Finite Group Databasebeta

← C3×#25497 #10644104

Subgroups — C3×#25497 #10644104

Order 768.

1194 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×C3×#465C212yesmaximal
3C3×#2854C211yesmaximal, characteristic
4C3×#2058C211yesmaximal, characteristic, complemented
5C3×#2043C211yesmaximal, characteristic, complemented
6C3×#2023C211yesmaximal, characteristic, complemented
7C3×#2013C211yesmaximal, characteristic, complemented
8C3×#1892C212yesmaximal, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1865C212yesmaximal, complemented
11C2×C3×#336C211yesmaximal, characteristic, complemented
12#25497C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C23×C3×C8C2212yes—
15C3×#533—28no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125—21no—
19C2×C3×#123—21no—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×#121C2211yescharacteristic
23C2×C3×C4×Q8—21no—
24C2×C3×C4×Q8C2212yes—
25C2×C3×#117—21no—
26C22×C3×C42C2211yescharacteristic
27C3×#465—28no—
28C2×C3×#106C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#102C2212yes—
33C3×#390C2211yescharacteristic
34C3×#390C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#389C2211yescharacteristic
37C3×#388—21no—
38C3×#388—21no—
39C3×#386—21no—
40C3×#380C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C3×#336C2212yes—
46C2×#533C2×C311yescharacteristic, complemented
47C2×#465C2×C312yes—
48#2854C2×C311yescharacteristic
49#2058C2×C311yescharacteristic, complemented
50#2043C2×C311yescharacteristic, complemented
51#2023C2×C311yescharacteristic, complemented
52#2013C2×C311yescharacteristic, complemented
53#1892C2×C312yescomplemented
54#1880C2×C312yescomplemented
55#1865C2×C312yescomplemented
56C2×#336C2×C311yescharacteristic, complemented
57C22×C3×Q8—21no—
58C22×C3×Q8C2312yes—
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4D411yescharacteristic
62C23×C3×C4D411yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C2×C3×SD16—24no—
65C2×C3×SD16—24no—
66C2×C3×SD16—42no—
67C22×C3×C8—24no—
68C22×C3×C8—24no—
69C22×C3×C8—42no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C3×C4⋊Q8—24no—
73C3×C4⋊Q8—24no—
74C3×#125—28no—
75C3×#123—44no—
76C3×C22⋊Q8—28no—
77C3×C22⋊Q8—24no—
78C3×C22⋊Q8D418yes—
79C3×#121—24no—
80C3×C4×Q8—28no—
81C3×C4×Q8—24no—
82C3×C4×Q8—24no—
83C3×#117—28no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—21no—
102C2×C3×C42—24no—
103C2×C3×C42—22no—
104C2×C3×C42D411yescharacteristic
105C2×C3×C42D411yescharacteristic
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2311yescharacteristic
108C3×#106—24no—
109C3×#103—24no—
110C3×#103—24no—
111C3×#103—42no—
112C3×#102—24no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C22×SD16C22×C312yes—
117C23×C8C22×C312yes—
118#533—28no—
119C2×C4⋊Q8C22×C311yescharacteristic
120C2×C4⋊Q8C22×C311yescharacteristic
121C2×#125—21no—
122C2×#123—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8C22×C312yes—
125C2×#121C22×C311yescharacteristic
126C2×C4×Q8—21no—
127C2×C4×Q8C22×C312yes—
128C2×#117—21no—
129C22×C42C22×C311yescharacteristic
130#465—28no—
131C2×#106C22×C312yes—
132C2×#103C22×C312yes—
133C2×#103C22×C312yes—
134C2×#103C22×C312yes—
135C2×#102C22×C312yes—
136#390C22×C311yescharacteristic
137#390C22×C311yescharacteristic
138#389C22×C311yescharacteristic
139#389C22×C311yescharacteristic
140#388—21no—
141#388—21no—
142#386—21no—
