—
321C23×C3C22×D414yes—
322C23×C3C22×D412yes—
323C23×C3C22×D411yescharacteristic
324C3×Q8—216no—
325C3×Q8—28no—
326C3×Q8—28no—
327C3×Q8—24no—
328C3×Q8—28no—
329C3×Q8—44no—
330C3×Q8—48no—
331C3×Q8C22×D418yes—
332C3×D4—232no—
333C3×D4—216no—
334C3×D4—216no—
335C3×D4—216no—
336C3×D4—216no—
337C3×D4—28no—
338C3×D4—28no—
339C3×D4—24no—
340C3×D4C22×D418yes—
341C2×C3×C4—216no—
342C2×C3×C4—232no—
343C2×C3×C4—28no—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—216no—
347C2×C3×C4—28no—
348C2×C3×C4—216no—
349C2×C3×C4—28no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—44no—
353C2×C3×C4—44no—
354C2×C3×C4—48no—
355C2×C3×C4#12018yes—
356C2×C3×C4#12014yes—
357C2×C3×C4C22×D414yes—
358C2×C3×C4C22×D412yes—
359C2×C3×C4C22×D411yescharacteristic
360C2×C3×C4C22×D414yes—
361C2×C3×C4C22×D414yes—
362C2×C3×C4C22×D414yes—
363C2×C3×C4C2511yescharacteristic
364C3×C8—28no—
365C24—24no—
366C24—22no—
367C24C2×C3×D412yes—
368C24C2×C3×D411yescharacteristic
369#41—232no—
370#41—264no—
371#41—232no—
372#41—232no—
373#41—232no—
374#41—216no—
375#41—216no—
376#41—216no—
377#41—216no—
378#41—216no—
379#41—28no—
380#41—28no—
381#41—28no—
382#41—28no—
383#41C2×C3×D4116yes—
384#41C2×C3×D4116yes—
385C2×Q8—232no—
386C2×Q8—216no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—48no—
397C2×Q8—44no—
398C2×Q8C2×C3×D418yes—
399C2×Q8C2×C3×D414yes—
400C2×Q8C24×C314yes—
401C2×Q8C24×C314yes—
402C2×D4—232no—
403C2×D4—264no—
404C2×D4—232no—
405C2×D4—216no—
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—216no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—24no—
415C2×D4C2×C3×D418yes—
416C2×D4C2×C3×D414yes—
417C2×D4C2×C3×D418yes—
418C2×D4C24×C314yes—
419C22×C4—232no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—216no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—44no—
431C22×C4C2×C3×D412yes—
432C22×C4C2×C3×D414yes—
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D412yes—
435C22×C4C2×C3×D418yes—
436C22×C4C24×C314yes—
437C22×C4C24×C311yescharacteristic
438C22×C4C24×C312yes—
439Q16—232no—
440SD16—232no—
441M16—28no—
442M16—28no—
443C2×C8—28no—
444C2×C8C24×C318yes—
445C4⋊C4—216no—
446C4⋊C4—22no—
447C4⋊C4C24×C314yes—
448C22⋊C4—24no—
449C22⋊C4—216no—
450C42—24no—
451C22×C3—22no—
452C22×C3—24no—
453C22×C3—28no—
454C22×C3—24no—
455C22×C3—28no—
456C22×C3—44no—
457C22×C3—48no—
458C22×C3—84no—
459C22×C3—88no—
460C22×C3—816no—
461C22×C3C2×#12014yes—
462C22×C3C2×#12012yes—
463C22×C3C23×D411yescharacteristic
464C3×C4—28no—
465C3×C4—24no—
466C3×C4—44no—
467C3×C4—44no—
468C3×C4C2×#12012yes—
469C3×C4C2×#12014yes—
470C3×C4C23×D412yes—
471C23—216no—
472C23—24no—
473C23—24no—
474C23—28no—
475C23—42no—
476C23—44no—
477C23—84no—
478C23—88no—
479C23C3×#12014yes—
480C23C22×C3×D414yes—
481C23C22×C3×D412yes—
482C23C22×C3×D411yescharacteristic
483Q8—216no—
484Q8—24no—
485Q8—28no—
486Q8—28no—
487Q8—28no—
