Finite Group Databasebeta

← C3×#29750 #10648357

Subgroups — C3×#29750 #10648357

Order 768.

930 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#465C211yesmaximal, characteristic, complemented
2C3×#3007C211yesmaximal, characteristic, complemented
3C3×#3006C211yesmaximal, characteristic
4C2×C3×#408C211yesmaximal, characteristic, complemented
5C3×#2079C212yesmaximal, complemented
6C3×#2052C212yesmaximal, complemented
7C3×#2051C212yesmaximal, complemented
8C3×#2019C212yesmaximal, complemented
9C3×#1901C211yesmaximal, characteristic, complemented
10C3×#1900C211yesmaximal, characteristic, complemented
11C3×#1810C211yesmaximal, characteristic, complemented
12#29750C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×#137C2212yes—
15C2×C3×AGL(1,ℤ/8)C2212yes—
16C22×C3×SD16C2211yescharacteristic, complemented
17C22×C3×M16C2211yescharacteristic
18C23×C3×C8C2211yescharacteristic
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#121C2211yescharacteristic
21C3×#465—22no—
22C3×#465—22no—
23C3×#452—22no—
24C3×#451—21no—
25C3×#450—21no—
26C3×#449—22no—
27C3×#448—21no—
28C3×#447—21no—
29C2×C3×#108C2212yescomplemented
30C2×C3×#106C2211yescharacteristic, complemented
31C3×#417C2212yes—
32C3×#417C2212yes—
33C2×C3×#103C2211yescharacteristic, complemented
34C2×C3×#102C2211yescharacteristic, complemented
35C2×C3×#101C2212yescomplemented
36C2×C3×#100C2212yescomplemented
37C3×#408—22no—
38C3×#408C2212yes—
39C3×#408C2212yes—
40C3×#407C2211yescharacteristic
41C3×#407C2211yescharacteristic
42C2×C3×C22⋊C8—22no—
43C3×#343C2212yescomplemented
44C3×#343C2212yescomplemented
45C3×#341C2212yescomplemented
46C2×#465C2×C311yescharacteristic, complemented
47#3007C2×C311yescharacteristic, complemented
48#3006C2×C311yescharacteristic
49C2×#408C2×C311yescharacteristic, complemented
50#2079C2×C312yescomplemented
51#2052C2×C312yescomplemented
52#2051C2×C312yescomplemented
53#2019C2×C312yescomplemented
54#1901C2×C311yescharacteristic, complemented
55#1900C2×C311yescharacteristic, complemented
56#1810C2×C311yescharacteristic, complemented
57C2×C3×#41—21no—
58C2×C3×#41—21no—
59C2×C3×#41—42no—
60C2×C3×#41—42no—
61C2×C3×#41D412yes—
62C2×C3×#41C2312yes—
63C22×C3×Q8D411yescharacteristic
64C22×C3×Q8C2311yescharacteristic
65C22×C3×D4—21no—
66C22×C3×D4—21no—
67C22×C3×D4D411yescharacteristic
68C22×C3×D4C2311yescharacteristic
69C23×C3×C4—21no—
70C23×C3×C4C2311yescharacteristic
71C3×#137—42no—
72C3×#137—42no—
73C3×AGL(1,ℤ/8)—42no—
74C3×AGL(1,ℤ/8)—42no—
75C2×C3×Q16—22no—
76C2×C3×SD16—22no—
77C2×C3×SD16—22no—
78C2×C3×SD16—22no—
79C2×C3×SD16—42no—
80C2×C3×D8—22no—
81C2×C3×M16—21no—
82C2×C3×M16—42no—
83C2×C3×M16C2312yes—
84C2×C3×M16C2312yes—
85C2×C3×M16C2312yes—
86C22×C3×C8—21no—
87C22×C3×C8—42no—
88C22×C3×C8D412yes—
89C22×C3×C8D412yes—
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C3×C22⋊Q8—21no—
93C3×C22⋊Q8—21no—
94C3×C22⋊Q8—21no—
95C3×C22⋊Q8—21no—
96C3×#121—21no—
97C3×#121—21no—
98C3×#121—21no—
99C3×#121—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C22⋊C4—21no—
104C2×C3×C22⋊C4—21no—
105C3×#108—42no—
106C3×#106—22no—
107C3×#103—22no—
108C3×#103—22no—
109C3×#103—22no—
110C3×#102—22no—
111C3×#102—22no—
112C3×#102—22no—
113C3×#101—42no—
114C3×#100—42no—
115C3×C22⋊C8—21no—
116C3×C22⋊C8—21no—
117C3×C22⋊C8—21no—
118C3×C22⋊C8—21no—
119C3×C22⋊C8—24no—
120C3×C22⋊C8—24no—
121C22×#41C22×C311yescharacteristic
122C2×#137C22×C312yes—
123C2×AGL(1,ℤ/8)C22×C312yes—
124C22×SD16C22×C311yescharacteristic, complemented
125C22×M16C22×C311yescharacteristic
126C23×C8C22×C311yescharacteristic
127C2×C22⋊Q8C22×C311yescharacteristic
128C2×#121C22×C311yescharacteristic
129#465—22no—
130#465—22no—
131#452—22no—
132#451—21no—
133#450—21no—
134#449—22no—
135#448—21no—
136#447—21no—
137C2×#108C22×C312yescomplemented
138C2×#106C22×C311yescharacteristic, complemented
139#417C22×C312yes—
140#417C22×C312yes—
141C2×#103C22×C311yescharacteristic, complemented
142C2×#102C22×C311yescharacteristic, complemented
