Finite Group Databasebeta

← C3×#33000 #10651607

Subgroups — C3×#33000 #10651607

Order 768.

1300 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic
2C3×#3095C211yesmaximal, characteristic, complemented
3C3×#3007C211yesmaximal, characteristic
4C3×#2502C211yesmaximal, characteristic, complemented
5C3×#1882C214yesmaximal, complemented
6C3×#1859C214yesmaximal, complemented
7C3×#1826C211yesmaximal, characteristic, complemented
8C3×#1821C211yesmaximal, characteristic, complemented
9C3×#1789C211yesmaximal, characteristic
10#33000C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2211yescharacteristic
12C2×C3×#137C2212yes—
13C2×C3×C22⋊Q8C2211yescharacteristic
14C3×#520—22no—
15C3×#520—22no—
16C3×#520—22no—
17C3×#520C2214yes—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×D4—21no—
20C2×C3×C4×D4—22no—
21C2×C3×#117—21no—
22C2×C3×#117C2211yescharacteristic
23C22×C3×C4⋊C4—21no—
24C22×C3×C4⋊C4C2211yescharacteristic
25C3×#485—22no—
26C3×#484—21no—
27C3×#483—21no—
28C3×#452—22no—
29C3×#451—21no—
30C3×#450—21no—
31C3×#428C2212yescomplemented
32C2×C3×#104C2214yes—
33C3×#419C2212yescomplemented
34C3×#417C2212yes—
35C3×#407C2211yescharacteristic
36C3×#407C2211yescharacteristic
37C3×#398—22no—
38C3×#391C2212yescomplemented
39C3×#385—22no—
40C3×#384C2212yescomplemented
41C3×C4×C22⋊C4C2211yescharacteristic
42C3×#343C2214yescomplemented
43C3×#339C2214yescomplemented
44C2×#520C2×C311yescharacteristic
45#3095C2×C311yescharacteristic, complemented
46#3007C2×C311yescharacteristic
47#2502C2×C311yescharacteristic, complemented
48#1882C2×C314yescomplemented
49#1859C2×C314yescomplemented
50#1826C2×C311yescharacteristic, complemented
51#1821C2×C311yescharacteristic, complemented
52#1789C2×C311yescharacteristic
53C2×C3×#41—28no—
54C2×C3×#41—21no—
55C2×C3×#41—21no—
56C2×C3×#41D412yes—
57C2×C3×#41C2312yes—
58C22×C3×Q8D411yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4D411yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4—22no—
64C23×C3×C4D412yes—
65C23×C3×C4C2311yescharacteristic
66C3×#137—28no—
67C2×C3×Q16—22no—
68C2×C3×SD16—22no—
69C2×C3×M16C2312yes—
70C2×C3×M16C2312yes—
71C3×C22⋊Q8—21no—
72C3×C22⋊Q8—21no—
73C3×C22⋊Q8—21no—
74C3×C22⋊Q8—21no—
75C3×C4×Q8—24no—
76C3×C4×Q8—24no—
77C3×C4×D4—28no—
78C3×C4×D4—24no—
79C3×C4×D4—24no—
80C3×C4×D4—28no—
81C3×#117—24no—
82C3×#117—21no—
83C3×#117—21no—
84C3×#117—24no—
85C3×#117D412yes—
86C3×#117C2312yes—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—24no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—24no—
94C2×C3×C4⋊C4—21no—
95C2×C3×C4⋊C4—24no—
96C2×C3×C4⋊C4D412yes—
97C2×C3×C4⋊C4D411yescharacteristic
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2312yes—
101C2×C3×C22⋊C4—22no—
102C2×C3×C22⋊C4—21no—
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C2×C3×C42—22no—
106C2×C3×C42—21no—
107C2×C3×C42C2312yes—
108C3×#107—22no—
109C3×#106—22no—
110C3×#104—28no—
111C3×#103—22no—
112C3×#103—22no—
113C3×#103—22no—
114C3×#102—22no—
115C3×C22⋊C8—21no—
116C3×C22⋊C8—21no—
117C3×C22⋊C8—21no—
118C3×C22⋊C8—21no—
119C3×#95—21no—
120C3×#95—24no—
121C22×#41C22×C311yescharacteristic
122C2×#137C22×C312yes—
123C2×C22⋊Q8C22×C311yescharacteristic
124#520—22no—
125#520—22no—
126#520—22no—
127#520C22×C314yes—
128C2×C4×Q8—21no—
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131C2×#117—21no—
132C2×#117C22×C311yescharacteristic
133C22×C4⋊C4—21no—
134C22×C4⋊C4C22×C311yescharacteristic
135#485—22no—
136#484—21no—
137#483—21no—
138#452—22no—
139#451—21no—
