Finite Group Databasebeta

← C3×#60520 #10679127

Subgroups — C3×#60520 #10679127

Order 768.

2170 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#537C211yesmaximal, characteristic
2C3×#3428C211yesmaximal, characteristic, complemented
3C3×#3099C218yesmaximal, complemented
4C3×#3098C218yesmaximal, complemented
5C3×#3095C214yesmaximal, complemented
6C3×#3094C214yesmaximal, complemented
7C2×C3×#428C212yesmaximal, complemented
8C3×#2896C212yesmaximal, complemented
9C3×#2879C211yesmaximal, characteristic, complemented
10#60520C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×C3×#41C2211yescharacteristic
13C22×C3×M16C2212yes—
14C3×#537—24no—
15C3×#537C2218yes—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×#121—21no—
19C2×C3×#121—22no—
20C2×C3×#121C2212yes—
21C3×#520C2214yescomplemented
22C3×#520C2214yescomplemented
23C3×#519—21no—
24C3×#518—21no—
25C3×#518—22no—
26C3×C4×#41—24no—
27C2×C3×#117C2212yes—
28C2×C3×#117C2214yes—
29C22×C3×C4⋊C4C2211yescharacteristic
30C3×#485C22116yescomplemented
31C3×#484C2218yescomplemented
32C3×#483C2218yescomplemented
33C3×#482C22116yescomplemented
34C3×#481C2218yescomplemented
35C3×#480C2218yescomplemented
36C3×#428C2218yes—
37C3×#428C2218yescomplemented
38C2×C3×#107C2214yescomplemented
39C2×C3×#106C2214yescomplemented
40C3×#419C2214yes—
41C3×#418C2214yes—
42C3×#417C2214yescomplemented
43C3×#417C2214yescomplemented
44C3×#416C2218yescomplemented
45C3×#408C2218yescomplemented
46C3×#407C2214yes—
47C2×#537C2×C311yescharacteristic
48#3428C2×C311yescharacteristic, complemented
49#3099C2×C318yescomplemented
50#3098C2×C318yescomplemented
51#3095C2×C314yescomplemented
52#3094C2×C314yescomplemented
53C2×#428C2×C312yescomplemented
54#2896C2×C312yescomplemented
55#2879C2×C311yescharacteristic, complemented
56C2×C3×#41—22no—
57C2×C3×#41—28no—
58C2×C3×#41—28no—
59C2×C3×#41D414yes—
60C2×C3×#41C2314yes—
61C2×C3×#41C2314yes—
62C22×C3×Q8D411yescharacteristic
63C22×C3×Q8C2311yescharacteristic
64C22×C3×D4—21no—
65C22×C3×D4—22no—
66C22×C3×D4D412yes—
67C22×C3×D4D411yescharacteristic
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4—21no—
71C23×C3×C4C2312yes—
72C23×C3×C4C2311yescharacteristic
73C2×C3×M16—28no—
74C2×C3×M16C2314yes—
75C2×C3×M16C2314yes—
76C22×C3×C8C2314yes—
77C3×C22⋊Q8—28no—
78C3×C22⋊Q8—24no—
79C3×C22⋊Q8C2318yes—
80C3×#121—28no—
81C3×#121—216no—
82C3×#121—24no—
83C3×#121C2318yes—
84C3×C4×Q8—24no—
85C3×C4×Q8—24no—
86C3×C4×D4—28no—
87C3×C4×D4—24no—
88C3×C4×D4—28no—
89C3×C4×D4—24no—
90C3×#117—24no—
91C3×#117—22no—
92C3×#117—24no—
93C3×#117—22no—
94C3×#117C2314yes—
95C3×#117C2314yes—
96C3×#117C2318yes—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—22no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2314yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2314yes—
105C2×C3×C4⋊C4C2314yes—
106C2×C3×C22⋊C4—24no—
107C2×C3×C22⋊C4—22no—
108C2×C3×C22⋊C4—22no—
109C2×C3×C22⋊C4—22no—
110C2×C3×C42—24no—
111C2×C3×C42—22no—
112C3×#107C23116yes—
