Finite Group Databasebeta

← C3×#29719 #10648326

Subgroups — C3×#29719 #10648326

Order 768.

1038 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#451C211yesmaximal, characteristic, complemented
2C2×C3×#448C211yesmaximal, characteristic
3C3×#2850C211yesmaximal, characteristic
4C2×C3×#391C211yesmaximal, characteristic, complemented
5C3×#2074C212yesmaximal, complemented
6C3×#2056C212yesmaximal, complemented
7C3×#2055C212yesmaximal, complemented
8C3×#2015C212yesmaximal, complemented
9C3×#1874C211yesmaximal, characteristic
10C3×#1872C211yesmaximal, characteristic, complemented
11C3×#1797C211yesmaximal, characteristic, complemented
12#29719C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C2×C3×#124C2211yescharacteristic
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C3×#451—24no—
22C3×#448—24no—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#391—21no—
31C3×#391—21no—
32C3×#391C2212yes—
33C3×#391C2212yes—
34C3×#391C2211yescharacteristic
35C3×#390C2211yescharacteristic
36C3×#388—21no—
37C3×#388—21no—
38C3×#386—21no—
39C3×#386—21no—
40C3×#382C2211yescharacteristic
41C2×C3×#95—21no—
42C2×C3×#95—21no—
43C3×#340C2212yes—
44C3×#340C2212yes—
45C3×#336C2212yes—
46C2×#451C2×C311yescharacteristic, complemented
47C2×#448C2×C311yescharacteristic
48#2850C2×C311yescharacteristic
49C2×#391C2×C311yescharacteristic, complemented
50#2074C2×C312yescomplemented
51#2056C2×C312yescomplemented
52#2055C2×C312yescomplemented
53#2015C2×C312yescomplemented
54#1874C2×C311yescharacteristic
55#1872C2×C311yescharacteristic, complemented
56#1797C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8—22no—
59C22×C3×Q8—42no—
60C22×C3×Q8—42no—
61C22×C3×Q8D414yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×Q8C2312yes—
64C22×C3×D4—21no—
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C23×C3×C4D411yescharacteristic
69C23×C3×C4D411yescharacteristic
70C23×C3×C4C2311yescharacteristic
71C2×C3×Q16—24no—
72C2×C3×Q16—42no—
73C2×C3×Q16—42no—
74C2×C3×SD16—24no—
75C2×C3×SD16—42no—
76C2×C3×SD16—42no—
77C22×C3×C8C2312yes—
78C22×C3×C8C2312yes—
79C3×#124—42no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C3×#121—22no—
83C3×#121—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—42no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42—42no—
97C2×C3×C42D411yescharacteristic
98C2×C3×C42D411yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C3×C4⋊C8—42no—
102C3×#103—24no—
103C3×#103—24no—
104C3×#103—42no—
105C3×#102—42no—
106C3×C22⋊C8—22no—
107C3×C22⋊C8—22no—
108C3×#95—22no—
109C3×#95—22no—
110C3×#95—22no—
111C3×#95—22no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C23×Q8C22×C311yescharacteristic
117C22×Q16C22×C312yes—
118C22×SD16C22×C312yes—
119C2×#124C22×C311yescharacteristic
120C2×C22⋊Q8C22×C311yescharacteristic
121C2×#121C22×C311yescharacteristic
122C22×C4⋊C4C22×C311yescharacteristic
123C22×C42C22×C311yescharacteristic
124#451—24no—
125#448—24no—
126C2×C4⋊C8C22×C312yes—
127C2×#103C22×C312yes—
128C2×#103C22×C312yes—
129C2×#103C22×C312yes—
130C2×#102C22×C312yes—
131C2×C22⋊C8C22×C311yescharacteristic
132C2×C22⋊C8C22×C311yescharacteristic
133#391—21no—
134#391—21no—
135#391C22×C312yes—
136#391C22×C311yescharacteristic
137#391C22×C312yes—
138#390C22×C311yescharacteristic
139#388—21no—
140#388—21no—
141#386—21no—
142#386—21no—
143#382C22×C311yescharacteristic
144C2×#95—21no—
145C2×#95—21no—
146#340C22×C312yes—
147#340C22×C312yes—
148#336C22×C312yes—
149C24×C3—21no—
150C24×C3C2×D411yescharacteristic
151C2×C3×Q8—24no—
152C2×C3×Q8—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—22no—
161C2×C3×Q8—48no—
162C2×C3×Q8—42no—
163C2×C3×Q8—42no—
164C2×C3×Q8—48no—
165C2×C3×Q8C2×D414yes—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—22no—
170C2×C3×D4—42no—
171C2×C3×D4—42no—
172C2×C3×D4—42no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—41no—
185C22×C3×C4—41no—
186C22×C3×C4—41no—
187C22×C3×C4—42no—
188C22×C3×C4—42no—
189C22×C3×C4—42no—
190C22×C3×C4—42no—
191C22×C3×C4—42no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4#4112yes—
199C22×C3×C4C2411yescharacteristic
200C3×Q16—48no—
201C3×SD16—48no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C2×C3×C8—22no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4—42no—
216C3×C22⋊C4—42no—
217C3×C22⋊C4—42no—
218C3×C22⋊C4—42no—
219C3×C22⋊C4—42no—
220C3×C22⋊C4—42no—
221C3×C22⋊C4—42no—
222C3×C42—22no—
223C3×C42—22no—
224C3×C42—41no—
225C3×C42—41no—
226C22×Q8—21no—
227C22×Q8—22no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8C3×D414yes—
231C22×Q8C23×C311yescharacteristic
232C22×Q8C23×C312yes—
233C22×D4—21no—
234C22×D4C23×C311yescharacteristic
235C23×C4—21no—
236C23×C4—21no—
237C23×C4C3×D411yescharacteristic
238C23×C4C3×D411yescharacteristic
