Finite Group Databasebeta

← C3×#29735 #10648342

Subgroups — C3×#29735 #10648342

Order 768.

1200 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#465C211yesmaximal, characteristic, complemented
2C22×C3×C22⋊C8C211yesmaximal, characteristic, complemented
3C3×#2848C211yesmaximal, characteristic, complemented
4C3×#2847C211yesmaximal, characteristic
5C3×#2077C212yesmaximal, complemented
6C3×#2036C212yesmaximal, complemented
7C3×#2033C212yesmaximal, complemented
8C3×#2000C212yesmaximal, complemented
9C3×#1893C211yesmaximal, characteristic, complemented
10C3×#1892C211yesmaximal, characteristic, complemented
11C3×#1808C211yesmaximal, characteristic, complemented
12#29735C311yesmaximal, characteristic, direct product factor, Sylow
13C24×C3×C4C2211yescharacteristic
14C22×C3×SD16C2211yescharacteristic, complemented
15C23×C3×C8C2211yescharacteristic
16C23×C3×C8C2211yescharacteristic
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×#121C2211yescharacteristic
20C2×C3×#121C2212yes—
21C2×C3×#120—21no—
22C3×#465—24no—
23C2×C3×#107C2212yescomplemented
24C2×C3×#106C2211yescharacteristic, complemented
25C2×C3×#103C2211yescharacteristic, complemented
26C2×C3×#103C2212yescomplemented
27C2×C3×#102C2211yescharacteristic, complemented
28C2×C3×#102C2212yescomplemented
29C2×C3×C22⋊C8—24no—
30C2×C3×C22⋊C8C2212yes—
31C2×C3×C22⋊C8C2212yes—
32C3×#386—21no—
33C3×#386—22no—
34C3×#385—21no—
35C3×#385—22no—
36C3×#380C2212yes—
37C3×#380C2212yes—
38C3×#380C2211yescharacteristic
39C3×#380C2211yescharacteristic
40C3×#379—21no—
41C3×#340C2212yescomplemented
42C3×#340C2212yescomplemented
43C3×#340C2212yescomplemented
44C2×#465C2×C311yescharacteristic, complemented
45C22×C22⋊C8C2×C311yescharacteristic, complemented
46#2848C2×C311yescharacteristic, complemented
47#2847C2×C311yescharacteristic
48#2077C2×C312yescomplemented
49#2036C2×C312yescomplemented
50#2033C2×C312yescomplemented
51#2000C2×C312yescomplemented
52#1893C2×C311yescharacteristic, complemented
53#1892C2×C311yescharacteristic, complemented
54#1808C2×C311yescharacteristic, complemented
55C25×C3D411yescharacteristic
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4—22no—
58C22×C3×D4—21no—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—22no—
61C23×C3×C4—22no—
62C23×C3×C4—42no—
63C23×C3×C4—42no—
64C23×C3×C4D411yescharacteristic
65C23×C3×C4D412yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C2×C3×SD16—22no—
69C2×C3×SD16—42no—
70C22×C3×C8—22no—
71C22×C3×C8—42no—
72C22×C3×C8—42no—
73C22×C3×C8D414yescomplemented
74C22×C3×C8C2312yes—
75C22×C3×C8C2311yescharacteristic
76C22×C3×C8C2311yescharacteristic
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—42no—
80C3×C22⋊Q8—42no—
81C3×#121—22no—
82C3×#121—22no—
83C3×#121—42no—
84C3×#121—42no—
85C3×#120—44no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—22no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C3×#107—42no—
101C3×#106—22no—
102C3×#103—22no—
103C3×#103—42no—
104C3×#102—22no—
105C3×#102—42no—
106C3×C22⋊C8—24no—
107C3×C22⋊C8—24no—
108C3×#95—22no—
109C3×#95—21no—
110C3×#95—22no—
111C3×#95—21no—
112C24×C4C22×C311yescharacteristic
113C22×SD16C22×C311yescharacteristic, complemented
114C23×C8C22×C311yescharacteristic
115C23×C8C22×C311yescharacteristic
116C2×C22⋊Q8C22×C311yescharacteristic
117C2×C22⋊Q8C22×C312yes—
118C2×#121C22×C311yescharacteristic
119C2×#121C22×C312yes—
120C2×#120—21no—
121#465—24no—
122C2×#107C22×C312yescomplemented
123C2×#106C22×C311yescharacteristic, complemented
124C2×#103C22×C311yescharacteristic, complemented
125C2×#103C22×C312yescomplemented
126C2×#102C22×C311yescharacteristic, complemented
127C2×#102C22×C312yescomplemented
128C2×C22⋊C8—24no—
129C2×C22⋊C8C22×C312yes—
130C2×C22⋊C8C22×C312yes—
