Finite Group Databasebeta

← C3×#25501 #10644108

Subgroups — C3×#25501 #10644108

Order 768.

1388 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic
2C2×C3×#468C212yesmaximal, complemented
3C3×#2855C211yesmaximal, characteristic, complemented
4C3×#2057C211yesmaximal, characteristic, complemented
5C3×#2041C211yesmaximal, characteristic, complemented
6C3×#2024C211yesmaximal, characteristic, complemented
7C3×#2014C211yesmaximal, characteristic, complemented
8C3×#1895C212yesmaximal, complemented
9C3×#1877C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1728C211yesmaximal, characteristic, complemented
12#25501C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41—21no—
14C22×C3×SD16C2212yes—
15C22×C3×M16C2212yes—
16C3×#532—28no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#127C2211yescharacteristic
19C2×C3×#124—21no—
20C2×C3×C22⋊Q8C2211yescharacteristic
21C2×C3×#121—21no—
22C2×C3×#121C2212yes—
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2212yes—
25C22×C3×C42C2211yescharacteristic
26C3×#468—28no—
27C2×C3×#107C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×#102C2212yes—
31C2×C3×#102C2212yes—
32C3×#391C2211yescharacteristic
33C3×#391C2211yescharacteristic
34C3×#389C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#387—21no—
37C3×#387—21no—
38C3×#385—21no—
39C3×#381—21no—
40C3×#380C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2211yescharacteristic
44C3×#336C2211yescharacteristic
45C3×#336C2212yes—
46C2×#532C2×C311yescharacteristic
47C2×#468C2×C312yescomplemented
48#2855C2×C311yescharacteristic, complemented
49#2057C2×C311yescharacteristic, complemented
50#2041C2×C311yescharacteristic, complemented
51#2024C2×C311yescharacteristic, complemented
52#2014C2×C311yescharacteristic, complemented
53#1895C2×C312yescomplemented
54#1877C2×C312yescomplemented
55#1862C2×C312yescomplemented
56#1728C2×C311yescharacteristic, complemented
57C2×C3×#41—28no—
58C2×C3×#41—44no—
59C2×C3×#41—44no—
60C22×C3×Q8—21no—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—22no—
63C22×C3×D4—21no—
64C22×C3×D4—21no—
65C22×C3×D4C2312yes—
66C23×C3×C4—22no—
67C23×C3×C4D411yescharacteristic
68C23×C3×C4D411yescharacteristic
69C23×C3×C4C2311yescharacteristic
70C2×C3×SD16—24no—
71C2×C3×SD16—24no—
72C2×C3×SD16—42no—
73C2×C3×M16—24no—
74C2×C3×M16—24no—
75C2×C3×M16—42no—
76C22×C3×C8C2312yes—
77C22×C3×C8C2312yes—
78C3×C4⋊Q8—24no—
79C3×#127—24no—
80C3×#124—28no—
81C3×C22⋊Q8—24no—
82C3×#121—28no—
83C3×#121—24no—
84C3×#121D418yes—
85C3×C4×D4—28no—
86C3×C4×D4—24no—
87C3×C4×D4—24no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C42—24no—
101C2×C3×C42D411yescharacteristic
102C2×C3×C42D411yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C2×C3×C42C2311yescharacteristic
105C3×#107—24no—
106C3×#103—24no—
107C3×#102—24no—
108C3×#102—24no—
109C3×#102—42no—
110C3×#95—21no—
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C22×#41—21no—
118C22×SD16C22×C312yes—
119C22×M16C22×C312yes—
120#532—28no—
121C2×C4⋊Q8C22×C311yescharacteristic
122C2×#127C22×C311yescharacteristic
123C2×#124—21no—
124C2×C22⋊Q8C22×C311yescharacteristic
125C2×#121—21no—
126C2×#121C22×C312yes—
127C2×C4×D4—21no—
128C2×C4×D4C22×C312yes—
129C22×C42C22×C311yescharacteristic
130#468—28no—
131C2×#107C22×C312yes—
132C2×#103C22×C312yes—
133C2×#102C22×C312yes—
134C2×#102C22×C312yes—
135C2×#102C22×C312yes—
136#391C22×C311yescharacteristic
137#391C22×C311yescharacteristic
138#389C22×C311yescharacteristic
139#389C22×C311yescharacteristic
140#387—21no—
141#387—21no—
142#385—21no—
143#381—21no—