239C23×C4C23×C311yescharacteristic
240C2×Q16—24no—
241C2×Q16—42no—
242C2×Q16—42no—
243C2×SD16—24no—
244C2×SD16—42no—
245C2×SD16—42no—
246C22×C8C23×C312yes—
247C22×C8C23×C312yes—
248#124—42no—
249C22⋊Q8—22no—
250C22⋊Q8—22no—
251#121—22no—
252#121—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C23×C312yes—
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C42—42no—
266C2×C42C3×D411yescharacteristic
267C2×C42C3×D411yescharacteristic
268C2×C42C23×C311yescharacteristic
269C2×C42C23×C311yescharacteristic
270C4⋊C8—42no—
271#103—24no—
272#103—24no—
273#103—42no—
274#102—42no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—41no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3#12112yes—
297C23×C3#13512yes—
298C23×C3#13712yes—
299C23×C3C22×D411yescharacteristic
300C3×Q8—24no—
301C3×Q8—24no—
302C3×Q8—48no—
303C3×Q8—42no—
304C3×Q8—42no—
305C3×Q8—42no—
306C3×Q8—42no—
307C3×D4—42no—
308C3×D4—42no—
309C3×D4—84no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—24no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—44no—
327C2×C3×C4—44no—
328C2×C3×C4—48no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—42no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—84no—
344C2×C3×C4#12112yes—
345C2×C3×C4#13712yes—
346C2×C3×C4C22×D411yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4H(5,𝔽2)12yes—
349C2×C3×C4H(5,𝔽2)12yes—
350C3×C8—42no—
351C3×C8—42no—
352C24—21no—
353C24C2×C3×D411yescharacteristic
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—48no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—48no—
368C2×Q8C2×C3×D414yes—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C2×C3×D412yes—
400C22×C4C2×C3×D411yescharacteristic
401C22×C4C3×#4112yes—
402C22×C4C24×C311yescharacteristic
403Q16—48no—
404SD16—48no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C42—22no—
426C42—22no—
427C42—41no—
428C42—41no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3—81no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3#46414yes—
443C22×C3C2×#12111yescharacteristic
444C22×C3#53411yescharacteristic
445C22×C3#53511yescharacteristic
446C22×C3C2×#13511yescharacteristic
447C22×C3C2×#13711yescharacteristic
448C22×C3C2×#13711yescharacteristic
449C22×C3#57511yescharacteristic
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—44no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—81no—
457C3×C4—82no—
458C3×C4—81no—
459C3×C4—81no—
460C3×C4#53412yes—
461C3×C4#53812yes—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23C3×#12112yes—
474C23C3×#13512yes—
475C23C3×#13712yes—
476C23C22×C3×D411yescharacteristic
477Q8—24no—
478Q8—24no—
479Q8—48no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484D4—42no—
485D4—42no—
486D4—84no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—44no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—84no—
521C2×C4C3×#12112yes—
522C2×C4C3×#13712yes—
523C2×C4C22×C3×D411yescharacteristic
524C2×C4C2×C3×#4111yescharacteristic
525C2×C4C3×H(5,𝔽2)12yes—
526C2×C4C3×H(5,𝔽2)12yes—
527C8—42no—
528C8—42no—
529C2×C3—24no—
530C2×C3—161no—
531C2×C3#261711yescharacteristic
532C2×C3C2×#46411yescharacteristic
533C2×C3#303711yescharacteristic
534C2×C3#319011yescharacteristic
535C2×C3#319611yescharacteristic
536C2×C3#321111yescharacteristic
537C2×C3#321811yescharacteristic
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22C3×#46414yes—
552C22C2×C3×#12111yescharacteristic
553C22C3×#53411yescharacteristic
554C22C3×#53511yescharacteristic
555C22C2×C3×#13511yescharacteristic
556C22C2×C3×#13711yescharacteristic
557C22C2×C3×#13711yescharacteristic
558C22C3×#57511yescharacteristic
559C4—22no—
560C4—22no—
561C4—44no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—82no—
567C4—81no—
568C4—81no—
569C4C3×#53412yes—
570C4C3×#53812yes—
571C3#2971911yescharacteristic, direct product factor, Sylow
572C2—24no—
573C2—161no—
574C2C3×#261711yescharacteristic
575C2C2×C3×#46411yescharacteristic
576C2C3×#303711yescharacteristic
577C2C3×#319011yescharacteristic
578C2C3×#319611yescharacteristic
579C2C3×#321111yescharacteristic
580C2C3×#321811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#451C2semidirect
4. C2×C3×#391C2semidirect
5. C3×#2074C2semidirect
6. C3×#2056C2semidirect
7. C3×#2055C2semidirect
8. C3×#2015C2semidirect
10. C3×#1872C2semidirect
11. C3×#1797C2semidirect
12. #29719C3direct (N × H)
46. C2×#451C2×C3semidirect
49. C2×#391C2×C3semidirect
50. #2074C2×C3semidirect
51. #2056C2×C3semidirect
52. #2055C2×C3semidirect
53. #2015C2×C3semidirect
55. #1872C2×C3semidirect
56. #1797C2×C3semidirect
571. C3#29719direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4512
C2×C3×#4482
C3×#28502
C2×C3×#3912
C3×#20742
C3×#20562
C3×#20552
C3×#20152
C3×#18742
C3×#18722
C3×#17972
#297193