Finite Group Databasebeta

← C3×#25641 #10644248

Subgroups — C3×#25641 #10644248

Order 768.

1172 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×#559C211yesmaximal, characteristic
2C2×C3×#497C211yesmaximal, characteristic, complemented
3C2×C3×#461C211yesmaximal, characteristic, complemented
4C3×#2568C211yesmaximal, characteristic, complemented
5C3×#2015C212yesmaximal, complemented
6C3×#2014C212yesmaximal
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1983C211yesmaximal, characteristic, complemented
9C3×#1878C212yesmaximal, complemented
10C3×#1877C212yesmaximal, complemented
11C3×#1762C211yesmaximal, characteristic, complemented
12#25641C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C22×C3×SD16C2212yes—
16C3×#559—24no—
17C3×#559—24no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×C3×#124—21no—
21C2×C3×#124C2211yescharacteristic
22C2×C3×#124C2211yescharacteristic
23C2×C3×#123—21no—
24C2×C3×#123—21no—
25C2×C3×C4×Q8C2211yescharacteristic
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×D4—21no—
28C2×C3×C4×D4C2211yescharacteristic
29C2×C3×C4×D4C2211yescharacteristic
30C3×#497—24no—
31C3×#461—24no—
32C2×C3×C4⋊C8C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#102C2212yes—
36C2×C3×#102C2212yes—
37C2×C3×#97C2211yescharacteristic
38C3×#400—21no—
39C3×#400—21no—
40C3×#391—21no—
41C3×#391C2212yes—
42C3×#391C2212yes—
43C3×#382—21no—
44C3×C4×C4⋊C4C2211yescharacteristic
45C3×#340C2212yes—
46C3×#336C2212yes—
47C3×#336C2212yes—
48C2×#559C2×C311yescharacteristic
49C2×#497C2×C311yescharacteristic, complemented
50C2×#461C2×C311yescharacteristic, complemented
51#2568C2×C311yescharacteristic, complemented
52#2015C2×C312yescomplemented
53#2014C2×C312yes—
54#1985C2×C311yescharacteristic, complemented
55#1983C2×C311yescharacteristic, complemented
56#1878C2×C312yescomplemented
57#1877C2×C312yescomplemented
58#1762C2×C311yescharacteristic, complemented
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2312yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C22×C3×D4C2311yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4—21no—
68C2×C3×Q16—24no—
69C2×C3×Q16—24no—
70C2×C3×Q16—42no—
71C2×C3×SD16—24no—
72C2×C3×SD16—24no—
73C2×C3×SD16—24no—
74C2×C3×SD16—42no—
75C2×C3×SD16—42no—
76C2×C3×SD16—42no—
77C22×C3×C8C2312yes—
78C22×C3×C8C2312yes—
79C3×C4⋊Q8—22no—
80C3×C4⋊Q8—22no—
81C3×#127—22no—
82C3×#127—22no—
83C3×#124—28no—
84C3×#124—22no—
85C3×#124—22no—
86C3×#124—22no—
87C3×#124D414yes—
88C3×#123—28no—
89C3×#123—28no—
90C3×C4×Q8—22no—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8—22no—
94C3×C4×D4—28no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4D414yes—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4D411yescharacteristic
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C42—21no—
114C2×C3×C42—21no—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C2×C3×C42C2311yescharacteristic
118C3×C4⋊C8—22no—
119C3×#103—24no—
120C3×#103—42no—
121C3×#102—24no—
122C3×#102—42no—
123C3×#97—22no—
124C3×#95—21no—
125C3×#95—22no—
126C3×#95—22no—
127C3×#95—22no—
128C3×#95—22no—
129C22×Q16C22×C312yes—
130C22×SD16C22×C312yes—
131C22×SD16C22×C312yes—
132#559—24no—
133#559—24no—
134C2×C4⋊Q8C22×C311yescharacteristic
135C2×#127C22×C311yescharacteristic
136C2×#124—21no—
137C2×#124C22×C311yescharacteristic
138C2×#124C22×C311yescharacteristic
139C2×#123—21no—
140C2×#123—21no—
141C2×C4×Q8C22×C311yescharacteristic