143C2×#101C22×C312yescomplemented
144C2×#100C22×C312yescomplemented
145#408—22no—
146#408C22×C312yes—
147#408C22×C312yes—
148#407C22×C311yescharacteristic
149#407C22×C311yescharacteristic
150C2×C22⋊C8—22no—
151#343C22×C312yescomplemented
152#343C22×C312yescomplemented
153#341C22×C312yescomplemented
154C24×C3—21no—
155C24×C3—21no—
156C24×C3#4111yescharacteristic
157C3×#41—22no—
158C3×#41—22no—
159C3×#41—22no—
160C3×#41—22no—
161C3×#41—42no—
162C3×#41—42no—
163C3×#41—44no—
164C3×#41—42no—
165C3×#41—42no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—42no—
172C2×C3×Q8C2×D412yes—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—42no—
181C2×C3×D4—44no—
182C2×C3×D4—41no—
183C2×C3×D4—41no—
184C2×C3×D4—82no—
185C2×C3×D4C2×D412yes—
186C22×C3×C4—22no—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4#4111yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C3×Q16—44no—
203C3×SD16—44no—
204C3×SD16—44no—
205C3×SD16—44no—
206C3×D8—44no—
207C3×M16—22no—
208C3×M16—22no—
209C3×M16—22no—
210C3×M16—22no—
211C3×M16—22no—
212C3×M16—22no—
213C3×M16—42no—
214C2×C3×C8—22no—
215C2×C3×C8—22no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—42no—
220C2×C3×C8—42no—
221C2×C3×C8—42no—
222C2×C3×C8C2×D412yes—
223C2×C3×C8C2×D412yes—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—41no—
229C3×C4⋊C4—41no—
230C3×C22⋊C4—41no—
231C3×C22⋊C4—41no—
232C3×C22⋊C4—41no—
233C3×C22⋊C4—41no—
234C2×#41—21no—
235C2×#41—21no—
236C2×#41—42no—
237C2×#41—42no—
238C2×#41C3×D412yes—
239C2×#41C23×C312yes—
240C22×Q8C3×D411yescharacteristic
241C22×Q8C23×C311yescharacteristic
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C3×D411yescharacteristic
245C22×D4C23×C311yescharacteristic
246C23×C4—21no—
247C23×C4C23×C311yescharacteristic
248#137—42no—
249#137—42no—
250AGL(1,ℤ/8)—42no—
251AGL(1,ℤ/8)—42no—
252C2×Q16—22no—
253C2×SD16—22no—
254C2×SD16—22no—
255C2×SD16—22no—
256C2×SD16—42no—
257C2×D8—22no—
258C2×M16—21no—
259C2×M16—42no—
260C2×M16C23×C312yes—
261C2×M16C23×C312yes—
262C2×M16C23×C312yes—
263C22×C8—21no—
264C22×C8—42no—
265C22×C8C3×D412yes—
266C22×C8C3×D412yes—
267C22×C8C23×C311yescharacteristic
268C22×C8C23×C311yescharacteristic
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
273#121—21no—
274#121—21no—
275#121—21no—
276#121—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4C23×C311yescharacteristic
279C2×C4⋊C4C23×C311yescharacteristic
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282#108—42no—
283#106—22no—
284#103—22no—
285#103—22no—
286#103—22no—
287#102—22no—
288#102—22no—
289#102—22no—
290#101—42no—
291#100—42no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—24no—
297C22⋊C8—24no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—41no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3—42no—
305C23×C3—41no—
306C23×C3—42no—
307C23×C3—81no—
308C23×C3—81no—
309C23×C3#12111yescharacteristic
310C23×C3#12111yescharacteristic
311C23×C3C2×#4111yescharacteristic
312C3×Q8—22no—
313C3×Q8—22no—
314C3×Q8—42no—
315C3×Q8—41no—
316C3×Q8—42no—
317C3×Q8—41no—
318C3×D4—22no—
319C3×D4—22no—
320C3×D4—42no—
321C3×D4—44no—
322C3×D4—42no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—41no—
326C3×D4—41no—
327C3×D4—82no—
328C3×D4—82no—
329C2×C3×C4—22no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—22no—
337C2×C3×C4—44no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—81no—
346C2×C3×C4—81no—
347C2×C3×C4#12111yescharacteristic
348C2×C3×C4#12111yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4H(5,𝔽2)12yes—
352C2×C3×C4H(5,𝔽2)12yes—
353C3×C8—22no—
354C3×C8—22no—
355C3×C8—42no—
356C3×C8—42no—
357C3×C8—42no—
358C24—21no—
359C24—21no—
360C24C3×#4111yescharacteristic
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—42no—
366#41—42no—
367#41—44no—
368#41—42no—
369#41—42no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—42no—
376C2×Q8C2×C3×D412yes—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—42no—