131#386—21no—
132#386—22no—
133#385—21no—
134#385—22no—
135#380C22×C311yescharacteristic
136#380C22×C311yescharacteristic
137#380C22×C312yes—
138#380C22×C312yes—
139#379—21no—
140#340C22×C312yescomplemented
141#340C22×C312yescomplemented
142#340C22×C312yescomplemented
143C24×C3—22no—
144C24×C3—24no—
145C24×C3—22no—
146C24×C3—22no—
147C24×C3—21no—
148C24×C3—42no—
149C24×C3C2×D412yes—
150C24×C3C2×D411yescharacteristic
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—44no—
160C2×C3×D4—44no—
161C2×C3×D4—42no—
162C2×C3×D4—44no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—42no—
181C22×C3×C4—42no—
182C22×C3×C4—42no—
183C22×C3×C4—44no—
184C22×C3×C4—44no—
185C22×C3×C4—48no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D412yes—
189C22×C3×C4#4112yes—
190C22×C3×C4C2411yescharacteristic
191C3×SD16—44no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—44no—
198C2×C3×C8—44no—
199C2×C3×C8C2×D412yes—
200C2×C3×C8C2×D412yes—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—44no—
210C3×C4⋊C4—42no—
211C3×C22⋊C4—44no—
212C3×C22⋊C4—44no—
213C3×C22⋊C4—44no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—44no—
216C3×C22⋊C4—42no—
217C3×C22⋊C4—42no—
218C3×C22⋊C4—42no—
219C25C3×D411yescharacteristic
220C22×Q8C23×C311yescharacteristic
221C22×D4—22no—
222C22×D4—21no—
223C22×D4C23×C311yescharacteristic
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4C3×D411yescharacteristic
229C23×C4C3×D412yes—
230C23×C4C23×C312yes—
231C23×C4C23×C311yescharacteristic
232C2×SD16—22no—
233C2×SD16—42no—
234C22×C8—22no—
235C22×C8—42no—
236C22×C8—42no—
237C22×C8C3×D414yescomplemented
238C22×C8C23×C312yes—
239C22×C8C23×C311yescharacteristic
240C22×C8C23×C311yescharacteristic
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—42no—
244C22⋊Q8—42no—
245#121—22no—
246#121—22no—
247#121—42no—
248#121—42no—
249#120—44no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C23×C311yescharacteristic
253C2×C4⋊C4C23×C312yes—
254C2×C4⋊C4C23×C311yescharacteristic
255C2×C4⋊C4C23×C312yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264#107—42no—
265#106—22no—
266#103—22no—
267#103—42no—
268#102—22no—
269#102—42no—
270C22⋊C8—24no—
271C22⋊C8—24no—
272#95—22no—
273#95—21no—
274#95—22no—
275#95—21no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—28no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—48no—
286C23×C3—48no—
287C23×C3—44no—
288C23×C3—44no—
289C23×C3—41no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3#12112yes—
295C23×C3#13514yes—
296C23×C3AGL(1,ℤ/8)12yes—
297C23×C3#13712yes—
298C23×C3C22×D411yescharacteristic
299C3×Q8—41no—
300C3×Q8—41no—
301C3×Q8—42no—
302C3×D4—41no—
303C3×D4—41no—
304C3×D4—42no—
305C3×D4—88no—
306C3×D4—84no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—48no—
315C2×C3×C4—44no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4#12112yes—
334C2×C3×C4C22×D411yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4H(5,𝔽2)12yes—
337C2×C3×C4H(5,𝔽2)12yes—
338C3×C8—22no—
339C3×C8—22no—
340C3×C8—42no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—21no—
346C24—42no—
347C24C2×C3×D412yes—
348C24C2×C3×D411yescharacteristic
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—42no—
362C2×D4—42no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—48no—
384C22×C4C2×C3×D412yes—
385C22×C4C2×C3×D411yescharacteristic
386C22×C4C2×C3×D412yes—
387C22×C4C3×#4112yes—
388C22×C4C24×C311yescharacteristic
389SD16—44no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—44no—
396C2×C8—44no—
397C2×C8C2×C3×D412yes—
398C2×C8C2×C3×D412yes—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—42no—
413C22⋊C4—44no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22×C3—22no—
418C22×C3—24no—
419C22×C3—24no—