142C2×C4×Q8C22×C311yescharacteristic
143C2×C4×D4—21no—
144C2×C4×D4C22×C311yescharacteristic
145C2×C4×D4C22×C311yescharacteristic
146#497—24no—
147#461—24no—
148C2×C4⋊C8C22×C311yescharacteristic
149C2×#103C22×C312yes—
150C2×#103C22×C312yes—
151C2×#102C22×C312yes—
152C2×#102C22×C312yes—
153C2×#97C22×C311yescharacteristic
154#400—21no—
155#400—21no—
156#391—21no—
157#391C22×C312yes—
158#391C22×C312yes—
159#382—21no—
160C4×C4⋊C4C22×C311yescharacteristic
161#340C22×C312yes—
162#336C22×C312yes—
163#336C22×C312yes—
164C24×C3—21no—
165C24×C3—21no—
166C2×C3×Q8—22no—
167C2×C3×Q8—22no—
168C2×C3×Q8—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—41no—
178C2×C3×Q8C2×D412yes—
179C2×C3×D4—24no—
180C2×C3×D4—24no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—22no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—82no—
190C2×C3×D4C2×D412yes—
191C2×C3×D4C2×D412yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—82no—
199C22×C3×C4—82no—
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C3×Q16—48no—
208C3×SD16—48no—
209C3×SD16—48no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—22no—
214C2×C3×C8—22no—
215C2×C3×C8#4114yes—
216C3×C4⋊C4—24no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—42no—
225C3×C4⋊C4C2×D412yes—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—24no—
231C3×C22⋊C4—24no—
232C3×C42—24no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237C3×C42—42no—
238C3×C42C2×D412yes—
239C3×C42C2×D412yes—
240C22×Q8C3×D411yescharacteristic
241C22×Q8C23×C312yes—
242C22×Q8C23×C311yescharacteristic
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C23×C311yescharacteristic
246C22×D4C23×C311yescharacteristic
247C23×C4—21no—
248C23×C4—21no—
249C2×Q16—24no—
250C2×Q16—24no—
251C2×Q16—42no—
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16—24no—
255C2×SD16—42no—
256C2×SD16—42no—
257C2×SD16—42no—
258C22×C8C23×C312yes—
259C22×C8C23×C312yes—
260C4⋊Q8—22no—
261C4⋊Q8—22no—
262#127—22no—
263#127—22no—
264#124—28no—
265#124—22no—
266#124—22no—
267#124—22no—
268#124C3×D414yes—
269#123—28no—
270#123—28no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×D4—28no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4C3×D414yes—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4C3×D411yescharacteristic
284C2×C4⋊C4C3×D411yescharacteristic
285C2×C4⋊C4C3×D411yescharacteristic
286C2×C4⋊C4C23×C312yes—
287C2×C4⋊C4C23×C311yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42C23×C311yescharacteristic
297C2×C42C23×C311yescharacteristic
298C2×C42C23×C311yescharacteristic
299C4⋊C8—22no—
300#103—24no—
301#103—42no—
302#102—24no—
303#102—42no—
304#97—22no—
305#95—21no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310C23×C3—22no—
311C23×C3—22no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3—42no—
319C23×C3—42no—
320C23×C3C22×D411yescharacteristic
321C3×Q8—24no—
322C3×Q8—44no—
323C3×Q8—41no—
324C3×Q8—41no—
325C3×Q8—42no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×Q8—42no—
329C3×D4—22no—
330C3×D4—22no—
331C3×D4—24no—
332C3×D4—48no—
333C3×D4—41no—
334C3×D4—41no—
335C3×D4—42no—
336C3×D4—84no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—84no—
360C2×C3×C4—84no—
361C2×C3×C4#12012yes—
362C2×C3×C4#12012yes—
363C2×C3×C4#13712yes—
364C2×C3×C4C22×D411yescharacteristic
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4112yes—
367C2×C3×C4H(5,𝔽2)12yes—
368C3×C8—24no—
369C3×C8—42no—
370C24—21no—