140#450—21no—
141#428C22×C312yescomplemented
142C2×#104C22×C314yes—
143#419C22×C312yescomplemented
144#417C22×C312yes—
145#407C22×C311yescharacteristic
146#407C22×C311yescharacteristic
147#398—22no—
148#391C22×C312yescomplemented
149#385—22no—
150#384C22×C312yescomplemented
151C4×C22⋊C4C22×C311yescharacteristic
152#343C22×C314yescomplemented
153#339C22×C314yescomplemented
154C24×C3—21no—
155C24×C3C2×D411yescharacteristic
156C3×#41—28no—
157C3×#41—28no—
158C3×#41—24no—
159C3×#41—24no—
160C3×#41—24no—
161C3×#41#4118yes—
162C2×C3×Q8—24no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—24no—
167C2×C3×Q8C2×D412yes—
168C2×C3×D4—28no—
169C2×C3×D4—22no—
170C2×C3×D4—22no—
171C2×C3×D4—24no—
172C2×C3×D4—21no—
173C2×C3×D4C2×D412yes—
174C22×C3×C4—28no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—24no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C22×C3×C4—22no—
183C22×C3×C4—28no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—82no—
188C22×C3×C4—84no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2411yescharacteristic
196C3×Q16—44no—
197C3×SD16—44no—
198C3×M16—24no—
199C3×M16—24no—
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C3×C4⋊C4—28no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—41no—
212C3×C4⋊C4—41no—
213C3×C4⋊C4—84no—
214C3×C4⋊C4C2×D412yes—
215C3×C4⋊C4C2×D412yes—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—28no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—41no—
221C3×C22⋊C4—41no—
222C3×C42—28no—
223C3×C42—22no—
224C3×C42—22no—
225C3×C42—24no—
226C2×#41—28no—
227C2×#41—21no—
228C2×#41—21no—
229C2×#41C3×D412yes—
230C2×#41C23×C312yes—
231C22×Q8C3×D411yescharacteristic
232C22×Q8C23×C311yescharacteristic
233C22×D4—21no—
234C22×D4C3×D411yescharacteristic
235C23×C4—21no—
236C23×C4—22no—
237C23×C4C3×D412yes—
238C23×C4C23×C311yescharacteristic
239#137—28no—
240C2×Q16—22no—
241C2×SD16—22no—
242C2×M16C23×C312yes—
243C2×M16C23×C312yes—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×D4—28no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—28no—
254#117—24no—
255#117—21no—
256#117—21no—
257#117—24no—
258#117C3×D412yes—
259#117C23×C312yes—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4C3×D412yes—
270C2×C4⋊C4C3×D411yescharacteristic
271C2×C4⋊C4C3×D411yescharacteristic
272C2×C4⋊C4C23×C311yescharacteristic
273C2×C4⋊C4C23×C312yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42C23×C312yes—
281#107—22no—
282#106—22no—
283#104—28no—
284#103—22no—
285#103—22no—
286#103—22no—
287#102—22no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292#95—21no—
293#95—24no—
294C23×C3—24no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—42no—
301C23×C3#12011yescharacteristic
302C23×C3#12011yescharacteristic
303C23×C3C22×D411yescharacteristic
304C3×Q8—24no—
305C3×Q8—22no—
306C3×Q8—42no—
307C3×Q8—42no—
308C3×Q8#12314yes—
309C3×D4—28no—
310C3×D4—24no—
311C3×D4—28no—
312C3×D4—22no—
313C3×D4#12314yes—
314C2×C3×C4—24no—
315C2×C3×C4—28no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—28no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—42no—
328C2×C3×C4—44no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—44no—
333C2×C3×C4—42no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—42no—
337C2×C3×C4—82no—
338C2×C3×C4—88no—
339C2×C3×C4#12011yescharacteristic
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C2×#4114yes—
344C3×C8—42no—
345C3×C8—42no—
346C24—21no—
347C24C2×C3×D411yescharacteristic
348#41—28no—
349#41—28no—