113C3×#106C23116yes—
114C3×#103C2318yes—
115C3×#103C2318yes—
116C3×#102C2318yes—
117C3×#102C2318yes—
118C3×C22⋊C8C2318yes—
119C3×C22⋊C8C2318yes—
120C22×#41C22×C311yescharacteristic
121C22×#41C22×C311yescharacteristic
122C22×M16C22×C312yes—
123#537—24no—
124#537C22×C318yes—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8C22×C312yes—
127C2×#121—21no—
128C2×#121—22no—
129C2×#121C22×C312yes—
130#520C22×C314yescomplemented
131#520C22×C314yescomplemented
132#519—21no—
133#518—21no—
134#518—22no—
135C4×#41—24no—
136C2×#117C22×C314yes—
137C2×#117C22×C312yes—
138C22×C4⋊C4C22×C311yescharacteristic
139#485C22×C3116yescomplemented
140#484C22×C318yescomplemented
141#483C22×C318yescomplemented
142#482C22×C3116yescomplemented
143#481C22×C318yescomplemented
144#480C22×C318yescomplemented
145#428C22×C318yes—
146#428C22×C318yescomplemented
147C2×#107C22×C314yescomplemented
148C2×#106C22×C314yescomplemented
149#419C22×C314yes—
150#418C22×C314yes—
151#417C22×C314yescomplemented
152#417C22×C314yescomplemented
153#416C22×C318yescomplemented
154#408C22×C318yescomplemented
155#407C22×C314yes—
156C24×C3—21no—
157C24×C3C2×D412yes—
158C24×C3C2×D411yescharacteristic
159C3×#41—216no—
160C3×#41—28no—
161C3×#41—28no—
162C3×#41—28no—
163C3×#41—28no—
164C3×#41—48no—
165C2×C3×Q8—24no—
166C2×C3×Q8—21no—
167C2×C3×Q8—24no—
168C2×C3×Q8C2×D414yes—
169C2×C3×Q8C2412yes—
170C2×C3×D4—28no—
171C2×C3×D4—24no—
172C2×C3×D4—28no—
173C2×C3×D4—24no—
174C2×C3×D4—21no—
175C2×C3×D4—24no—
176C2×C3×D4—44no—
177C2×C3×D4—84no—
178C2×C3×D4C2×D418yes—
179C2×C3×D4C2×D414yes—
180C2×C3×D4C2412yes—
181C22×C3×C4—28no—
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—24no—
185C22×C3×C4—24no—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—24no—
189C22×C3×C4—24no—
190C22×C3×C4—44no—
191C22×C3×C4—82no—
192C22×C3×C4C2×D414yes—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4#4118yes—
197C22×C3×C4C2414yes—
198C22×C3×C4C2412yes—
199C22×C3×C4C2411yescharacteristic
200C3×M16—28no—
201C3×M16—28no—
202C2×C3×C8—28no—
203C2×C3×C8C2418yes—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—28no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4C2414yes—
212C3×C4⋊C4C2414yes—
213C3×C4⋊C4C2414yes—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—28no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—28no—
218C3×C22⋊C4—28no—
219C3×C42—24no—
220C3×C42—28no—
221C3×C42—24no—
222C2×#41—22no—
223C2×#41—28no—
224C2×#41—28no—
225C2×#41C3×D414yes—
226C2×#41C23×C314yes—
227C2×#41C23×C314yes—
228C22×Q8C3×D411yescharacteristic
229C22×Q8C23×C311yescharacteristic
230C22×D4—21no—
231C22×D4—22no—
232C22×D4C3×D412yes—
233C22×D4C3×D411yescharacteristic
234C22×D4C23×C311yescharacteristic
235C23×C4—21no—
236C23×C4—21no—
237C23×C4C23×C312yes—
238C23×C4C23×C311yescharacteristic
239C2×M16—28no—
240C2×M16C23×C314yes—
241C2×M16C23×C314yes—
242C22×C8C23×C314yes—
243C22⋊Q8—28no—
244C22⋊Q8—24no—
245C22⋊Q8C23×C318yes—
246#121—28no—
247#121—216no—
248#121—24no—