371C24—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—41no—
384C2×Q8C2×C3×D412yes—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—82no—
396C2×D4C2×C3×D412yes—
397C2×D4C2×C3×D412yes—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4C2×C3×D411yescharacteristic
407C22×C4C2×C3×D411yescharacteristic
408C22×C4C2×C3×D411yescharacteristic
409C22×C4C2×C3×D411yescharacteristic
410C22×C4C2×C3×D411yescharacteristic
411C22×C4C2×C3×D411yescharacteristic
412C22×C4C24×C311yescharacteristic
413Q16—48no—
414SD16—48no—
415SD16—48no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8C3×#4114yes—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4C2×C3×D412yes—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C42—24no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—42no—
444C42C2×C3×D412yes—
445C42C2×C3×D412yes—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—82no—
453C22×C3—82no—
454C22×C3—82no—
455C22×C3—82no—
456C22×C3C2×#12011yescharacteristic
457C22×C3#55012yes—
458C22×C3C2×#13711yescharacteristic
459C22×C3#57511yescharacteristic
460C22×C3#57712yes—
461C3×C4—22no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—42no—
465C3×C4—41no—
466C3×C4—41no—
467C3×C4—41no—
468C3×C4—42no—
469C3×C4—82no—
470C3×C4—82no—
471C3×C4#53211yescharacteristic
472C3×C4#53511yescharacteristic
473C3×C4#54612yes—
474C23—22no—
475C23—22no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23C22×C3×D411yescharacteristic
485Q8—24no—
486Q8—44no—
487Q8—41no—
488Q8—41no—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493D4—22no—
494D4—22no—
495D4—24no—
496D4—48no—
497D4—41no—
498D4—41no—
499D4—42no—
500D4—84no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4C3×#12012yes—
526C2×C4C3×#12012yes—
527C2×C4C3×#13712yes—
528C2×C4C22×C3×D411yescharacteristic
529C2×C4C22×C3×D411yescharacteristic
530C2×C4C2×C3×#4112yes—
531C2×C4C3×H(5,𝔽2)12yes—
532C8—24no—
533C8—42no—
534C2×C3—82no—
535C2×C3—82no—
536C2×C3#243111yescharacteristic
537C2×C3#300711yescharacteristic
538C2×C3#300911yescharacteristic
539C2×C3#340012yes—
540C2×C3#340112yes—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22C2×C3×#12011yescharacteristic
552C22C3×#55012yes—
553C22C2×C3×#13711yescharacteristic
554C22C3×#57511yescharacteristic
555C22C3×#57712yes—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—42no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—42no—
564C4—82no—
565C4—82no—
566C4C3×#53211yescharacteristic
567C4C3×#53511yescharacteristic
568C4C3×#54612yes—
569C3#2564111yescharacteristic, direct product factor, Sylow
570C2—82no—
571C2—82no—
572C2C3×#243111yescharacteristic
573C2C3×#300711yescharacteristic
574C2C3×#300911yescharacteristic
575C2C3×#340012yes—
576C2C3×#340112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#497C2semidirect
3. C2×C3×#461C2semidirect
4. C3×#2568C2semidirect
5. C3×#2015C2semidirect
7. C3×#1985C2semidirect
8. C3×#1983C2semidirect
9. C3×#1878C2semidirect
10. C3×#1877C2semidirect
11. C3×#1762C2semidirect
12. #25641C3direct (N × H)
49. C2×#497C2×C3semidirect
50. C2×#461C2×C3semidirect
51. #2568C2×C3semidirect
52. #2015C2×C3semidirect
54. #1985C2×C3semidirect
55. #1983C2×C3semidirect
56. #1878C2×C3semidirect
57. #1877C2×C3semidirect
58. #1762C2×C3semidirect
569. C3#25641direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×C3×#4972
C2×C3×#4612
C3×#25682
C3×#20152
C3×#20142
C3×#19852
C3×#19832
C3×#18782
C3×#18772
C3×#17622
#256413