488Q8—44no—
489Q8—48no—
490Q8C22×C3×D418yes—
491D4—232no—
492D4—216no—
493D4—216no—
494D4—216no—
495D4—216no—
496D4—24no—
497D4—28no—
498D4—28no—
499D4C22×C3×D418yes—
500C2×C4—216no—
501C2×C4—232no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—216no—
507C2×C4—28no—
508C2×C4—216no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—48no—
514C2×C4C3×#12018yes—
515C2×C4C3×#12014yes—
516C2×C4C22×C3×D414yes—
517C2×C4C22×C3×D414yes—
518C2×C4C22×C3×D414yes—
519C2×C4C22×C3×D414yes—
520C2×C4C22×C3×D412yes—
521C2×C4C22×C3×D411yescharacteristic
522C2×C4C25×C311yescharacteristic
523C8—28no—
524C2×C3—22no—
525C2×C3—24no—
526C2×C3—44no—
527C2×C3—48no—
528C2×C3C22×#12011yescharacteristic
529C2×C3#358612yes—
530C22—22no—
531C22—24no—
532C22—24no—
533C22—28no—
534C22—28no—
535C22—44no—
536C22—48no—
537C22—84no—
538C22—88no—
539C22—816no—
540C22C2×C3×#12014yes—
541C22C2×C3×#12012yes—
542C22C23×C3×D411yescharacteristic
543C4—28no—
544C4—24no—
545C4—44no—
546C4—44no—
547C4C2×C3×#12012yes—
548C4C2×C3×#12014yes—
549C4C23×C3×D412yes—
550C3#6037011yescharacteristic, direct product factor, Sylow
551C2—22no—
552C2—24no—
553C2—44no—
554C2—48no—
555C2C22×C3×#12011yescharacteristic
556C2C3×#358612yes—
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC2×C3×#585C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×#137C2semidirect
3. C2×C3×#536C2semidirect
4. C3×#3011C2semidirect
5. C3×#3009C2semidirect
6. C3×#3007C2semidirect
7. C3×#2896C2semidirect
8. C3×#2879C2semidirect
9. #60370C3direct (N × H)
20. C23×C3×Q8C22semidirect
22. C2×C3×#137C22semidirect
23. C2×C3×#137C22semidirect
24. C22×C3×Q16C22semidirect
25. C22×C3×SD16C22semidirect
26. C22×C3×M16C22semidirect
28. C3×#536C22semidirect
32. C2×C3×C22⋊Q8C22semidirect
33. C2×C3×#117C22semidirect
34. C3×#452C22semidirect
35. C3×#451C22semidirect
36. C3×#450C22semidirect
37. C3×#419C22semidirect
38. C3×#418C22semidirect
39. C3×#417C22semidirect
40. C3×#416C22semidirect
41. C3×#408C22semidirect
42. C3×#407C22semidirect
44. C22×#137C2×C3semidirect
45. C2×#536C2×C3semidirect
46. #3011C2×C3semidirect
47. #3009C2×C3semidirect
48. #3007C2×C3semidirect
49. #2896C2×C3semidirect
50. #2879C2×C3semidirect
132. C23×Q8C22×C3semidirect
134. C2×#137C22×C3semidirect
135. C2×#137C22×C3semidirect
136. C22×Q16C22×C3semidirect
137. C22×SD16C22×C3semidirect
138. C22×M16C22×C3semidirect
140. #536C22×C3semidirect
144. C2×C22⋊Q8C22×C3semidirect
145. C2×#117C22×C3semidirect
146. #452C22×C3semidirect
147. #451C22×C3semidirect
148. #450C22×C3semidirect
149. #419C22×C3semidirect
150. #418C22×C3semidirect
151. #417C22×C3semidirect
152. #416C22×C3semidirect
153. #408C22×C3semidirect
154. #407C22×C3semidirect
550. C3#60370direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5852
C22×C3×#1372
C2×C3×#5362
C3×#30112
C3×#30092
C3×#30072
C3×#28962
C3×#28792
#603703