143#380C22×C311yescharacteristic
144#340C22×C312yes—
145#340C22×C312yes—
146#336C22×C311yescharacteristic
147#336C22×C312yes—
148#336C22×C311yescharacteristic
149C24×C3—21no—
150C24×C3C2×D411yescharacteristic
151C2×C3×Q8—24no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—42no—
158C2×C3×Q8—42no—
159C2×C3×Q8C2×D414yes—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C2×C3×D4—42no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—41no—
180C22×C3×C4SD1614yescomplemented
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D414yes—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C3×SD16—48no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—48no—
195C2×C3×C8#4114yes—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—28no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—28no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4C2×D414yes—
209C3×C22⋊C4—28no—
210C3×C22⋊C4—42no—
211C3×C22⋊C4—42no—
212C3×C22⋊C4—42no—
213C3×C22⋊C4—42no—
214C3×C22⋊C4—42no—
215C3×C22⋊C4—42no—
216C3×C42—28no—
217C3×C42—22no—
218C3×C42—22no—
219C3×C42—22no—
220C3×C42—22no—
221C22×Q8—21no—
222C22×Q8C23×C312yes—
223C22×D4—21no—
224C22×D4C23×C311yescharacteristic
225C23×C4C3×D411yescharacteristic
226C23×C4C3×D411yescharacteristic
227C23×C4C23×C311yescharacteristic
228C2×SD16—24no—
229C2×SD16—24no—
230C2×SD16—42no—
231C22×C8—24no—
232C22×C8—24no—
233C22×C8—42no—
234C22×C8C23×C312yes—
235C22×C8C23×C312yes—
236C4⋊Q8—24no—
237C4⋊Q8—24no—
238#125—28no—
239#123—44no—
240C22⋊Q8—28no—
241C22⋊Q8—24no—
242C22⋊Q8C3×D418yes—
243#121—24no—
244C4×Q8—28no—
245C4×Q8—24no—
246C4×Q8—24no—
247#117—28no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4C23×C311yescharacteristic
256C2×C4⋊C4C23×C311yescharacteristic
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C4⋊C4C23×C312yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42C3×D411yescharacteristic
269C2×C42C3×D411yescharacteristic
270C2×C42C23×C311yescharacteristic
271C2×C42C23×C311yescharacteristic
272#106—24no—
273#103—24no—
274#103—24no—
275#103—42no—
276#102—24no—
277#95—21no—
278#95—21no—
279#95—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—21no—
284C23×C3—41no—
285C23×C3—41no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3#12012yes—
292C23×C3C2×SD1612yes—
293C23×C3#13712yes—
294C23×C3C22×D411yescharacteristic
295C3×Q8—24no—
296C3×Q8—24no—
297C3×Q8—48no—
298C3×Q8—42no—
299C3×Q8—42no—
300C3×D4—42no—
301C3×D4—42no—
302C3×D4—84no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—28no—
312C2×C3×C4—24no—
313C2×C3×C4—24no—
314C2×C3×C4—21no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—44no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4#12012yes—
334C2×C3×C4C2×SD1612yes—
335C2×C3×C4C2×SD1612yes—
336C2×C3×C4#13512yes—
337C2×C3×C4C22×D412yes—
338C2×C3×C4C2×#4112yes—
339C2×C3×C4H(5,𝔽2)11yescharacteristic
340C2×C3×C42−1+411yescharacteristic
341C3×C8—24no—
342C3×C8—42no—
343C24—21no—
344C24C2×C3×D411yescharacteristic
345C2×Q8—24no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8C2×C3×D414yes—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—41no—
374C22×C4C3×SD1614yescomplemented
375C22×C4C2×C3×D411yescharacteristic
376C22×C4C2×C3×D411yescharacteristic
377C22×C4C2×C3×D414yes—
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D411yescharacteristic
381C22×C4C24×C311yescharacteristic