420C22×C3—22no—
421C22×C3—24no—
422C22×C3—24no—
423C22×C3—24no—
424C22×C3—416no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—81no—
430C22×C3—82no—
431C22×C3—82no—
432C22×C3—81no—
433C22×C3—81no—
434C22×C3#46814yes—
435C22×C3C2×#12111yescharacteristic
436C22×C3#53412yes—
437C22×C3C2×#13512yes—
438C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
439C22×C3C2×#13711yescharacteristic
440C3×C4—22no—
441C3×C4—42no—
442C3×C4—81no—
443C3×C4—82no—
444C3×C4—82no—
445C3×C4—81no—
446C3×C4—81no—
447C3×C4#53411yescharacteristic
448C3×C4#53511yescharacteristic
449C3×C4#53812yes—
450C23—22no—
451C23—22no—
452C23—28no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—48no—
460C23—48no—
461C23—44no—
462C23—44no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23C3×#12112yes—
469C23C3×#13514yes—
470C23C3×AGL(1,ℤ/8)12yes—
471C23C3×#13712yes—
472C23C22×C3×D411yescharacteristic
473Q8—41no—
474Q8—41no—
475Q8—42no—
476D4—41no—
477D4—41no—
478D4—42no—
479D4—88no—
480D4—84no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—48no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4C3×#12112yes—
508C2×C4C22×C3×D411yescharacteristic
509C2×C4C2×C3×#4111yescharacteristic
510C2×C4C3×H(5,𝔽2)12yes—
511C2×C4C3×H(5,𝔽2)12yes—
512C8—22no—
513C8—22no—
514C8—42no—
515C2×C3—24no—
516C2×C3—44no—
517C2×C3—161no—
518C2×C3#261711yescharacteristic
519C2×C3#305011yescharacteristic
520C2×C3C2×#46811yescharacteristic
521C2×C3#318812yes—
522C2×C3#319012yes—
523C22—22no—
524C22—24no—
525C22—24no—
526C22—22no—
527C22—24no—
528C22—24no—
529C22—24no—
530C22—416no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—81no—
536C22—82no—
537C22—82no—
538C22—81no—
539C22—81no—
540C22C3×#46814yes—
541C22C2×C3×#12111yescharacteristic
542C22C3×#53412yes—
543C22C2×C3×#13512yes—
544C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
545C22C2×C3×#13711yescharacteristic
546C4—22no—
547C4—42no—
548C4—81no—
549C4—82no—
550C4—82no—
551C4—81no—
552C4—81no—
553C4C3×#53411yescharacteristic
554C4C3×#53511yescharacteristic
555C4C3×#53812yes—
556C3#2973511yescharacteristic, direct product factor, Sylow
557C2—24no—
558C2—44no—
559C2—161no—
560C2C3×#261711yescharacteristic
561C2C3×#305011yescharacteristic
562C2C2×C3×#46811yescharacteristic
563C2C3×#318812yes—
564C2C3×#319012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2847C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#465C2semidirect
2. C22×C3×C22⋊C8C2semidirect
3. C3×#2848C2semidirect
5. C3×#2077C2semidirect
6. C3×#2036C2semidirect
7. C3×#2033C2semidirect
8. C3×#2000C2semidirect
9. C3×#1893C2semidirect
10. C3×#1892C2semidirect
11. C3×#1808C2semidirect
12. #29735C3direct (N × H)
14. C22×C3×SD16C22semidirect
23. C2×C3×#107C22semidirect
24. C2×C3×#106C22semidirect
25. C2×C3×#103C22semidirect
26. C2×C3×#103C22semidirect
27. C2×C3×#102C22semidirect
28. C2×C3×#102C22semidirect
41. C3×#340C22semidirect
42. C3×#340C22semidirect
43. C3×#340C22semidirect
44. C2×#465C2×C3semidirect
45. C22×C22⋊C8C2×C3semidirect
46. #2848C2×C3semidirect
48. #2077C2×C3semidirect
49. #2036C2×C3semidirect
50. #2033C2×C3semidirect
51. #2000C2×C3semidirect
52. #1893C2×C3semidirect
53. #1892C2×C3semidirect
54. #1808C2×C3semidirect
73. C22×C3×C8D4semidirect
113. C22×SD16C22×C3semidirect
122. C2×#107C22×C3semidirect
123. C2×#106C22×C3semidirect
124. C2×#103C22×C3semidirect
125. C2×#103C22×C3semidirect
126. C2×#102C22×C3semidirect
127. C2×#102C22×C3semidirect
140. #340C22×C3semidirect
141. #340C22×C3semidirect
142. #340C22×C3semidirect
237. C22×C8C3×D4semidirect
556. C3#29735direct (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
C22×C3×C22⋊C82
C3×#28482
C3×#28472
C3×#20772
C3×#20362
C3×#20332
C3×#20002
C3×#18932
C3×#18922
C3×#18082
#297353