385C2×D4—44no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—82no—
389C2×D4C2×C3×D412yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C2×C3×D411yescharacteristic
402C22×C4C2×C3×D411yescharacteristic
403C22×C4C2×C3×D411yescharacteristic
404C22×C4C3×#4111yescharacteristic
405C22×C4C24×C311yescharacteristic
406Q16—44no—
407SD16—44no—
408SD16—44no—
409SD16—44no—
410D8—44no—
411M16—22no—
412M16—22no—
413M16—22no—
414M16—22no—
415M16—22no—
416M16—22no—
417M16—42no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8C2×C3×D412yes—
427C2×C8C2×C3×D412yes—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—44no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—84no—
450C22×C3—81no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—162no—
454C22×C3#47112yes—
455C22×C3C2×#12111yescharacteristic
456C22×C3#53812yes—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—42no—
460C3×C4—81no—
461C3×C4—81no—
462C3×C4—81no—
463C3×C4#53411yescharacteristic
464C3×C4#53412yes—
465C3×C4#53511yescharacteristic
466C23—22no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—81no—
476C23—81no—
477C23C3×#12111yescharacteristic
478C23C3×#12111yescharacteristic
479C23C2×C3×#4111yescharacteristic
480Q8—22no—
481Q8—22no—
482Q8—42no—
483Q8—41no—
484Q8—42no—
485Q8—41no—
486D4—22no—
487D4—22no—
488D4—42no—
489D4—44no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—41no—
494D4—41no—
495D4—82no—
496D4—82no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4C3×#12111yescharacteristic
516C2×C4C3×#12111yescharacteristic
517C2×C4C22×C3×D411yescharacteristic
518C2×C4C22×C3×D411yescharacteristic
519C2×C4C3×H(5,𝔽2)12yes—
520C2×C4C3×H(5,𝔽2)12yes—
521C8—22no—
522C8—22no—
523C8—42no—
524C8—42no—
525C8—42no—
526C2×C3—22no—
527C2×C3—22no—
528C2×C3—41no—
529C2×C3—82no—
530C2×C3—161no—
531C2×C3#261711yescharacteristic
532C2×C3C2×#47111yescharacteristic
533C2×C3#306711yescharacteristic
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—44no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—84no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—162no—
550C22C3×#47112yes—
551C22C2×C3×#12111yescharacteristic
552C22C3×#53812yes—
553C4—21no—
554C4—21no—
555C4—42no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4C3×#53411yescharacteristic
560C4C3×#53412yes—
561C4C3×#53511yescharacteristic
562C3#2975011yescharacteristic, direct product factor, Sylow
563C2—22no—
564C2—22no—
565C2—41no—
566C2—82no—
567C2—161no—
568C2C3×#261711yescharacteristic
569C2C2×C3×#47111yescharacteristic
570C2C3×#306711yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3006C2
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
1. C2×C3×#465C2semidirect
2. C3×#3007C2semidirect
4. C2×C3×#408C2semidirect
5. C3×#2079C2semidirect
6. C3×#2052C2semidirect
7. C3×#2051C2semidirect
8. C3×#2019C2semidirect
9. C3×#1901C2semidirect
10. C3×#1900C2semidirect
11. C3×#1810C2semidirect
12. #29750C3direct (N × H)
16. C22×C3×SD16C22semidirect
29. C2×C3×#108C22semidirect
30. C2×C3×#106C22semidirect
33. C2×C3×#103C22semidirect
34. C2×C3×#102C22semidirect
35. C2×C3×#101C22semidirect
36. C2×C3×#100C22semidirect
43. C3×#343C22semidirect
44. C3×#343C22semidirect
45. C3×#341C22semidirect
46. C2×#465C2×C3semidirect
47. #3007C2×C3semidirect
49. C2×#408C2×C3semidirect
50. #2079C2×C3semidirect
51. #2052C2×C3semidirect
52. #2051C2×C3semidirect
53. #2019C2×C3semidirect
54. #1901C2×C3semidirect
55. #1900C2×C3semidirect
56. #1810C2×C3semidirect
124. C22×SD16C22×C3semidirect
137. C2×#108C22×C3semidirect
138. C2×#106C22×C3semidirect
141. C2×#103C22×C3semidirect
142. C2×#102C22×C3semidirect
143. C2×#101C22×C3semidirect
144. C2×#100C22×C3semidirect
151. #343C22×C3semidirect
152. #343C22×C3semidirect
153. #341C22×C3semidirect
562. C3#29750direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4652
C3×#30072
C3×#30062
C2×C3×#4082
C3×#20792
C3×#20522
C3×#20512
C3×#20192
C3×#19012
C3×#19002
C3×#18102
#297503