382SD16—48no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—48no—
389C2×C8C3×#4114yes—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—28no—
393C4⋊C4—24no—
394C4⋊C4—28no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4C2×C3×D414yes—
403C22⋊C4—28no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C42—28no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—81no—
422C22×C3—81no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3—81no—
426C22×C3—81no—
427C22×C3—81no—
428C22×C3#44814yes—
429C22×C3C2×#12011yescharacteristic
430C22×C3#53211yescharacteristic
431C22×C3#53711yescharacteristic
432C22×C3C22×SD1611yescharacteristic
433C22×C3C2×#13511yescharacteristic
434C22×C3C2×#13711yescharacteristic
435C22×C3#57511yescharacteristic
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—44no—
441C3×C4—41no—
442C3×C4—44no—
443C3×C4—81no—
444C3×C4—81no—
445C3×C4—81no—
446C3×C4#54612yes—
447C3×C4#54712yes—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23C3×#12012yes—
460C23C2×C3×SD1612yes—
461C23C3×#13712yes—
462C23C22×C3×D411yescharacteristic
463Q8—24no—
464Q8—24no—
465Q8—48no—
466Q8—42no—
467Q8—42no—
468D4—42no—
469D4—42no—
470D4—84no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—28no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4C3×#12012yes—
502C2×C4C2×C3×SD1612yes—
503C2×C4C2×C3×SD1612yes—
504C2×C4C3×#13512yes—
505C2×C4C22×C3×D412yes—
506C2×C4C2×C3×#4112yes—
507C2×C4C3×H(5,𝔽2)11yescharacteristic
508C2×C4C3×2−1+411yescharacteristic
509C8—24no—
510C8—42no—
511C2×C3—24no—
512C2×C3—161no—
513C2×C3#243011yescharacteristic
514C2×C3C2×#44811yescharacteristic
515C2×C3#300211yescharacteristic
516C2×C3#316711yescharacteristic
517C2×C3#318111yescharacteristic
518C2×C3#324611yescharacteristic
519C2×C3#324811yescharacteristic
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22C3×#44814yes—
534C22C2×C3×#12011yescharacteristic
535C22C3×#53211yescharacteristic
536C22C3×#53711yescharacteristic
537C22C22×C3×SD1611yescharacteristic
538C22C2×C3×#13511yescharacteristic
539C22C2×C3×#13711yescharacteristic
540C22C3×#57511yescharacteristic
541C4—22no—
542C4—22no—
543C4—22no—
544C4—22no—
545C4—44no—
546C4—41no—
547C4—44no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4C3×#54612yes—
552C4C3×#54712yes—
553C3#2549711yescharacteristic, direct product factor, Sylow
554C2—24no—
555C2—161no—
556C2C3×#243011yescharacteristic
557C2C2×C3×#44811yescharacteristic
558C2C3×#300211yescharacteristic
559C2C3×#316711yescharacteristic
560C2C3×#318111yescharacteristic
561C2C3×#324611yescharacteristic
562C2C3×#324811yescharacteristic
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×#533C2semidirect
4. C3×#2058C2semidirect
5. C3×#2043C2semidirect
6. C3×#2023C2semidirect
7. C3×#2013C2semidirect
8. C3×#1892C2semidirect
9. C3×#1880C2semidirect
10. C3×#1865C2semidirect
11. C2×C3×#336C2semidirect
12. #25497C3direct (N × H)
46. C2×#533C2×C3semidirect
49. #2058C2×C3semidirect
50. #2043C2×C3semidirect
51. #2023C2×C3semidirect
52. #2013C2×C3semidirect
53. #1892C2×C3semidirect
54. #1880C2×C3semidirect
55. #1865C2×C3semidirect
56. C2×#336C2×C3semidirect
180. C22×C3×C4SD16semidirect
374. C22×C4C3×SD16semidirect
553. C3#25497direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4652
C3×#28542
C3×#20582
C3×#20432
C3×#20232
C3×#20132
C3×#18922
C3×#18802
C3×#18652
C2×C3×#3362
#254973