249#121C23×C318yes—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×D4—28no—
253C4×D4—24no—
254C4×D4—28no—
255C4×D4—24no—
256#117—24no—
257#117—22no—
258#117—24no—
259#117—22no—
260#117C23×C314yes—
261#117C23×C314yes—
262#117C23×C318yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4C23×C312yes—
267C2×C4⋊C4C23×C314yes—
268C2×C4⋊C4C23×C312yes—
269C2×C4⋊C4C23×C312yes—
270C2×C4⋊C4C23×C314yes—
271C2×C4⋊C4C23×C314yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C42—24no—
277C2×C42—22no—
278#107C23×C3116yes—
279#106C23×C3116yes—
280#103C23×C318yes—
281#103C23×C318yes—
282#102C23×C318yes—
283#102C23×C318yes—
284C22⋊C8C23×C318yes—
285C22⋊C8C23×C318yes—
286C23×C3—24no—
287C23×C3—22no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3#12314yes—
296C23×C3C22×D414yes—
297C23×C3C22×D411yescharacteristic
298C23×C3C22×D412yes—
299C3×Q8—28no—
300C3×Q8—24no—
301C3×Q8—42no—
302C3×D4—216no—
303C3×D4—28no—
304C3×D4—24no—
305C3×D4—48no—
306C3×D4—42no—
307C3×D4—88no—
308C2×C3×C4—24no—
309C2×C3×C4—28no—
310C2×C3×C4—28no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—22no—
315C2×C3×C4—44no—
316C2×C3×C4—42no—
317C2×C3×C4—48no—
318C2×C3×C4—82no—
319C2×C3×C4—84no—
320C2×C3×C4—82no—
321C2×C3×C4—84no—
322C2×C3×C4—84no—
323C2×C3×C4#12314yes—
324C2×C3×C4#12318yes—
325C2×C3×C4C22×D414yes—
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C22×D412yes—
328C2×C3×C4C2×#4114yes—
329C2×C3×C4C2×#4114yes—
330C2×C3×C4C2×#4114yes—
331C2×C3×C4C2511yescharacteristic
332C3×C8—28no—
333C24—21no—
334C24C2×C3×D412yes—
335C24C2×C3×D411yescharacteristic
336#41—216no—
337#41—28no—
338#41—28no—
339#41—28no—
340#41—28no—
341#41—48no—
342C2×Q8—24no—
343C2×Q8—21no—
344C2×Q8—24no—
345C2×Q8C2×C3×D414yes—
346C2×Q8C24×C312yes—
347C2×D4—28no—
348C2×D4—24no—
349C2×D4—28no—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4—24no—
353C2×D4—44no—
354C2×D4—84no—
355C2×D4C2×C3×D418yes—
356C2×D4C2×C3×D414yes—
357C2×D4C24×C312yes—
358C22×C4—28no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—44no—
368C22×C4—82no—
369C22×C4C2×C3×D414yes—
370C22×C4C2×C3×D412yes—
371C22×C4C2×C3×D412yes—
372C22×C4C2×C3×D411yescharacteristic
373C22×C4C3×#4118yes—
374C22×C4C24×C314yes—
375C22×C4C24×C312yes—
376C22×C4C24×C311yescharacteristic
377M16—28no—
378M16—28no—
379C2×C8—28no—
380C2×C8C24×C318yes—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—28no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4C24×C314yes—
389C4⋊C4C24×C314yes—
390C4⋊C4C24×C314yes—
391C22⋊C4—24no—
392C22⋊C4—28no—
393C22⋊C4—24no—
394C22⋊C4—28no—
395C22⋊C4—28no—
396C42—24no—
397C42—28no—
398C42—24no—
399C22×C3—21no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—21no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—44no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—84no—
411C22×C3—84no—
412C22×C3C2×#12312yes—
413C22×C3C2×#12314yes—
414C22×C3C23×D411yescharacteristic
415C3×C4—24no—
416C3×C4—44no—
417C3×C4—44no—
418C3×C4—44no—
419C3×C4—42no—
420C3×C4C2×#12312yes—