144#380C22×C311yescharacteristic
145#340C22×C312yes—
146#340C22×C312yes—
147#336C22×C311yescharacteristic
148#336C22×C312yes—
149#336C22×C311yescharacteristic
150C24×C3—22no—
151C24×C3C2×D411yescharacteristic
152C3×#41—432no—
153C2×C3×Q8—24no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—42no—
159C2×C3×Q8—42no—
160C2×C3×D4—24no—
161C2×C3×D4—24no—
162C2×C3×D4—28no—
163C2×C3×D4—22no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—42no—
169C2×C3×D4—42no—
170C2×C3×D4—44no—
171C2×C3×D4—44no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4C2×D414yes—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—28no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—41no—
190C22×C3×C4—44no—
191C22×C3×C4—44no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2411yescharacteristic
199C3×SD16—48no—
200C3×M16—48no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8#4114yescomplemented
207C3×C4⋊C4—24no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—42no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4C2×D414yes—
217C3×C22⋊C4—28no—
218C3×C22⋊C4—28no—
219C3×C22⋊C4—42no—
220C3×C22⋊C4—42no—
221C3×C42—22no—
222C3×C42—22no—
223C3×C42—22no—
224C3×C42—22no—
225C2×#41—28no—
226C2×#41—44no—
227C2×#41—44no—
228C22×Q8—21no—
229C22×Q8C23×C311yescharacteristic
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C23×C312yes—
234C23×C4—22no—
235C23×C4C3×D411yescharacteristic
236C23×C4C3×D411yescharacteristic
237C23×C4C23×C311yescharacteristic
238C2×SD16—24no—
239C2×SD16—24no—
240C2×SD16—42no—
241C2×M16—24no—
242C2×M16—24no—
243C2×M16—42no—
244C22×C8C23×C312yes—
245C22×C8C23×C312yes—
246C4⋊Q8—24no—
247#127—24no—
248#124—28no—
249C22⋊Q8—24no—
250#121—28no—
251#121—24no—
252#121C3×D418yes—
253C4×D4—28no—
254C4×D4—24no—
255C4×D4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C23×C311yescharacteristic
261C2×C4⋊C4C23×C311yescharacteristic
262C2×C4⋊C4C23×C311yescharacteristic
263C2×C4⋊C4C23×C312yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C42—24no—
269C2×C42C3×D411yescharacteristic
270C2×C42C3×D411yescharacteristic
271C2×C42C23×C311yescharacteristic
272C2×C42C23×C311yescharacteristic
273#107—24no—
274#103—24no—
275#102—24no—
276#102—24no—
277#102—42no—
278#95—21no—
279#95—21no—
280#95—21no—
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—21no—
290C23×C3—21no—
291C23×C3—24no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3#12012yes—
299C23×C3C22×D411yescharacteristic
300C3×Q8—48no—
301C3×Q8—42no—
302C3×Q8—42no—
303C3×D4—24no—
304C3×D4—24no—
305C3×D4—48no—
306C3×D4—48no—
307C3×D4—416no—
308C3×D4—42no—
309C3×D4—42no—
310C2×C3×C4—28no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—21no—
322C2×C3×C4—24no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—44no—
326C2×C3×C4—416no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4#12012yes—
342C2×C3×C4AGL(1,ℤ/8)12yes—
343C2×C3×C4AGL(1,ℤ/8)12yes—
344C2×C3×C4#13712yes—
345C2×C3×C4C22×D412yes—
346C2×C3×C4C2×#4112yes—
347C2×C3×C4H(5,𝔽2)11yescharacteristic
348C2×C3×C42−1+411yescharacteristic
349C3×C8—24no—
350C3×C8—42no—
351C24—22no—
352C24C2×C3×D411yescharacteristic
353#41—432no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—28no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4C2×C3×D414yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—41no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4C2×C3×D411yescharacteristic
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C2×C3×D414yes—
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D411yescharacteristic