350#41—24no—
351#41—24no—
352#41—24no—
353#41C3×#4118yes—
354C2×Q8—24no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—24no—
359C2×Q8C2×C3×D412yes—
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—21no—
365C2×D4C2×C3×D412yes—
366C22×C4—28no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—82no—
380C22×C4—84no—
381C22×C4C2×C3×D411yescharacteristic
382C22×C4C2×C3×D411yescharacteristic
383C22×C4C2×C3×D411yescharacteristic
384C22×C4C2×C3×D411yescharacteristic
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D414yes—
387C22×C4C24×C311yescharacteristic
388Q16—44no—
389SD16—44no—
390M16—24no—
391M16—24no—
392C2×C8—22no—
393C2×C8—22no—
394C4⋊C4—28no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—84no—
406C4⋊C4C2×C3×D412yes—
407C4⋊C4C2×C3×D412yes—
408C22⋊C4—28no—
409C22⋊C4—24no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C42—28no—
415C42—22no—
416C42—22no—
417C42—24no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—24no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—44no—
427C22×C3—48no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3#45612yes—
431C22×C3C2×#12011yescharacteristic
432C22×C3#52512yes—
433C3×C4—21no—
434C3×C4—24no—
435C3×C4—21no—
436C3×C4—44no—
437C3×C4—42no—
438C3×C4—44no—
439C3×C4—42no—
440C3×C4—81no—
441C3×C4—81no—
442C3×C4C2×#12312yes—
443C3×C4#52512yes—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23C3×#12011yescharacteristic
452C23C3×#12011yescharacteristic
453C23C22×C3×D411yescharacteristic
454Q8—24no—
455Q8—22no—
456Q8—42no—
457Q8—42no—
458Q8C3×#12314yes—
459D4—28no—
460D4—24no—
461D4—28no—
462D4—22no—
463D4C3×#12314yes—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—28no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—82no—
488C2×C4—88no—
489C2×C4C3×#12011yescharacteristic
490C2×C4C3×#12011yescharacteristic
491C2×C4C22×C3×D411yescharacteristic
492C2×C4C22×C3×D411yescharacteristic
493C2×C4C2×C3×#4114yes—
494C8—42no—
495C8—42no—
496C2×C3—22no—
497C2×C3—22no—
498C2×C3—41no—
499C2×C3—44no—
500C2×C3#284911yescharacteristic
501C2×C3C2×#45611yescharacteristic
502C2×C3#301911yescharacteristic
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—24no—
511C22—44no—
512C22—48no—
513C22—42no—
514C22—42no—
515C22C3×#45612yes—
516C22C2×C3×#12011yescharacteristic
517C22C3×#52512yes—
518C4—21no—
519C4—24no—
520C4—21no—
521C4—44no—
522C4—42no—
523C4—44no—
524C4—42no—
525C4—81no—
526C4—81no—
527C4C2×C3×#12312yes—
528C4C3×#52512yes—
529C3#3300011yescharacteristic, direct product factor, Sylow
530C2—22no—
531C2—22no—
532C2—41no—
533C2—44no—
534C2C3×#284911yescharacteristic
535C2C2×C3×#45611yescharacteristic
536C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3095C2semidirect
4. C3×#2502C2semidirect
5. C3×#1882C2semidirect
6. C3×#1859C2semidirect
7. C3×#1826C2semidirect
8. C3×#1821C2semidirect
10. #33000C3direct (N × H)
31. C3×#428C22semidirect
33. C3×#419C22semidirect
38. C3×#391C22semidirect
40. C3×#384C22semidirect
42. C3×#343C22semidirect
43. C3×#339C22semidirect
45. #3095C2×C3semidirect
47. #2502C2×C3semidirect
48. #1882C2×C3semidirect
49. #1859C2×C3semidirect
50. #1826C2×C3semidirect
51. #1821C2×C3semidirect
141. #428C22×C3semidirect
143. #419C22×C3semidirect
148. #391C22×C3semidirect
150. #384C22×C3semidirect
152. #343C22×C3semidirect
153. #339C22×C3semidirect
529. C3#33000direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C3×#30952
C3×#30072
C3×#25022
C3×#18822
C3×#18592
C3×#18262
C3×#18212
C3×#17892
#330003