421C3×C4C2×#12314yes—
422C3×C4C22×#4112yes—
423C23—24no—
424C23—22no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23C3×#12314yes—
433C23C22×C3×D414yes—
434C23C22×C3×D411yescharacteristic
435C23C22×C3×D412yes—
436Q8—28no—
437Q8—24no—
438Q8—42no—
439D4—216no—
440D4—24no—
441D4—28no—
442D4—48no—
443D4—42no—
444D4—88no—
445C2×C4—24no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—48no—
455C2×C4—84no—
456C2×C4—82no—
457C2×C4—84no—
458C2×C4—82no—
459C2×C4—84no—
460C2×C4C3×#12314yes—
461C2×C4C3×#12318yes—
462C2×C4C22×C3×D414yes—
463C2×C4C22×C3×D411yescharacteristic
464C2×C4C22×C3×D412yes—
465C2×C4C2×C3×#4114yes—
466C2×C4C2×C3×#4114yes—
467C2×C4C2×C3×#4114yes—
468C2×C4C25×C311yescharacteristic
469C8—28no—
470C2×C3—22no—
471C2×C3—24no—
472C2×C3—44no—
473C2×C3—82no—
474C2×C3C22×#12311yescharacteristic
475C2×C3#358511yescharacteristic
476C2×C3#358611yescharacteristic
477C22—21no—
478C22—22no—
479C22—22no—
480C22—21no—
481C22—24no—
482C22—24no—
483C22—24no—
484C22—44no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—84no—
489C22—84no—
490C22C2×C3×#12312yes—
491C22C2×C3×#12314yes—
492C22C23×C3×D411yescharacteristic
493C4—24no—
494C4—42no—
495C4—44no—
496C4—44no—
497C4—44no—
498C4C2×C3×#12312yes—
499C4C2×C3×#12314yes—
500C4C22×C3×#4112yes—
501C3#6052011yescharacteristic, direct product factor, Sylow
502C2—22no—
503C2—24no—
504C2—44no—
505C2—82no—
506C2C22×C3×#12311yescharacteristic
507C2C3×#358511yescharacteristic
508C2C3×#358611yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC2×C3×#537C2
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. C3×#3428C2semidirect
3. C3×#3099C2semidirect
4. C3×#3098C2semidirect
5. C3×#3095C2semidirect
6. C3×#3094C2semidirect
7. C2×C3×#428C2semidirect
8. C3×#2896C2semidirect
9. C3×#2879C2semidirect
10. #60520C3direct (N × H)
21. C3×#520C22semidirect
22. C3×#520C22semidirect
30. C3×#485C22semidirect
31. C3×#484C22semidirect
32. C3×#483C22semidirect
33. C3×#482C22semidirect
34. C3×#481C22semidirect
35. C3×#480C22semidirect
37. C3×#428C22semidirect
38. C2×C3×#107C22semidirect
39. C2×C3×#106C22semidirect
42. C3×#417C22semidirect
43. C3×#417C22semidirect
44. C3×#416C22semidirect
45. C3×#408C22semidirect
48. #3428C2×C3semidirect
49. #3099C2×C3semidirect
50. #3098C2×C3semidirect
51. #3095C2×C3semidirect
52. #3094C2×C3semidirect
53. C2×#428C2×C3semidirect
54. #2896C2×C3semidirect
55. #2879C2×C3semidirect
130. #520C22×C3semidirect
131. #520C22×C3semidirect
139. #485C22×C3semidirect
140. #484C22×C3semidirect
141. #483C22×C3semidirect
142. #482C22×C3semidirect
143. #481C22×C3semidirect
144. #480C22×C3semidirect
146. #428C22×C3semidirect
147. C2×#107C22×C3semidirect
148. C2×#106C22×C3semidirect
151. #417C22×C3semidirect
152. #417C22×C3semidirect
153. #416C22×C3semidirect
154. #408C22×C3semidirect
501. C3#60520direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5372
C3×#34282
C3×#30992
C3×#30982
C3×#30952
C3×#30942
C2×C3×#4282
C3×#28962
C3×#28792
#605203