398C22×C4C2×C3×D411yescharacteristic
399C22×C4C24×C311yescharacteristic
400SD16—48no—
401M16—48no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8C3×#4114yescomplemented
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4C2×C3×D414yes—
418C22⋊C4—28no—
419C22⋊C4—28no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—41no—
433C22×C3—44no—
434C22×C3—44no—
435C22×C3—44no—
436C22×C3—82no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3C2×#12011yescharacteristic
441C22×C3#53211yescharacteristic
442C22×C3#53711yescharacteristic
443C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
444C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
445C22×C3C2×#13711yescharacteristic
446C22×C3#57511yescharacteristic
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—44no—
452C3×C4—41no—
453C3×C4—81no—
454C3×C4—81no—
455C3×C4—81no—
456C3×C4—81no—
457C3×C4#54612yes—
458C3×C4#54712yes—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—24no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23C3×#12012yes—
473C23C22×C3×D411yescharacteristic
474Q8—48no—
475Q8—42no—
476Q8—42no—
477D4—24no—
478D4—24no—
479D4—48no—
480D4—48no—
481D4—416no—
482D4—42no—
483D4—42no—
484C2×C4—28no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—44no—
500C2×C4—416no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4C3×#12012yes—
516C2×C4C3×AGL(1,ℤ/8)12yes—
517C2×C4C3×AGL(1,ℤ/8)12yes—
518C2×C4C3×#13712yes—
519C2×C4C22×C3×D412yes—
520C2×C4C2×C3×#4112yes—
521C2×C4C3×H(5,𝔽2)11yescharacteristic
522C2×C4C3×2−1+411yescharacteristic
523C8—24no—
524C8—42no—
525C2×C3—42no—
526C2×C3—84no—
527C2×C3#243011yescharacteristic
528C2×C3#300311yescharacteristic
529C2×C3#300711yescharacteristic
530C2×C3#317111yescharacteristic
531C2×C3#318211yescharacteristic
532C2×C3#325111yescharacteristic
533C2×C3#325411yescharacteristic
534C22—22no—
535C22—22no—
536C22—22no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22C2×C3×#12011yescharacteristic
549C22C3×#53211yescharacteristic
550C22C3×#53711yescharacteristic
551C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
552C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
553C22C2×C3×#13711yescharacteristic
554C22C3×#57511yescharacteristic
555C4—22no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—44no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4C3×#54612yes—
566C4C3×#54712yes—
567C3#2550111yescharacteristic, direct product factor, Sylow
568C2—42no—
569C2—84no—
570C2C3×#243011yescharacteristic
571C2C3×#300311yescharacteristic
572C2C3×#300711yescharacteristic
573C2C3×#317111yescharacteristic
574C2C3×#318211yescharacteristic
575C2C3×#325111yescharacteristic
576C2C3×#325411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#532C2
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
2. C2×C3×#468C2semidirect
3. C3×#2855C2semidirect
4. C3×#2057C2semidirect
5. C3×#2041C2semidirect
6. C3×#2024C2semidirect
7. C3×#2014C2semidirect
8. C3×#1895C2semidirect
9. C3×#1877C2semidirect
10. C3×#1862C2semidirect
11. C3×#1728C2semidirect
12. #25501C3direct (N × H)
47. C2×#468C2×C3semidirect
48. #2855C2×C3semidirect
49. #2057C2×C3semidirect
50. #2041C2×C3semidirect
51. #2024C2×C3semidirect
52. #2014C2×C3semidirect
53. #1895C2×C3semidirect
54. #1877C2×C3semidirect
55. #1862C2×C3semidirect
56. #1728C2×C3semidirect
206. C2×C3×C8#41semidirect
407. C2×C8C3×#41semidirect
567. C3#25501direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×C3×#4682
C3×#28552
C3×#20572
C3×#20412
C3×#20242
C3×#20142
C3×#18952
C3×#18772
C3×#18622
C3×#17282
#255013